# Research on Energy-Saving Operation Strategy for Multiple Trains on the Urban Subway Line

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Introduction of the Multi-Train System and the Previous Study for the Single Train and Two-Train System

#### 2.1. Parameters of the Subway Train 100% Low Floor Train

_{train}is 37.34 m. The maximum speed V

_{max}is 70 km/h. The mass of the train in the AW2 case is 70 T.

_{t}(v) with respect to the speed v is described as:

_{b}(v) with respect to the speed v is described as:

#### 2.2. Parameters of the Traction Substation—Beijing Subway Line 4

_{g}= gi, where g is the gravitational acceleration and i is the gradient. The unit curve resistance is w

_{c}= 600/R, where R is the curve radius.

#### 2.3. Energy-Saving Operation Strategy for the Single Train

#### 2.4. Energy-Saving Operation Strategy for the Two-Train System

## 3. The Energy-Saving Operation Strategy for Multi-Train System

#### 3.1. The Model of Multi-Train Cooperative Operation

^{j}

_{trac}is tractive energy consumption for the jth train, and E

^{j}

_{regen}is the available regenerative braking energy for the jth train that can be reused by other trains.

_{unit_i}(i = 0, 1, 2, …, n). n is the total number of unit times during the travel time (for example, if travel time is 109 s, and unit time T

_{unit_i}is 0.01 s for the algorithm, n is 10,900).

_{i}≤ V

_{limit_i}, v

_{0}= 0, v

_{n}= 0.

_{i}≤ T, t

_{0}= 0, t

_{n}= T.

_{i}≤ S, s

_{0}= 0, s

_{n}= S.

_{0}, t

_{0}and s

_{0}are the speed, travel time and travel distance when the train starts at one station, and v

_{n}, t

_{n}and s

_{n}are the speed, travel time and travel distance when the train arrives at a new station, respectively.

_{t}≤ 1.

_{b}≤ 1.

_{trac_j_i}can be calculated as:

_{b_j_i}can be calculated as:

_{regen_j_i}) equals to the total traction energy, and vice versa. Moreover, e

_{regen_j_i}can be calculated as:

^{j}

_{trac}and E

^{j}

_{regen}of Equation (4) can be calculated as:

#### 3.2. Energy-Saving Driving Strategy of Multiple Trains

_{trm_k}is the kth switching point for train m. T

_{A}is the travel time from Station A to Station B, and T

_{B}is the travel time from Station B to Station C. T

_{DB}is the dwell time of Station B. T

_{I2}is the departure interval for the 2nd train, and T

_{I3}is the departure interval for the 3rd train. If the departure interval for the 3rd train t

_{tr3_1}(T

_{I3}) is little before t

_{tr2_5}and enables the 3rd train to accelerate with the regenerate energy from the 2nd train, four-mode operation is suitable for the 3rd train. In addition, if the departure interval for the 3rd train t

_{tr3_1}(T

_{I3}) is after t

_{tr2_5}, four-mode operation is also suitable for the 3rd train, and the 3rd train can use the regenerated energy as much as possible.

_{tr3_1}(T

_{I3}) is far before t

_{tr2_5}, as shown in Figure 6, the 3rd train can operate in coasting condition for a while to make the second maximum power condition overlap with the braking condition for the 2nd train. Thus, the 3rd train operates under five modes (maximum power, coasting, maximum power, coasting and maximum braking) during the trip. For this circumstance, five-mode operation is suitable for the 3rd train. After starts at t

_{tr3_1}, the 3rd train coasts from t

_{tr3_2}to t

_{tr3_3}, and traction from t

_{tr3_3}(t

_{tr2_5}) to t

_{tr3_4}(t

_{tr2_6}).

_{tr1_5}equals to the travel time from Station A to Station B plus dwell time at Station B (T

_{A}+ T

_{DB}). There are four situations for the 1st train to use the regenerative braking energy produced by the 2nd train. Define the t

_{tr1_start}and t

_{tr1_end}to be the start time and end time for the 1st train to use the regenerative energy. The 2nd train’s switching points are chosen for discussion.

