# Multi-Objective Optimal Design of SPMSM for Electric Compressor Using Analytical Method and NSGA-II Algorithm

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## Abstract

**:**

## 1. Introduction

## 2. Analytical Method

- The end effects are negligible;
- The magnetic flux density only has radial and tangential components;
- The stator and rotor cores have infinite permeability;
- The relative permeability of permanent magnet (PM) is the same as that of air;
- The slotting effect is accounted for using Carter’s coefficient.

#### 2.1. Magnetic Field Solution

_{θ}represent radial, and tangential flux density, respectively. Finally, the unknown coefficient is derived from the boundary conditions, as shown in (6), and (7).

_{n}

_{1}, C

_{n}

_{2}, D

_{n}

_{1}, and D

_{n}

_{2}represent the harmonics, radius from the center, and undetermined coefficient, respectively.

_{a}represent radius from center, stator reference angle, and stator axial length, respectively. The back EMF can be calculated through Faraday’s law. Figure 2 compares the air gap magnetic flux density and back EMF derived through the analytical method with FEM.

#### 2.2. Copper Loss

_{ph}and I

_{phrms}are the phase resistance and RMS phase current, respectively.

#### 2.3. Core Loss

_{h}, K

_{c}, and K

_{e}denote the hysteresis, eddy current, and excess loss coefficients, respectively. These coefficients are obtained from experimental data provided by electrical steel manufacturers. B

_{m}represents the mth harmonic of the magnetic flux density. The stator can be divided into teeth and yoke areas. Since the teeth and yoke are distributed symmetrically according to the number of slots, the core loss can be calculated by substituting B after deriving the magnetic flux density for one tooth and the yoke. Figure 3a depicts the magnetic flux flow in the stator. The flux flowing through the teeth region flows through the yoke region. The flux linkage to the teeth can be obtained by integrating the air-gap flux density with respect to the tooth angle. As the ratio of magnetic flux flowing through the teeth to the magnetic flux flowing through the yoke is constant, based on pole slot combination, the magnetic flux density of the yoke can be obtained using TYratio. Figure 3b shows the FEM and analytical results of the flux according to the rotor position. Analytical teeth flux provides accurate results with the FEM. The yoke flux can be calculated by multiplying the teeth flux with 1.62. In this model, (6p27s) the TYratio is 1.62.

#### 2.4. Eddy Current Loss

_{eddy}flowing in the PM can be derived by substituting the eddy current density, expressed in (14), into (15) [12].

_{e}, σ

_{m}, I

_{e}, and R

_{pm}represent the eddy current density, conductivity of the PM, eddy current RMS value, and PM electrical resistance, respectively. J

_{s}is the RMS value of the eddy current per pole pair, S is the surface of eddy current flow, and l

_{a}is the PM axial length. The PM eddy current loss can be calculated by substituting the analytically calculated A

_{I}into the eddy current equation.

## 3. Optimal Design Process

_{w}, B

_{g}

_{1}, ac, D

_{g}, L

_{stk}, represent the winding factor, flux density per pole, electrical loading, rotor outer diameter, rotor axial length, respectively. The pole-slot combination and winding layout selection are the first stages of motor design. Generally, these stages are determined based on the design specifications. Subsequently, the winding coefficient is automatically determined. The magnetic torque consists of three components: electric loading, magnetic loading, and rotor size. Therefore, the design of an SPMSM is determined mainly by the selection of each component. However, such selection is challenging because each component influences the others.

#### 3.1. Stator Design Based on a Given Rotor

_{c}. According to the voltage limit, N

_{c}is calculated using (17), where E

_{fmax}, Φ

_{fmax}, and w

_{e}represent the back-EMF 1st wave, flux-linkage 1st wave, and electric angular speed, respectively.

_{re}, is determined from (18). The teeth thickness T

_{t}, yoke thickness T

_{y}, and rotor core thickness T

_{rc}are determined from (19)

_{tmax}and Φ

_{p}represent the peak value of magnetic flux in the teeth and the magnetic flux per pole, respectively. B

_{tm}and B

_{rc}are the tooth and rotor core saturation limits, respectively, which are set at 1.4–1.5 [T] to prevent stator saturation. As mentioned in Section II, the magnetic flux through the teeth and yoke has a constant ratio according to the pole-slot combination. Therefore, multiplying TYratio by the teeth thickness determines the thickness of the yoke, which has the same saturation level as that of teeth.

_{s}

_{2}, which can be calculated using the slot fill factor limit K

_{sf}and current density limit J

_{lim}.

_{cu}is the coil area that satisfies the current-density limit. A

_{dw}is the area of a strand. Using (20), the number of strands, n

_{sn}is determined.

