France



The finite element method (FEM) is widely used to model electromagnetic devices. This method ensures precise results at the expense of a consequent computation time, especially when dealing with parametric problems. Model order reduction methods had proven their efficiency in reducing the computation time for parametric problems. The proper orthogonal decomposition (POD) is a popular model order reduction approach [1]. Based on the solutions of the FE model for different values of parameters (called snapshots), the POD approximates the solution of the FE model in a reduced basis. The parametric FE model is projected onto a reduced basis, decreasing the order of the numerical model to be solved. Another approach consists in constructing a metamodel by interpolation of the parametric FE solution expressed into a reduced basis. A hybrid approach has also been proposed combining the POD with an interpolation-based method such as the radial basis functions (RBF) interpolation method [2].&lt;br/&gt;Moreover, when the problem involves geometric parameters, an adaption of the FEM mesh is required. Remeshing the geometry at each modification can deteriorate the precision of the FEM model by introducing significant numerical noise. A solution to preserve the problem&#39;s consistency is to deform the initial mesh using the RBF interpolation [3][4]. The method consists of imposing a displacement on a set of nodes and interpolating the remaining nodes&#39; position. Then, a metamodel of a FE solution taking into account geometric and electric parameters, can be defined by using the POD approach and the RBF interpolation.&lt;br/&gt;In this work, we propose to build a metamodel of a nonlinear magnetostatic parametric problem involving geometric parameters using the POD-RBF approach. The study case is a single-phase EI inductance, with one electrical parameter and two geometrical parameters. An initial geometry is meshed with 36859 elements; the number of nodes is 18553. The geometric parameters are the thickness of the airgap varying from 0.1 mm to 0.9 mm, the width of the central column of the magnetic core varying from 20 mm to 40 mm. The phase current varying from 0 to 50 A is also a parameter of the model. The mesh deformation is modeled using the RBF interpolation with a multi-quadratic function for two consecutive times to consider the two variating geometric parameters. The results of the original FEM model are taken as a reference.&lt;br/&gt;The POD-RBF approach is applied to the parametric FEM model. The method consists of first gathering a set of data, using the method of snapshots, by solving the original FEM model for a set of inputs. The snapshots matrix is decomposed using the singular value decomposition (SVD). The left orthogonal matrix of the SVD decomposition corresponds to the reduced basis, and the right orthogonal matrix is composed of the FE solution projected into the reduced basis for each snapshot parameter. An RBF interpolation is then performed on the right orthogonal matrix to interpolate the solution expressed in the reduced basis for the new inputs set. Finally, an approximation of the FE solution is defined by projecting the reduced solution into the FE basis.&lt;br/&gt;The choice of snapshots is made using a greedy algorithm [2], which consists of an iterative process. The metamodel is then improved at each iteration until the required precision is reached.&lt;br/&gt;The stop criterion to the greedy algorithm is set to an error between the FE solution and the metamodel solution lower than 0.1%. The evolution of error represented by Frobenius norm as a function of the iterations&#39; number of the greedy algorithm is represented in &quot;Fig. 1&quot;.&lt;br/&gt;In order to satisfy the selected criterion, 158 snapshots are required. The magnetic flux cartography as a function of phase current and the central column width for an airgap width fixed at 0.1 mm is represented in &quot;Fig.2&quot;. The global mean absolute error between the flux cartographies obtained using the metamodel and the original FEM model is 1.7%, while the maximal local error is 4.96%.&lt;br/&gt;Then, the metamodel of the parametric FE solution can be used in an optimization process for different applications or to be coupled with other numerical models in order to simulate a system.

**References:**

[1] L. Sirovich, Quarterly of applied mathematics, vol. 45, no 3, pp.561-571, (1987).&lt;br/&gt;[2] T. Henneron, A. Pierquin and S. Clénet, IEEE Transactions on Magnetics, vol. 56, no. 1, pp. 1-4, (2020).&lt;br/&gt;[3] A. de Boer, M.S. van der Schoot and H. Bijl, Conference on Computational Fluid and Solid Mechanics, vol. 85, pp. 784-795, (2007).&lt;br/&gt;[4] T. Henneron, A. Pierquin and S. Clénet, IEEE Transactions on Magnetics, vol. 55, no. 6, pp. 1-4, (2019).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/66ade7d4382d68fb036d292ea79e24d1.jpg)
Fig.1 Error as a function of the number of iterations

![](https://assets.underline.io/uploads/markdown_image/1/image/d0ae73399462ce7b097771fa6cc378a2.jpg)
Fig. 2 Flux cartography as a function of current and central column width

INTERMAG 2021

 Parametric Geometric Metamodel of Nonlinear Magnetostatic Problem Based on POD and RBF Approaches

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The finite element method (FEM) is widely used to model electromagnetic devices. This method ensures precise results at the expense of a consequent computation time, especially when dealing with parametric problems. Model order reduction methods had proven their efficiency in reducing the computation time for parametric problems. The proper orthogonal decomposition (POD) is a popular model order reduction approach [1]. Based on the solutions of the FE model for different values of parameters (called snapshots), the POD approximates the solution of the FE model in a reduced basis. The parametric FE model is projected onto a reduced basis, decreasing the order of the numerical model to be solved. Another approach consists in constructing a metamodel by interpolation of the parametric FE solution expressed into a reduced basis. A hybrid approach has also been proposed combining the POD with an interpolation-based method such as the radial basis functions (RBF) interpolation method [2]. Moreover, when the problem involves geometric parameters, an adaption of the FEM mesh is required. Remeshing the geometry at each modification can deteriorate the precision of the FEM model by introducing significant numerical noise. A solution to preserve the problem's consistency is to deform the initial mesh using the RBF interpolation [3][4]. The method consists of imposing a displacement on a set of nodes and interpolating the remaining nodes' position. Then, a metamodel of a FE solution taking into account geometric and electric parameters, can be defined by using the POD approach and the RBF interpolation. In this work, we propose to build a metamodel of a nonlinear magnetostatic parametric problem involving geometric parameters using the POD-RBF approach. The study case is a single-phase EI inductance, with one electrical parameter and two geometrical parameters. An initial geometry is meshed with 36859 elements; the number of nodes is 18553. The geometric parameters are the thickness of the airgap varying from 0.1 mm to 0.9 mm, the width of the central column of the magnetic core varying from 20 mm to 40 mm. The phase current varying from 0 to 50 A is also a parameter of the model. The mesh deformation is modeled using the RBF interpolation with a multi-quadratic function for two consecutive times to consider the two variating geometric parameters. The results of the original FEM model are taken as a reference. The POD-RBF approach is applied to the parametric FEM model. The method consists of first gathering a set of data, using the method of snapshots, by solving the original FEM model for a set of inputs. The snapshots matrix is decomposed using the singular value decomposition (SVD). The left orthogonal matrix of the SVD decomposition corresponds to the reduced basis, and the right orthogonal matrix is composed of the FE solution projected into the reduced basis for each snapshot parameter. An RBF interpolation is then performed on the right orthogonal matrix to interpolate the solution expressed in the reduced basis for the new inputs set. Finally, an approximation of the FE solution is defined by projecting the reduced solution into the FE basis. The choice of snapshots is made using a greedy algorithm [2], which consists of an iterative process. The metamodel is then improved at each iteration until the required precision is reached. The stop criterion to the greedy algorithm is set to an error between the FE solution and the metamodel solution lower than 0.1%. The evolution of error represented by Frobenius norm as a function of the iterations' number of the greedy algorithm is represented in "Fig. 1". In order to satisfy the selected criterion, 158 snapshots are required. The magnetic flux cartography as a function of phase current and the central column width for an airgap width fixed at 0.1 mm is represented in "Fig.2". The global mean absolute error between the flux cartographies obtained using the metamodel and the original FEM model is 1.7%, while the maximal local error is 4.96%. Then, the metamodel of the parametric FE solution can be used in an optimization process for different applications or to be coupled with other numerical models in order to simulate a system.

