# Modular Bus Unit Scheduling for an Autonomous Transit System under Range and Charging Constraints

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## Abstract

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## 1. Introduction

- The optimization model for determining the optimal formation and trip sequences of MB units is developed. In particular, given that the vehicles are electrically powered, battery range limits and charging plans are considered in the system scheduling process.
- A column generation-based heuristic algorithm is designed to efficiently solve this model. The constraints of trip demand and charging station capacity are included in the main problem, and the problem of the mileage of modular units under a limited range is solved by subproblems.
- Taking real data from transit operations for numerical examples, the proposed model performs well in terms of both algorithmic performance and practical applications, enabling strategic support for the promotion of modular bus technology in transit systems.

## 2. Modular Bus Scheduling Model

#### 2.1. Preliminaries

#### 2.2. Mathematical Formulation

## 3. Solution Algorithm

#### 3.1. Master Problem

#### 3.2. Pricing Subproblems

#### 3.3. Solution Procedure

## 4. Computational Results

#### 4.1. Case Setup

#### 4.2. Algorithm Efficiency

#### 4.3. Results and Analyses

#### 4.4. Effects of the Dispatch Cost and Battery Capacity

## 5. Concluding Remarks

- The proposed column generation-based heuristic algorithm, which decomposes the original problem into a master problem and subproblems, outperforms widely used solvers in terms of time and computation speed. Even in a network of 214 nodes, the scheduling strategy can be obtained in about 10 min with a gap of less than 0.3%.
- The optimal modular bus scheduling scheme can reduce the overall system cost from $1534.31 to $1144.26, a reduction of approximately 25%, while accommodating uneven trip demand and embracing battery and charging station capacity constraints.
- Sensitivity analysis highlights the impact of dispatch cost and battery capacity of modular buses on system total costs. Compared to the traditional bus, operators are recommended to consider applying modular units in scenarios with low or volatile demand; there may still be scope for profitability even if the dispatching cost is high. Additionally, procuring modular buses with 25 kWh–35 kWh capacity can avoid frequent charging.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

Sets | |

$N$: | The set of nodes, index by $i$ or $j$ |

$A$: | The set of arcs, index by $\left(i,j\right)$ |

$K$: | The set of trips, index by $i$ or $j$ |

$D$: | The set of dummy nodes providing temporary stay for modular vehicles, index by $i$ or $j$ |

$F$: | The set of time-expanded charging stations, index by $i$ or $j$ |

${N}^{+}\left(i\right)$: | The set of nodes succeeding node $i$ on the directed graph, G, that is ${N}^{+}\left(i\right)=\left\{j\in N|\left(i,j\right)\in A\right\}$, index by $i$ or $j$ |

${N}^{-}\left(i\right)$: | The set of nodes preceding node $i$ on the directed graph $G$, that is ${N}^{-}\left(i\right)=\left\{j\in N|\left(j,i\right)\in A\right\}$, index by $i$ or $j$ |

$H$: | The set of modular bus platoons, index by $h$ |

$R$: | The set of trip sequence, index by $r$ |

Parameters | |

$o$: | The source node (depot departure) |

$s$: | The sink node (depot arrival) |

${v}_{i}$: | Start time of node $i$ |

${w}_{i}$: | End time of node $i$ |

${d}_{i}$: | Passenger demand of node $i$ |

${l}_{i}$: | Distance of node $i$ |

${p}_{i}$: | Energy consumption of node $i$ |

${p}_{ij}$: | Energy consumption of arc $\left(i,j\right)\in A$ |

${t}_{ij}$: | Idle time between node $i$ and node $j$ |

${t}_{a}$: | Constant threshold to ensure the smooth coupling/decoupling action between the modular bus units. |

${t}_{b}$: | Constant threshold introduced to prevent the nodes from being visited prematurely |

${c}_{ij}$: | Cost of a modular bus passing through arc $\left(i,j\right)$ |

${c}_{m}$: | Dispatch cost of one modular unit |

${c}_{p}$: | Charging cost of the modular bus |

${c}_{t}$: | Idling cost per unit time of the modular bus |

${c}_{s}$: | Operating cost per unit time of the modular bus |

${c}_{w}$: | Waiting cost per unit time of the modular bus |

${\delta}_{h}$: | The number of module units carried by module bus platoon $h$. ${\delta}_{h}\in \left\{1,2,\dots {g}_{max}\right\}$, where ${g}_{max}$ is the maximum number of module units allowed to be carried |

$M$: | Capacity of single modular bus |

$U$: | Charging station capacity |

$Q$: | Battery capacity of single modular bus |

$\mu $: | Battery loss rate |

${t}_{c}$: | Time interval between adjacent time-expanded charging nodes |

${Y}_{i}^{a}$: | The $a$-th node ahead of charging node $i$, where $a=1,\dots ,{a}_{max}$, ${a}_{max}=\frac{{w}_{i}-{v}_{i}}{{t}_{c}}$ |

${C}_{r}$: | Cost of the trip sequence $r$ |

${V}_{ri}$: | ${V}_{ri}=1$ if node $i$ is covered by the trip sequence $r$ provided by the subproblem and 0 otherwise |

${\theta}_{i}$: | Values of the dual variables associated with constraints (11) |

${\omega}_{i}$: | Values of the dual variables associated with constraints (12) |

Variables | |

${x}_{ijh}$: | Binary decision variable that equals 1 if module bus platoon $h\in H$ traverses arc $\left(i,j\right)$, and 0 otherwise |

${q}_{ih}$: | Intermediate variables, denoting the cumulative energy consumption of the module bus platoon $h$ at node $i$ |

${Z}_{r}$: | Integer variable, reveals the number of modular bus units assigned to the sequence $r$ |

${x}_{ij}$: | Binary decision variable that equals 1 if the modular bus traverses arc $\left(i,j\right)$ and 0 otherwise |

${q}_{i}$: | Intermediate variables, denoting the cumulative energy consumption of the module bus at node $i$ |

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**Figure 1.**Modular bus unit (Source: http://www.next-future-mobility.com/ (accessed on 5 February 2023)).

