# Charging Station Allocation for Electric Vehicle Network Using Stochastic Modeling and Grey Wolf Optimization

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**Location:**Several candidate locations for installing CS were suggested in previous studies which can be categorized into:**Level:**Three charging levels were investigated and considered in the infrastructure planning designs and algorithms in a few studies [8]:- Level 1 (standard): operating at 120 V, 15 or 20 A, and 1.44 kW.
- Level 2 (commercial): operating at 240 V, single-phase, 40 A and 6 kW.
- Level 3 (Fast Charger): operating at high-voltage, high-current (D.C.) and 90 kW.

**Size and Capacity**: While many papers refer to the size of CS as CS capacity, the two terms are defined differently in this paper. CS capacity is defined as the number of EVs the CS can handle at a time including both EVs that are being served at the moment in addition to the EVs waiting in the queue. Whereas, CS size is determined by the number of available charging sockets at the station [9]. The CS Level affects the service time, while the capacity and the size affects the queuing time at the CS [10]. Many constraints need to be taken into consideration when developing the optimization algorithm of allocating EVCSs. Based on related work in the literature, these constraints can be classified as: (I) Budget-related constraints including demand and cost constraints, and (II) Route-related constraints including available routes between candidate locations, the distance EV can go before next charge, traffic, weather, etc.

## 2. Literature Review

- (I)
- The proposed algorithm simulates the stochastic behavior of EVCS infrastructure including demand and cost uncertainty using Markov-chain processes.
- (II)
- The parameters affecting the CS quality of service and the achieved profit of EV network were investigated using birth-and-death model and sensitivity analysis.
- (III)
- GWO is proposed to optimize the NP-hard allocation problem. GWO is known for lower complexity and higher flexibility compared to other solutions proposed in the literature to address the EVCS infrastructure problem.
- (IV)
- Both budget and routing constraints are considered in the optimization model.
- (V)
- Unlike FLRM, state of charge (SOC) uncertainty is included in the proposed model.

