# Kriging-Assisted Multi-Objective Optimization Framework for Electric Motors Using Predetermined Driving Strategy

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. The Calculation of the Mass

#### 2.2. The Calculation of the Losses

#### 2.3. The Optimization

#### 2.4. The Multi-Objective Optimization

#### 2.5. The Genetic Algorithm

#### 2.6. The Kriging Surrogate Model

## 3. The Optimization Framework

#### 3.1. The Workflow of the Optimization Framework

#### 3.2. The Motor Model

#### Validation of Motor Model

#### 3.3. The Validation of the Optimization Framework through an Application

#### 3.3.1. Motor Model for Validation

#### 3.3.2. The Design Variables and the Operating Points

#### 3.3.3. The Constraints

#### 3.3.4. The Parametrization of Optimization Algorithm

#### 3.3.5. The Results of Optimization

#### 3.3.6. The Validation of the Optimization Framework

## 4. The Application of Optimization Framework

#### 4.1. The Objective Functions and the Design Variables

#### 4.2. The Simulation Results of the SZEVOL

#### 4.3. The Operating Points of the Optimization

#### 4.4. The Constraints of the Optimization

#### 4.5. Results

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

FEM | finite element method |

GA | genetic algorithm |

GSA | gravitational search algorithm |

PMSM | permanent magnet synchronous machines |

PSO | particle swarm optimization |

SEM | Shell Eco-marathon |

SZEVOL | the electric motor to be improved in the example application |

## References

- Naik, S.; Bag, B.; Chandrasekaran, K. Comparative Analysis of Surface Mounted and Interior Permanent Magnet Synchronous Motor for Low Rating Power Application. J. Phys. Conf. Ser.
**2021**, 2070, 012119. [Google Scholar] [CrossRef] - Istenes, G.; Horvath, Z. Multi-objective Optimization of Electric Motors with a Kriging Surrogate Model. In Proceedings of the IEEE 2022 22nd International Symposium on Electrical Apparatus and Technologies (SIELA), Bourgas, Bulgaria, 1–4 June 2022; pp. 1–4. [Google Scholar] [CrossRef]
- Elkholy, M.M.; Abd El-Hameed, M. Minimization of Starting Energy Loss of Three Phase Induction Motors Based on Particle Swarm Optimization and Neuro Fuzzy Network. Int. J. Power Electron. Drive Syst.
**2016**, 7, 1038–1048. [Google Scholar] [CrossRef] - Das, S.; Mondal, S.; Saha, S.; Sarkar, C. A GSA Based Torque and Loss Optimisation of an Induction Motor. Int. J. Adv. Res. Electr. Electron. Instrum. Energy
**2013**, 26, 3717–3725. [Google Scholar] - Bittner, F.; Hahn, I. Kriging-Assisted Multi-Objective Particle Swarm Optimization of Permanent Magnet Synchronous Machine for Hybrid and Electric Cars. In Proceedings of the 2013 International Electric Machines & Drives Conference, Chicago, IL, USA, 12–15 May 2013; pp. 15–22. [Google Scholar] [CrossRef]
- Sun, I.; Shi, Z.; Cai, Y.; Lei, G.; Guo, Y.; Zhu, J. Driving-Cycle-Oriented Design Optimization of a Permanent Magnet Hub Motor Drive System for a Four-Wheel-Drive Electric Vehicle. IEEE Trans. Transp. Electrif.
**2020**, 6, 1115–1125. [Google Scholar] [CrossRef] - Sun, I.; Xu, N.; Yao, M. Sequential Subspace Optimization Design of a Dual Three-Phase Permanent Magnet Synchronous Hub Motor Based on NSGA III. IEEE Trans. Transp. Electrif.
**2023**, 9, 622–630. [Google Scholar] [CrossRef] - Shi, Z.; Sun, X.; Cai, Y.; Yang, Z. Robust Design Optimization of a Five-Phase PM Hub Motor for Fault-Tolerant Operation Based on Taguchi Method. IEEE Trans. Energy Convers.
**2020**, 35, 2036–2044. [Google Scholar] [CrossRef] - Kalyanmoy, D. Multi-Objective Optimization Using Evolutionary Algorithms; Indian Institute of Technology Kanpur: Kanpur, India, 2011; p. 24. [Google Scholar]
- Gong, W.; Duan, Q.; Li, J.; Wang, C.; Di, Z.; Ye, A.; Miao, C.; Dai, Y. Multiobjective Adaptive Surrogate Modeling-Based Optimization for Parameter Estimation of Large, Complex Geophysical Models. Water Resour. Res.
**2016**, 52, 1984–2008. [Google Scholar] [CrossRef] [Green Version] - Maxwell 2D: ANSYS Maxwell tutorial on the 2004 PriusIPM Motor; Study of a Permanent Magnet Motor with MAXWELL 2D, ANSYS Maxwell. 2004, p. 95. Available online: https://www.academia.edu/35037357/Topic_Motor_Application_Note_Study_of_a_Permanent_Magnet_Motor_with_MAXWELL_2D_Example_of_the_2004_Prius_IPM_Motor (accessed on 15 May 2023).
- Ortiz García, G. Identificación de Sistemas Estructurales Histeréticos Usando Algoritmos de Optimización Multi-Objetivo; Universidad Nacional de Colombia, Industrial Automation: Manizales, Colombia, 2013; p. 112. [Google Scholar]
- Pusztai, Z.; Kőrös, P.; Szauter, F.; Friedler, F. Vehicle Model-Based Driving Strategy Optimization for Lightweight Vehicle. Energies
**2022**, 15, 3631. [Google Scholar] [CrossRef] - Pusztai, Z.; Kőrös, P.; Szauter, F.; Friedler, F. Implementation of Optimized Regenerative Braking in Energy Efficient Driving Strategies. Energies
**2023**, 16, 2682. [Google Scholar] [CrossRef]

