# Optimal Planning of Electric Vehicle Charging Stations Considering Traffic Load for Smart Cities

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## Abstract

**:**

## 1. Introduction

## 2. Related Works

#### 2.1. Assignment and Location Problem

#### 2.2. Resource Allocation under Optimal Criteria

#### 2.3. Planning of Charging Infrastructure for Batteries in Electric Mobility

_{2}emissions by conventional private and public transport. Today there are multiple electric mobility alternatives, such as bicycles, cars, and public transport buses. However, it has not been possible to introduce EVs massively into the land transport system because of variables that do not make large-scale purchases attractive to potential users in urban and rural areas. These unattractive variables to consumers could be limited autonomy, long charging times, battery life, high costs, and lack of charging infrastructure for EVs.

## 3. Problem Description

#### 3.1. Stochastic Analysis of Vehicular Traffic and Resource Allocation at ICSEC

#### 3.2. Proposed Strategy and Methodology

#### 3.3. Problem Formulation

- ${\varphi}_{j}$ location variable $\epsilon $, is 1 when it is found any node that is part of the set j, $j\in \kappa $, 0 in any other way.
- ${\theta}_{k}$ subset of assigned arcs, if the arc $k\in \mathcal{M}$ it is used, otherwise 0.
- ${\tau}_{e}^{j}$ path enabled, is 1 if the edge $e\in \mathcal{L}$ and $j\in \kappa $, otherwise 0.

#### Column Generator

## 4. Analysis of Results

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Flowchart for solving the Multiple-Product Problem in Location and Resource Allocation by considering Capacity (MP-LRAC).

**Figure 4.**Case A, Grover Beach with an area of 2.13 km

^{2}, a city in San Luis Obispo County, US state of California.

**Figure 9.**Case A: Performance of the algorithm considering scalability of 4 years with a growth rate of 3%.

**Figure 10.**Case B: Weights assigned to each track section with microscopic analysis in an area of 0.9887 km

^{2}.

**Figure 13.**Case A. Energy consumption and density of electric vehicles per expected charging station.

Vehicle Electric | Autonomy (km) | Voltage Terminal (V) | Engine (kW) | Maximum Speed (km/h) | System Regenerative | Vehicle Category Dimensions (m) |
---|---|---|---|---|---|---|

Tazzari Zero | 200 | 230 | 15–25 | 90 | ✓ | |

Renault Twizy | 100 | 230 | 17 | 80 | ✓ | height < 2.5 |

Audi Urban | 73 | 230–400 | 15 | 100 | - | width < 1.5 |

Peugeot BB1 | 120 | 230 | 15 | 60 | - | length < 3.7 |

Author, Year | Model | Problem | Obj. Function | Constraints | Solution | Trajectories | Other | ||||
---|---|---|---|---|---|---|---|---|---|---|---|

Theoretical | Experimental | Location Allocation | Route | Exact | Heuristic/Meta | Density Traffic | Urban Study | ||||

Gan, 2020 [18] | ✓ | - | ✓ | ✓ | Maximize Profit Service | Distance Standy time Power grid capacity | - | ✓ | - | - | Elastic demand Stochastic Newton-Raphson NILP |

Inga, 2019 [19] | ✓ | - | ✓ | ✓ | Minimize Costs Flows | Capacity Distance | - | ✓ | - | ✓ | Scalable multi-commodity capacities network |

Demir, 2019 [20] | - | ✓ | - | ✓ | Minimize Travel distance Costs | Flows # Vehicles | - | ✓ | - | - | Deterministic ILP Demand fluctuation NP-hard |

Azadi, 2019 [21] | ✓ | - | - | ✓ | Maximize User Routes Service | Capacity Operation Schedule | ✓ | - | ✓ | - | Multiple commodity flows network Heterogenous traffic MILP Branch and Bound Algorithm |

Bevrani, 2019 [22] | ✓ | - | - | ✓ | Minimize Flow Routs | Congestion Expanstion tree Conservation flow | ✓ | - | - | ✓ | Multi-commodity network flow Flow reduction functions NLP Multiple nodes |

Campaña, 2019 [23] | ✓ | - | ✓ | ✓ | Minimize # Charging stations Trajectory Costs | Distance Minimal tree Coverage | - | ✓ | ✓ | ✓ | ILP Graph theory Homogenous nodes |

Liu, 2019 [24] | ✓ | - | - | ✓ | Minimize Power losses in distribution Maximize Flow | Capacity Congestion | ✓ | - | ✓ | - | Heterogeneous nodes Optimal Solution Charging congestion during fuzzy multi-objective model |

Ghasemi, 2019 [25] | ✓ | - | ✓ | ✓ | Min. Transportation cost Minimize shortages Max. humanitarian | Demand Heterogenous flow Capacity Distance | ✓ | ✓ | ✓ | - | Probability approach Planning periods Multiple-objective PSO Stochastic demand |