**References:**

[1] L. Sirovich, Quarterly of applied mathematics, vol. 45, no 3, pp.561-571, (1987). [2] T. Henneron, A. Pierquin and S. Clénet, IEEE Transactions on Magnetics, vol. 56, no. 1, pp. 1-4, (2020). [3] A. de Boer, M.S. van der Schoot and H. Bijl, Conference on Computational Fluid and Solid Mechanics, vol. 85, pp. 784-795, (2007). [4] T. Henneron, A. Pierquin and S. Clénet, IEEE Transactions on Magnetics, vol. 55, no. 6, pp. 1-4, (2019).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/66ade7d4382d68fb036d292ea79e24d1.jpg)
Fig.1 Error as a function of the number of iterations

![](https://assets.underline.io/uploads/markdown_image/1/image/d0ae73399462ce7b097771fa6cc378a2.jpg)
Fig. 2 Flux cartography as a function of current and central column width

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Interior permanent magnet synchronous motor (IPMSM) are widely used for industrial and traction applications. IPMSM's are known for high efficiency, reliability, and power density. For certain applications, like traction in hybrid electric vehicles, a high torque and power density with wide range of speed is of high importance. The efficiency of the motors can be increased in two ways: (1) increasing the output torque of the motor or (2) reducing the losses of the motor. The torque of various motors has been improved by introducing dual rotor structure and dual excitation using the unused space in electrical machine [1, 2]. The authors in [3] compares number of electrical motors. A quantitative analysis of three types of dual stator flux modulation permanent magnet (PM) motors including magnetic geared motor, and vernier machine. High torque density has been achieved in [4] for double stator brushless PM motors. The design has also been optimized based on split ratio to achieve maximum torque density. Recently, a lot of research has been carried out on vernier permanent magnet motors (VPMMs). The authors proposed a dual stator VPMMs and performs a comparative analysis of proposed design with regular VPM [5]. The proposed design achieved high efficiency and torque density and lower cogging torque. Similarly, power factor based analytical and FEM analysis have been provided in [6] on dual stator VPMM. A high torque density VPMM with dual stator topology has been proposed in [7]. This draft proposes a drum type configuration of dual stator IPMSM with improvement in the torque density, rotor alignment and inner winding temperature compared to single stator IPMSM by utilizing the unused space in the motor. The new design of dual stator IPMSM with 16-poles and 72-slots in each stator are presented with the aim of improving the average torque and torque ripple for industrial and traction applications. The presented design is then compared with the commercial design of single stator IPMSM to show the direct and fair comparison. Fig. 1 (a) shows the proposed design (PD) and conventional design (CD). The CD uses tangential type rotor with rare-earth PMs mounted inside the surface of rotor. The conventional IPMSM offers a good torque capability, but this torque capability can be improved by utilizing the unused space of the rotor. For this purpose, another stator with the same number of stator slots has been added to replace the unused space of rotor resulting in a proposed dual stator IPMSM. To increase the torque of the proposed models, the inner stator is shifted by a half slot pitch to aid the flux lines of outer stator with the inner stator. As the slots of inner stator are aligned with the outer stator-teeth, hence the slot area of an inner stator is somewhat smaller compared to the outer stator-slots. The number of turns of the inner stator-slots are double compared to the outer stator-slots to reduce the current density and inner winding temperature. Due to the addition of an inner stator, the cup rotor is the suitable. However, the cup type rotor configuration decreases the rotor alignment because of only one bearing towards the rotor shaft. Hence, in this draft, the drum type rotor is introduced comprising of two bearing on both sides of stator to support the inner stator shaft and to improve the alignment of the rotor. Furthermore, to reduce the inner winding temperature, the curve-windows are introduced in the drum configuration for the air inlet to reduce the inner winding temperature as presented in Fig. 1 (b). The analysis is performed on the conventional and proposed designs to show the competency of the proposed model. The Fig. 2 (a), and (b) shows the back EMF of the inner and outer stator. Moreover, the back EMF of the conventional design is identical to the outer stator back EMF of the proposed model due to same geometry of outer stator. Moreover, the Fig. 2 (b) shows the substantial improvement in the average torque and torque ripple of the proposed model compared to the conventional design.

**References:**

[1] A. Toba and T. A. Lipo, "Generic torque-maximizing design methodology of surface permanent-magnet vernier machine," vol. 36, no. 6, pp. 1539-1546, 2000. [2] F. Zhao, T. A. Lipo, and B. Kwon, "Magnet flux focusing design of double stator permanent magnet vernier machine," in Proc. COMPUMAG, 2013. [3] C. Liu, K. Chau, J. Zhong, W. Li, and F. J. I. T. o. a. s. Li, "Quantitative comparison of double-stator permanent magnet vernier machines with and without HTS bulks," vol. 22, no. 3, pp. 5202405-5202405, 2011. [4] Y. Wang, M. Cheng, M. Chen, Y. Du, and K. J. I. E. p. a. Chau, "Design of high-torque-density double-stator permanent magnet brushless motors," vol. 5, no. 3, pp. 317-323, 2011. [5] Y. Gao, R. Qu, D. Li, H. Fang, J. Li, and W. J. I. T. o. M. Kong, "A novel dual-stator Vernier permanent magnet machine," vol. 53, no. 11, pp. 1-5, 2017. [6] Y. Shi and T. J. I. T. o. M. Ching, "Power Factor Analysis of Dual-Stator Permanent Magnet Vernier Motor with Consideration on Turn-Number Assignment of Inner and Outer Stator Windings," 2020. [7] N. Baloch, B.-i. Kwon, and Y. J. I. T. o. M. Gao, "Low-cost high-torque-density dual-stator consequent-pole permanent magnet Vernier machine," vol. 54, no. 11, pp. 1-5, 2018.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/e08dd9b8ef2a303357bf995cb72dbe36.png)
Fig. 1. (a) Geometry of the proposed and conventional designs; (b) Proposed dual stator drum type rotor; (c) Configuration of the inner and outer stator with the inverter.

![](https://assets.underline.io/uploads/markdown_image/1/image/5d21bfb823abd43232cbdc236793432a.jpg)
Fig. 2. Back EMF of the (a) Inner Stator; (b) Outer Stator; and (c) torque comparison.

