# Optimal Scheduling to Manage an Electric Bus Fleet Overnight Charging

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## Abstract

**:**

## 1. Introduction

- A methodological approach to manage an EBs fleet overnight charging based on nonlinear programing was carried out. The resulting smart charging strategy aims to minimize the battery aging cost by making optimal charging decisions. This approach can handle several operating constraints and the proposed algorithm could be extended to hundreds of buses with an acceptable computation time.
- An optimization tool was developed in the Matlab/Simulink environment. The tool optimizes a large-scale EBs fleet charging considering the grid technical constraints and real operating constraints. It includes an Electro-Thermal and Aging coupled battery model of a given EB to simulate the dynamic response of the battery.
- A case study was performed where the simulation of the proposed approach was conducted taking into account some real operating constraints. The potential economic gain of an optimal EBs charging for 10 years operation was compared to three typical non-optimal charging strategies to show the potential economic gain.

## 2. Methodology and System Modeling

#### 2.1. Electric Vehicle Supply Equipment and Communication Protocol

- The overnight charging where EBs batteries are charged at the depot overnight with slow chargers (typically 40–120 kW).
- The opportunity charging where EBs batteries are charged at bus stops (up to 600 kW) or terminals (usually between 150 to 500 kW) mainly using over-head pantographs.

#### 2.2. Optimization Tool for the Management of the EBs Fleet Charging

#### 2.3. Electro-Thermal and Aging Coupled Model

#### 2.3.1. Battery Electrical Model

#### 2.3.2. Battery Thermal Model

#### 2.3.3. Battery Aging Model

#### 2.3.4. Electro-Thermal and Aging Coupled Battery Model

#### 2.3.5. Converter Model

## 3. Optimization Problem Formulation

#### 3.1. Literature Review of Large-Scale EBs Smart Charging Algorithms

#### 3.2. Nonlinear Programming Optimization (NLP)

#### 3.3. Pre-Optimization Process

#### 3.4. Optimization Design Variable

#### 3.5. Objective Function

#### 3.6. Linear Equality and Inequality Constraints

#### 3.7. Nonlinear Equality and Inequality Constraints

_{1}, p

_{2}) during a time slot, in such a way that the CV phase charging allows the SoC to increase from 95% to 100%. This charging limitation can be expressed as a nonlinear equality:

Code 1 Proposed optimization steps |

1- Initialize the number of buses $Eb=1:n$ |

2- Define the linear (in)equalities and lower/upper bounds: lb, ub, A, B, C |

$lb=0|ub={p}_{max}$; $A\times {P}_{\left\{j\right\}}\le {p}_{subscribed}$; $B\times {P}_{\left\{i\right\}}{}^{T}=0$ |

$C\times {P}_{\left\{i\right\}}{}^{T}=\frac{\left(So{C}_{final\left(i\right)}-So{C}_{initial\left(i\right)}\right)\times {Q}_{bat\left(i\right)}\times {V}_{bat\left(i\right)}}{100\times \Delta T\times {\eta}_{ch\left(i\right)}\times {\eta}_{bat\left(i\right)}}$ (Pre-optimization process) |

3- Define the nonlinear equalities: $\mathrm{Ceq}\left(1\right)={p}_{i,j+1}-{p}_{1}$ $\mathrm{Ceq}\left(2\right)={p}_{i,j}-{p}_{2}$ |

4- Define randomly an initial point ${P}_{0}$ (n×m) matrix |

${P}_{0}=\left(\begin{array}{cccc}{p}_{1,1}& {p}_{1,2}& \cdots & {p}_{1,m}\\ \cdots & \cdots & \ddots & \cdots \\ {p}_{n,1}& {p}_{n,2}& \cdots & {p}_{n,m}\end{array}\right)$ |

5- Define the objective function $\mathrm{fun}={\displaystyle \sum}_{i=1}^{n}{\displaystyle \sum}_{j=1}^{m}\frac{\Delta {Q}_{\mathit{loss}}{}_{i,j}}{{Q}_{\mathit{EoL}}}\times {\mathit{Bat}}_{\mathit{price}}\times {E}_{\mathit{bat}\text{}i,j}$ |

6- Optimization process |

for i = 1: MaxIter |

Calculate Gradient around current point ${P}_{i}$ |

Generate the next solution ${P}_{i+1}$ |

Evaluate the next solution fun (${P}_{i+1}$) |

Report the optimum solution |

end |

7- Optimization results: ${P}_{opt}$ |

An optimal charging power depending on objectives and respecting all the constraints |

## 4. Case Study

^{−1}. The reference [44] proposed a Li-ion battery capital cost ranges between 500–2500 $·kWh

^{−1}. Other research [45] proposed a Li-ion battery pack cost for EV ranges between 250–360 $·kWh

^{−1}for 2015, 200–250 $·kWh

^{−1}for 2020 and 150–200 $·kWh

^{−1}for 2025. The cost estimation uncertainty of lithium-ion cells depends on several factors. Different values were found in the literature, thus, a mean value of 500 €·kWh

^{−1}was chosen in this study as a realistic example (which seems more acceptable).