## 3. System Model

#### 3.1. Markov-Chain of EV Charging Station

#### 3.2. Grey Wolf Optimization (GWO)

#### 3.3. GWO for EV Charging Station Allocation

## 4. Experimental Results

#### 4.1. Sensitivity Analysis

#### 4.1.1. Single Charging Station

#### 4.1.2. Multiple Charging Stations

#### 4.2. CS Allocation Using GWO

## 5. Research Significance and Limitations

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Andrenacci, N.; Ragona, R.; Valenti, G. A demand-side approach to the optimal deployment of electric vehicle charging stations in metropolitan areas. Appl. Energy
**2016**, 182, 39–46. [Google Scholar] [CrossRef] - Bayram, I.S.; Michailidis, G.; Devetsikiotis, M.; Granelli, F. Electric Power Allocation in a Network of Fast Charging Stations. IEEE J. Sel. Areas Commun.
**2013**, 31, 1235–1246. [Google Scholar] [CrossRef] [Green Version] - Berman, O.; Larson, R.C.; Fouska, N. Optimal Location of Discretionary Service Facilities. Transp. Sci.
**1992**, 26, 201–211. [Google Scholar] [CrossRef] - Bokeh Development Team. Bokeh: Python Library for Interactive Visualization. 2018. Available online: https://bokeh.pydata.org/en/latest/ (accessed on 18 April 2018).
- Cai, H.; Jia, X.; Chiu, A.S.; Hu, X.; Xu, M. Siting public electric vehicle charging stations in Beijing using big-data informed travel patterns of the taxi fleet. Transp. Res. Part D Transp. Environ.
**2014**, 33, 39–46. [Google Scholar] [CrossRef] [Green Version] - Chen, T.D.; Kockelman, K.M.; Khan, M. The Electric Vehicle Charging Station Location Problem: A Parking-Based Assignment Method for Seattle. In Proceedings of the 92nd Annual Meeting of the Transportation Research Board, Washington, DC, USA, 13–17 January 2013. [Google Scholar]
- Chung, S.H.; Kwon, C. Multi-period planning for electric car charging station locations: A case of Korean Expressways. Eur. J. Oper. Res.
**2015**, 242, 677–687. [Google Scholar] [CrossRef] - Davis, M.A.; Oliner, S.D.; Pinto, E.J.; Bokka, S. Residential land values in the Washington, DC metro area: New insights from big data. Reg. Sci. Urban Econ.
**2017**, 66, 224–246. [Google Scholar] [CrossRef] - District of Columbia Open Data. Gas Stations. 2015. Available online: https://opendata.dc.gov/ (accessed on 15 May 2019).
- Dong, J.; Liu, C.; Lin, Z. Charging infrastructure planning for promoting battery electric vehicles: An activity-based approach using multiday travel data. Transp. Res. Part C Emerg. Technol.
**2014**, 38, 44–55. [Google Scholar] [CrossRef] [Green Version] - Kuby, M.; Lim, S. The flow-refueling location problem for alternative-fuel vehicles. Socio-Econ. Plan. Sci.
**2005**, 39, 125–145. [Google Scholar] [CrossRef] - Eberle, U.; Von Helmolt, R. Sustainable transportation based on electric vehicle concepts: A brief overview. Energy Environ. Sci.
**2010**, 3, 689–699. [Google Scholar] [CrossRef] - Emary, E.; Zawbaa, H.M.; Hassanien, A.E. Binary grey wolf optimization approaches for feature selection. Neurocomputing
**2016**, 172, 371–381. [Google Scholar] [CrossRef] - Farkas, C.; Prikler, L. Stochastic modelling of EV charging at charging stations. Renew. Energy Power Qual. J.
**2012**, 1046–1051. [Google Scholar] [CrossRef] - Franke, T.; Krems, J.F. Interacting with limited mobility resources: Psychological range levels in electric vehicle use. Transp. Res. Part A Policy Pr.
**2013**, 48, 109–122. [Google Scholar] [CrossRef] - Google Developers. Getting Started with the Google Places API for Work. 2017. Available online: https://developers.google.com/ (accessed on 27 July 2017).
- Liu, J. Electric vehicle charging infrastructure assignment and power grid impacts assessment in Beijing. Energy Policy
**2012**, 51, 544–557. [Google Scholar] [CrossRef] - Shi, Q.S.; Zheng, X.Z. Electric Vehicle Charging Stations Optimal Location Based on Fuzzy C-Means Clustering. Appl. Mech. Mater.
**2014**, 556-562, 3972–3975. [Google Scholar] [CrossRef] - Hodgson, M.J. A Flow-Capturing Location-Allocation Model. Geogr. Anal.
**2010**, 22, 270–279. [Google Scholar] [CrossRef] - Vazifeh, M.M.; Zhang, H.; Santi, P.; Ratti, C. Optimizing the deployment of electric vehicle charging stations using pervasive mobility data. Transp. Res. Part A Policy Pr.
**2019**, 121, 75–91. [Google Scholar] [CrossRef] [Green Version] - Jeff Desjardins. Visualizing the Rise of the Electric Vehicle. 2018. Available online: https://www.visualcapitalist.com/rise-electric-vehicle/ (accessed on 2 July 2019).
- Rahman, I.; Vasant, P.M.; Singh, B.S.M.; Abdullah-Al-Wadud, M. Swarm Intelligence-Based Smart Energy Allocation Strategy for Charging Stations of Plug-In Hybrid Electric Vehicles. Math. Probl. Eng.
