# Optimal Design of a Short Primary Double-Sided Linear Induction Motor for Urban Rail Transit

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Equivalent Circuit of the Short Primary Double-Sided Linear Induction Motor

_{s}represents slot width, t

_{s}is the tooth width, τ

_{s}is slot pitch, h

_{1}denotes slot height, d is the thickness of the secondary action sheet, g is the length between the surface of two primary cores, l_ce is the length of the end connection of the primary winding per phase, and v

_{x}is the mechanical speed of the secondary plate.

_{s}is slot width, h

_{1}is slot height and h

_{0}is slot open height. Notably, 0.05 h

_{1}is the predefined slot open height in this paper, which can be selected based on different requirements.

_{e}is equivalent air gap length, and the coefficient ${k}_{\beta}$ can be looked up from the handbooks of rotated induction motors, which is set to 0.0644 in this article. Further, ${m}_{1}$ denotes the number of phases, and q is the number of slots per pole per phase.

_{e}) can be modified based on mechanical gap length (g) by two coefficients. One is the Carter’s factor, another is the fringing coefficient. The fringing coefficient may be considered because the magnetic airgap to per pole pitch ratio of SP-DLIMs is much larger than that of rotary induction motors [10].

## 3. Design Optimization Process

#### 3.1. Main Indicators of SP-DLIM

#### 3.2. Parameter Search Range of the Optimal Design

#### 3.2.1. Initialization

_{j}and ub

_{j}represent the lower bound and upper bound of the j-th variable. Then, solution vectors execute the DE operation to update.

#### 3.2.2. Mutation

_{i}

^{G}is the i-th newly generated solution vector in the G-th generation, which concludes all the design parameters that are randomly preset in the search range, i ∈ [1, NV]. X is the current population in the G-th generation. NV represents the number of total vectors in every iteration. “r1, r2, r3” denote three different integers with different values of i. What’s more, the search range will be proposed in the later section. Additionally, after mutation strategy, the variables that search out of border should be amended according to the boundary.

#### 3.2.3. Crossover

_{rand}represents a random number in the range of [1, D], where D is the dimension of the problem. Specifically, D is equal to the number of design parameters, the value of which is 11 in this paper.

#### 3.2.4. Selection

#### 3.3. Process of the Optimal Design

## 4. Optimal Results and FEM validation

#### 4.1. Optimal Results

#### 4.1.1. Operating Rules of DE Algorithm

^{4}× D, where D is number of dimensions of the problem [25]. In this paper, D = 11. Therefore, maxFEs is set as 1.1 × 10

^{5}. What’s more, DE is executed for 51 independent times to examine the stability. Further, other pivotal parameters of DE are as follows:

- Crossover rate is 0.5;
- Scaling factor is set as a randomly distributed number for every individual;
- Population size is set as 50.

#### 4.1.2. Optimal Results

#### 4.1.3. Comparison Results between DE, PSO, and GA

#### 4.2. 2-D transient FEM Simulation Results

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 5.**Convergence and statistics analysis of DE, PSO and GA within 50 runs. (