**Figure 4.**The results of the Prius IPM FEM model: (

**a**) the results of the technical report Ref. [11]; (

**b**) the results of Ansys Maxwell.

**Figure 13.**The distribution of operating points in optimized driving strategy—SZEVOL driven at TT Circuit Assen.

Parameters | Description |
---|---|

E | energy loss |

Ep | previous energy loss |

Ee | estimated energy loss |

p | design variables |

pp | previous design variables |

t | time weight of operating point |

Mr | required torque of operating point |

n | rotation speed of operating point |

M | calculated torque |

I | required current amplitude |

pi | additional variables |

P | power loss |

Name | Description | SZEVOL | Lower Limit | Upper Limit |
---|---|---|---|---|

x1 | Outer radius of the stator. | 61.75 mm | 50 mm | 61.75 mm |

x2 | Thickness of the back-iron. | 7 mm | 5 mm | 10 mm |

x3 | Depth of the slots. | 16.3 mm | 5 mm | 25 mm |

x4 | Thickness of the tooth tangs. | 2.5 mm | 2 mm | 5 mm |

x5 | Gap between the tooth tangs. | 1.5 mm | 1 mm | 10 mm |

x6 | Thickness of the tooths. | 8 mm | 2 mm | 10 mm |

x7 | Airgap. | 0.15 mm | 0.1 mm | 1 mm |

x8 | Maximum thickness of the magnets. | 4.6 mm | 2 mm | 10 mm |

x9 | Gap between the magnets. | 3 mm | 2 mm | 10 mm |

x10 | Width of the rotor cuts. | 8 mm | 3 mm | 15 mm |

x11 | Height of the rotor cuts. | 6.5 mm | 3 mm | 20 mm |

x12 | Distance between the cuts and the origin of the axis. | 21.15 mm | 18.75 mm | 27.75 mm |

x13 | Ratio of the magnet groove (x8/x13b). | 0.3577 | 0.1 | 0.9 |

x14 | Number of turns. | 17 | 10 | 30 |

x15 | Diameter of the wire. | 1.128 mm | 0.65 mm | 1.38 mm |

x16 | Length of the motor. | 52.5 mm | 40 mm | 52.5 mm |

Rotational Speed (rpm) | Required Torque (Nm) | Time Weighting (s) |
---|---|---|

267 | 32 | 30.94 |

315 | 7 | 20.44 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Istenes, G.; Pusztai, Z.; Kőrös, P.; Horváth, Z.; Friedler, F.
Kriging-Assisted Multi-Objective Optimization Framework for Electric Motors Using Predetermined Driving Strategy. *Energies* **2023**, *16*, 4713.
https://doi.org/10.3390/en16124713

**AMA Style**

Istenes G, Pusztai Z, Kőrös P, Horváth Z, Friedler F.
Kriging-Assisted Multi-Objective Optimization Framework for Electric Motors Using Predetermined Driving Strategy. *Energies*. 2023; 16(12):4713.
https://doi.org/10.3390/en16124713

**Chicago/Turabian Style**

Istenes, György, Zoltán Pusztai, Péter Kőrös, Zoltán Horváth, and Ferenc Friedler.
2023. "Kriging-Assisted Multi-Objective Optimization Framework for Electric Motors Using Predetermined Driving Strategy" *Energies* 16, no. 12: 4713.
https://doi.org/10.3390/en16124713