# Energy Distribution Diagram Used for Cost-Effective Battery Sizing of Electric Trucks

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## Abstract

**:**

## 1. Introduction

## 2. Designing Electromobility Systems Using Energy Distribution Diagrams

## 3. Cost of a Battery Electric Truck

- Energy from private charging;
- Energy from public charging;
- Battery;
- Charger;
- Grid fees;

## 4. The Energy Distribution Diagram

- The total energy consumption is the area under the curve.
- The total number of operational days is found from the curve’s intersection with the x-axis.
- The mean consumed energy for the days when the truck is used equals the total energy divided by the number of operational days.
- The highest consumed energy can be read off from the curve’s intersection with the y-axis.
- The number of days that need fast charging and the amount of fast charging needed for a particular useful battery capacity.
- The shape of the EDD gives a quick picture of the size of the variation in daily energy consumption and the respective share of days with high, medium or low energy consumption.

## 5. A Rectangular Energy Distribution Diagram

## 6. A Triangular Energy Distribution Diagram

## 7. A Two-Step Rectangular Energy Distribution Diagram

## 8. Importance of the Number of Operational Days

## 9. An Algorithm for Sizing the Battery of a General EDD

**Algorithm Description**

- Choose a sufficiently small battery capacity value, such as ${B}_{c1}$.
- Compute the total energy, ${E}_{tot}=\underset{0}{\overset{{N}_{op}}{\int}}E\left(N\right)dN$.
- Find the smallest ${N}_{1}$ so that ${r}_{SoC}\xb7{B}_{c1}=E\left({N}_{1}\right)$. The number of days needed to fast charge is ${N}_{1}$.
- Compute
- (a)
- ${r}_{ch}=\frac{{r}_{SoC}\xb7{B}_{c1}\xb7{N}_{1}+\underset{{N}_{1}}{\overset{{N}_{op}}{\int}}E\left(N\right)dN}{{E}_{tot}}$,
- (b)
- ${\Gamma}_{b}=\frac{{E}_{tot}}{{B}_{c1}}$and
- (c)
- ${\Gamma}_{ch}=\frac{{r}_{ch}\xb7{E}_{tot}}{\frac{{r}_{SoC}\xb7{B}_{c1}}{{T}_{ch}}\xb7T}$.

- Compute ${f}_{BEV1}=f\left({B}_{c1}\right)$ from Equation (5).
- Set ${B}_{c2}={B}_{c1}+\Delta {B}_{c}$, $\Delta {B}_{c}$ should be sufficiently small.
- Repeat steps 3–6 but increase each index by one until ${r}_{SoC}\xb7{B}_{c(n+1)}>{E}_{max}$, then set ${r}_{SoC}\xb7{B}_{c(n+1)}={E}_{max}$, repeat steps 3–5 and then jump to step 8.
- One now have the cost per kWh of useful energy expressed as a function of battery capacity. Select the battery capacity that gives the lowest cost.

## 10. Conclusions

**How Usage Pattern Influences the Cost-Effectiveness of Battery Electric Vehicles**

**These conclusions are drawn under the following assumptions:**

- Battery capacity and charger power are continuously variable, from zero to arbitrary high values.
- Charging is always possible, precisely when needed.
- Charging is done on planned breaks and requires no extra time.
- A battery electric truck has the same payload capacity as a diesel truck.
- Parameter values are selected according to Table 1, the vehicle is operated for 1750 days, the home charger can be used for up to 14 hours per night and there is 80% of the battery capacity available.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

EDD | Energy distribution diagram |

Nomenclature | |

${B}_{c}$ | Battery capacity [kWh] |

${C}_{b}$ | Battery cost [€/kWh] |

${C}_{ch}$ | Combined price for charger and grid [€/kW/year] |

${C}_{charger}$ | Price of charger [€/kW] |

${C}_{d}$ | Diesel cost [€/kWh] |

${C}_{e}$ | Electricity cost, private charging [€/kWh] |

${C}_{epub}$ | Electricity cost, public fast charging [€/kWh] |

${C}_{g}$ | Grid fee [€/kW/year] |

E | Energy function [kWh] |

${E}_{max}$ | Highest daily energy consumption in the truck’s service life [kWh] |

${E}_{min}$ | Lowest daily energy consumption in the truck’s service life [kWh] |

${E}_{tot}$ | Total propulsion energy consumed over the truck’s Service life [kWh] |