^{2}/K. These values were compared with experimental values from the literature. The reference [21] used a Cp value of 854 J/kg/K given by the battery manufacturer for a lithium iron phosphate battery (LIFePO4 or LFP). In the same study, the average value of 15.86 W/m

^{2}/K was used for h including a thermal insulation. The battery thermal characteristics are strongly dependent on the battery pack configuration. We assumed that we have similar configuration as the Citroen C-Zero battery pack with no thermal insulation, no forced air convection and no cooling system.

## 5. Results and Discussions

#### 5.1. Optimization of the Aging Cost for One EB Charging

#### 5.2. Optimization of the Aging Cost for Two EBs Charging

**EB1**and

**EB2**during all the possible charging periods (when each bus is available). The algorithm ensures that the EBs charging is done progressively without exceeding the subscribed power. At ${t}_{0}$ + 5 h, the algorithm will manage the

**EB2**arrival by decreasing the charging power of

**EB1**. As a result of several constraints, the algorithm was forced to decreasingly charge

**EB2**, otherwise the power would exceed the subscribed power.

#### 5.3. Baseline Comparison and Annual Cost Review

- The “Greedy” baseline represents one typical behavior where the EB is charged with the maximum power as soon as possible, ignoring charging cost, until it is fully charged.
- The “Medium” baseline represents one typical behavior where the EB is charged with an average power during the full charging time.
- The “Postponed” baseline represents one typical behavior where the charging of the EB is postponed as late as possible.

^{−1}. It must be noted that the optimization results are highly dependent on the battery electro-thermal aging model.

## 6. Conclusions and Future Works

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

EVs | Electric vehicles. |

EBs | Electric buses. |

EVSE | Electric vehicle supply equipment. |

V2G | Vehicle to grid |

SoC | State of charge. |

OCV | Open circuit voltage. |

EOL | End of life. |

CV | Constant voltage. |

CC | Constant current. |

DC | Direct current. |

CCS | Combined charging system. |

PLC | Power-line communication. |

NLP | Nonlinear programming. |

NSGA | Non-dominated Sorting Genetic Algorithm. |

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**Table 1.**Simulation parameters input, operating constraints and electro-thermal-aging battery characteristics.

Parameters | Value | Parameters | Value |
---|---|---|---|

Number of buses | 1 to 10 | Battery type | LIFePO4 or LFP |

Number of simulated days | 1 day | Nominal energy/capacity | 311 kWh/540 Ah |

Charging time period | 13 h 30 | Pack surface for thermal exchange | 18.79 m^{2} |

Charging slot $\Delta \mathrm{t}$ | 30 min | Battery pack weight | 2500 kg |

Charging power for 1 charger | P_{max} | Specific heat capacity | 900 J·kg^{−1}·K^{−1} |

Number of charging time slots | 27 | Heat transfer coefficient | 5 W·m^{−2}·K^{−1} |

Initial state of charge | 10% | A: pre-exponential factor | 4.35 $\times $ 10^{7} p.u.day^{−1} |

Target state of charge | 100% | ${\mathrm{E}}_{\mathrm{a}}$: activation energy | 0.719 eV |

Arrival & departure time | t_{0}$\to $t_{0} + 14 h | k: Boltzmann constant | 8.617 $\times $ 10^{−5} eV·K^{−1} |

Initial battery temperature | 25 °C | B: quantity of charge factor | 1.104 |

Initial battery capacity fade | 0% | ${\mathrm{Q}}_{\mathrm{EoL}}$: Capacity loss at end of life | 0.2 p.u |

Fixed outside temperature | 25 °C | ${\mathrm{Bat}}_{\mathrm{price}}$: Battery price | 500 €·kWh^{−1} |

EBs Fleet | 1 EB | 4 EBs | 10 EBs | 50 EBs | 100 EBs |
---|---|---|---|---|---|

NLP | 1–10 s | 10–15 s | 20 s–1 min | 5–15 min | 5–15 min |

NSGA-II | 2 min | 20 min | 1 h | No convergence | No convergence |

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## Share and Cite

**MDPI and ACS Style**

Houbbadi, A.; Trigui, R.; Pelissier, S.; Redondo-Iglesias, E.; Bouton, T.
Optimal Scheduling to Manage an Electric Bus Fleet Overnight Charging. *Energies* **2019**, *12*, 2727.
https://doi.org/10.3390/en12142727

**AMA Style**

Houbbadi A, Trigui R, Pelissier S, Redondo-Iglesias E, Bouton T.
Optimal Scheduling to Manage an Electric Bus Fleet Overnight Charging. *Energies*. 2019; 12(14):2727.
https://doi.org/10.3390/en12142727

**Chicago/Turabian Style**

Houbbadi, Adnane, Rochdi Trigui, Serge Pelissier, Eduardo Redondo-Iglesias, and Tanguy Bouton.
2019. "Optimal Scheduling to Manage an Electric Bus Fleet Overnight Charging" *Energies* 12, no. 14: 2727.
https://doi.org/10.3390/en12142727