$j$ | $o$ | $D$ | $K$ | $F$ | $s$ | |

$i$ | ||||||

$o$ | × | × | ${w}_{o}+{t}_{oj}\le {v}_{j}$ | × | × | |

$D$ | × | ${w}_{i}={v}_{j}$ | ${v}_{j}-{t}_{b}\le {w}_{i}+{t}_{ij}\le {v}_{j}-{t}_{a}$ | × | ||

$K$ | × | ${v}_{j}-{t}_{b}\le {w}_{i}+{t}_{ij}\le {v}_{j}-{t}_{a}$ | ${w}_{i}+{t}_{is}\le {v}_{s}$ | |||

$F$ | × | ${v}_{j}-{t}_{b}\le {w}_{i}+{t}_{ij}\le {v}_{j}-{t}_{a}$ | × | |||

$s$ | × | × | × | × | × |

Instance | GUROBI | CGBH | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

Number | Trip | Node | LB ($) | Obj ($) | Time (s) | Gap (%) | LB ($) | Obj ($) | Time (s) | Gap (%) |

(1) | 10 | 52 | 375.45 | 375.45 | 43.13 | 0.00 | 375.45 | 375.45 | 0.89 | 0.00 |

(2) | 30 | 92 | 1106.22 | 1147.16 | >7200 | 3.57 | 1144.26 | 1144.26 | 18.93 | 0.00 |

(3) | 60 | 184 | 2166.27 | 2438.63 | >7200 | 11.17 | 2213.68 | 2217.35 | 464.87 | 0.17 |

(4) | 90 | 214 | 3094.25 | 4113.42 | >7200 | 24.78 | 3207.08 | 3215.24 | 824.40 | 0.25 |

Number | Trip Sequence | Number of Units Equipped with MB Platoon | Cumulative Energy Consumption (kWh) | Cost ($) |
---|---|---|---|---|

1 | $o$-1-7-14-84-21-27-$s$ | 3 | 45.37 | 155.23 |

2 | $o$-3-9-80-18-26-30-$s$ | 1 | 15.76 | 53.25 |

3 | $o$-2-7-48-51-54-88-25-29-$s$ | 1 | 15.01 | 48.18 |

4 | $o$-4-70-10-17-59-26-30-$s$ | 1 | 18.13 | 50.89 |

5 | $o$-6-11-18-24-$s$ | 1 | 18.34 | 39.57 |

6 | $o$-3-41-13-19-$s$ | 1 | 17.37 | 34.6 |

7 | $o$-3-70-10-17-23-28-$s$ | 3 | 52.26 | 152.08 |

8 | $o$-5-11-52-55-22-$s$ | 1 | 15.87 | 32.79 |

9 | $o$-6-12-54-22-$s$ | 2 | 33.6 | 67.38 |

10 | $o$-5-11-18-$s$ | 1 | 15.76 | 32.91 |

11 | $o$-15-20-26-30-$s$ | 3 | 57.16 | 116.52 |

12 | $o$-9-16-22-$s$ | 1 | 16.46 | 32.02 |

13 | $o$-2-8-14-84-57-25-29-$s$ | 1 | 15.01 | 53.34 |

14 | $o$-5-11-18-24-$s$ | 1 | 19.05 | 40.02 |

15 | $o$-6-12-54-57-60-26-30-$s$ | 1 | 19.05 | 41.27 |

16 | $o$-2-8-77-17-23-28-$s$ | 1 | 15.16 | 50.17 |

17 | $o$-1-68-9-16-22-$s$ | 1 | 16.46 | 43.25 |

18 | $o$-4-41-13-19-$s$ | 3 | 52.13 | 100.79 |

Total | Satisfying the demands of 30 trips | 27 | Not exceeding the battery capacity | 1144.26 |

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## Share and Cite

**MDPI and ACS Style**

Gao, H.; Liu, K.; Wang, J.; Guo, F.
Modular Bus Unit Scheduling for an Autonomous Transit System under Range and Charging Constraints. *Appl. Sci.* **2023**, *13*, 7661.
https://doi.org/10.3390/app13137661

**AMA Style**

Gao H, Liu K, Wang J, Guo F.
Modular Bus Unit Scheduling for an Autonomous Transit System under Range and Charging Constraints. *Applied Sciences*. 2023; 13(13):7661.
https://doi.org/10.3390/app13137661

**Chicago/Turabian Style**

Gao, Hong, Kai Liu, Jiangbo Wang, and Fangce Guo.
2023. "Modular Bus Unit Scheduling for an Autonomous Transit System under Range and Charging Constraints" *Applied Sciences* 13, no. 13: 7661.
https://doi.org/10.3390/app13137661