- When t
_{tr1_5}≤ t_{tr2_5}≤ t_{tr1_6}, the 1st train can use regenerative braking energy provided by the 2nd train. Meanwhile, the overlapping period is from t_{tr1_start}= t_{tr2_5}to t_{tr1_end}= t_{tr1_6}. - When t
_{tr1_5}≤ t_{tr2_5}≤ t_{tr2_6}≤ t_{tr1_6}, the 1st train can use regenerative braking energy provided by the 2nd train. Meanwhile, the overlapping period is from t_{tr1_start}= t_{tr2_5}to t_{tr1_end}= t_{tr2_6}. - When t
_{tr2_5}≤ t_{tr1_5}≤ t_{tr2_6}, the 1st train can use regenerative braking energy provided by the 2nd train. Meanwhile, the overlapping period is from t_{tr1_start}= t_{tr1_5}to t_{tr1_end}= t_{tr2_6}. - Otherwise, there is no available regenerative braking energy when the 1st train departs from Station B again.

- When t
_{tr1_5}≤ t_{tr3_3}≤ t_{tr1_6}, the 1st train can use regenerative braking energy provided by the 2nd train. Meanwhile, the overlapping period is from t_{tr1_start}= t_{tr3_3}to t_{tr1_end}= t_{tr1_6}. - When t
_{tr1_5}≤ t_{tr3_3}≤ t_{tr3_4}≤ t_{tr1_6}, the 1st train can use regenerative braking energy provided by the 2nd train. Meanwhile, the overlapping period is from t_{tr1_start}= t_{tr3_3}to t_{tr1_end}= t_{tr3_4}. - When t
_{tr3_3}≤ t_{tr1_5}≤ t_{tr3_4}, the 1st train can use regenerative braking energy provided by the 2nd train. Meanwhile, the overlapping period is from t_{tr1_start}= t_{tr1_5}to t_{tr1_end}= t_{tr3_4.} - Otherwise, there is no available regenerative braking energy when the 1st train departs from Station B again.

- The 3rd train operates under four-mode operation or five-mode operation and is ensured to use the regenerate energy firstly.
- The 1st train under different dwell time tries to use the regenerate energy of the 2nd train as much as possible.
- The energy-saving operation strategy only provides the available operation modes (four-mode operation and five-mode operation) for the multi-train system, which can be regarded as the guidance for the computing method. The computing method solves the switching points and the drive curves under different modes and finally chooses the optimum drive curves and the optimum operation mode for the multi-train system.

#### 3.3. Computing Method for Energy-Saving Driving Strategy of Multiple Trains

- Initialize. Determine the parameters of urban subway train and traction substation.
- Import the line conditions (including the limiting velocity, the unit curve and slope resistance).
- Import the driving curves of the 1st train and the 2nd train which can be obtained based on the energy-saving operation strategy for the two-train system.
- Find the switching points t
_{tr2_5}and t_{tr2_6}(directly obtain from the driving curves). - Calculate E
_{total}. The calculating progress can be realize by Equations (4)–(9). Choose minimum energy consumption (E_{min}) and determine the optimum driving mode and driving curves for the 1st train and the 3rd train.

_{trm_k}is the velocity at t

_{trm_k}for train m. S

_{trm_ij}is the distance traveled by the mth train in the period (t

_{trm_i}, t

_{trm_j}).

- Determine the known switching points of the 3rd train. t
_{tr3_1}equals to the departure interval of the 3rd train (T_{I3}), t_{tr3_3}is the switching point when the 2nd train starts braking (t_{tr2_5}), t_{tr3_4}is the switching point when the 2nd train stops braking (t_{tr3_6}), and t_{tr3_6}equals to the departure interval plus the travel time from Station A to Station B (T_{I3}+ T_{AB}). The unknown switching points are t_{tr3_2}and t_{tr3_5}. - Two loops are built to gain the unknown switching points. Firstly, define the v
_{tr3_2}. Then, t_{tr3_2}can be calculated based on the line conditions. Since the t_{tr3_1}, t_{tr3_2}, t_{tr3_3}and t_{tr3_4}are known, the S_{tr3_12}, S_{tr3_23}and S_{tr3_34}can be calculated. - Under the circumstance that the v
_{tr3_2}is determined, define v_{tr3_5}. The t_{tr3_5}can be calculated. Since the t_{tr3_4}, t_{tr3_5}and t_{tr3_6}are known, the S_{tr3_45}and S_{tr3_56}can be calculated. - If the total travel distance S
_{tr3_16}(S_{tr3_12}+ S_{tr3_23}+ S_{tr3_34}+ S_{tr3_45}+ S_{tr3_56}) does not equals to the distance S_{AB}between Station A and Station B, the v_{tr3_5}will be added with a small increment and back to the Step 3 of the process. Moreover, if S_{tr3_16}equals the S, the current (t_{tr3_2}, v_{tr3_2}) and (t_{tr3_5}, v_{tr3_5}) will be saved as one solution for the five-mode operation. - Calculate total energy consumption of the 3rd train E
^{3}_{total}. - A small increment is added to v
_{tr3_2}, and back to step 2, 3, 4 again. - Choose minimum energy consumption of the 3rd train, and the corresponding driving curve is defined as the optimum driving curve for the 3rd train under five-mode operation.

