# Energy-Efficient Speed Profile Optimization and Sliding Mode Speed Tracking for Metros

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Dynamic Characteristics of Metro Trains

## 3. Fitness Calculation Function about Speed Profile Optimization

#### 3.1. Energy Consumption Calculation Function

#### 3.2. Punctuality Penalty Function

#### 3.3. Comfortableness Penalty Function

#### 3.4. Fitness Calculation Function Based on Double Penalty Mechanism

## 4. Speed Profile Optimization with RR-ABC

#### 4.1. Emergency Braking Intervention (EBI) Curve and Warning Speed Limit

- At first, the train is in the process of maximum acceleration. After the train issued an emergency braking command, the train still maintained the maximum acceleration during the transmission delay and traction cutoff delay, because the traction cannot be cut off instantly and transmission delay exists objectively at any moment.
- Since the braking system needs some time to receive instructions and gradually generate braking force, it usually takes some time to reach the maximum force from 0. During this period, the train is considered to keep coasting.

#### 4.2. Energy-Saving Strategies and Coast Choices

#### 4.2.1. Coast Interval ${x}_{1}$

- Features:
- (1)
- Driving from high speed limit zone to low speed limit zone
- (2)
- Maintain constant speed at the next low speed limit

- Strategy selection:

- (1)
- If ${v}_{S1}>{v}_{3}$, then brake to ${v}_{3}$;
- (2)
- If ${v}_{S1}<{v}_{3}$, then when $v={v}_{3}$, let the train drive at a constant speed ${v}_{3}$.

#### 4.2.2. Coast Interval ${x}_{2}$

- Features:
- (1)
- Driving from high speed limit zone to low speed limit zone;
- (2)
- Braking in the next low speed limit.

- Strategy selection:

- (1)
- If ${v}_{S2}>{v}_{4}$, then brake to ${v}_{4}$;
- (2)
- If ${v}_{S2}\le {v}_{4}$, then keep coasting to maintain the coast condition until it contacts the braking curve.

#### 4.3. Regional Reinforcement Artificial Bee Colony (RR-ABC) Algorithm

**(1)****Model initialization: Chaos mirror initialization**

**(2)****Employed bee stage: Evolutionary dimension adjustment strategy**

**(3)****Onlooker bees stage: Search radius adjustment strategy**

**Remark**

**1.**

#### 4.4. Speed Profile Optimization Model

## 5. Adaptive Terminal Sliding Mode Controller

#### 5.1. Dynamics Model of Speed Tracking

#### 5.2. Design of Sliding Mode Terminal Controller

#### 5.3. Disturbance Observer

## 6. Experimental Result

#### 6.1. Basic Test of RR-ABC

- (1)
- The initial value of the RR-ABC algorithm is better than that of the standard ABC algorithm;
- (2)
- Compared with the standard ABC algorithm, RR-ABC maintained good evolutionary performance, both in the early and late stages; especially in the late stage, the evolution efficiency of RR-ABC is significantly higher than that of ABC. This proves that the RR-ABC algorithm has good local search capabilities.

- (1)
- Running with the safety speed limit curve, the train can safely travel under the fixed speed limit curve;
- (2)
- The speed profile contains two coast sections;
- (3)
- When the train is in traction section and cruising section, the energy consumption increases. Furthermore, the rate during traction is higher than that in cruise. In contrast, the energy consumption remains a level-out during coast section and braking section.

#### 6.2. Advanced Test about Different Interval Time and Different Mode

#### 6.3. Advanced Test about Different Driving Mode

- (1)
- The energy consumption of optimal mode is 9.55% lower than that of cruising mode, so optimal mode is at energy saving;
- (2)
- The average acceleration rate of cruising mode is much lower than that of optimal mode, and although the index of optimal mode is qualified, the cruising mode is much better at comfortableness.

#### 6.4. Basic Tracking Test about ATSMC+DOB Controller under Different Fixed Delays

#### 6.5. Tracking Test about ATSMC+DOB Controller

## 7. Conclusions

- (1)
- A multi-objective optimization model that considers energy consumption calculation and takes punctuality and comfortableness as penalty factors is established to optimize the train speed profile. This model takes the comfortableness of the metro and the punctuality into consideration when optimizing energy saving.
- (2)
- An optimization strategy that considers the metro EBI speed limit and the actual volume of the train (warning speed limit) is proposed. An improved ABC algorithm named RR-ABC algorithm is proposed for speed profile optimization. Compared with ordinary algorithms, the RR-ABC algorithm not only has good global search ability to avoid the local optimal solutions, but it also has excellent local search ability to improve the evolutionary efficiency.
- (3)
- A terminal sliding mode controller with disturbance observer (ATSMC+DOB) is designed by introducing parameter adaptation mechanism and disturbance observer. The controller has better robustness and anti-disturbance which brings minor speed tracking error and good energy saving.
- (4)
- The real data from Qingdao Metro Line 6 were used for the verification of the research. The simulation test results prove that the research on speed-profile optimization and speed-tracking control is effective.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 14.**Comparison of evolutionary performance: (