 Dual Stator Drum Rotor Interior Permanent Magnet Synchronous Motor with Improvement in Torque Density, and Rotor Alignment

I. Introduction Conventional radial flux permanent magnet machines (RFPMMs) and axial flux permanent magnet machines (AFPMMs) are always the research hotpot high performance drive systems. RFPMMs have simple structure and magnetic circuit, which makes them easy to design and analyze [1]. As for the AFPMMs, they are paid attention for their compactness [2]. The merit makes them more suitable for direct-drive system [3][4]. However, for some special applications such as industrial robotic arms, machines with higher torque density are required [5]. In this case, the axial-radial flux permanent magnet machines (ARFPMMs) may meet the requirements. Compared with the AFPMM and RFPMM, the proposed ARFPMM has the merits of compactness, high utilization of windings and high torque density. In this paper, The structures of ARFPMM, AFPMM and RFPMM are introduced. A detailed performance comparison between the three topologies is also conducted. 3D models of different topologies are built and 3D finite element method (FEM) is used to analyze them. The performances such as back electromotive force (EMF), torque density, and torque ripple are compared to specify the best topology for particular applications. II. Three Topologies In order to exclude factors which may affect the comparison results, some parameters of the machines are kept consistent. The inner/outer diameter and axial length of three machines are the same. As shown in Fig. 1, the pole/slot combinations of three topologies are all 20 poles 24 slots. In addition, toroidal windings are also adopted in the RFPMM to keep consistent with the AFPMM and ARFPMM. They are all placed in the opening slots. Permanent magnets which are mounted on the surface of rotors are arranged as Halbach array. It can suppress the harmonics of air-gap flux density [6]. Each pole pair contains six segments, so the angle of magnetization direction between permanent magnets is 60°. Fig. 1 shows that the numbers of rotors of three machines are different. The RFPMM and the AFPMM has a radial rotor and two axial rotors, respectively, while the ARFPMM has a radial rotor and two axial rotors. It makes the active part of windings of the ARFPMM the longest and that of the RFPMM the shortest. Those differences lead to different performances. III. Operation Principles of Three Topologies Due to the different configurations, the equivalent magnetic circuits of three machines are different. The magnetic flux of RFPMM goes from rotor to stator in radial direction. There are circumferential and radial components of magnetic flux in the stator core. The electromagnetic torque is generated by the reaction between the radial magnetic field and the outside components of windings. As for the AFPMM, the magnetic flux goes through air gap in axial direction. There are circumferential and axial components in the stator core. The electromagnetic torque is due to the reaction between the axial magnetic field and the radial components of windings. The magnetic flux of ARFPMM is the combination of that of RFPMM and AFPMM. It flows in the stator core in circumferential, radial, and axial direction. Similarly, the electromagnetic torque is also the combination of that of RFPMM and AFPMM. IV. Performance Comparison The three machines are analyzed through FEM. Considering the ending effect, a 3D model is also built for the RFPMM. Fig. 2(a) shows the back EMFs of three machines obtained through 3D finite element analysis. The amplitude of ARFPMM is the largest, while that of RFPMM is the smallest. The waveforms show that the distortion of back EMF in the ARFPMM is the most serious. However, as shown in Fig. 2(b), the largest harmonic of back EMF in the ARFPMM is the third harmonic, which can be neglected due to the star connection of three phase windings. V. Conclusion This paper has described the structures and operation principles of ARFPMM, AFPMM, and RFPMM. The results of 3D FEA show that the proposed ARFPMM has the highest torque density. Because the slots and PMs of RFPMM are all rectangular, while the PMs of AFPMM are fan-shaped and slots are rectangular, the percentage of torque ripple to average torque of the RFPMM is the largest. The ARFPMM is equipped with the characteristics of both AFPMM and RFPMM. Thus, the percentage of torque ripple of ARFPMM is in the middle. Acknowledge This work is supported in part by NSFC (52077186&51677159), in part by STIC of Shenzhen Municipality (JCYJ20180307123918658), and in part by ITF (ITP/027/19AP), HK.

**References:**

[1] H. Zhao, C. Liu, Z. Song and W. Wang, "Exact Modeling and Multi-Objective Optimization of Vernier Machines," in IEEE Transactions on Industrial Electronics, early access. [2] M. Aydin, Surong Huang and T. A. Lipo, "Torque quality and comparison of internal and external rotor axial flux surface-magnet disc machines," in IEEE Transactions on Industrial Electronics, vol. 53, no. 3, pp. 822-830, June 2006. [3] C. Liu, K. T. Chau, and C. H. T. Lee, "A Critical Review of Advanced Electric Machines and Control Strategies for Electric Vehicles," in Proceedings of the IEEE, early access. [4] M. Wang, C. Tong, Z. Song, J. Liu and P. Zheng, "Performance Analysis of an Axial Magnetic-Field-Modulated Brushless Double-Rotor Machine for Hybrid Electric Vehicles," in IEEE Transactions on Industrial Electronics, vol. 66, no. 1, pp. 806-817, Jan. 2019. [5] A. Dwivedi, S. k. Singh and R. K. Srivastava, "Analysis and Performance Evaluation of Axial Flux Permanent Magnet Motors," in IEEE Transactions on Industry Applications, vol. 54, no. 2, pp. 1765-1772, March-April 2018. [6] Z. Song, C. Liu, K. Feng, H. Zhao and J. Yu, "Field Prediction and Validation of a Slotless Segmented-Halbach Permanent Magnet Synchronous Machine for More Electric Aircraft," in IEEE Transactions on Transportation Electrification, vol. 6, no. 4, pp. 1577-1591, Dec. 2020.

**Images:**

Fig. 1 Topologies of three machines. (a) ARFPMM, (b) AFPMM, (c) RFPMM. 1-Stator, 2-Windings, 3-Radial PMs, 4-Radial back iron, 5-Axial PMs, 6-Axial back iron.

Fig. 2 Back EMF and torque analyzed through 3D FEM. (a) Back EMF, (b) Fast Fourier transform of back EMF, (c) Electromagnetic torque.

 Comparative Analysis of Axial-Radial, Axial, and Radial Flux Permanent Magnet Machines

I.Introduction The rare-earth permanent magnet (RE-PM) machines are ideally suited for electric vehicles (EVs) and hybrid electric vehicles (HEVs) due to high power density and efficiency. However, with the accelerated pace of industrialization, the RE-PMs resources are becoming less and less, and the price of RE-PMs are becoming higher and higher. Less resources and higher price have restricted the application of REPM machines. Therefore, the less-rare-earth PMSMs has become a research issue [1]. This paper proposes and designs a novel less-rare-earth spoke-type PM machine for EVs applications, where a parallel placement of the rare-earth PM (SmCo) and non-rare-earth PM (ferrite) is involved. The rotor key parameters of traditional spoke-type PM machine and novel spoke-type PM machine are optimized based on finite-element analysis (FEA) and genetic algorithm (GA). In addition, the electromagnetic performances are compared between two optimized spoke-type PM machines by the finite-element analysis (FEA) model. Finally, the results of FEA are confirmed by the experiments on a 50kW traditional spoke-type PM machine. The contribution of this paper is to propose a novel low cost spoke-type PM machine replacing PM machine in EVs applications. II.Spoke-type PM Machine with Different Rotors Fig. 1 (a) shows the 12-slot stator core and winding distribution. The traditional spoke-type PM rotor and novel low cost spoke-type PM rotor are shown in Fig. 1(b) and Fig. 1(c). In addition, the basic parameters are listed in Table I, which are the same for two machine stator diameter, axial length and air-gap length. In order to obtain better performance, the FEM coupled with multi-objective GA is adopted to optimize two machines, and the key parameters of the two machines rotor are listed in Fig. 3 and Table II. Fig. 4 shows the optimization results of average torque and torque ripple for two machines. It can be seen that the novel spoke-type PM rotor and traditional spoke-type PM rotor can obtain similar torque. However, the novel spoke-type PM rotor can obtain lower torque ripple. III.Electromagnetic Performance Comparison and Analysis Fig. 5 shows the two machines no-load distribution of air-gap flux density and its harmonic content. Furthermore, FE predicted average torque and torque ripple under rated current are shown in Fig. 6. Fig. 7 shows the variation of average torque with phase current for two machines. It can be seen that the novel spoke-type PM rotor has similar torque compared with the traditional spoke-type PM rotor, but it has the lower torque ripple. Considering the limit of dc voltage (540V), the efficiency contour maps and torque-speed curves of two machines under rated current are shown in Fig. 8. The measured back-EMF waveforms of traditional spoke-type PM machine at rated speed is shown in Fig. 9, and the measured average torque versus phase current is shown in Fig. 10. In addition, the tested current waveform is shown in Fig. 11. The novel spoke-type PM machine will be supplemented in the later formal paper. IV.CONCLUSION In this paper, a novel low cost spoke-type PM machine based on traditional spoke-type PM machine is proposed and designed for EVs. The rotor key parameters of two machines are optimized based on GA. In addition, the electromagnetic performance of two machines are compared, including flux density, torque, torque ripple, efficiency and cost. Both the simulation and experimental results demonstrate the effectiveness of the proposed machine and the design optimization method.

