# Sensitivity Analysis for Multi-Objective Optimization of Switched Reluctance Motors

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Switched Reluctance Motor

#### 2.1. SRM Working Principle

#### 2.2. Torque Ripple

#### 2.3. Geometrical Parameters of SRM

_{RT}is the diameter of the rotor additional teeth and D

_{SI}is the stator inner diameter.

_{min}, the duration of the maximum inductance θ

_{max}, and the duration of the growing θ

_{inc}and falling inductance θ

_{dec}phases can be defined as follows.

_{ag}) with a bigger value at the beginning of the overlap and a smaller value at the end of the overlap. Applying an appropriate value of the air-gap shift allows one to obtain the torque, which grows together with the teeth overlap.

## 3. Case Study

#### 3.1. Design Model of SRM

^{2}.

#### 3.2. Taguchi Method

Specification/Geometry Characteristic | Symbol | Value |
---|---|---|

Phase resistance | R | 5 Ω |

Stack length | L | 60 mm |

Stator outer diameter | D_{SO} | 103 mm |

Stator inner diameter | D_{SI} | 60 mm |

Stator slot depth | S_{D} | 14.8 mm |

Rotor inner diameter | D_{RI} | 22 mm |

Stator/rotor pole angle | β_{S}/β_{R} | 28.5/29.1° |

Air-gap length | a | 0.25 mm |

Average three-phase torque | T_{av} | 1.34 Nm |

Torque ripple coefficient | K | 2.03 |

_{25}(6

^{5}) of twenty-five experiments was utilized. In case of the full factorial experiment, the 7776 experiments would have been required. The orthogonal array or matrix of experiments is presented in Table 3.

_{f}is the total mean value of the objective function.

## 4. Results

#### 4.1. Results of the Sensitivity Analysis

_{1}) and airgap shift (X

_{5}) are 42% and 28%, respectively. Figure 13 suggests that the growth of the stator pole angle makes the average torque increase rapidly. This fact correlates with Equation (2), which defines θ

_{inc}and the duration of torque production as a direct function of the stator pole angle. Meanwhile, the growth in the airgap shift reduces the average torque. One of the reasons could be a torque fall at the beginning of the overlap. On the other hand, the average torque has a nonlinear response to the change in the stator pole angle at the core and additional tooth angle on the rotor. There was a slight reaction noticed from the change in the rotor pole angle and the rotor tooth angle at the core.

_{3}) is more than 50%. Particularly, with the growth of the rotor pole angle, the torque ripple swiftly increases. The same tendency can be noticed for the growth of the additional tooth angle. This fact correlates with Equation (2), which illustrates that the duration of the maximum inductance θ

_{max}grows together with the rotor pole angle. One of the reasons that torque ripple increases along with the rotor pole angle could be that the SRM reaches the maximum inductance too fast and moves to a phase with no positive torque production. The air gap shift had a positive influence on the torque ripple reduction, as was expected. On the other hand, the advance in the stator pole angle and the reduction in the stator angle on the core slightly reduced the torque ripple.

_{1}, airgap shift X

_{5}, and rotor pole angle X

_{3}on the average torque. For that reason, Figure 16 presents the static torque curve for designs 1, 6, 11, 16, and 21 with the stator pole angle levels 1, 2, 3, 4, and 5, respectively (see Table 3).

_{3}and rotor additional angle X

_{6}, which are higher in designs 6, 11, and 16.

_{5}and rotor pole angle X

_{3}. It is apparent from the figure that the average torque decreases. On the other hand, the area of the torque production shifted to the beginning of the overlap due to the growth of the rotor pole angle. Therefore, the area of zero torque enlarges at the end of the growing inductance phase when θ

_{max}= [30, 45]. It is important to highlight that the biggest improvement in the width of the torque and constancy of the torque at its maximum can be noticed between Design 1 and Design 2; θ

_{max}= [18, 30] and θ

_{max}= [22, 32], respectively. Due to the introduction of the airgap shift, the torque in design 2 became more stable at its maximum. However, further increase in the air-gap shift did not have a positive influence on the torque characteristic due to the possible influences of other parameters, such as additional tooth angle X

_{6}.

- Stator pole angle has a major influence on the average torque. The increase in the stator pole angle leads to a wider torque production region and higher average torque.
- The growth of the airgap shift has a positive influence on the torque ripple reduction and a negative influence on the average torque. The airgap shift raises the torque at the end of the growing inductance phase. However, the air-gap shift is a sensitive parameter, and its value should be selected carefully.
- The increase in the rotor pole angle has a negative effect on the torque ripple. With the growth of the rotor pole angle, the torque curve shifts to the beginning of the torque production phase. To avoid the torque ripple with the increase in the rotor pole angle, the control turn-on and turn-off angles should be adjusted according to the area of torque production.
- The introduction of the rotor pole angle within this SRM design leads to an increase in torque ripples. The possible explanation can be an improper selection of its values.

#### 4.2. Topology Optimization Design Space Definition

- Stator pole angle was set to 40°, pursuing the best combination of maximum average torque and minimum torque ripple.
- Stator pole angle at the core was set to 30°, insuring almost the lowest torque ripple and reasonably high torque.
- The rotor pole angle and rotor pole angle at the core were set to 28.75° and 60°, respectively, trying to achieve the minimum torque ripple and keep the average torque ripple at the average level.
- Due to the high influence of the airgap shift and the low influence of the core angles on the objective functions, the depth of the TO domain was set to 5 mm.
- The possible additional angle for the rotor teeth was set to ±1.875° due to the highly negative influence of the additional teeth after 1.875 on both the average torque and torque ripple.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**SRM drive: (

**a**) SRM drive concept; (

**b**) asymmetric bridge converter. L

_{A–C}denote to the phase inductances.

