# Mathematical Model for the Electric Vehicle Routing Problem Considering the State of Charge of the Batteries

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## Abstract

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## 1. Introduction

## 2. Literature Review

- Capacity fading: When the capacity of a battery under reference conditions has been reduced to 80% compared to the nominal capacity.
- Energy fading: The maximum power delivered by the battery in the reference conditions is reduced to 80% compared to the nominal power.

## 3. Proposed Methodology

#### 3.1. Electric Vehicle Routing Problem Considering the State of Charge of the Batteries

#### 3.2. Proposed Mathematical Formulations

#### 3.2.1. Model That Only Allows Batteries to Be Fully Recharged at Stations

$V=$ | Customers |

${V}_{0}=$ | Customers including the depot node $0$, ${V}_{0}=V\text{}{{\displaystyle \cup}}^{\text{}}\left\{0\right\}$ |

${F}_{R}=$ | Real charging stations |

${F}_{F}=$ | Fictitious charging stations, to visit other stations more than once |

$F=$ | Real and fictitious charging stations $F={F}_{R\text{}}{{\displaystyle \cup}}^{\text{}}\text{}{F}_{\text{}}{}_{F}\text{}$ |

${V}^{\prime}=$ | Customers including the charging stations ${V}^{\prime}=V{{\displaystyle \cup}}^{\text{}}F$ |

${V}_{0}^{\prime}=$ | Customers including the charging stations and the depot node $0$, ${V}_{0}^{\prime}=V{{\displaystyle \cup}}^{\text{}}F{{\displaystyle \cup}}^{\text{}}\left\{0\right\}$ |

${V}_{N}^{\prime}=$ | Customers including the charging stations and the depot node $N$, ${V}_{N}^{\prime}=V{{\displaystyle \cup}}^{\text{}}F{{\displaystyle \cup}}^{\text{}}\left\{N\right\}$ |

${V}_{0,N}^{\prime}=$ | Customers including the charging stations and the depots node $0$ and $N$, ${V}_{0,N}^{\prime}=V{{\displaystyle \cup}}^{\text{}}F{{\displaystyle \cup}}^{\text{}}\left\{0\right\}{{\displaystyle \cup}}^{\text{}}\left\{N\right\}$ |

${d}_{ij}=$ | Euclidean distance from vertex $i$ to $j$ ($i\in {V}_{0}^{\prime}\text{},\text{}j\in {V}_{N\text{}}^{\prime}$) |

${t}_{ij}=$ | Travel time from vertex $i$ to $j$ ($i\in {V}_{0}^{\prime}\text{},\text{}j\in {V}_{N\text{}}^{\prime}$) |

${z}_{ij}=$ | Consumption energy from vertex $i$ to $j$ ($i\in {V}_{0}^{\prime}\text{},\text{}j\in {V}_{N\text{}}^{\prime}$) $\left({z}_{ij}=h\ast {d}_{ij}\right)$ |

$C=$ | Vehicle capacity |

$Q=$ | Battery capacity |

$h=$ | Consumption energy rate of battery |

$g=$ | Charging rate of battery at the station |

$m=$ | Fleet size (number of electric vehicles) |

$vel=$ | Average driver speed |

${q}_{i}=$ | Demand of the node $\text{}i$ ($i\in V\text{};$ is 0 if $i\notin V\text{})$ |

${e}_{i}=$ | Earliest time window of node $i$ ($i\in {V}_{0,N}^{\prime}$) |

${l}_{i}=$ | Latest time window of node i ($i\in {V}_{0,N}^{\prime}$) |

${S}_{i}=$ | Service time of node $i$ (${S}_{0},{\text{}S}_{N}=0;\text{}i\text{}\in {V}_{0,N}^{\prime}$) |

${x}_{ij}\{\begin{array}{c}1\\ \\ \\ 0\end{array}$ | If node $i$ immediately precedes node $j$ ($i\in {V}_{0}^{\prime}\text{},\text{}j\in {V}_{N\text{}}^{\prime}$) |

Otherwise | |

${\tau}_{i}$ | Decision variable that indicates the arrival time at node $i$ ($i\in {V}_{0,N}^{\prime}$) |

${u}_{i}$ | Decision variable that indicates the remaining capacity of the vehicle battery upon reaching the node $i$ ($i\in {V}_{0,N}^{\prime}$). |

${y}_{i}$ | Decision variable that indicates the status or remaining capacity of the battery in node $i$ ($i\in {V}_{0,N}^{\prime}$). |

#### 3.2.2. Mathematical Model That Allows Partial Recharging of the Battery at Stations

#### 3.2.3. Model That Allows Partial Recharging and Also Limits Battery Discharge

#### 3.2.4. Model That Allows Partial Recharging, Defines a Maximum Charge Threshold, and Limits Battery Discharge

## 4. Computational Experiments

^{®}Core™ i5-2430M CPU @ 2.4 GHz processor with an 8 GB RAM and a 64-bit operating system. A resolution time limit of 7200 s was considered.