**References:**

X. Zhu, W. Zhu, L. Quan and W. Gu,. IEEE Trans. on Energy Convers., vol. 34, pp. 1178(2019)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/d2547d5d49efed1fb451a36711a17887.jpg)
Fig.1

 Design and Optimization of a Novel Low-Cost Hybrid-Pole Rotor for Spoke-type Permanent Magnet Machine

Fractional-slot concentrated-winding permanent magnet synchronous motors (FSCW-PMSMs) have been widely used in the field of electric vehicles (EVs) due to their high-power density, high efficiency, short end turns, low cogging toque, good flux weakening capability and so on [1]. However, these machines usually exhibit high amount of sub-harmonic spatial fields, which will lead to high rotor eddy current loss, particularly in NdFeB magnets. This part of loss will increase significantly as the speed goes up, causing the rising temperature and hence the potential irreversible demagnetization of NdFeB magnets [2]. Aiming at the reduction of magnet loss, a hybrid rotor structure FSCW-PMSM is proposed in the paper. The key is to introduce additional ferrite magnets in D and Q axis magnetic flux paths, respectively. Compared with traditional all-NdFeB structure, the proposed design can effectively curb the magnet loss due to the low conductivity of ferrite magnets. Fig.1 shows the comparison of existing all-NdFeB rotor structure and proposed hybrid rotor design for 12 slots/8 poles IPMSM with only the cross section of one pole illustrated, from which it can be seen that part of the tangentially magnetized NdFeB magnets are replaced with ferrite magnets and additional ferrite magnet is introduced to fill up the air at the bottom between the two tangentially magnetized magnets. Fig.2 shows the comparison of total magnet loss and the magnet loss in the tangentially magnetized magnets between existing and proposed FSCW-PMSM with respect to rotor speed. It can be seen that as the rotor speed goes beyond the rated value of 3700rpm with field-weakening operation, the total magnet loss of traditional all-NdFeB structure increases rapidly with the contribution mainly coming from tangentially magnetized NdFeB magnets. It can be well explained by the fact that as a result of increasing leading angle of current vector against Q-axis under field-weakening operation, more and more armature harmonic flux will cross-link with the tangentially magnetized NdFeB magnets and hence lead to high magnet loss in them. Despite of the significant loss reduction by partially replacing the tangentially magnetized NdFeB with ferrite magnets, the magnet loss in the radially magnetized NdFeB magnets still remain high. Therefore, additional ferrite magnets should also be introduced in the d-axis flux-path just as shown in Fig.1. As a result, the introduced ferrite magnets can not only effectively curb the magnet loss in the radial magnets but also contribute to the air-gap flux density and hence improve the output torque. Although the proposed design effectively diminishes the magnet loss, it inevitably leads to the decrease of output torque due to the reduced usage of NdFeB magnets as well as increased torque pulsation. Further optimization is still needed for the proposed structure to improve its competitive edge against the traditional design. Fortunately, the hybrid rotor structure leaves more scope for the improvement of saliency ratio so that it can be possible to enhance the reluctance torque. Furthermore, by adopting auxiliary slots design on the rotor surface, the torque pulsation is effectively curbed and the magnet loss in the radially magnetized NdFeB magnets is further reduced. As a result of the Finite Element analysis, compared with traditional NdFeB IPMSM with the maximum torque of 172N.m, the prototype with the proposed design can accomplish a maximum torque of 158N.m, nearly 92% of that of traditional one due to the contribution of enhanced reluctance torque through optimization, as well as reduced torque ripple from 10.6% to 6.9%, while the total cost of magnets is reduced by 39%. As a conclusion, the proposed hybrid rotor structure can not only solve the problem of excessive magnet loss of traditional NdFeB motor at high speed, but also present a cost-effective design approach for IPMSM for EV applications with less usage of expensive NdFeB magnets as well as enhanced reluctance torque capability.

**References:**

[1] Ayman M. EL-Refaie, Fractional-Slot Concentrated-Windings Synchronous Permanent Magnet Machines: Opportunities and Challenges. IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 57, NO. 1(2010) [2] Xinggang Fan, Bin Zhang, Ronghai Qu, Comparative Thermal Analysis of IPMSMs WithIntegral-Slot Distributed-Winding (ISDW) and Fractional-Slot Concentrated-Winding (FSCW) for Electric Vehicle Application. IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 55, NO. 4(2019)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/555c3005c2af53d40b2f3bef59b8e5c8.jpg)
Fig.1 Comparison of the two rotor structures. Left. The traditional rotor structure. Right. The hybrid rotor structure.

![](https://assets.underline.io/uploads/markdown_image/1/image/dc7fc0c12d1d9743419c8c6e2c07534a.jpg)
Fig.2 Comparison of magnet eddy current loss (MECL) between the traditional and hybrid rotor structure in full speed range

 A New IPMSM with Hybrid Rotor Structure for Electrical Vehicle with Reduced Magnet Loss