_{trace}can be obtained as:

_{max}is the maximum speed, and b is the deceleration coefficient of the train, its unit is m/s

^{2}. L

_{safety}is the safe distance between the parking target position for the following train and the parking position of the leading train, L

_{train}is the length of the train.

_{last}and v

_{current}are the velocities of last unit time and current unit time, and the S

_{last}and S

_{current}are the travel distances of last unit time and current unit time. The ΔS is the distance between two trains. The v

_{current}and ΔS in the calculating process are verified to modify the velocity.

_{current}can be calculated as:

_{t}(v), braking force F

_{b}(v), running resistance w(v), unit gradient resistance w

_{g}and unit curve resistance w

_{c}can be obtained from Section 2.1 and Section 2.2. The travel distance of current unit time S

_{current}can be calculated as:

#### 3.4. Conclusion of the Energy-Saving Operation Strategy

## 4. Simulation Results and Analysis

#### 4.1. Results of Simulation

#### 4.2. Analysis

- When the departure interval and the dwell time are certain, the optimum operation mode and driving curves for the multi-train system can be obtained by the proposed energy-saving operation strategy. For example, when the dwell time of the 1st train is 30 s, and the departure interval is between 110 s and 186 s, the five-mode operation is suitable for the 3rd train, and, when the departure interval is after 186 s, four-mode operation is suitable for the 3rd train. The driving curves can be obtained by the corresponding computing method.
- When the dwell time is specified and the departure interval is adjustable, the optimum departure interval can be obtained to realize minimum energy consumption for the multi-train system by the proposed energy-saving operation strategy. For example, when the dwell time of the 1st train is 30 s, the optimum departure interval for the 3rd train is 182 s and the current energy-saving efficiency is 18.15%.
- When the dwell time is adjustable, the optimum dwell time for the 1st train can be obtained to make full use of the 2nd train’s regenerate energy. For example, when the dwell time is 50 s, the total energy consumption is the minimum, and the energy-saving efficiency is 18.24%.
- The computing method can work under complex circumstance considering the line conditions and the moving automatic block system. For example, the 3rd train is not allowed to depart when the 1st train starts accelerating.
- The proposed energy-saving operation strategy is suitable for both online and offline operation. When the departure interval and the dwell time are given by train dispatch, the operation strategy with velocity adjustment and moving automatic block system can gain the driving curves quickly. Moreover, the strategy is also suitable for offline calculation because energy consumption results under different departure intervals and dwell times can be calculated to guide and adjust the train dispatch.
- The proposed energy-saving operation strategy not only can be applied to the three-train system but can also be generalized to other multi-train systems. As in the previous example, one traction substation contains at most three trains, so the three trains can be regarded as a group. All the trains on one subway line depart by groups. For the three-train group, after the 1st train departs again from Station B, the whole system can be regarded to operate for a new circulation, so the energy-saving operation strategy is fit for the 2nd train and the 3rd train departing from station B again. For other multi-train systems that one traction substation contains more than three trains, the computing method is also able to calculate the driving curves for rest of the trains under different dwell times and departure intervals.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

F_{t}(v) | Traction force |

w(v) | Running resistance |

w_{c} | Unit curve resistance |

E^{j}_{trac} | Tractive energy consumption for the jth train |

T_{unit_i} | Unit time |

v_{i} | Speed |

s_{i} | Travel distance |

n_{b} | Braking force coefficient |

e_{b_j_i} | Braking energy of jth train in unit time |

T_{X} | Travel time from Station X to next station |

T_{Im} | Departure interval for the train m |

S_{trm_ij} | Distance traveled by the mth train in the period (t_{trm_i}, t_{trm_j}) |