**a**–

**c**) show the convergence of DE, PSO, and GA, respectively. (

**d**) suggests the distribution of the optimal OFV provided by DE, PSO and GA in 51 runs.

**Figure 7.**Distribution of flux lines and flux density when SP-DLIM is running in a steady state at (

**a**) t = 0; (

**b**) t = 0.25 T; (

**c**) t = 0.5 T; (

**d**) t = 0.75 T; (

**e**) t = T. (

**a**) Flux lines and flux density distribution at t = 0. (

**b**) Flux lines and flux density distribution at t = 0.25 T. (

**c**) Flux lines and flux density distribution at t = 0.5 T. (

**d**) Flux lines and flux density distribution at t = 0.75 T. (

**e**) Flux lines and flux density distribution at t = T.

Line Voltage | 500 V |

Rated thrust | 1.1 kN ± 5% |

Rated speed | 45 Km/h |

Winding layer | 1 |

Design Parameters | Min. Value | Max. Value | Unit |
---|---|---|---|

$a$ | 0.05 | 0.15 | m |

${b}_{s}/{\tau}_{s}$ | 0.3 | 0.5 | - |

${d}_{con}$ | 0.5 | 1.5 | mm |

$g$ | 10 | 30 | mm |

$d$ | 4 | 10 | mm |

$s$ | 0.1 | 0.5 | - |

$Jc$ | 3 | 6 | A/mm^{2} |

${f}_{1}$ | 1 | 200 | Hz |

$p$ | 3 | 10 | - |

$q$ | 1 | 3 | - |

$Nz$ | 10 | 100 | - |

Type | Parameters | Unit | Optimal Results | |
---|---|---|---|---|

Design parameters | Primary stack width (a) | mm | 85.8 | |

Slot width/slot pitch (b_{s}/τ_{s}) | - | 0.5 | ||

Conductor diameter (d_{con}) | mm | 1.5 | ||

Number of turns per slot (N_{z}) | - | 52 | ||

Number of slots/pole/phase (q) | - | 3 | ||

Number of pole pairs (p) | - | 3 | ||

Supply power frequency (f_{1}) | Hz | 76.18 | ||

Primary current density (J_{c}) | A/m^{2} | 6 × 10^{6} | ||

Slip (s) | - | 0.24 | ||

Secondary thickness (d) | mm | 4 | ||

Air gap length (g) | mm | 10 | ||

Equivalent circuit parameters | Slot width (b_{s}) | mm | 6 | |

Tooth width (t_{s}) | mm | 6 | ||

Pole pitch (τ) | m | 0.108 | ||

Line voltage (U_{1}) | V | 500 | ||

Primary turns per phase (N_{ph}) | - | 468 | ||

Primary current (I_{1}) | A | 21.21 | ||

Phase winding resistance (R_{1}) | Ω | 0.73 | ||

Phase winding leakage reactance (X_{1}) | mH | 12.5 | ||

Slot height (h_{1}) | mm | 21.7 | ||

Goodness factor (G) | - | 10.1 | ||

Output characteristics | Efficiency (η) | % | Mean | 71.87 |

Best/Optimal | 73.29/71.74 | |||

Power factor (pf) | % | Mean | 61.01 | |

Best/Optimal | 63.16/61.11 | |||

Thrust (Fem) | N × 10^{3} | Mean | 1.12 | |

Best/Optimal | 1.18/1.12 | |||

Tooth weight | kg | Mean | 9.51 | |

Best/Optimal | 9.25/9.45 | |||

Objective function value | - | Mean Best/Optimal | 51.53 52.66/51.91 |

OFV | DE | PSO | GA | F = 0.2 | F = 0.3 | F = 0.4 | F = 0.5 | F = 0.6 | F = 0.7 | F = 0.8 | F = 0.9 |

Successful rate | 100% | 64.71% | 43.14% | 94% | 100% | 100% | 47.06% | 100% | 100% | 100% | 100% |

Best | 52.66 | 50.89 | 48.86 | 53.13 | 52.24 | 52.15 | 51.76 | 51.91 | 52.66 | 52.66 | 53.14 |

Worst | 39.81 | 24.40 | 30.16 | 40.40 | 39.81 | 49.84 | 24.36 | 51.91 | 51.91 | 51.91 | 51.91 |

mean | 51.68 | 43.59 | 40.33 | 50.77 | 51.17 | 51.74 | 42.46 | 51.91 | 51.94 | 52.00 | 52.00 |

Std | 1.70 | 7.35 | 4.14 | 1.99 | 1.87 | 0.59 | 7.33 | 0 | 0.15 | 0.24 | 0.29 |

OFV | F = 1 | CR = 0.1 | CR = 0.2 | CR = 0.3 | CR = 0.4 | CR = 0.5 | CR = 0.6 | CR = 0.7 | CR = 0.8 | CR = 0.9 | CR = 1 |

Successful rate | 100% | 98% | 98% | 100% | 100% | 100% | 98% | 100% | 100% | 100% | 94% |

Best | 53.12 | 52.25 | 52.31 | 52.64 | 52.66 | 52.66 | 52.91 | 53.14 | 53.14 | 53.14 | 53.14 |

Worst | 48.33 | 49.02 | 51.70 | 51.81 | 51.91 | 51.91 | 39.81 | 51.91 | 39.81 | 39.81 | 33.47 |

mean | 51.99 | 51.4 | 51.91 | 51.97 | 51.95 | 51.95 | 51.72 | 51.96 | 50.78 | 50.35 | 47.35 |

Std | 0.58 | 0.84 | 0.11 | 0.19 | 0.18 | 0.18 | 1.71 | 0.22 | 3.89 | 4.54 | 6.43 |

Characteristics | Optimal Results | FEM | Relative Error |

Phase current | 21.21 A | 21.5 A | 1.3% |

Thrust | 1117 N | 1076 kN | 3.8% |

Efficiency | 71.84% | 71.09% | 1.1% |

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**MDPI and ACS Style**

Wang, H.; Zhao, J.; Xiong, Y.; Xu, H.; Yan, S.
Optimal Design of a Short Primary Double-Sided Linear Induction Motor for Urban Rail Transit. *World Electr. Veh. J.* **2022**, *13*, 30.
https://doi.org/10.3390/wevj13020030

**AMA Style**

Wang H, Zhao J, Xiong Y, Xu H, Yan S.
Optimal Design of a Short Primary Double-Sided Linear Induction Motor for Urban Rail Transit. *World Electric Vehicle Journal*. 2022; 13(2):30.
https://doi.org/10.3390/wevj13020030

**Chicago/Turabian Style**

Wang, Hanming, Jinghong Zhao, Yiyong Xiong, Hao Xu, and Sinian Yan.
2022. "Optimal Design of a Short Primary Double-Sided Linear Induction Motor for Urban Rail Transit" *World Electric Vehicle Journal* 13, no. 2: 30.
https://doi.org/10.3390/wevj13020030