I. INTRODUCTION In order to achieve reliable and high-performance control of permanent magnet synchronous machines (PMSMs), the accurate model of electromagnetic characteristics of PMSMs under different load conditions is critical [1]. The conventional method is to use the finite element analysis (FEA) to derive the magnetic model of PMSM [2]. However, this method requires accurate geometric parameters and material information of the machine and the results must be validated by experiments. Therefore, it is desirable to obtain accurate machine model from measurable terminal signals. To date, considerable research work has been conducted to deal with the flux linkage identification problem for PMSMs [3]-[5]. However, either magnetic saturation or spatial harmonics effects have been ignored during the flux linkage identification in the existing methods. Therefore, this paper proposes a novel two-step flux linkage identification method for PMSMs considering both magnetic saturation and spatial harmonics. Step one is an offline estimation of the flux linkages considering magnetic saturation based on genetic algorithm (GA) , while the spatial harmonics effect has been mitigated compared with the existing method. In the second step, a state-space observer is employed to estimate the spatial harmonics online with the estimated offline data in step one. In addition, Kalman filter is applied in dq-axis inductances harmonics estimation to reduce the influence of noise and high order harmonics in the motor drive system. Therefore, the proposed method achieves to consider both magnetic saturation and spatial harmonics for PMSM flux linkage identification. The proposed approach is validated on both the surface-mounted and interior PMSMs. II. Magnetic Saturation Modeling and Parameter Estimation using GA To consider magnetic saturation in the machine model, the dq-axis flux linkages are modeled using the quadratic equation as in (1). Ψd(id, iq)= α0+α1id+α2iq+α3id2+α4iq2+α5idiq; Ψq(id, iq)= β0+β1id+β2iq+β3id2+β4iq2+β5idiq. (1) Where id, iq, Ψd, Ψq are dq-axis currents and flux linkages; α0 denotes the fundamental PM flux linkage; β0 is adopted to generalize the quadratic form of the model and compensate the offset for Ψq; α1 and β1 denote self-inductances in the dq-axis; α2,…, α5 and β2,…, β5 denote the cross-coupling effect and the influence of magnetic saturation. There are 12 parameters to be estimated, while the rank number is two. Therefore, the coefficients of flux linkages equations cannot be solved directly as it is rank-deficient. To solve this issue, a GA based estimation approach is proposed. Suppose that, for a given pair of (id, iq), the (ud, uq) under a certain speed ωm during the period T seconds (T=2πn/ωm, where n=1,2, …,N represents the number of rotations) are recorded in order to mitigate the influence of spatial harmonics. To estimate the coefficients of flux linkages, two fitness functions are designed as follows: Fd = ∑i=1..I∑j=1..J[α0 + α1id(i) + α2iq(j) + α3id2(i) +α4iq2(j) + α5id(i)iq(j)-Ψd(id(i), iq(j))]; Fq = ∑i=1..I∑j=1..J[β0 + β1id(i) + β2iq(j) + β3id2(i) +β4iq2(j) + β5id(i)iq(j)-Ψq(id(i), iq(j))], (2) where i=1,…, I and j=1,…, J are directories of (id, iq). Based on (2), the coefficients of dq-axis flux linkages at each pair of (id, iq) can be obtained by using GA to achieve the minimization of Fd and Fq. III. Online Parameter Estimation Considering Spatial Harmonics Considering spatial harmonics, dq-axis voltage equations of a PMSM can be written as (3) under the steady state. ud = Rid-ωeLqq,0iq+ωeΨd6sin(6pθ)+sin(6pθ)(-6pωeLdd,6id-ωeLdq,6id)+cos(6pθ)(6pωeLdq,6iq-ωeLqq,6iq)+εd ; uq = Riq+ωeLdd,0id+ωeΨd1+ωeΨq6cos(6pθ)+sin(6pθ)(-6pωeLqq,6iq-ωeLdq,6iq)+cos(6pθ)(6pωeLdq,6id-ωeLdd,6id)+εq, (3) where ud, uq are dq-axis voltages; θ is the electrical rotor position; p is the number of pole pairs; R is the resistance; ωe is the rotor speed; Ψd6 and Ψq6 are the magnitudes of the 6th permanent magnet (PM) flux-linkage harmonics in dq-axis. Ldd,0,Lqq,0,Ldd,6,Lqq,6 are the magnitudes of fundamental and the 6th harmonic of self-inductances in the dq-axis; Ldq,6 is the magnitude of the 6th harmonic of mutual-inductance; εd and εq are high order harmonics and high frequency noise components in the dq-axis. The harmonic magnitudes of the PM flux linkage can be easily obtained by using FFT analysis of the back-electromotive force. To achieve an online identification of the dq-axis inductances harmonics, a state observer is employed. The state observer model is firstly built based on (3) to get the magnitude of the 6th harmonic of the self-inductance and mutual-inductance in the dq-axis. The state vector is built as: x=[ Ls6cos(6pθ); Lm6sin(6pθ); -6pωeLs6sin(6pθ); 6pωeLm6cos(6pθ)], (4) with Ls6=Ldd,6 and Lm6=Ldq,6. The estimated dq-axis flux linkage maps are shown in Fig.1 and Fig.2. Detail analysis and experimental results will be given in the full paper. IV.CONCLUSION This paper proposes a novel two-step method to identify flux linkages in PMSMs considering both the effects of magnetic saturation and spatial harmonics during the estimation process. In step one, GA is used to estimate the magnetic saturation coefficients of flux linkage. With the results obtained from GA, the Kalman filter based online method is proposed to estimate the spatial harmonics to achieve an accurate estimation of the flux linkage map. The co-simulation results show that the proposed approach is effective for flux linkage estimation.

**References:**

[1] E. Armando, R. I. Bojoi, P. Guglielmi, G. Pellegrino and M. Pastorelli, in IEEE Transactions on Industry Applications, vol. 49, no. 5, pp. 2116-2125, Sept.-Oct. 2013. [2] N. Bianchi and S. Bolognani, in Proc. Int. Conf. IEEE Ind. Appl., Oct. 1998, pp. 27–34. [3] Z. Li, G. Feng, C. Lai, W. Li and N. C. Kar, in IEEE Transactions on Transportation Electrification, vol. 5, no. 4, pp. 1430-1440, Dec. 2019. [4] Q. Chen, W. Zhao, G. Liu, Z. Lin, in IEEE Transactions on Industrial Electronics., vol. 66, no. 2, pp. 944-955, Feb. 2019. [5] K. Liu, J. Feng, S. Guo, L. Xiao and Z. Zhu, in IEEE Transactions on Industrial Informatics, vol. 14, no. 2, pp. 556-568, Feb. 2018.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/cdb1bbe318f847c1cdda6bffcdd7b846.jpg)
Fig. 1. SPMSM q-axis flux linkages estimation results.

![](https://assets.underline.io/uploads/markdown_image/1/image/a4a335ed2eb49867f6497eb0dde4319d.jpg)
Fig. 2. IPMSM q-axis flux linkages estimation results.

 A Novel Two-Step Flux Linkage Identification for PMSMs Considering Magnetic Saturation and Spatial Harmonics

I. INTRODUCTION The permanent magnet (PM) machine has attracted wide attention in many application areas such as electric vehicles (EVs) for high output torque, high efficiency and high reliability. For the traditional PM machine, its maximum output torque is constricted by the current density supplied from the power source, which may lead to the insufficient output torque, especially at low speed [1-2]. Furthermore, in the direct-drive PM machines, the number of the pole-pair is usually relatively high to maintain the large output torque within the low speed range. However, due to the high pole-pair number, the PM dosage and the production cost are both increased accordingly. Therefore, the consequent-pole structure seems to be an appropriate choice to reduce the rare-earth materials and increase the pole-pair number in the limited space [3-4]. In this paper, a new consequent-pole segmented dual-stator permanent magnet (CPSDSPM) machine is proposed for some high-torque & low-speed applications. Compared with the traditional PM machines, this proposed CPSDSPM machine, which reasonably utilizes the internal space of the intermediate rotor, realizes the low-demagnetization, the low dosage of the rare earth, the flexible control strategy and the large output torque by a particular coupling effect between two stators. II. MACHINE TOPOLOGY As shown in Fig. 1(a), the topology of the proposed CPSDSPM machine mainly includes three parts: the external stator, the internal stator and the intermediate rotor. First, the external stator is divided into three sub-parts, in which the phase difference of two adjacent sub-parts of the external stator is a complete 360 electrical angle. Second, for convenient maintenance and low cost, the internal stator located inside the intermediate rotor is designed as an integrative structure. Third, the intermediate rotor, which adopts 12 V-type PMs and 6 iron cores, is sandwiched between the external stator and the internal stator. Forth, all the PMs and the iron cores located in the intermediate rotor are alternately distributed as the sequence of N-iron-N-S-iron-S. Fifth, two sets of windings, which are mounted in the external and the internal stator respectively, are adopted in this proposed topology. The five-phase winding is composed of three sub-windings connected in series. It's placed in the external stator and designed to produce the main output torque. The three-phase winding, which is housed in the internal stator, can not only be used as the field coil, but also offers an additional output torque directly when it's needed. III. WORKING PRINCIPLE According to the analysis result mentioned above, the flux paths in the proposed CPSDSPM machine are summarized in Fig. 1(b). These flux paths can be mainly divided into three types. First, the flux path 1 includes two PMs in different polarity. It can be seen that the flux path 1 exists in the external stator and the intermediate rotor. Second, the flux path 2 includes one PM and one iron core. Compared with the flux path 1, the ability of the PM excitation in the flux path 2 is weakened, but the total magnetic conductance is improved. Third, the flux path 3 needs to cross both the inner air gap and the outer air gap. It can be easily obtained that the flux density of both iron cores and PMs in the outer air gap can be adjusted by the three-phase winding. Therefore, the three-phase winding can also be regarded as the supplemental excitation source for the external stator. At last, the flux directions of these two iron cores marked in Fig. 1(b) are different. Because of the downward direction of the flux, the iron core on the left can be regard as S-pole. Similarly, the iron core on the right can be defined as N-pole. Therefore, these two iron cores can be regarded as one pole-pair equivalently in the intermediate rotor. IV. ELECTROMAGNETIC PERFORMANCE ANALYSIS First, the no-load phase back-EMF waveforms at rated speed of the five-phase winding presented in Fig. 2(a) are calculated by finite element method (FEM). It can be seen that the back-EMF of the proposed machine is extremely sinusoidal owing to the segmented structure, which can contribute to the electromagnetic performance of the external stator of the proposed CPSDSPM machine. Second, for the proposed machine, the cogging torque is mainly caused by the external stator. As shown in Fig. 2(b), it can be obviously found that the total cogging torque generated from the whole machine is decreased considerably, compared with the single part of the external stator. Third, due to the segmented structure, it can be obtained in Fig. 2(c) that the torque ripple generated by the external stator is quite low (around 0.4%). Furthermore, under the vector control strategy, the maximum total output torque waveform generated by both the external stator and the internal stator is shown in Fig. 2(c). V. CONCLUSION In this paper, a new CPSDSPM machine is proposed and analyzed. First, the configuration is introduced. Second, the working principle and the main flux paths of this machine are analyzed and classified. Third, the electromagnetic properties, such as the no-load phase back-EMF waveforms, the cogging torques and the output torques are all calculated by FEM, which can verify the correctness of the theoretical analysis and the superiority of the proposed CPSDSPM machine.

