# Optimal Charging Schedule Planning for Electric Buses Using Aggregated Day-Ahead Auction Bids

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## Abstract

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## 1. Introduction

#### 1.1. The Purpose, Objectives, and Hypothesis of Research

- It is possible to define minutely requirements and constraints of the charging equipment and individual buses at the aggregate level in such a way that taking them into account in an hourly model allows obtaining the feasible disaggregated charging schedules.
- The hourly charging model can be incorporated into the auction-clearing problem, extending the standard formulation with new bidding parameters adopted for storage-based participants.

#### 1.2. Literature Review

#### 1.3. Contribution

- We formulate new mixed-integer linear programming aggregation models regarding the detailed, minutely characteristics and constraints of the charging equipment and the buses. They determine charging availability expressed as minimum and maximum hourly energy requirements.
- We propose only a few aggregated bids parameters that lead to incorporating linear variables and constraints into the standard auction model. The extended auction model determines aggregated hourly charging plan.
- We formulate the new mixed-integer linear programming disaggregation model linking the auction-based hourly plan and the detailed charging characteristics.
- We demonstrate that the bus fleet can be economically scheduled with an auction-clearing model with suitable bid parameters.

## 2. Materials and Methods

#### 2.1. Aggregation Model

#### 2.1.1. Minimum Hourly SoC Level

#### 2.1.2. Maximum Hourly Energy Requirements

#### 2.2. Auction Model

#### 2.3. Disaggregation Model

#### 2.4. Summary

## 3. Results

#### 3.1. Data

#### 3.2. Results of the Proposed Approach

#### 3.2.1. Aggregation Results

#### 3.2.2. Auction Results

#### 3.2.3. Disaggregation Results

#### 3.3. Cost Savings Analysis

#### 3.4. Sensitivity Analysis of the Number of Chargers

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

Indices | |

b | Index of electric buses |

h | Index of time periods (hours) |

t | Index of time periods (minutes) |

${i}_{b}$ | Index of minutes when bus b ends the idle period |

Parameters | |

$\delta $ | idle time (in minutes) |

${A}_{bt}$ | 0/1 indicator if bus b is available for charging in minute t |

C | Number of chargers |

c | Maximum unit price for the excess amount of energy set by bus aggregator |

${E}_{b}^{\mathrm{cycle}}$ | Per-cycle trip energy demand of bus b |

${E}_{bt}^{\mathrm{Trip}}$ | Per-minute trip energy demand of bus b (per-cycle energy use/cycle time) |

${\overline{\mathrm{SoC}}}_{b}$ | Maximum battery State of Charge of bus b |

${\underline{\mathrm{SoC}}}_{b}$ | Minimum battery State of Charge of bus b |

${\mathrm{SoC}}_{b}^{0}$ | Initial battery State of Charge of bus b |

r | Charging rate of a charger |

Variables used specifically in aggregation/disaggregation model | |

${p}_{bt}$ | Charging energy of bus b scheduled in minute t |

${s}_{bt}$ | Binary charging status of bus b in minute t |

${\mathrm{SoC}}_{bt}$ | Battery State of Charge of bus b in minute t |

${\mathrm{SoC}}_{bh}^{min}$ | Minimum battery State of Charge of bus b at the end of hour h |

Variables in Aggregation model used as Parameters in Auction model | |

${E}^{1}$ | Total energy that must be loaded by buses |

${E}^{2}$ | Additional energy that can be loaded by buses |

${E}_{h}^{\mathrm{Agg},\mathrm{Trip}}$ | Summary hourly trip energy of buses |

${\mathrm{P}}_{h}^{\mathrm{Agg},max}$ | Aggregated maximum amount of energy that can be loaded in hour h |

${\mathrm{SoC}}_{h}^{\mathrm{Agg},min}$ | Aggregated minimum battery State of Charge in hour h |

Variables used specifically in auction model | |

$\mathit{Cost}\left({\mathit{NetSupply}}_{\mathit{h}}\right)$ | Net supply cost in hour h |

${\mathit{NetSupply}}_{h}$ | Net supply in hour h |

${\mathrm{SoC}}_{h}^{\mathrm{Agg}}$ | Aggregated battery State of Charge in hour h |

${\pi}_{h}$ | Auction energy price in hour h derived as the shadow prices to balance constraint () |

Variables in Auction model used as Parameters in Disaggregation model | |

${p}_{h}^{\mathrm{Agg}}$ | Charging energy scheduled in hour h |

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**Figure 4.**Illustration of the Asap recharging strategy that aims in refilling batteries to maximum SoC during each idle period.