**Figure 15.**Control factors’ effects on the objectives: (

**a**) average torque; (

**b**) most influential parameters on average torque; (

**c**) torque ripple; (

**d**) most influential parameters on torque ripple.

**Figure 17.**Static torque curve for Taguchi designs with different levels of air-gap shift and rotor pole angle.

**Figure 19.**(

**a**) Topology optimization design domain; (

**b**) Torque comparison of initial and trade-off designs.

Design Parameter/Control Factor | Symbol | Factor | Level | ||||
---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | |||

Stator pole angle (˚) | β_{s} = X_{1} | X_{1} | 25 | 28.75 | 32.5 | 36.25 | 40 |

Stator pole angle at core (˚) | α_{s}= β_{s} – X_{2} | X_{2} | 5 | 7.5 | 10 | 12.5 | 15 |

Rotor pole angle (˚) | β_{r} = X_{3} | X_{3} | 25 | 28.75 | 32.5 | 36.25 | 40 |

Rotor pole angle at core (˚) | α_{r}= β_{r} + X_{4} | X_{4} | 25 | 31.25 | 37.5 | 43.75 | 50 |

Airgap shift, mm | ɛ =X_{5} | X_{5} | 0 | 0.25 | 0.5 | 0.75 | 1 |

Additional tooth angle (˚) | β_{r+} = X_{6}/2 | X_{6} | 0 | 3.75 | 7.5 | 11.25 | 15 |

**Table 3.**Taguchi orthogonal array L

_{25}(6

^{5}) for 6 control factors and 5 levels and simulation results.

Experiment | Control Factor | Average Torque | Torque Ripple Coefficient | |||||
---|---|---|---|---|---|---|---|---|

X_{1} | X_{2} | X_{3} | X_{4} | X_{5} | X_{6} | |||

1 | 1 | 1 | 1 | 1 | 1 | 1 | 0.851 | 1.22 |

2 | 1 | 2 | 2 | 2 | 2 | 2 | 0.838 | 1.06 |

3 | 1 | 3 | 3 | 3 | 3 | 3 | 0.790 | 1.36 |

4 | 1 | 4 | 4 | 4 | 4 | 4 | 0.716 | 1.55 |

5 | 1 | 5 | 5 | 5 | 5 | 5 | 0.614 | 1.73 |

6 | 2 | 1 | 2 | 3 | 4 | 5 | 0.808 | 1.16 |

7 | 2 | 2 | 3 | 4 | 5 | 1 | 0.792 | 1.06 |

8 | 2 | 3 | 4 | 5 | 1 | 2 | 0.899 | 1.37 |

9 | 2 | 4 | 5 | 1 | 2 | 3 | 0.850 | 1.39 |

10 | 2 | 5 | 1 | 2 | 3 | 4 | 0.812 | 0.95 |

11 | 3 | 1 | 3 | 5 | 2 | 4 | 0.887 | 1.53 |

12 | 3 | 2 | 4 | 1 | 3 | 5 | 0.829 | 1.55 |

13 | 3 | 3 | 5 | 2 | 4 | 1 | 0.837 | 1.19 |

14 | 3 | 4 | 1 | 3 | 5 | 2 | 0.796 | 0.92 |

15 | 3 | 5 | 2 | 4 | 1 | 3 | 0.930 | 1.25 |

16 | 4 | 1 | 4 | 2 | 5 | 3 | 0.936 | 1.52 |

17 | 4 | 2 | 5 | 3 | 1 | 4 | 0.878 | 1.72 |

18 | 4 | 3 | 1 | 4 | 2 | 5 | 0.907 | 1.30 |

19 | 4 | 4 | 2 | 5 | 3 | 1 | 0.855 | 0.92 |

20 | 4 | 5 | 3 | 1 | 4 | 2 | 0.856 | 1.02 |

21 | 5 | 1 | 5 | 4 | 3 | 2 | 0.935 | 1.52 |

22 | 5 | 2 | 1 | 5 | 4 | 3 | 0.847 | 0.94 |

23 | 5 | 3 | 2 | 1 | 5 | 4 | 0.826 | 1.00 |

24 | 5 | 4 | 3 | 2 | 1 | 5 | 0.901 | 1.43 |

25 | 5 | 5 | 4 | 3 | 2 | 1 | 0.910 | 1.26 |

Control Factor | 1st Optimal Design | 2nd Optimal Design | Trade-Off Design |
---|---|---|---|

X1 | 4 | 2 | 5 |

X2 | 1 | 5 | 3 |

X3 | 4 | 1 | 2 |

X4 | 2 | 2 | 2 |

X5 | 1 | 4 | 2 |

X6 | 3 | 1 | 2 |

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**MDPI and ACS Style**

Andriushchenko, E.; Kallaste, A.; Mohammadi, M.H.; Lowther, D.A.; Heidari, H.
Sensitivity Analysis for Multi-Objective Optimization of Switched Reluctance Motors. *Machines* **2022**, *10*, 559.
https://doi.org/10.3390/machines10070559

**AMA Style**

Andriushchenko E, Kallaste A, Mohammadi MH, Lowther DA, Heidari H.
Sensitivity Analysis for Multi-Objective Optimization of Switched Reluctance Motors. *Machines*. 2022; 10(7):559.
https://doi.org/10.3390/machines10070559

**Chicago/Turabian Style**

Andriushchenko, Ekaterina, Ants Kallaste, Mohammad Hossain Mohammadi, David A. Lowther, and Hamidreza Heidari.
2022. "Sensitivity Analysis for Multi-Objective Optimization of Switched Reluctance Motors" *Machines* 10, no. 7: 559.
https://doi.org/10.3390/machines10070559