**References:**

[1] S. Niu, S. L. Ho and W. N. Fu, "A Novel Direct-Drive Dual-Structure Permanent Magnet Machine," IEEE Trans. Magn., vol. 46, no. 6, pp. 2036-2039, June 2010. [2] H. Yang et al., "Novel Dual-Stator Machines With Biased Permanent Magnet Excitation," IEEE Trans. Energy Convers., vol. 33, no. 4, pp. 2070-2080, Dec. 2018. [3] W. Li, T. W. Ching and K. T. Chau, "A New High-Temperature Superconducting Vernier Permanent-Magnet Machine for Wind Turbines," IEEE Trans. Appl. Supercond., vol. 27, no. 4, pp. 1-5, June 2017. [4] H. Hua, Z. Q. Zhu and H. Zhan, "Novel Consequent-Pole Hybrid Excited Machine With Separated Excitation Stator," IEEE Trans. Ind. Electron., vol. 63, no. 8, pp. 4718-4728, Aug. 2016.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/b99f8a29e0d766e89b9567b2626da29f.jpg)
Fig. 1. CPSDSPM machine. (a) Topology. (b) Magnetic paths.

![](https://assets.underline.io/uploads/markdown_image/1/image/fd107b6246fda9106441b1c62138c772.jpg)
Fig. 2. Electromagnetic field analysis of the proposed CPSDSPM machine. (a) No-load phase back-EMF waveforms. (b) Cogging torques. (c) Output torques.

 Design of a New Consequent-Pole Segmented Dual-Stator Permanent Magnet Machine

– Electric motors, magnetic resonance imaging and many other modern applications are built out of systems of multiple permanent magnets, which are arranged so that the magnetic field that they produce has the required properties. Here, starting from a given magnetic field, we invert the design procedure to predict a matching magnetic structure using deep learning. A convolutional neural network (CNN) learns the mapping from a desired magnetic field to a simple magnetic structure with an error of 0.22%. We further extend the method to work with multiple permanent magnets. Introduction Systems of multiple magnets are the basis for many modern applications like magnetic resonance imaging, frictionless bearings in flywheels or electric motors. In these applications permanent magnets of different shapes and magnetic properties are assembled to produce a magnetic field, whose field strength and homogeneity match the requirements for the given application. In order to provide better medical devices, store energy or pave the way to green mobility with electric vehicles, optimized structures of multiple magnets are essential. Traditionally, the design procedure for producing optimized structures of permanent magnets is by selecting a known design that produces a field close to the desired field and then optimizing the structure through modeling. As an example, consider the Halbach cylinder [1] where in real-world applications manufacturing tolerances mean that its actual field will not be exactly equal to the analytically calculated field. In this case one can e.g. use a genetic algorithm to incorporate the manufacturing variations of the single magnetic segments in a traditional design optimization approach [2]. However, this optimization approach is limited in scope, as it relies on a design structure to start from and is computationally intense. Problem Here we consider the general inverse problem of determining a magnetic structure that generates a desired magnetic field. We investigate this through the use of a deep neural network, with the network learning the mapping of Maxwell’s equations in an inverse manner. We want to predict the properties of a system of multiple permanent magnets from a part of the magnetic field, as illustrated in Fig. 1. We consider complicated structures of permanent magnets formed out of different simple geometries, e.g. prisms, tetrahedrons or cylindrical tiles. Previously, artificial intelligence has been used to approximate the solution of Maxwell’s equations for electromagnetic structures [3], in order to calculate the magnetic field in a computationally cheap manner. Here, however we invert this problem to use deep learning to predict magnet structures from the magnetic field. A major challenge is to find an invariant representation for systems of a varying number of magnets. Each of the permanent magnets is specified by its geometric appearance, location, and magnetic properties. The output space easily explodes with an increasing number of magnets, when a poor representation is chosen. Additionally, the problem becomes complex when predicting the properties of the magnetic system from a magnetic field, which does not contain the structure itself. There may exist several solutions for the given desired field. Method The magnetic field in a so-called “design area” is calculated point-wise, which results in a matrix of field values in each of the spatial dimensions. That tensor is fed into a neural network of multiple convolutional layers. As baseline model, we use a convolutional neural network (CNN) inspired by the ResNet architecture [4]. The output shape of the structural design depends heavily on the chosen magnetic system representation. The model parameters are then updated by backpropagation of the difference between the ground-truth and the predicted target structure. To train such models of millions of tunable parameters artificial neural networks require a large amount of data. We produce accurate field-structure-pairs fast and efficiently with MagTense [5], an open-source magnetostatic and a micromagnetism calculation framework. The learned network parameters are then able to infer magnetic structures of unseen magnetic fields. Initially we limit the input space to be a two-dimensional magnetic field by setting the z-component of the single magnet magnetizations to 0 T. We further only use prisms with a constant height and omit rotations of the geometries. The primarily two-dimensional “design area” lies in the plane, which cuts the single tiles in their center point. Results We show in [6] that the prediction of the properties of a single permanent magnet from its magnetic field has an average error rate of 0.22%, which is much below magnet manufacturing tolerances. Predictions of the trained CNN are hardly distinguishable from its ground-truth, when given a previously unseen magnetic field, as exemplified in Fig. 2. We discuss the error rates in extending this to multiple magnets and finally discuss how to represent magnetic structures with multiple permanent magnets of different shapes.

**References:**

[1] K. Halbach, Nuclear Instruments and Methods, Vol. 169, pp. 1-10 (1980) [2] K.K. Nielsen et al., Journal of Magnetism and Magnetic Materials, Vol. 514, p. 167175 (2020) [3] A. Khan, V. Ghorbanian and D. Lowther, IEEE Transaction on Magnetics, Vol. 55, no. 6 (2019) [4] K. He et al., CVPR, ”Deep Residual Learning for Image Recognition” (2016) [5] R. Bjørk and K.K. Nielsen, www.magtense.org, https://doi.org/10.11581/DTU:00000071 (2019) [6] S. Pollok, R. Bjørk and P.S. Jørgensen, “Inverse Design of Magnetic Fields using Deep Learning”, Under review (2021)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/67b4dca71c7e6adebff4b6367ae16294.jpg)
Fig. 1. Deep learning pipeline for inverse design of magnetic structures. The magnetic field in a so-called "design area" serves as input to the neural network, which predicts a systems of permanent magnets required to produce such a field.