**a**) Comparison about RR-ABC and ABC; (

**b**) Comparison about RR-ABC and GA.

**Figure 15.**Optimal speed profile: (

**a**) The optimal speed profile made by optimization; (

**b**) Cumulative energy consumption growth curve.

**Figure 16.**Optimal speed profile. (

**a**) The optimal speed profile made by optimization; (

**b**) Cumulative energy consumption growth curve.

**Figure 17.**Optimal speed profile. (

**a**) The optimal speed profile made by optimization; (

**b**) Cumulative energy consumption growth curve.

**Figure 18.**Comparison of different driving modes: (

**a**) Comparison about coast mode and cruise mode in speed profile; (

**b**) Comparison about coast mode and cruise mode in cumulative energy consumption curve.

**Figure 21.**Comparison of adaptive terminal sliding mode controller (ATSMC) and ATSMC+disturbance observer (DOB): (

**a**) Comparison in speed curve; (

**b**) Comparison in speed error.

**Figure 22.**Comparison of PID and ATSMC+DOB: (

**a**) Comparison in tracking curve; (

**b**) Comparison in speed error.

Symbol | Description |
---|---|

${F}_{n}{}^{\prime}$ | traction force |

${E}_{n}{}^{\prime}$ | energy consumption |

${a}_{n}{}^{\prime}$ | acceleration |

${X}_{n}$ | length of the subinterval |

Comfort Level | Acceleration Rate | Evaluation |
---|---|---|

Level 1 | <0.315 | Very comfortable |

Level 2 | 0.315–0.63 | Comfortable |

Level 3 | 0.63–1.0 | Relatively comfortable |

Level 4 | 1.0–1.6 | Uncomfortable |

Level 5 | 1.6–2.5 | Very uncomfortable |

Level 6 | >2.5 | Extremely uncomfortable |

Vehicle Parameters | Evaluation |
---|---|

Formation | 6 trains |

Mass-AW2 (ton) | 339.6 |

Length (m) | 120 |

Basic resistance parameters (${r}_{1}$) | 9.067 |

Basic resistance parameters (${r}_{2}$) | 0 |

Basic resistance parameters (${r}_{3}$) | 0.001334 |

Location (m) | Slope (‰) |
---|---|

0–97 | 0 |

97–377 | 2.357 |

377–1087 | 10 |

1087–1926 | 17.682 |

1926–2045 | 0 |

Location (m) | Speed Limit (km/h) |
---|---|

0–175 | 60 |

175–900 | 80 |

900–1831 | 70 |

1831–2030 | 65 |

Given Time (s) | Real Time (s) | Consumption (kJ) | Average Acceleration Rate (m/s^{3}) |
---|---|---|---|

141 | 143.6 | 118,147 | 0.0456 |

147 | 148.1 | 106,923 | 0.0331 |

153 | 155.3 | 101,240 | 0.0237 |

Mode | Given Time (s) | Real Time (s) | Consumption (kJ) | Average Acceleration Rate (m/s^{3}) |
---|---|---|---|---|

Optimal | 147 | 144.6 | 118,147 | 0.0456 |

Cruising | 147 | 146.3 | 130,626 | 0.0171 |

Experiment Number | $\mathit{T}$ (seconds) | $\mathit{\tau}$ (seconds) |
---|---|---|

1 | 0.5 | 0.1 |

2 | 0.8 | 0.2 |

3 | 1 | 0.3 |

4 | 1.2 | 0.4 |

Performance Indices | ATSMC+DOB | ATSMC | PID |
---|---|---|---|

Arrival time error (s) | 1.2 | 1.4 | 6.7 |

Total consumption (kJ) | 102,528 | 103,231 | 110,207 |

Average accelerated change rate (m/s^{3}) | 0.0371 | 0.0399 | 0.55 |

Average speed error (m/s) | 0.05276 | 0.1933 | 0.8731 |

Stop error (m) | 0.268 | 0.941 | 2.459 |

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## Share and Cite

**MDPI and ACS Style**

Wang, X.; Xiao, Z.; Chen, M.; Sun, P.; Wang, Q.; Feng, X.
Energy-Efficient Speed Profile Optimization and Sliding Mode Speed Tracking for Metros. *Energies* **2020**, *13*, 6093.
https://doi.org/10.3390/en13226093

**AMA Style**

Wang X, Xiao Z, Chen M, Sun P, Wang Q, Feng X.
Energy-Efficient Speed Profile Optimization and Sliding Mode Speed Tracking for Metros. *Energies*. 2020; 13(22):6093.
https://doi.org/10.3390/en13226093

**Chicago/Turabian Style**

Wang, Xiaowen, Zhuang Xiao, Mo Chen, Pengfei Sun, Qingyuan Wang, and Xiaoyun Feng.
2020. "Energy-Efficient Speed Profile Optimization and Sliding Mode Speed Tracking for Metros" *Energies* 13, no. 22: 6093.
https://doi.org/10.3390/en13226093