- Review the literature related to electric vehicles, their physical composition, the behavior of the battery, the types of charge, and their fundamental components. A review of the electric vehicle routing problem with time windows (E-VRPWT) was conducted.
- Define the considered problem.
- Formulate the optimization models mathematically and implement them in the Julia programming language.
- Generate structured instances used as input data for the models.
- Adapt instances of the literature according to the considered problem.
- Run experiments with the Gurobi commercial solver for each formulated instance.
- Compare the results obtained between the implemented models, considering the computing time according to the size of the instance, the objective function value, the number of used electric vehicles, the number of times the vehicle arrives at a charging station, and the quantity of energy supplied to recharge the battery.
- Carry out an analysis of the impact generated by allowing partial recharging and restricting the state of charge of the car batteries and how they influenced the results.

#### 4.1. Discussion Results: Instance with 5 Customers—3 Recharging Stations

#### 4.2. Discussion Results: Adapted Instance with 10 Customers—5 Recharging Stations

## 5. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 1.**Battery capacity with respect to charge and discharge cycles. Source [15].

**Figure 2.**Typical charging curve, where $i,\text{}u\text{},$ and $SoC$ represent the current, terminal voltage, and state of charge, respectively. The $SoC$ is equivalent to the battery level. $t\text{}$ is the period. Source [19].

**Figure 3.**Real data vs. piecewise linear approximation for a charging station of 22 kW charging a battery of 16 kWh. The battery level Source: [19].

Paper | Allows Partial Recharge | Considers Battery Charging Stations | Forces a Full Recharge | Time Window for Customers | Limits Complete Discharge | Penalizes Overcharging at Stations | Considers the Nonlinear Recharge Curve | Allows Exchange Batteries | Grants Multiple Recharge Options | Considers a Heterogeneous Fleet |
---|---|---|---|---|---|---|---|---|---|---|

Conrad and Figliozzi [27] | x | x | x | |||||||

Sweda and Klabjan [40] | x | x | x | |||||||

Barco et al. [54] | x | x | x | |||||||

Schneider et al. [21] | x | x | x | |||||||

Felipe et al. [32] | x | x | x | x | ||||||

Goeke and Schneider [39] | x | x | x | |||||||

Yang and Sun [43] | x | |||||||||

Sweda et al. [41] | x | x | x | |||||||

Hiermann et al. [26] | x | x | x | X | ||||||

Desaulniers et al. [30] | x | x | x | x | ||||||

Keskin and Çatay [31] | x | x | x | x | ||||||

Hof et al. [45] | x | x | ||||||||

Montoya et al. [18] | x | x | x | x | x | |||||

Raeesi et al. [46] | x | |||||||||

Futalef et al. [47] | x | x | x | |||||||

Schiffer et al. [48] | x | x | x | x | ||||||

Zhang et al. [49] | x | x | x | X | ||||||

Xu et al. [50] | x | x | X | |||||||

Zang et al. [51] | x | x | x | x | ||||||

Our proposal | x | x | x | x | x | x |

Instance | Coordinates | Time Windows | |||||
---|---|---|---|---|---|---|---|

ID | Type | X-Axis | Y-Axis | Earliest Time Windows ${\mathit{e}}_{\mathit{i}}\text{}\left(\mathbf{s}\right)$ | Latest Time Windows ${\mathit{l}}_{\mathit{i}}\text{}\left(\mathbf{s}\right)$ | Demand ${\mathit{q}}_{\mathit{i}}$ (Units) | Service Time ${\mathit{S}}_{\mathit{i}}$ (s) |

