# Daytime and Overnight Joint Charging Scheduling for Battery Electric Buses Considering Time-Varying Charging Power

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Battery Electric Vehicle Charging Scheduling

#### 2.2. Optimization of BEB Charging Station Locations

#### 2.3. BEB Charging Scheduling

## 3. Problem Description

#### 3.1. Bus Transit Network with a Single Charging Station

#### 3.2. Peak–Valley Electricity Price

#### 3.3. BEB Operation Timetable

## 4. Model Formulation

#### 4.1. Notations

**Sets:**

- J:
- Set of BEBs, indexed by j, i.e., $j\in J=\{1,2,...,|J\left|\right\}$;
- T:
- Set of times, indexed by t, i.e., $t\in T=\{1,2,...,|T\left|\right\}$;

**Parameters:**

- ${L}_{jt}$:
- Location of BEBs, 1 if BEB $j\in J$ is at the terminal at time slot $t\in T$;
- ${f}_{t}$:
- Electricity cost at time slot $t\in T$ in CNY/kWh;
- ${C}_{j}^{int}$:
- Initial SOC of BEB j in kWh;
- ${C}_{min}$
- Minimum state of charge required by BEB $j\in J$ during daytime operations in kWh;
- ${C}_{max}$
- Maximum battery capacity in kWh;
- ${p}_{a}$:
- Maximum power allowed by the battery in kW;
- ${P}_{b}$:
- Maximum transmit power of each charging pile in kW;
- ${e}_{t}$:
- Electricity consumption per time slot during the operation of a BEB in kWh;
- $\Delta t$:
- Duration of each slot;
- ${P}_{max}$
- Total output power, upper limit of the charging station in kW;
- N:
- Number of charging piles in the charging station;

**Decision Variables:**

- ${x}_{jt}$:
- Binary variable, equal to 1 if BEB $j\in J$ is charging at time slot $t\in T$, 0 otherwise;
- ${I}_{jt}$:
- Binary variable, auxiliary variable for controlling the continuity of charge between time slot t and $t+1$, ${I}_{jt}\in \{0,1\}$;
- ${V}_{jt}$:
- Binary variable, auxiliary variable for controlling the continuity of charge between time slot t and $t-1$, ${V}_{jt}\in \{0,1\}$;
- ${C}_{jt}$:
- Continuous variable, SOC of BEB $j\in J$ at time slot $t\in T$ in kWh;
- $\Delta {C}_{jt}$:
- Continuous variable, amount of SOC variations of BEB $j\in J$ at time slot $t\in T$ in kWh;
- ${p}_{jt}$:
- Continuous variable, charging power of BEB $j\in J$ at time slot t;
- Q:
- Continuous variable, total charging cost.

#### 4.2. Mathematical Model

## 5. Case Studies

#### 5.1. Data Description

_{4}batteries. The maximum charging power allowed for the battery ${p}_{a}$ is 90 kW. According to the peak–valley electricity price in Table 2, we can derive the electricity price ${f}_{t}$ at any time slot t. In addition, referring to the research by Liu et al. (2021) [12], we set the power consumption of the BEB to 0.25 kWh per minute during operation. In the case study, we define the length of one time slot as one minute (i.e., $\Delta t=1$), and one day from 5:30 to 23:30 is divided into 1080 time slots. Therefore, the power consumption ${e}_{t}$ of one time slot during the driving of any BEB is 0.25 kWh. We define the start time of each cycle as 5:30 every morning.

#### 5.2. Single-Line Bus Network

#### 5.3. Multi-Line Bus Network

_{4}batteries. There are 6 charging piles in the charging station, and each charging pile is equipped with a single connector. The maximum transmitting power ${p}_{b}$ of each charging pile is 80 kW, and the total power limit ${p}_{max}$ allowed for the bus transit network is 420 kW. Basic data of the four bus lines are given in Table 5. The operating timetable of each BEB in the multi-line bus network is shown in Table 6.