_{sr}is the slot area required to satisfy the slot fill factor limit. By substituting A

_{sr}to (21), the teeth length H

_{s}

_{2}is determined.

#### 3.2. NSGA—II

#### 3.3. Flowchart of the Proposed SPMSM Optimal Design

- -
- Pre-process

- -
- Analytical model

- -
- NSGA-II

- -
- Optimization

#### 3.4. Verification

## 4. Prototype Experiment

_{rms}/mm

^{2}, and the cooling method was a compressor-integrated refrigerant cooling method. However, in this experiment, only air cooling was performed, owing to the limitation of the test set. Therefore, the copper loss increased owing to an increase in the winding temperature. Second, based on the abnormal noise during no-load operation in the used bearing, it could be assumed that the mechanical loss owing to the bearing was larger than in the general case. Additionally, distinct from the sinusoidal current analysis, in reality, harmonic currents are generated due to the inverter, reducing torque and increasing electromagnetic losses. Thus, the efficiency decreases.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Surface permanent magnet synchronous motor (SPMSM) for electric vehicle (EV) air conditioner compressor (

**a**) Analysis model (

**b**) Simplified slot less analytical model.

**Figure 3.**(

**a**) Conceptual diagram of the magnetic flux flow, and (

**b**) the tooth and yoke magnetic flux according to rotor position.

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

Rated Power | [kW] | 4.5 |

Rated RPM | [rpm] | 6540 |

Rated Torque | [Nm] | 6.6 |

Pole/Slot | - | 6/27 |

DC Link Voltage | [Vdc] | 288 |

Current Density | [${A}_{rms}/{\mathrm{mm}}^{2}]$ | 15–20 |

Slot Fill Factor | [%] | 38 |

Parameter | Initial Model | Optimal Model | ||||
---|---|---|---|---|---|---|

Analytical | FEM | Error [%] | Analytical | FEM | Error [%] | |

${H}_{m}$ [mm] | 4.7 | 4 | ||||

${L}_{stk}$ [mm] | 50 | 50 | ||||

${D}_{so}$ [mm] | 96.97 | 100 | 3.03 | 93.22 | 96 | 2.9 |

Embrace | 0.8 | 0.79 | ||||

${I}_{ph}$ [${A}_{peak}$] | 21.69 | 22.18 | 2.24 | 20.74 | 20.68 | −0.66 |

Average Torque [$\mathrm{Nm}]$ | 6.6 | 6.6 | ||||

Back EMF 1st [${V}_{peak}$] | 126.79 | 126.4 | −0.31 | 132.79 | 136.81 | 2.94 |

Back-EMF THD [%] | 6.54 | 7 | 6.53 | 6.08 | 6.29 | 3.24 |

Coreloss [W] | 56.65 | 54.36 | −4.23 | 53.54 | 55.9 | 4.23 |

Eddy-current Loss [W] | 2.84 | 3.04 | 6.48 | 3.39 | 34 | 0.28 |

Copperloss [W] | 203.42 | 198.31 | −2.57 | 210.41 | 195.64 | −7.55 |

Power Density [kW/kg] | 1.77 | 1.7 | −4.61 | 1.93 | 1.87 | −3.5 |

Efficiency [%] | 93.59 | 93.7 | 0.11 | 93.52 | 93.71 | 0.2 |

PM Consumption [kg] | 0.23 | 0.19 | ||||

Analysis Time [s] | 2 | 134 | - | 2 | 134 | - |

Parameter | Unit | FEM | Experiment | Error [%] |
---|---|---|---|---|

Efficiency | [%] | 93.7 | 90.1 | 3.8 |

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**MDPI and ACS Style**

Jo, S.-T.; Kim, W.-H.; Lee, Y.-K.; Kim, Y.-J.; Choi, J.-Y.
Multi-Objective Optimal Design of SPMSM for Electric Compressor Using Analytical Method and NSGA-II Algorithm. *Energies* **2022**, *15*, 7510.
https://doi.org/10.3390/en15207510

**AMA Style**

Jo S-T, Kim W-H, Lee Y-K, Kim Y-J, Choi J-Y.
Multi-Objective Optimal Design of SPMSM for Electric Compressor Using Analytical Method and NSGA-II Algorithm. *Energies*. 2022; 15(20):7510.
https://doi.org/10.3390/en15207510

**Chicago/Turabian Style**

Jo, Seong-Tae, Woo-Hyeon Kim, Young-Keun Lee, Yong-Joo Kim, and Jang-Young Choi.
2022. "Multi-Objective Optimal Design of SPMSM for Electric Compressor Using Analytical Method and NSGA-II Algorithm" *Energies* 15, no. 20: 7510.
https://doi.org/10.3390/en15207510