D0 | Depot | 40 | 50 | 0 | 240 | 0 | 0 |

S1 | Station | 10 | 28 | 0 | 240 | 0 | 10 |

S2 | Station | 27 | 10 | 0 | 240 | 0 | 10 |

S3 | Station | 77 | 30 | 0 | 240 | 0 | 10 |

C1 | Customer | 85 | 35 | 68 | 182 | 30 | 10 |

C2 | Customer | 40 | 5 | 55 | 185 | 10 | 10 |

C3 | Customer | 4 | 18 | 58 | 131 | 35 | 10 |

C4 | Customer | 65 | 55 | 26 | 111 | 14 | 10 |

C5 | Customer | 2 | 40 | 96 | 190 | 20 | 10 |

Description | Value |
---|---|

Battery Capacity (Q) | 77.75 kWh |

Vehicle Capacity (C) | 200 Units |

Energy Consumption Rate (h) | 1 kWh/m |

Battery Charging Rate (g) | 0.39 s/kWh |

Average driving speed | 1 m/s |

Model | (a) | (b) | (c) | (d) |
---|---|---|---|---|

Fleet Size (vehicles) | 3 | 2 | 3 | 3 |

Objective Function (s) | 452.07 | 372.34 | 429.93 | 444.55 |

Computing Time (s) | <1 | <1 | <1 | <1 |

Visits to charging stations | Route 1 = 1 | Route 1 = 1 | Route 1 = 1 | Route 1 = 1 |

Route 2 = 1 | Route 2 = 2 | Route 2 = 1 | Route 2 = 1 | |

Route 3 = 1 | Route 3 = 1 | Route 3 = 2 | ||

Routing Time | Route 1 = 159.9 | Route 1 = 146.0 | Route 1 = 146.0 | Route 1 = 146.0 |

(s) | Route 2 = 139.4 | Route 2 = 226.3 | Route 2 = 131.0 | Route 2 = 131.0 |

Route 3 = 145.7 | Route 3 = 152.9 | Route 3 = 167.5 | ||

Minimum Threshold | 0 | 0 | 77.75 kWh | 77.75 kWh |

(SoC = 25%) | (SoC = 25%) | |||

Maximum Threshold | 77.75 kWh | 77.75 kWh | 77.75 kWh | 66.08 kWh |

(SoC = 100%) | (SoC = 100%) | (SoC = 100%) | (SoC = 85%) | |

Minimum Value | 14.53 kWh | 18.82 kWh | 25.49 kWh | 25.49 kWh |

(Customer or Station) | (SoC = 18.70%) | (SoC = 24.21%) | (SoC = 32.79%) | (SoC = 32.79%) |

Maximum Value | 77.75 kWh | 64.25 kWh | 73.04 kWh | 63.20 kWh |

(Customer or Station) | (SoC = 100.00%) | (SoC = 82.64%) | (SoC = 93.95%) | (SoC = 81.30%) |

Charge Status (SoC) at the end of the route | Route 1 = 45% | Route 1 = 0% | Route 1 = 0% | Route 1 = 0% |

Route 2 = 24% | Route 2 = 0% | Route 2 = 0% | Route 2 = 0% | |

Route 3 = 6.2% | Route 3 = 0% | Route 3 = 0% | Route 3 = 0% |

Instance | Number of Charging Stations | Number of Customers | Type of Customer |
---|---|---|---|

C101C5_S3 | 3 | 5 | C |

R104C5_S3 | 3 | 5 | R |

R105C5_S3 | 3 | 5 | R |

RC204C5_S4 | 4 | 5 | R |

C202C10_S5 | 5 | 10 | C |

R102C10_S4 | 4 | 10 | R |

R203C10_S5 | 5 | 10 | R |

RC108C10_S4 | 4 | 10 | RC |

C106C15_S3 | 3 | 15 | C |

C208C15_S4 | 4 | 15 | C |

Fleet Size (Number of Vehicles) | ||||
---|---|---|---|---|

Instance | (a) | (b) | (c) | (d) |

C101C5S3 | 2 | 2 | 2 | 3 |

R104C5S3 | 2 | 2 | 2 | 2 |

R105C5S3 | 2 | 2 | 3 | 3 |

RC204C5S4 | 1 | 1 | 1 | 2 |

C202C10S5 | 1 | 1 | 2 | 2 |

R102C10S4 | 4 | 4 | 4 | 4 |

R203C10S5 | 1 | 1 | 1 | 3 |

RC108C10S4 | 3 | 3 | 4 | 4 |

C106C15S3 | 3 | 3 | 3 | 3 |

C208C15S4 | 4 | 4 | 4 | 4 |

Model | Number of Customers | 5 Customers | 10 Customers | 15 Customers | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Instances | C101C5S3 | R104C5S3 | R105C5S3 | RC204C5S4 | C202C10S5 | R102C10S4 | R203C10S5 | RC108C10S4 | C106C15S3 | C208C15S4 | |