## 6. Sensitivity Analysis

#### 6.1. Impact of Battery Capacity

#### 6.2. Impact of the Number of Charging Piles

#### 6.3. Impact of the Minimum SOC

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Peak Period | Parity Period | Valley Period | |
---|---|---|---|

Time interval | 08:00∼11:00 | 06:00∼08:00 | 22:00∼6:00 |

18:00∼22:00 | 11:00∼18:00 | - |

Electricity Classification | Electricity Price (CNY/kW·h) | ||||
---|---|---|---|---|---|

Less than 1 kV | 10 kV | 35 kV | More than 110 kV | ||

Commercial, industrial electricity | Peak period | 1.074 | 1.049 | 1.024 | 0.999 |

Flat valley period | 0.671 | 0.646 | 0.621 | 0.596 | |

Valley period | 0.316 | 0.310 | 0.304 | 0.298 | |

Agriculture, residential electricity | Peak period | - | 0.730 | - | - |

Flat valley period | - | 0.448 | - | - | |

Valley period | - | 0.242 | - | - |

Bus Line | Number of BEBs | Mileage | Number of Trips | Trip Duration |
---|---|---|---|---|

Line S1 | 18 | 74.8 km | 3 | 140 min |

Line S2 | 17 | 65.4 km | 4 | 130 min |

Bus Line | Objective | Solution Time | FCFS | GAP |
---|---|---|---|---|

Line S1 | CNY 1174.12 | 1947.3 s | CNY 1268.91 | 7.5% |

Line S2 | CNY 1086.78 | 1766.5 s | CNY 1172.37 | 7.3% |

Bus Line | Number of BEBs | Mileage | Number of Trips | Trip Duration |
---|---|---|---|---|

Line 1 | 7 | 24.2 km | 7 | 90 min |

Line 2 | 8 | 28.2 km | 6 | 100 min |

Line 3 | 7 | 26.6 km | 7 | 90 min |

Line 4 | 7 | 25.6 km | 7 | 90 min |

Bus Number | Time Slots on the Trip |
---|---|

Line 1–1 | 5:30–7:00; 7:40–9:10; 9:50–11:20; 12:00–13:30; 14:10–15:40; 16:20–17:50; 18:30–20:00 |

Line 1–2 | 5:50–7:20; 8:00–9:30; 10:10–11:40; 12:20–13:50; 14:30–16:00; 16:40–18:10; 18:50–20:20 |

Line 1–3 | 6:10–7:40; 8:20–9:50; 10:30–12:00; 12:40–14:10; 14:50–16:20; 17:00–18:30; 19:10–20:40 |

Line 1–4 | 6:30–8:00; 8:40–10:10; 10:50–12:20; 13:00–14:30; 15:10–16:40; 17:20–18:50; 19:30–21:00 |

Line 1–5 | 6:50–8:20; 9:00–10:30; 11:10–12:40; 13:20–14:50; 15:30–17:00; 17:40–19:10; 19:50–21:20 |

Line 1–6 | 7:10–8:40; 9:20–10:50; 11:30–13:00; 13:40–15:10; 15:50–17:20; 18:00–19:30; 20:10–21:40 |

Line 1–7 | 7:30–9:00; 9:40–11:10; 11:50–13:20; 14:00–15:30; 16:10–17:40; 18:20–19:50; 20:30–22:00 |

Line 2–1 | 6:00–7:40; 8:40–10:20; 11:20–13:00; 14:00–15:40; 16:40–18:20; 19:20–21:00 |

Line 2–2 | 6:20–8:00; 9:00–10:40; 11:40–13:20; 14:20–16:00; 17:00–18:40; 19:40–21:20 |

Line 2–3 | 6:40–8:20; 9:20–11:00; 12:00–13:40; 14:40–16:20; 17:20–19:00; 20:00–21:40 |

Line 2–4 | 7:00–8:40; 9:40–11:20; 12:20–14:00; 15:00–16:40; 17:40–19:20; 20:20–22:00 |

Line 2–5 | 7:20–9:00; 10:00–11:40; 12:40–14:20; 15:20–17:00; 18:00–19:40; 20:40–22:20 |

