# Multi-Objective Optimization Design of Permanent Magnet Torque Motor

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Determination of Optimization Objectives

#### 2.1.1. Structure of the PMTM

#### 2.1.2. Analysis of Torque Ripple

_{rip}is motor torque ripple, T

_{avg}is the average value of output torque, T

_{max}the maximum value of output torque, T

_{min}is the minimum value of output torque. Due to problems of meshing, the finite element software has the problem of non-convergence of a certain unit, which can cause inaccurate measurements of the maximum or minimum value. Therefore, it is necessary to analyze the cause of motor torque ripple and find out the substitute index of torque ripple. The torque formulas in permanent mag-net motor are as follows [26]:

_{cog}is cogging torque, T

_{em}is electromagnetic torque. N

_{p}is the number of pole pairs of the motor, T

_{pm}is permanent magnet torque, T

_{r}is reluctance torque. After the above analysis, it is easy to know that torque ripple is related to cogging torque, harmonics of permanent magnets, and reluctance torque.

#### 2.2. Multi-Objective Optimization

#### 2.2.1. Sensitive Analysis

_{i}is the ith optimization objective, X

_{i}is the design parameters, and N is sample size.

#### 2.2.2. Optimization of First Level

_{i}is the regression parameter vector, K(x) is kernel function. n is the number of support vector, x

_{i}is support vector, x is predicted vector, b* is the Bias factor. The structure of SVM is shown in Figure 4.

^{2}is the determining factor, l is the number of samples in the test set, ${y}_{i}$ is the true value of the ith sample, and ${\widehat{y}}_{i}$ is the predicted value of the ith sample. The value of MSE and R

^{2}can be used to judge the accuracy of SVM models. According to the above formula, MSE and R

^{2}can be calculated and be shown in the following Table 5.

_{1}(X) is the fitness function of torque, f

_{2}(X) is the fitness function of cogging torque, f

_{3}(X) is the fitness function of THD, f

_{4}(X) is the fitness function of efficiency. Trated(X), THD(X), Cogging(X), and Efficiency(X) are the SVM model of four optimization objectives respectively.

_{1}and c

_{2}are self-learning factor and social learning factor, r

_{1}and r

_{2}are two random numbers respectively, ω

_{max}and ω

_{min}are the maximum and minimum values of the inertia weight respectively, k and k

_{max}are the current number of iterations and maximum number of iterations respectively.

#### 2.2.3. Optimization of Second Level

## 3. Results

## 4. Conclusions

- (1)
- Through the analysis of the causes of torque ripple, the optimization objectives are determined. The simulation results show that the optimization of THD and cogging torque is helpful to reduce torque ripple and improve torque performance of the motor.
- (2)
- Aiming at the coupling problem between parameters and performance of motor, the Pearson formula is used to calculate the sensitivity of the design parameters for the optimization objectives. The design parameters are divided into two levels, which can achieve decoupling to a certain extent.
- (3)
- Aiming at the long time-consuming problem of finite element simulation, SVM is used to fit mathematical models which can replace the finite element models. The models can have high accuracy and meet engineering needs.
- (4)
- Compared with the initial design, the output torque of the motor becomes larger and the torque ripple becomes smaller, which can prove the effective-ness of the proposed method.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviation

Symbol | Unit |

T_MAG | ° |

H_MAG | Mm |

GAP | Mm |

W_T | Mm |

H_SLOT | Mm |

N_COIL | turn |

SO | mm |

H_SO | Mm |

H_PS | Mm |

OFFSET | Mm |

Trated | Nm |

Efficiency | % |

THD | % |

Cogging | mN·m |

Trip | % |

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**Figure 10.**Variation curve of cogging torque and THD with various parameters. (