${f}_{BEV}$ | Cost function [€/kWh] |

${f}_{BEV}^{rec}$ | Lowest value of the cost functions when the EDD is rectangular [€/kWh] |

${\Gamma}_{b}$ | Battery utilisation factor [equivalent full cycles] |

${\Gamma}_{ch}$ | Charger utilisation factor $[-]$ |

M | No. of high energy consumption days for a two-step rectangular EDD $[-]$ |

${N}_{ch}$ | Number of fast charging days $[-]$ |

${N}_{op}$ | Number of operational days $[-]$ |

${N}_{tot}$ | Number of days in the truck’s service life $[-]$ |

${P}_{ch}$ | Charger power [kW] |

${r}_{ch}$ | Ratio of private charging to total amount of energy $[-]$ |

${r}_{SoC}$ | Share of battery capacity that could be used $[-]$ |

T | Service life of truck, charger and battery [year] |

${T}_{ch}$ | Available time for charging during night [hours] |

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**Figure 1.**The daily energy consumption for five working days appears on the left and the energy distribution diagram for a truck’s full service life is on the right. In the right-hand part of the figure, the number of days that needs fast charging and the amount of fast charging needed are marked in red, while the energy from private chargers for a given useful battery capacity is marked with blue.

**Figure 2.**A rectangular EDD. In the figure, the number of days that need fast charging and the amount of fast charging needed is marked in red. The energy delivered by private chargers for a given useful battery capacity is marked in blue.

**Figure 3.**A triangular EDD. The available energy stored in a full battery is marked on the y-axis in blue and the number of days charging with a public fast charger is marked as ${N}_{ch}$ in red on the x-axis. The amount of fast charging needed is marked in red and the energy delivered by private chargers is marked in blue for a given useful battery capacity.

**Figure 4.**The charging strategy for a triangular EDD with ${N}_{op}=1750$. The figure illustrates that fast charging must take place on 55% of the days and that there must be fast charging twice on 10% of the days.

**Figure 5.**A two-step rectangular EDD. In the figure, the number of days requiring fast charging and the amount of fast charging needed is marked in red, while the energy delivered by private chargers is marked in blue for a given useful battery capacity.

**Figure 6.**The cost function as a function of M and r in the left-hand part of the figure. The battery electric trucks are more expensive than the commercial diesel ones in the area enclosed by the red curve and the M-axis. The right-hand part of the figure shows how the shape of the EDD depends on the parameters M and r. If there is an EDD on or to the left of the dashed vertical black line, the smaller battery is the most cost-efficient and, to the right of it, the larger battery.

**Table 1.**Notations for the different costs and their assumed values. Please keep in mind that the cost refers to the cost of propulsion energy.

Costs | Notation | Typical Value |
---|---|---|

Diesel Cost | ${C}_{d}$ | 0.30 €/kWh |

Electricity Cost, Private Charging | ${C}_{e}$ | 0.08 €/kWh |

Electricity Cost, Public Fast Charging | ${C}_{epub}$ | 0.4 €/kWh |

Battery Cost | ${C}_{b}$ | 200 €/kWh |

Price of Charger | ${C}_{charger}$ | 400 €/kW |

Grid Fee | ${C}_{g}$ | 60 €/kW/year |

Service Life of Truck, Charger and Battery | T | 7 years |

Combined Price of Charger and Grid | ${C}_{ch}=\frac{{C}_{charger}}{T}+{C}_{g}$ | 117 €/kW/year |

Parameters | Notation |
---|---|

Total Propulsion Energy Consumed Over the Trucks Service Life | ${E}_{tot}$ |

Ratio of Private Charging to the Total Amount of Energy | ${r}_{ch}$ |

Battery Capacity | ${B}_{c}$ |

Charger Power | ${P}_{ch}$ |

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**MDPI and ACS Style**

Karlsson, J.; Grauers, A.
Energy Distribution Diagram Used for Cost-Effective Battery Sizing of Electric Trucks. *Energies* **2023**, *16*, 779.
https://doi.org/10.3390/en16020779

**AMA Style**

Karlsson J, Grauers A.
Energy Distribution Diagram Used for Cost-Effective Battery Sizing of Electric Trucks. *Energies*. 2023; 16(2):779.
https://doi.org/10.3390/en16020779

**Chicago/Turabian Style**

Karlsson, Johannes, and Anders Grauers.
2023. "Energy Distribution Diagram Used for Cost-Effective Battery Sizing of Electric Trucks" *Energies* 16, no. 2: 779.
https://doi.org/10.3390/en16020779