V_{max} | Maximum speed |

L_{train} | Length of the train |

v_{last} | Velocity of last unit time |

S_{last} | Travel distance of last unit time |

ΔS | Distance between two trains |

F_{b}(v) | Braking force |

w_{g} | Unit gradient resistance |

E_{total} | Total energy consumption |

E^{j}_{regen} | Available regenerative braking energy for the jth train |

n | Total number of unit times |

t_{i} | Travel time |

n_{t} | Traction coefficient |

e_{trac_j_i} | Traction energy consumption of jth train in unit time |

t_{trm_k} | kth switching point for train m |

T_{DX} | Dwell time of Station X |

v_{trm_k} | Velocity at t_{trm_k} for train m |

L_{trace} | Tracing distance |

L_{safety} | Safe distance |

b | Deceleration coefficient |

v_{current} | Velocity of current unit time |

S_{current} | Travel distance of current unit time |

M | Mass of the train |

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**Figure 5.**The driving curves for multi-train system when the 3rd train operates under four-mode operation.

**Figure 6.**The driving curves for multi-train system when the 3rd train operates under five-mode operation.

**Figure 15.**Results under five-mode operation: (

**a**) energy consumption of the 1st train and the 3rd train; and (

**b**) total energy consumption and energy-saving efficiency.

**Figure 16.**Results under four-mode operation: (

**a**) energy consumption of the 1st train and the 3rd train; and (

**b**) total energy consumption and energy-saving efficiency.

Slope Conditions | Curve Conditions | |||||
---|---|---|---|---|---|---|

Station | Section (km) | Gradient (‰) | Travel Time (s) | Section (km) | Curve Radius (m) | Limiting Velocity (km/h) |

Anheqiao North | 0.000–0.030 | −8.00 | 109 | 0.100–0.268 | 400 | 63.7792 |

0.030–0.200 | 23.00 | 0.860–0.974 | 2000 | 77.0208 | ||

0.20–0.860 | 2.00 | 0.994–1.143 | 1000 | 74.2191 | ||

0.860–1.060 | 3.00 | 1.164–1.298 | 2000 | 77.0208 | ||

1.060–1.363 | −4.00 | |||||

Beigongmen | 1.363–1.420 | −4.00 | 93 | 1.613–1.670 | 3000 | 61.7540 |

1.420–1.640 | 2.00 | 1.772–1.958 | 2000 | 77.0208 | ||

1.640–1.880 | 3.25 | 1.983–2.131 | 2000 | 77.0208 | ||

1.880–2.245 | −3.00 | 2.469–2.614 | 600 | 67.6481 | ||

2.245–2.465 | 2.00 | |||||

2.465–2.614 | −15.00 | |||||

Xiyuan | 2.614–2.950 | −15.00 | - | 2.614–2.993 | 600 | 67.6481 |

Station | Tractive Energy Consumption (MJ) | Regenerative Braking Energy (MJ) | Total Energy Consumption (MJ) |
---|---|---|---|

A–B | 14.330454 | 4.861847 | 9.468607 |

B–C | 12.446502 | 5.634088 | 6.812414 |

Energy Consumption | Two-Train Strategy | Single Train Strategy |
---|---|---|

energy consumption of the 2nd (MJ) | 9.040636 | 9.587708 |

total energy consumption of two trains (MJ) | 23.371090 | 23.918162 |

Five-Mode Operation | Four-Mode Operation | ||||
---|---|---|---|---|---|

Dwell Time (s) | Total Energy Consumption (MJ) | Efficiency (%) | Dwell Time (s) | Total Energy Consumption (MJ) | Efficiency (%) |

30 | 45.37766 | 18.15 | 30 | 45.89939 | 17.21 |

40 | 45.37766 | 18.15 | 40 | 45.89939 | 17.21 |

50 | 45.32823 | 18.24 | 50 | 45.80053 | 17.31 |

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## Share and Cite

**MDPI and ACS Style**

Liu, J.; Zhao, N.
Research on Energy-Saving Operation Strategy for Multiple Trains on the Urban Subway Line. *Energies* **2017**, *10*, 2156.
https://doi.org/10.3390/en10122156

**AMA Style**

Liu J, Zhao N.
Research on Energy-Saving Operation Strategy for Multiple Trains on the Urban Subway Line. *Energies*. 2017; 10(12):2156.
https://doi.org/10.3390/en10122156

**Chicago/Turabian Style**

Liu, Jianqiang, and Nan Zhao.
2017. "Research on Energy-Saving Operation Strategy for Multiple Trains on the Urban Subway Line" *Energies* 10, no. 12: 2156.
https://doi.org/10.3390/en10122156