Campaña, 2019 [26] | ✓ | - | ✓ | ✓ | Min. Trajectories Min. Charging Station Min. Congestion road | Coverage Distance Capacity | - | ✓ | ✓ | ✓ | Segmentation ILP Need for Users Neighborhood analysis |

Current study | ✓ | - | ✓ | ✓ | Min. Trajectories Min. flow Min. Charging stations Max. Humanitarian | Capacity Distance Coverage Demand | ✓ | ✓ | ✓ | ✓ | Revised-Simplex MILP Multi-commodity network flow Heterogeneous nodes Scalable Multi-objective NP-Hard Non convex |

**Table 3.**Types of charging terminals for ICSEC [23].

Type | Current (Amp) Type | Time (h) | Recharge (%) | Power kW | Owner | INEC Standard |
---|---|---|---|---|---|---|

Slow | 16 AC | 8 | 100 | 4–8 | Public-Private | |

Semi-fast | 32 AC | 1.150 | 50–80 | 22 | Public | |

Fast | 63 AC | 0.500 | 50–80 | 50 | Public | 61,851 |

Ultra-fast | 250–400 DC | 0.170 | 50–80 | 350 | Public | |

Change-Battery | AC-DC | 0.033 | 100 | - | Public |

Nomenclature | Description |
---|---|

m | Number of nodes $\{\nu \cup \epsilon \cup \zeta \cup \beta \}$ |

${G}_{m\times m}$ | Directed graph matrix |

${\phi}_{1,2,\cdots ,m}$, $\phi $ | EV autonomy vector |

$\mathcal{V}$ | Nodes vector |

$\mathcal{A}$ | Edges Vector |

$\nu $ | EV nodes |

$\epsilon $ | CS nodes |

$u,v$ | Origin and destination edges, respectively |

$i,j$ | Origin and destination transversal intersection edges |

$\iota $ | Road cross length |

$\alpha $ | Number of heterogeneous vehicles in $\iota $ |

$\varpi $ | Discrete probability of the number of vehicles |

$\beta $ | Intersections vector |

${W}_{\beta}$ | Total vehicle concentration |

${\mathcal{S}}_{1,2,3\cdots}$ | Set of paths $u,v$ |

${f}_{\beta}$ | Flow rate |

$\kappa $ | CS candidate sites vector |

${\delta}_{\kappa}$ | Fixed cost to install EDS |

${n}_{j}$ | Number of vehicles to cover capacity $\psi $ |

$\psi $ | CS capacity |

$\mathcal{M}$ | Set of enabled paths edges |

$\mathcal{L}$ | Set of edges of a path $u,v$ |

${\varphi}_{j}$ | Location variable |

${\theta}_{k}$ | Subset of assigned arcs |

${\tau}_{e}^{j}$ | Route enabled |

$q,Q$ | Partial and total vehicular flow rate |

${V}_{e}$ | Constant travel speed |

${t}_{s}$ | Time in hours |

t | Number of commodities |

$\Theta $ | Incremental annual rate given as a percentage |

s | Average spacing between vehicles |

k | Index vector |

${\chi}_{\ast}^{k}$ | Revised simplex reduced negative value |

${\mathsf{\Psi}}_{p}$ | Partial solution to the primary problem |

P | Identification for each iteration |

${\pi}_{1,2,\cdots}$ | Dual variables |

$np$ | Number of periods in years |

Approach | Description | Consideration |
---|---|---|

Deployment | Density EV | Variable |

Study cases | A–B | |

Study areas | 0.99 y 2.13 km^{2} | |

Geographical area | Urban | |

Geographic reference | Latitude–Longitude | |

Allocation | Annual rate | Variable |

Scalability | 4 years | |

Spacing | Variable | |

Application | Charging terminal | AC–DC |

Safety distance | 3–6 m | |

Vehicular flow | Variable | |

Vehicular concentration | Variable | |

Traffic speed | [20 40 60] km/h | |

Traffic | Light | 4.3 (m) ≤ |

Buses | 15 (m) | |

Trucks | 12 (m) |

**Table 6.**Case A: Maximum partial demand in each CS at hour 19:00 with 32.97 km of road network in contingency N-0 corresponding to year 0.