Item | |||||||||||

(a) | Fleet Size | 3 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 3 | 4 |

Objective Function (s) | 925.60 | 210.54 | 245.08 | 262.90 | 1571.70 | 416.96 | 424.21 | 572.86 | 2029.50 | 2134.50 | |

Computing Time (s) | <1 | <1 | <1 | <1 | 51 | 3 | 6, 7 | 9, 5 | 1240 | . | |

Visited Stations | 2 | 1 | 1 | 1 | 2 | 2 | 3 | 2 | 2 | 2 | |

State of Charge (SoC) at Exiting Stations (%) | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | |

(b) | Fleet Size | 3 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 3 | 4 |

Objective Function (s) | 876, 2 | 206, 16 | 238, 29 | 258, 9 | 1.507, 4 | 415, 75 | 419, 39 | 557, 98 | 2.006, 9 | 2.033, 6 | |

Computing Time (s) | <1 | <1 | <1 | <1 | 242 | 3, 2 | 11, 4 | 84, 2 | 3.727 | 770 | |

Visited Stations | 2 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 2 | 3 | |

State of Charge (SoC) at Exiting Stations (%) | 79; 87 | 85 | 77 | 58 | 85; 91 | 98; 92 | 98; 76; 76 | 72; 72; 79 | 96; 95 | 81; 67; 67 | |

(c) | Fleet Size | 3 | 2 | 3 | 2 | 2 | 4 | 2 | 4 | 3 | 4 |

Objective Function (s) | 876, 2 | 206, 16 | 255, 89 | 258, 9 | 1.556, 4 | 415, 75 | 443, 55 | 577, 71 | 2.016, 2 | 2.072, 5 | |

Computing Time (s) | <1 | <1 | <1 | <1 | 273 | 3, 2 | 13, 2 | 22, 35 | 3.427 | 1.676 | |

Visited Stations | 2 | 1 | 1 | 1 | 3 | 2 | 4 | 3 | 2 | 3 | |

State of Charge (SoC) at Exiting Stations (%) | 79; 87 | 85 | 91 | 58 | 84; 77; 91 | 98; 92 | 96; 98; 90; 89 | 80; 89; 89 | 79; 95 | 80; 66; 82 | |

(d) | Fleet Size | 3 | 2 | 3 | 2 | 2 | 4 | 3 | 4 | 4 | 4 |

Objective Function (s) | 877, 45 | 225, 43 | 279, 10 | 272, 2 | 1.605, 2 | 450, 27 | 508, 17 | 605, 51 | 2.375, 6 | 2.072, 5 | |

Computing Time (s) | <1 | <1 | <1 | <1 | 19 | 2, 7 | 12 | 3, 8 | 3.162 | 51, 54 | |

Visited Stations | 2 | 2 | 2 | 2 | 4 | 4 | 5 | 4 | 5 | 3 | |

State of Charge (SoC) at Exiting Stations (%) | 76; 73 | 66; 66 | 66; 77 | 65; 85 | 75; 73; 73; 85 | 53; 53; 81; 65 | 84; 76; 76; 76; 85 | 79; 79; 85; 85 | 82; 85; 82; 80; 80 | 80; 66; 82 |

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## Share and Cite

**MDPI and ACS Style**

Cataldo-Díaz, C.; Linfati, R.; Escobar, J.W.
Mathematical Model for the Electric Vehicle Routing Problem Considering the State of Charge of the Batteries. *Sustainability* **2022**, *14*, 1645.
https://doi.org/10.3390/su14031645

**AMA Style**

Cataldo-Díaz C, Linfati R, Escobar JW.
Mathematical Model for the Electric Vehicle Routing Problem Considering the State of Charge of the Batteries. *Sustainability*. 2022; 14(3):1645.
https://doi.org/10.3390/su14031645

**Chicago/Turabian Style**

Cataldo-Díaz, Cristian, Rodrigo Linfati, and John Willmer Escobar.
2022. "Mathematical Model for the Electric Vehicle Routing Problem Considering the State of Charge of the Batteries" *Sustainability* 14, no. 3: 1645.
https://doi.org/10.3390/su14031645