![](https://assets.underline.io/uploads/markdown_image/1/image/f82853dcf9e607e8bc8bf12268bbdd30.jpg)
Fig. 2. Prediction of a trained convolutional neural network from an unseen magnetic field. When plotting the predicted single permanent magnet it is visually indistinguishable from its ground-truth.

 Convolutional Neural Networks for Inverse Design of Magnetic Structures

I. INTRODUCTION Hybrid-magnetic-circuit variable flux memory machine (HMC-VFMM) [1] can combine the advantages of wide flux adjusting range in parallel type [2] and beneficial on-load demagnetization withstand capability in series type [3]. The rotor pole number plays an essential role in the electromagnetic performances of conventional PM machines [4]-[6]. In addition, the design process of HMC-VFMM needs to consider not only the basic performance metrics, such as the torque capability, but also flux regulation range and on-load magnetic stability of low coercive force (LCF) magnet. Thus, it is crucial to provide and generalize a utility design guideline for choosing proper pole number for VFMM. This paper attempts to analyze the electromagnetic performances of HMC-VFMMs with different rotor pole numbers. The influence rules of different rotor pole numbers on the performance metrics of the HMC-VFMMs are then provided, and a 4-pole machine prototype is manufactured and tested to experimentally validate the theoretical analyses. II. TOPOLOGIES AND WINDING LAYOUTS OF HMC-VFMM The investigated objectives of this paper refer to HMC-VFMMs with 36-slot/4-pole, 36-slot/6-pole and 48-slot/8-pole combinations, which are all characterized by integer slot structures. The double-layer and full-pitch windings are utilized in all machine models, as shown in Fig. 1. For the parallel branch, the “U”-shaped hybrid magnets are employed. For the series branch, the HCF PMs feature a spoke-type configuration. And only A-phase winding is shown here. III. ELECTROMAGNETIC PERFORMANCE AND COMPARISON Fig. 2 (a) shows the rated-load torque waveforms of HMC-VFMMs with different rotor pole numbers under the conditions of flux-enhanced and flux-weakened states, respectively. It can be seen that with the increase of rotor pole number, the average torque shows an increasing trend, while the torque ripple decreases. These results indicate that the increase of rotor pole number can improve the torque performance to a certain extent. The variations of LCF PM working point subject to an intermediate on-load operation in HMC-VFMM are illustrated in Fig. 2 (b). The excellent on-load demagnetization withstand ability can be observed in all machines. The variations of the back-EMF fundamental magnitudes with the remagnetizing and demagnetizing currents are plotted in Fig. 2 (c). It can be seen that the increase of rotor pole number enlarges the adjusting range of the fundamental back EMF. However, higher current magnitudes are required for full magnetization in the high rotor pole cases at the same time. The detailed characteristics analysis, comparison results and experimental verification will be presented in the full paper.

**References:**

[1] H. Yang, S. Lyu, H. Lin, Z. Zhu, H. Zheng and T. Wang, "A Novel Hybrid-Magnetic-Circuit Variable Flux Memory Machine," IEEE Trans. Ind. Electron., vol. 67, no. 7, pp. 5258-5268, July 2020 [2] K. Sakai, K. Yuki, Y. Hashiba, N. Takahashi, and K. Yasui, “Principle of the variable-magnetic-force memory motor,” in Proc. Int. Conf. Electr. Mach. Syst., Tokyo, Japan, 2009, pp. 1-6. [3] A. Athavale, K. Sasaki, B. S. Gagas, T. Kato, and R. D. Lorenz, "Variable flux permanent magnet synchronous machine (VF-PMSM) design methodologies to meet electric vehicle traction requirements with reduced losses", IEEE Trans. Ind. Appl., vol. 53, no. 5, pp. 4318-4326, Sep./Oct. 2017. [4] J. T. Chen and Z. Q. Zhu, "Winding Configurations and Optimal Stator and Rotor Pole Combination of Flux-Switching PM Brushless AC Machines," IEEE Trans. Energy Convers., vol. 25, no. 2, pp. 293-302, Jun. 2010. [5] M. Ibrahim, L. Masisi and P. Pillay, "Design of Variable Flux Permanent-Magnet Machine for Reduced Inverter Rating," IEEE Trans. Ind. Appl., vol. 51, no. 5, pp. 3666-3674, Sep./Oct. 2015.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/52ccf8fa246dc06e66bbb6bc29976068.jpg)
Fig. 1. Topologies and winding layouts (A-phase) of three HMC-VFMMs.

![](https://assets.underline.io/uploads/markdown_image/1/image/8cecce5efa57003da28f93e3cdd6c375.jpg)
Fig. 2. (a) Steady-state torque waveform @ rated load. (b) Working point variations of the LCF PM in the HMC-VFMMs subject to an intermediate on-load operation. (c) Back-EMF magnitudes versus the current pulses (1500r/min).

 Influence of Rotor Pole Number on Electromagnetic Performance of Hybrid-Magnetic-Circuit Variable Flux Memory Machine

This paper proposes an efficient surrogate-assisted design optimization method based on multicriteria optimal Latin hypercube design (LHD) for multi-objective optimization of a permanent-magnet vernier machine (PMVM). By combining simulated annealing algorithm, the uniformity and orthogonality of spatial distribution of the LHD are optimized to efficiently capture data features over the optimization parameter ranges. Consequently, an accurate surrogate model with high generalization capability can be built with a small training set by using the optimal LHD. The optimal design of the PMVM is obtained by multi-objective design optimization based on the surrogate model. A prototype is made and tested to validate the proposed method. I. Introduction Due to high torque capability at low speeds, permanent-magnet vernier machines (PMVMs) are suitable for direct-drive applications such as wind power generation and electric drilling [1]. To deal with the issue of massive finite-element analysis (FEA) for multi-objective design optimization, this paper proposes a novel surrogate-assisted multi-objective design optimization method for PMVMs. A multicriteria optimal Latin hypercube design (LHD) employing simulated annealing (SA) algorithm is proposed for efficient date feature capturing that enables to use a small training dataset for obtaining an accurate surrogate model. The sparrow search algorithm (SSA)-optimized multi-output least squares support vector machine (MLS-SVM) [2] and the non-dominant sorted genetic algorithm-III (NSGA-III) [3] are combined to build the multi-output surrogate model and to perform the multi-objective design optimization. A prototype is made based on the obtained optimal design and tested for validation. II. PMVM and Sensitivity Analysis A surface-mounted PMVM with 18 slots and 15 pole pairs is studied as an example, whose topology is illustrated in Fig. 1 (a). The four optimization objectives are average torque, power factor, THD of back-EMF, and cogging torque. To improve the optimization efficiency, a global sensitivity analysis (GSA) of four optimization targets is performed using the Sobol’s method [4], the stack bars of total sensitivity coefficient (TSC) for optimization parameters are given in Fig. 1 (b), which shows the geometrical parameters that have the greatest influence on design objectives. Thus, five geometrical parameters including PM thickness (tPM) and width (wPM), yoke thickness (ty), slot width (ws), and stator split ratio (k=Rsi/Rso) are selected for optimization while stator outside diameter, axial length, air-gap length, and current density are fixed. III. Surrogate-Assisted Optimization with LHD The optimal LHD employs SA to minimize a composite indicator ψp that consists of the Morris-Mitchell index Φp and the correlation coefficient of the test factors ρ2 to guarantee the spatial distribution uniformity and orthogonality as given in (1), where ω1 and ω2 are the weight factors. Subsequently, a 270*5 LHD is built to capture the data feature over the optimization parameter ranges more efficiently as shown in Fig.2 (a), in which 20 sets of data are used for testing the regression results. minψp = ω1Φp + ω2ρ2(1) Finite element analysis (FEA) simulations are only performed within the optimal LHD to acquire the training data. The surrogate model based on MLS-SVM are trained using the data of the optimal LHD. The radial basis function (RBF) kernel is adopted for MLS-SVM and SSA is combined with MLS-SVM to find an optimal hyper-parameter set that produces a surrogate model with the highest average determination coefficient R2. The four-objectives optimization is performed by NSGA-III and an optimal design is selected from the Pareto front shown in Fig. 2 (b). A prototype is made based on the optimal design and tested, Figs. 2 (c) and (d), to validate the proposed method and the optimization. The proposed method needs only 270 FEA simulation cases during the whole design optimization procedure. For the conventional FEA-based parametric optimization method, even when only 5 levels are selected for each of the five geometrical parameters, 3125 FEA cases are required. Meanwhile, since the parameter space is discretized, the results of the conventional optimization method are likely to be locally optimal. For the method employing randomly generated LHDs [5], it requires nearly 500 FEA cases to achieve consistent R2s, which consumes 2 times the time of the proposed method. Thus, the proposed method can secure both efficient and accurate global design optimization of PMVMs. IV. Conclusions This paper proposes a novel optimal LHD-based surrogate assisted design optimization method for multi-objective optimization of PMVMs, which can perform high efficiency and high validity simultaneously. The proposed method and optimal design are validated by experiments on a prototype. Details of the design optimization and test results will be given in the full paper.