ID | Longitude | Latitude | Concentration Vehicular | Vehicles (#) |
---|---|---|---|---|

1 | −120.625225787882 | 35.1271587155963 | −0.77 | 25.41 |

2 | −120.622727167999 | 35.1269678899083 | −1.31 | 43.22 |

3 | −120.617829873029 | 35.1267623853211 | −0.64 | 21.22 |

4 | −120.609994201077 | 35.1261458715596 | −1.48 | 48.71 |

5 | −120.615391220024 | 35.1256174311927 | −1.29 | 42.44 |

6 | −120.620588349380 | 35.1256467889908 | −1.43 | 47.10 |

7 | −120.624546163274 | 35.1248688073394 | −1.09 | 36.03 |

8 | −120.624326284724 | 35.1243403669725 | −0.85 | 28.06 |

9 | −120.619488956631 | 35.1240174311927 | −1.43 | 47.11 |

10 | −120.610473936095 | 35.1246045871560 | −1.44 | 47.52 |

11 | −120.611033626948 | 35.1230192660550 | −0.67 | 22.00 |

12 | −120.620568360421 | 35.1230779816514 | −0.64 | 21.00 |

13 | −120.622167477145 | 35.1233128440367 | −0.85 | 27.90 |

14 | −120.625865434572 | 35.1238266055046 | −0.62 | 20.30 |

15 | −120.625805467695 | 35.1222559633028 | −0.61 | 20.19 |

16 | −120.623466759484 | 35.1219183486239 | −1.18 | 38.89 |

17 | −120.620868194807 | 35.1221678899083 | −0.92 | 30.32 |

18 | −120.617310160094 | 35.1218889908257 | −0.86 | 28.35 |

19 | −120.615671065451 | 35.1218743119266 | −0.99 | 32.73 |

20 | −120.613552235790 | 35.1218449541284 | −1.44 | 47.43 |

21 | −120.610833737358 | 35.1199220183486 | −1.21 | 39.74 |

22 | −120.614731584375 | 35.1193642201835 | −1.34 | 44.03 |

23 | −120.616150800468 | 35.1204357798165 | −0.96 | 31.70 |

24 | −120.619588901427 | 35.1209495412844 | −0.85 | 28.00 |

25 | −120.625925401449 | 35.1213018348624 | −0.97 | 32.00 |

26 | −120.623666649075 | 35.1200247706422 | −0.79 | 26.12 |

27 | −120.619688846222 | 35.1195550458716 | −0.81 | 26.78 |

28 | −120.624606130151 | 35.1191293577982 | −1.37 | 45.32 |

29 | −120.617350138012 | 35.1186743119266 | −1.19 | 39.35 |

30 | −120.617250193216 | 35.1176027522936 | −0.80 | 26.32 |

31 | −120.618829320982 | 35.1176321100917 | −1.09 | 36.00 |

32 | −120.622327388818 | 35.1168981651376 | −0.94 | 31.02 |

33 | −120.613552235790 | 35.1176027522936 | −0.61 | 20.19 |

34 | −120.616050855673 | 35.1232541284404 | −0.87 | 28.53 |

35 | −120.608654940820 | 35.1195550458716 | −0.62 | 20.31 |

Total | −34.92 | 1151.34 |

ID | Vehicle Concentration | # Vehicles |
---|---|---|

1 | −3.00 | 29.95 |

2 | −5.33 | 53.21 |

3 | −4.84 | 48.32 |

4 | −2.93 | 29.25 |

5 | −2.18 | 21.77 |

6 | −3.72 | 37.14 |

7 | −3.55 | 35.44 |

8 | −4.72 | 47.12 |

9 | −4.62 | 46.12 |

Total | −34.89 | 348.34 |

Contingency | Deactivated | ID | ID | ID | ID | ID | ID | ID | ID | ID |
---|---|---|---|---|---|---|---|---|---|---|

(ID) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |

N − 0 | − | −3.00 | −5.33 | −4.84 | −2.93 | −2.18 | −3.72 | −3.55 | −4.72 | −4.62 |

N − 1 | 1 | − | −5.60 | −4.38 | −4.97 | −4.08 | −3.90 | −4.20 | −4.16 | −5.08 |

2 | −4.01 | − | −5.98 | −4.99 | −3.65 | −5.96 | −4.48 | −5.03 | −4.93 | |

3 | −4.27 | −6.38 | − | −3.62 | −4.03 | −4.56 | −4.35 | −5.11 | −5.12 | |

4 | −4.42 | −4.32 | −4.06 | − | −3.55 | −5.34 | −4.72 | −5.24 | −5.68 | |

5 | −4.41 | −5.12 | −4.81 | −4.65 | − | −5.24 | −3.40 | −5.48 | −4.75 | |

6 | −3.74 | −4.37 | −4.99 | −6.05 | −3.15 | − | −4.34 | −5.15 | −5.79 | |

7 | −4.25 | −4.05 | −5.31 | −4.91 | −3.53 | −4.89 | − | −5.53 | −4.91 | |

8 | −3.15 | −4.98 | −4.79 | −4.73 | −3.88 | −5.12 | −3.94 | − | −4.85 | |

9 | −4.02 | −4.72 | −5.41 | −4.20 | −3.35 | −4.00 | −4.86 | −4.86 | − | |

Average ($\mathsf{\mu}$) | 4.03 | 4,94 | 4.97 | 4.77 | 3.65 | 4.88 | 4.29 | 5.07 | 5.14 | |