**Figure 5.**Total number of buses trying to recharge immediately after each cycle under Asap charging strategy.

**Figure 7.**Aggregated minimum battery State of Charge and aggregated maximum amount of energy that can be loaded in each hour.

**Figure 11.**Comparison of costs incurred in each scenario in reference to costs incurred in the case of charging buses as soon as possible (asap rule).

**Figure 12.**Histograms of costs savings achieved in each day of years 2018–2019 when comparing day-ahead auction-based charging to the Asap charging.

**Figure 13.**Comparison of computed parameters regarding aggregated minimum battery State of Charge and aggregated maximum amount of energy, assuming four (baseline case) or three fast charging stations.

**Table 1.**Description of bus lines at the Ohio campus [8].

Bus Line | Cycle Time [min] | Per-Cycle Energy Use [kWh] | Frequency [min] | No. of Buses |
---|---|---|---|---|

North Express | 23 | 8.41 | 9 | 5 |

Loop North | 31 | 10.91 | 9 | 4 |

Loop South | 31 | 11.08 | 9 | 4 |

Central Connector | 32 | 12.11 | 12 | 3 |

East Residential | 33 | 11.62 | 9 | 4 |

Buckeye Village | 30 | 12.71 | 15 | 2 |

$\mathit{\delta}$ | C | r | ${\underline{\mathbf{SoC}}}_{\mathit{b}}$ | ${\overline{\mathbf{SoC}}}_{\mathit{b}}$ | ${\mathbf{SoC}}_{\mathit{b}}^{0}$ | ${\mathit{E}}^{1}$ | ${\mathit{E}}^{2}$ |
---|---|---|---|---|---|---|---|

5 min | 4 | 250 kW | 11 kW | 52.25 kW | 52.25 kWh | 3854.98 kWh | 907.5 kWh |

h | ${\mathit{E}}_{\mathit{h}}^{\mathbf{Agg},\mathbf{Trip}}$ | ${\mathbf{P}}_{\mathit{h}}^{\mathbf{Agg},max}$ | ${\mathbf{SoC}}_{\mathit{h}}^{\mathbf{Agg},min}$ |
---|---|---|---|

1 | 336.8 | 186.0 | 242.0 |

2 | 410.3 | 597.7 | 285.3 |

3 | 413.3 | 740.2 | 366.4 |

4 | 415.0 | 665.0 | 336.3 |

5 | 412.1 | 692.7 | 351.3 |

6 | 411.4 | 700.6 | 357.9 |

7 | 411.4 | 736.3 | 360.5 |

8 | 414.6 | 692.7 | 348.3 |

9 | 408.5 | 649.2 | 375.5 |

10 | 416.1 | 641.3 | 347.5 |

11 | 414.3 | 732.3 | 340.2 |

12 | 298.6 | 843.1 | 242.0 |

**Table 4.**Comparison of hourly charging plans [kWh] obtained in each scenario, assuming four or three fast charging stations.

h | Scenario 1 | Scenario 2 | ||
---|---|---|---|---|

4 Chargers | 3 Chargers | 4 Chargers | 3 Chargers | |

1 | 186.0 | 186.0 | 0.0 | 0.0 |

2 | 411.0 | 411.0 | 43.3 | 43.3 |

3 | 299.9 | 250.4 | 334.0 | 333.9 |

4 | 0.0 | 0.0 | 384.9 | 386.2 |

5 | 292.3 | 341.8 | 427.1 | 425.8 |

6 | 556.0 | 500.6 | 418.1 | 418.0 |

7 | 504.8 | 491.0 | 517.7 | 532.3 |

8 | 439.3 | 430.4 | 458.1 | 451.0 |

9 | 397.1 | 370.9 | 441.0 | 381.2 |

10 | 415.2 | 421.9 | 414.6 | 427.0 |

11 | 353.3 | 451.1 | 416.3 | 454.2 |

12 | 0.0 | 0.0 | 0.0 | 2.1 |

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**MDPI and ACS Style**

Zoltowska, I.; Lin, J. Optimal Charging Schedule Planning for Electric Buses Using Aggregated Day-Ahead Auction Bids. *Energies* **2021**, *14*, 4727.
https://doi.org/10.3390/en14164727

**AMA Style**

Zoltowska I, Lin J. Optimal Charging Schedule Planning for Electric Buses Using Aggregated Day-Ahead Auction Bids. *Energies*. 2021; 14(16):4727.
https://doi.org/10.3390/en14164727

**Chicago/Turabian Style**

Zoltowska, Izabela, and Jeremy Lin. 2021. "Optimal Charging Schedule Planning for Electric Buses Using Aggregated Day-Ahead Auction Bids" *Energies* 14, no. 16: 4727.
https://doi.org/10.3390/en14164727