**References:**

[1] Y. Oner, Z. Q. Zhu and L. J. Wu, IEEE Trans. Ind. Electron., Vol. 63, No. 7, pp. 4105–4117 (2016). [2] S. Xu, X. An and X. Qiao, Pattern Recognit. Lett., Vol. 34, No. 9, pp. 1078–1084 (2013). [3] A. M. Usman, U. K. Yusof and S. Naim, IEEE Access, Vol. 8, pp. 76333-76356 (2020). [4] J. Yin, X. Zhu, L. Quan and C. Zhang, IET Electr. Power Appl., Vol. 11, pp. 72–79 (2017). [5] J. B. Kim, K. Y. Hwang and B. I. Kwon, IEEE Trans. Magn., Vol. 47, No. 5, pp. 1078-1081 (2011).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/a02e0d4508835ed9b436b082204c4f95.jpg)
Fig. 1 Topology of PMVM and GSA results. (a) PMVM. (b) GSA results characterized by TSCs.

![](https://assets.underline.io/uploads/markdown_image/1/image/0710c9f0c14fbc22cb16e33d2590d647.jpg)
Fig. 2 Surrogate-assisted optimization. (a) Optimal LHD. (b) Pareto front of multi-objective optimization. (c) Rotor and stator of the prototype. (d) Test platform.

 Multicriteria Optimal Latin Hypercube Design-Based Surrogate-Assisted Design Optimization for a Permanent-Magnet Vernier Machine

1. Introduction: Optimal operation of electromagnetic drive systems requires coupled analysis of electromagnetic equipment with their control units. Accurate equivalent circuit models of the electromagnetic equipment are essential and are expected to include the high-frequency eddy currents in nonlinear laminated iron cores. Model order reduction (MOR) is a powerful methodology providing reduced models based on finite element analysis. Some of the MOR approaches are capable of generating equivalent circuit models such as Cauer Ladder Network (CLN). The CLN method is one of the promising MORs proposed for eddy current problems in linear mediums [1]. It was extended to the magnetically nonlinear problems using the first-order approximation method (FO-CLN) [2]. Another extension concerns the analysis of laminated cores using homogenized eddy current field as a multi-scale MOR for linear mediums [3]. This paper makes a progress by introducing the nonlinear multi-scale MOR based on a new piece-wise linearization technique. Homogenization on its nonlinear form has been always a big challenge and is a fortiori so in multi-scale MOR. The main contribution of this work is to linearize peace-wise the nonlinear homogenization using the first-order approximation methodology of FO-CLN to find the equivalent circuit model for nonlinear laminated iron cores. 2. Multi-Scale MOR: The CLN method is a "Macroscopic-Scale" MOR, which aims for efficient electromagnetic field calculation over the whole machine on the macroscopic scale. On the other hand, "Microscopic-Scale" MOR deals with the homogenization and expressing the components of electromagnetic equipment with fine structures, such as laminated steel sheets. Fig. 1 illustrates how the microscopic-scale MOR approximates a stacked laminated-core Fig.1(a) into a homogenized bulk material Fig.1(b); see e.g. [4] for a homogenization method that considers eddy currents and skin effect in laminated stacks using the Legendre polynomials. The homogenized eddy-current field is expressed as K'a=-σ'da/dt+j0, where K', σ', a and, j0 are the homogenized stiffness and conductivity matrices, magnetic vector potential, and external current vectors, respectively. On the multi-scale MOR, the standard CLN procedure is adapted with K' and σ'. 3. FO-CLN: Using the CLN, the time-dependent electromagnetic fields in a linear eddy-current problem are decomposed into time-independent modes weighted by time-dependent coefficients. On linear case, these modes are constant and the coefficients are derived from the network currents and voltages in Fig. 1(c). When the core has nonlinear magnetic characteristics, the modes vary with respect to saturation in the core with field-dependent permeability μ(B) In the FO-CLN method, the first magnetic mode is considered as the main and only source of the core saturation. The saturation level is measured by the current h1 in Fig. 1(c). The CLN is performed at different degrees of saturation which results in parametric CLN elements as in Fig. 1(c). In Fig.1 the FO-CLN method is shown as the transition between Fig. 1(b) to Fig. 1(c). 4. Nonlinear Multi-Scale MOR: The FO-CLN methodology is applied to the multi-scale MOR since it is based on the frozen permeabilities obtained during the calculation of the first magnetic mode. More details will be given in the full paper. 5. Numerical Example: The accuracy of the proposed method is verified over a laminated inductor with sheet thickness of 0.5mm, shown in Fig. 2(a). The currents from the conventional transient finite element method and it's equivalent CLN compared with the 0th order homogenization in Fig. 2(b), demonstrates the accuracy of the proposed method.

**References:**

[1] A. Kameari, H. Ebrahimi and T. Matsuo, IEEE Trans. Magnetics., Vol. 54, no.11 (2018) [2] H. Eskandari and T. Matsuo, IEEE Trans. Magnetics., vol.56, no.2 (2020) [3] H. Eskandari, J. Gyselinck and T. Matsuo, CEFC Conference, TA_O2 (2020) [4] J. Gyselinck, P. Dular, and L. Krahenbuhl, IEEE Trans. Magnetics., vol.52, no.3 (2016)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/6912dd17a44d4e0739b952fdd708318c.jpg)
Fig.1 (a) Eddy-current problem with fine structure, (b) Nonlinear homogenized eddy-current, (c) Equivalent parametrized CLN.

![](https://assets.underline.io/uploads/markdown_image/1/image/16d36cf0e58884690f6fea3ac0a2aefa.jpg)
Fig.2 (a) Iron-core inductor (b) Transient responses to sinusoidal excitation.

Parametric Geometric Metamodel of Nonlinear Magnetostatic Problem Based on POD and RBF Approaches

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