Std. deviation ($\sigma $) | 0.40 | 0.72 | 0.57 | 0.65 | 0.31 | 0.65 | 0.43 | 0.40 | 0.36 | |

$\mu +2\ast \sigma $ | 4.83 | 6.37 | 6.10 | 6.07 | 4.26 | 6.18 | 5.14 | 5.87 | 5.86 | |

N − 2 | 1, 2 | − | − | −4.67 | −5.18 | −4.89 | −5.40 | −5.54 | −4.95 | −5.42 |

2, 3 | −4.05 | − | − | −4.25 | −4.59 | −5.42 | −4.43 | −6.08 | −5.02 | |

8, 9 | −4.13 | −5.40 | −5.49 | −5.56 | −5.12 | −5.76 | −5.46 | − | − | |

4, 5 | −4.46 | −4.89 | −5.09 | − | − | −6.08 | −5.64 | −5.71 | −6.11 | |

7, 9 | −4.83 | −4.01 | −7.10 | −5.73 | −4.31 | −5.70 | − | −5.93 | − | |

5, 7 | −4.38 | −4.20 | −4.71 | −4.25 | − | −5.62 | − | −6.31 | −5.16 | |

3, 7 | −5.37 | −4.68 | − | −5.09 | −4.97 | −5.13 | − | −5.90 | −6.14 | |

3, 5 | −4.15 | −5.03 | − | −5.51 | − | −4.95 | −4.36 | −5.80 | −4.78 | |

2, 4 | −5.20 | − | −6.08 | − | −5.22 | −4.91 | −5.06 | −5.41 | −6.00 | |

5, 6 | −4.19 | −4.94 | −5.38 | −6.36 | − | − | −4.89 | −6.11 | −5.79 | |

7, 8 | −4.30 | −6.30 | −4.50 | −4.78 | −5.27 | −5.06 | − | − | −5.46 | |

1, 3 | − | −5.22 | − | −4.25 | −3.51 | −5.89 | −4.41 | −5.69 | −5.09 | |

3, 4 | −4.77 | −6.39 | − | − | −3.84 | −5.58 | −5.07 | −5.50 | −6.08 | |

Average ($\mathsf{\mu}$) | 4.53 | 5.11 | 5.38 | 5.10 | 4.64 | 5.46 | 4.98 | 5.76 | 5.55 | |

Std. deviation ($\sigma $) | 0.43 | 0.74 | 0.81 | 0.68 | 0.59 | 0.36 | 0.47 | 0.36 | 0.47 | |

$\mu +2\ast \sigma $ | 5.39 | 6.58 | 7.00 | 6.46 | 5.82 | 6.19 | 5.93 | 6.48 | 6.50 | |

Increment N − 1 (%) | 61.0 | 19.5 | 26.0 | 107.2 | 95.4 | 66.1 | 44.8 | 24.4 | 26.8 | |

Increment N − 2 (%) | 79.7 | 23.5 | 44.6 | 120.5 | 167.0 | 66.4 | 67.0 | 37.3 | 40.7 |

**Table 9.**Maximum partial demand at each charging station with 9984 km of road network given a contingency.

Events | ID 1 | ID 2 | ID 3 | ID 4 | ID 5 | ID 6 | ID 7 | ID 8 | ID 9 |
---|---|---|---|---|---|---|---|---|---|

N-0 | 29.95 | 53.22 | 48.32 | 29.25 | 21.77 | 37.14 | 35.44 | 47.13 | 46.13 |

N-1 | 48.22 | 63.60 | 60.90 | 60.60 | 42.53 | 61.70 | 51.32 | 58.61 | 58.51 |

N-2 | 53.81 | 65.70 | 69.89 | 64.50 | 58.11 | 61.80 | 59.21 | 64.70 | 64.90 |

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**MDPI and ACS Style**

Campaña, M.; Inga, E.
Optimal Planning of Electric Vehicle Charging Stations Considering Traffic Load for Smart Cities. *World Electr. Veh. J.* **2023**, *14*, 104.
https://doi.org/10.3390/wevj14040104

**AMA Style**

Campaña M, Inga E.
Optimal Planning of Electric Vehicle Charging Stations Considering Traffic Load for Smart Cities. *World Electric Vehicle Journal*. 2023; 14(4):104.
https://doi.org/10.3390/wevj14040104

**Chicago/Turabian Style**

Campaña, Miguel, and Esteban Inga.
2023. "Optimal Planning of Electric Vehicle Charging Stations Considering Traffic Load for Smart Cities" *World Electric Vehicle Journal* 14, no. 4: 104.
https://doi.org/10.3390/wevj14040104