Line 2–6 | 7:40–9:20; 10:20–12:00; 13:00–14:40; 15:40–17:20; 18:20–20:00; 21:00–22:40 |

Line 2–7 | 8:00–9:40; 10:40–12:20; 13:20–15:00; 16:00–17:40; 18:40–20:20; 21:20–23:00 |

Line 2–8 | 8:20–10:00; 11:00–12:40; 13:40–15:20; 16:20–18:00; 19:00–20:40; 21:40–23:20 |

Line 3–1 | 5:30–7:00; 7:40–9:10; 9:50–11:20; 12:00–13:30; 14:10–15:40; 16:20–17:50; 18:30–20:00 |

Line 3–2 | 5:50–7:20; 8:00–9:30; 10:10–11:40; 12:20–13:50; 14:30-16:00; 16:40–18:10; 18:50–20:20 |

Line 3–3 | 6:10–7:40; 8:20–9:50; 10:30–12:00; 12:40–14:10; 14:50-16:20; 17:00–18:30; 19:10–20:40 |

Line 3–4 | 6:30–8:00; 8:40–10:10; 10:50–12:20; 13:00–14:30; 15:10–16:40; 17:20–18:50; 19:30–21:00 |

Line 3–5 | 6:50–8:20; 9:00–10:30; 11:10–12:40; 13:20–14:50; 15:30–17:00; 17:40–19:10; 19:50–21:20 |

Line 3–6 | 7:10–8:40; 9:20–10:50; 11:30–13:00; 13:40–15:10; 15:50–17:20; 18:00–19:30; 20:10–21:40 |

Line 3–7 | 7:30–9:00; 9:40-11:10; 11:50–13:20; 14:00–15:30; 16:10–17:40; 18:20–19:50; 20:30–22:00 |

Line 4–1 | 5:40–7:10; 8:00-9:30; 10:20–11:50; 12:40–14:10; 15:00–16:30; 17:20–18:50; 19:40–21:10 |

Line 4–2 | 6:00–7:30; 8:20–9:50; 10:40–12:10; 13:00–14:30; 15:20–16:50; 17:40–19:10; 20:00–21:30 |

Line 4–3 | 6:20–7:50; 8:40–10:10; 11:00–12:30; 13:20–14:50; 15:40–17:10; 18:00–19:30; 20:20–21:50 |

Line 4–4 | 6:40–8:10; 9:00–10:30; 11:20–12:50; 13:40–15:10; 16:00–17:30; 18:20–19:50; 20:40–22:10 |

Line 4–5 | 7:00–8:30; 9:20–10:50; 11:40–13:10; 14:00–15:30; 16:20–17:50; 18:40–20:10; 21:00–22:30 |

Line 4–6 | 7:20–8:50; 9:40–11:10; 12:00–13:30; 14:20–15:50; 16:40–18:10; 19:00–20:30; 21:20–22:50 |

Line 4–7 | 7:40–9:10; 10:00–11:30; 12:20–13:50; 14:40–16:10; 17:00–18:30; 19:20–20:50; 21:40–23:10 |

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## Share and Cite

**MDPI and ACS Style**

Zheng, F.; Wang, Z.; Wang, Z.; Liu, M.
Daytime and Overnight Joint Charging Scheduling for Battery Electric Buses Considering Time-Varying Charging Power. *Sustainability* **2023**, *15*, 10728.
https://doi.org/10.3390/su151310728

**AMA Style**

Zheng F, Wang Z, Wang Z, Liu M.
Daytime and Overnight Joint Charging Scheduling for Battery Electric Buses Considering Time-Varying Charging Power. *Sustainability*. 2023; 15(13):10728.
https://doi.org/10.3390/su151310728

**Chicago/Turabian Style**

Zheng, Feifeng, Zhixin Wang, Zhaojie Wang, and Ming Liu.
2023. "Daytime and Overnight Joint Charging Scheduling for Battery Electric Buses Considering Time-Varying Charging Power" *Sustainability* 15, no. 13: 10728.
https://doi.org/10.3390/su151310728