# Optimization of Static and Dynamic Charging Infrastructure for Electric Buses

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Methodology

#### 3.1. Physical Model

- The bus fleet can be heterogeneous, but a bus line is served by one bus type. Thus, the energy consumption and charging demand is the same for each turn on a bus line.
- Three bus types are distinguished in terms of charging: a bus can be charged either at static or dynamic charging units, or both (combined bus charging).
- Each turn of a bus line begins and ends at the same station.
- The runs to and from the depot are not considered. Thus, the cost of battery capacity is not considered.
- The state of charge of a vehicle is the same at the beginning and the end of a turn. Namely, the charging process belongs exclusively to runs. There is no “additional” charging.

- Sections: A bus line is segmented into several sections with various lengths. The division should be performed with a focus on the bus lines and branches. There is no overlap among the sections. The shorter are the sections, the greater is the number of sections. In many sections, the optimization result is more sophisticated, but the calculation takes more time. The segmentation is presented in Figure 2 with an example.

- Stops: Designated locations where buses frequently stop during the daily service (e.g., stop to board or alight). The energy consumption at stops is significantly smaller than during movement; accordingly, it is neglected.

- The cost of a charging unit is modeled by one value. This cost value may cover only the deployment cost or both the deployment cost and operational cost.
- The cost of a dynamic charger does not depend on the number of vehicles charging simultaneously.
- The vehicles are to be charged at maximum power at dynamic chargers regardless of the number of vehicles charging simultaneously.
- The cost of a static charger at a location does not depend on the number of charging units. Namely, the costs of the first and second charging stations at a specific location are equal.
- The electricity cost is the same at each location regardless the charging technology.
- The effect of electric bus technology on the charging infrastructure cost, and vice versa, were not considered. Namely, vehicles and charging units compatible with each other may be modeled.

#### 3.2. Mathematical Model

^{u}, p, and α), bus line specific (f), network element and bus line specific (e

^{−}, e

^{+}, x, and d), or system constant (c*) ones. Charging unit attributes are network element specific ones because the values may depend on the location. For example, the capacity coefficient may depend on the characteristic of a terminal. Factors affecting the charging power, such as batteries’ state of charge and electrical grid capacity, were not considered. Namely, an average charging power was considered. The network element specific attributes are assigned to row vectors, the bus line specific ones are assigned to column vectors, and the network element and bus line specific attributes are assigned to matrices in the model. The variable of the cost function is x (charged energy); the other input attributes are parameters.

- C
^{u}row vector of charging infrastructure cost; - c
_{j}^{u}infrastructure cost of a charging unit at network element j; - P row vector of charging unit power;
- p
_{j}charging power of charging unit at network element j; - α row vector of the capacity coefficient;
- α
_{j}capacity coefficient of charging unit at network element j; - n total number of line sections; and
- m total number of stops.

- ${E}^{-}$ matrix of energy consumption;
- ${E}^{+}$ matrix of chargeable energy;
- ${e}_{i,j}^{-}$ consumed energy of a bus on bus line i at network element j during one turn;
- ${e}_{i,j}^{+}$ chargeable energy of a bus on bus line i at network element j during one turn; and
- k number of bus lines.

- X matrix of charged energy; and
- x
_{i,j}charged energy of a bus on bus line i at network element j during one turn.

- F column vector of frequency; and
- f
_{i}number of departures on bus line i per hour.

- D matrix of total energy demand; and
- d
_{i,j}total energy demand of the operating buses on bus line i in the rush hour at network element j.

_{j}) is given in Equation (12). The variable of the summation is the index of the bus line.

#### 3.3. Constraints

_{j}) and the charging capacity coefficient (α

_{j}). The capacity coefficient is to be determined according to local specialties. For example, if an overhead wire can charge each simultaneously connected bus at maximum power, the capacity coefficient is infinite.

#### 3.4. Objective Function

_{j}= 0 (not recommended to be deployed) or cu if d

_{j}> 0 (recommended to be deployed). Namely, the cost function is not continuous at 0. Since the continuous functions are handled better by the optimization algorithms, substitute charging infrastructure cost functions are interpreted for two types of chargers. Charging unit with infinite capacity is interpreted in Equation (16) and charging unit with limited capacity is interpreted in Equation (17).

- ${c}_{j}^{s*}$ substitute cost of a charging unit where the capacity limitation is relevant; and
- ${p}_{j}{\alpha}_{j}$ energy capacity of the charging unit.

- High reserve charging capacity should be avoided. Hence, the substitute cost functions should be strictly increasing. Thus, the greater reserve charging capacity would cause a higher substitute cost value.
- It should be more beneficial to allocate extra charging demand to a network element where d ≠ 0 than “install” a new charger. Hence, the first derivative of the substitute cost functions should be strictly decreasing.
- $\underset{\sum d\to \infty}{\mathrm{lim}}{c}^{s}={c}^{u}$ and ${c}_{j}^{s*}\left(a{p}_{j}{\alpha}_{j}\right)\cong a{c}_{j}^{u}$ if a ϵ N.
- ${c}^{s}$ and ${c}^{s*}$ should be equal between 0 and p
_{j}α_{j}if the charging unit costs are equal.