**a**) Variation curve of cogging torque and THD with SO. (

**b**) Variation curve of cogging torque and THD with H_PS. (

**c**) Variation curve of cogging torque and THD with H_SO. (

**d**) Variation curve of cogging torque and THD with OFFSET.

Symbol | Quantity | Initial Value |
---|---|---|

T_MAG/° | Angle of PM | 20 |

H_MAG/mm | Thickness of PM | 4 |

GAP/mm | Length of air gap | 0.5 |

W_T/mm | Width of stator tooth | 10 |

H_SLOT/mm | Depth of slot | 15 |

SO/mm | Width of slot open | 1 |

H_SO/mm | Height of slot open | 0.8 |

H_PS/mm | Height of pole shoe | 1.5 |

OFFSET/mm | Magnetic pole eccentricity | 20 |

COIL | Turn of coil | 60 |

Symbol | Ranges |
---|---|

T_MAG/° | [17, 20] |

H_MAG/mm | [4.5, 6.5] |

GAP/mm | [0.5, 1.2] |

W_T/mm | [9.5, 12.5] |

H_SLOT/mm | [14, 16] |

COIL | [63, 67] |

Symbol | Ranges |
---|---|

SO/mm | [0.5, 1] |

H_SO/mm | [0.3, 0.7] |

H_PS/mm | [1.5, 2.5] |

OFFSET/mm | [20, 40] |

Label | Trated | Efficiency | THD | Cogging |
---|---|---|---|---|

1 | 36.38160 | 84.1 | 0.81 | 305.9 |

2 | 47.08480 | 83.6 | 1.32 | 122.8 |

3 | 39.11010 | 80.4 | 1.13 | 64.3 |

4 | 37.254500 | 84.1 | 0.86 | 245.5 |

5 | 37.88260 | 84.4 | 1.02 | 132.2 |

6 | 44.4506 | 80.2 | 1.37 | 57.1 |

7 | 44.10130 | 87.4 | 1.01 | 355.8 |

8 | 46.83510 | 83.2 | 1.31 | 351.0 |

9 | 45.30450 | 80.7 | 1.29 | 42.5 |

10 | 49.96070 | 83.5 | 1.11 | 246.1 |

11 | 42.90750 | 87.6 | 0.74 | 351.7 |

12 | 39.02500 | 86.80 | 1.23 | 146.4 |

13 | 42.056800 | 84.60 | 1.54 | 63.2 |

14 | 40.81640 | 84.8 | 1.42 | 217.7 |

15 | 43.95690 | 88 | 0.97 | 420.3 |

Symbol | MSE | R^{2} |
---|---|---|

Trated | 0.002 | 0.991 |

Efficiency | 0.001 | 0.994 |

THD | 0.012 | 0.893 |

Cogging | 0.011 | 0.914 |

Parameter | Value |
---|---|

Size | 30 |

k_{max} | 100 |

ω_{max} | 1 |

ω_{min} | 0.8 |

c_{1} | 1.5 |

c_{2} | 1 |

Symbol | Initial Value | Optimized Value |
---|---|---|

T_MAG/° | 20 | 18.6 |

H_MAG/mm | 4 | 5 |

GAP/mm | 0.5 | 1 |

W_T/mm | 10 | 12.2 |

H_SLOT/mm | 15 | 14.5 |

COIL | 60 | 65 |

Trated/N·m | 38.2 | 41.0 |

Trip/% | 3.7 | 2.5 |

Symbol | Initial Value | Optimized Value |
---|---|---|

SO/mm | 1 | 0.75 |

H_SO/mm | 0.8 | 0.5 |

H_PS/mm | 1.5 | 2 |

OFFET/mm | 20 | 30 |

Trated/N·m | 41.0 | 40.2 |

Trip/% | 2.5 | 1.7 |

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**MDPI and ACS Style**

Chai, J.; Zhao, T.; Gui, X.
Multi-Objective Optimization Design of Permanent Magnet Torque Motor. *World Electr. Veh. J.* **2021**, *12*, 131.
https://doi.org/10.3390/wevj12030131

**AMA Style**

Chai J, Zhao T, Gui X.
Multi-Objective Optimization Design of Permanent Magnet Torque Motor. *World Electric Vehicle Journal*. 2021; 12(3):131.
https://doi.org/10.3390/wevj12030131

**Chicago/Turabian Style**

Chai, Jiawei, Tianyi Zhao, and Xianguo Gui.
2021. "Multi-Objective Optimization Design of Permanent Magnet Torque Motor" *World Electric Vehicle Journal* 12, no. 3: 131.
https://doi.org/10.3390/wevj12030131