- ${c}^{s}\left(0\right)={c}^{s*}\left(0\right)=0$. Hence, the additions of 1 are applied.
- The cost of an additional charging unit and the first unit is considered the same. Therefore, c
^{s}* is divided into the cost of a charging unit with free capacity (left section) and the cost of charging unit (s) without free capacity (right section). In other words, the left section determines the substitute cost of the first or an additional charging unit, and its domain is between 0 and p_{j}α_{j}. - The difference between the substitute costs and ${c}^{u}$ should be low. Hence, the multiplication by 50 was applied. Therefore, the difference between ${c}^{s}$ or ${c}^{s*}$ and ${c}^{u}$is less than 2% if d
_{j}is numerically greater than 1.

^{t})) to the charging infrastructure cost. The cost of technology limitation is calculated for each bus line, according to Equation (19).

- ${c}_{i}^{t}$ cost of technology limitation for bus line i; and
- ${c}^{t*}$ technology limitation factor.

^{t}is there, where the combined bus charging is not realized. The total cost of technology limitation is given in Equation (20).

## 4. Case Study

_{68,j}= 0 for j = 41–45 (at terminals).

^{t}eliminates the combined charging of buses successfully.

- Without existing charging infrastructure.
- With existing dynamic charging infrastructure along a bus line because trolleybus services are spread in several European cities.

- A sensitivity analysis was made to reveal the relationship between the total length of the dynamic charging network and the charging power of dynamic charging units.
- The utilization of charging units was analyzed to validate the optimization.

^{u}of the existing charging infrastructure was €0. We hypothesized that the deployment of additional dynamic chargers would be more beneficial because of the existing dynamic charging infrastructure even though the cost is higher. The modeling parameters are summarized in Table 3.

## 5. Results and Discussion

#### 5.1. Without Existing Charging Infrastructure

#### 5.2. With Existing Dynamic Charging Infrastructure

## 6. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**Substitute charging unit cost as a function of charging demand for charging unit with infinite (c

^{s}) and limited capacity (c

^{s}*).

**Figure 5.**Structure of the public bus network of Kőbánya with the number of buses that run during the rush hour.

**Figure 7.**The total length of the dynamic charging network as a function of dynamic charging power (in Plan A).

Study | Static (1) | Dynamic (2) | Combined (3) | Network Approach (4) |
---|---|---|---|---|

Chen et al., 2018 [25] | yes | yes | no | no |

Ko and Jang, 2013 [26] | no | yes | no | no |

Jang et al., 2016 [27] | no | yes | no | no |

Jeong et al., 2015 [28] | yes | yes | no | no |

Lajunen, 2018 [29] | yes | no | no | no |

Rogge et al., 2018 [30] | yes | no | no | no |

Kunith et al., 2017 [31] | yes | no | no | yes |

Wang et al., 2017 [32] | yes | no | no | yes |

Xylia et al., 2017 [33] | yes | no | no | yes |

Wu et al., 2021 [34] | yes | no | no | yes |

Alwesabi et al., 2021 [35] | no | yes | no | yes |

Liu and Song, 2017 [36] | yes | yes | no | yes |

Bi et al., 2018 [37] | yes | no | no | yes |

Wei et al., 2018 [38] | yes | no | no | yes |

Component | Sign | Attribute | Description |
---|---|---|---|

Charging unit | c^{u} | unit cost | Cost of a charging unit (€) |

p | power | Maximum charging power (kW) | |

α | capacity coefficient | Efficiency rate because of technological specialties (connecting, reconnecting) (hour) | |

Energy consumption | e^{−} | energy consumption | The consumed energy of a bus at a section during one turn (kWh) |

Charging | e^{+} | chargeable energy | The maximum energy that a bus can charge at a charging unit during one turn (kWh/turn) |

x | charged energy | The energy flow from the charging unit to the bus during one turn (kWh). The variable of the cost function. | |

d | total energy demand | The total energy flow from charging unit to buses of a bus line during the rush hour (kWh/hour) | |

Bus line | f | frequency | The number of departures of a bus line during the rush hour (1/hour) |

Technology | c^{t}* | technology limitation factor | Its value is 0 if combined bus charging is allowed, otherwise not (€) |

Parameter | Component | Value |
---|---|---|

c^{u} | static charger | €200,000 |

dynamic charger | different for each optimization run, 50–250 €/m | |

p | static charger | 300 kW |

dynamic charger | 100 and 100–200 kW during sensitivity analysis | |

α | static charger | 0.7 h |

dynamic charger | infinite | |

e^{−} | solo bus | 1.2 Wh/m |

articulated bus | 1.5 Wh/m | |

e^{+} | according to the schedule and charging power | |

f | number of departures in rush hour | |

c^{t}* | €100,000 |

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**MDPI and ACS Style**

Csonka, B.
Optimization of Static and Dynamic Charging Infrastructure for Electric Buses. *Energies* **2021**, *14*, 3516.
https://doi.org/10.3390/en14123516

**AMA Style**

Csonka B.
Optimization of Static and Dynamic Charging Infrastructure for Electric Buses. *Energies*. 2021; 14(12):3516.
https://doi.org/10.3390/en14123516

**Chicago/Turabian Style**

Csonka, Bálint.
2021. "Optimization of Static and Dynamic Charging Infrastructure for Electric Buses" *Energies* 14, no. 12: 3516.
https://doi.org/10.3390/en14123516