**2015**, 2015, 1–10. [Google Scholar] [CrossRef] [Green Version] - Vasant, P.M.; Rahman, I.; Singh, B.S.M.; Abdullah-Al-Wadud, M. Optimal power allocation scheme for plug-in hybrid electric vehicles using swarm intelligence techniques. Cogent Eng.
**2016**, 3. [Google Scholar] [CrossRef] - Jing, W.; An, K.; Ramezani, M.; Kim, I. Location Design of Electric Vehicle Charging Facilities: A Path-Distance Constrained Stochastic User Equilibrium Approach. J. Adv. Transp.
**2017**, 2017, 1–15. [Google Scholar] [CrossRef] [Green Version] - Kendall, D.G. Stochastic Processes Occurring in the Theory of Queues and their Analysis by the Method of the Imbedded Markov Chain. Ann. Math. Stat.
**1953**, 24, 338–354. [Google Scholar] [CrossRef] - Kim, J.; Son, S.-Y.; Lee, J.-M.; Ha, H.-T. Scheduling and performance analysis under a stochastic model for electric vehicle charging stations. Omega
**2017**, 66, 278–289. [Google Scholar] [CrossRef] - Kuby, M.; Lim, S. Location of Alternative-Fuel Stations Using the Flow-Refueling Location Model and Dispersion of Candidate Sites on Arcs. Netw. Spat. Econ.
**2006**, 7, 129–152. [Google Scholar] [CrossRef] - Lee, Y.-G.; Kim, H.-S.; Kho, S.-Y.; Lee, C. User Equilibrium–Based Location Model of Rapid Charging Stations for Electric Vehicles with Batteries that have Different States of Charge. Transp. Res. Rec. J. Transp. Res. Board
**2014**, 2454, 97–106. [Google Scholar] [CrossRef] - Mirjalili, S.; Mirjalili, S.M.; Lewis, A. Grey Wolf Optimizer. Adv. Eng. Softw.
**2014**, 69, 46–61. [Google Scholar] [CrossRef] [Green Version] - Pinto, F.A.V.; Costa, L.H.M.K.; De Amorini, M.D. Modeling spare capacity reuse in EV charging stations based on the Li-ion battery profile. In Proceedings of the 2014 International Conference on Connected Vehicles and Expo (ICCVE), Vienna, Austria, 3–7 November 2014; pp. 92–98. [Google Scholar] [CrossRef]
- Python Software Foundation. Python Language Reference, Version 2.7. 2013. Available online: https://www.python.org/ (accessed on 20 July 2017).
- Ross, S.M. Introduction to Probability Models, 20th ed.; Academic Press: Oxford, UK, 2009. [Google Scholar] [CrossRef]
- Smith, M.; Castellano, J. Costs Associated with Non-Residential Electric Vehicle Supply Equipment: Factors to Consider in the Implementation of Electric Vehicle Charging Stations; NewWest Technologies: Portland, OR, USA, 2015. [Google Scholar]
- Socrata. About Gas Stations in Washington DC. 2011. Available online: https://opendata.socrata.com/dataset/Gas-Stations-in-Washington-DC/ (accessed on 27 July 2019).
- Tan, X.; Sun, B.; Tsang, D.H.K. Queueing network models for electric vehicle charging station with battery swapping. In Proceedings of the 2014 IEEE International Conference on Smart Grid Communications (SmartGridComm), Venice, Italy, 3–6 November 2014. [Google Scholar] [CrossRef]
- Tian, Z.; Hou, W.; Gu, X.; Gu, F.; Yao, B. The location optimization of electric vehicle charging stations considering charging behavior. Simulation
**2018**, 94, 625–636. [Google Scholar] [CrossRef] [Green Version] - Vardakas, J.S. Electric vehicles charging management in communication controlled fast charging stations. In Proceedings of the 2014 IEEE 19th International Workshop on Computer Aided Modeling and Design of Communication Links and Networks (CAMAD), Athens, Greece, 1–3 December 2014. [Google Scholar] [CrossRef]
- Wang, S.; Bi, S.; Zhang, Y.-J.A.; Huang, J. Electrical Vehicle Charging Station Profit Maximization: Admission, Pricing, and Online Scheduling. IEEE Trans. Sustain. Energy
**2018**, 9, 1722–1731. [Google Scholar] [CrossRef] [Green Version] - Wu, F.; Sioshansi, R. A stochastic flow-capturing model to optimize the location of fast-charging stations with uncertain electric vehicle flows. Transp. Res. Part D Transp. Environ.
**2017**, 53, 354–376. [Google Scholar] [CrossRef]

**Figure 3.**Birth-and-Death Markov-chain sensitivity analysis: (

**a**) Waiting Time ($c=5$); (

**b**) Queue Length ($c=5$); (

**c**) Net Profit ($c=5,\lambda =1/20$); (

**d**) Net Profit ($c=5,\lambda =1/10$); (

**e**) Blocking Probability ($\lambda =1/20$); (

**f**) Waiting Time ($\lambda =1/20$); (

**g**) Queue Length (c = 5, $\lambda =1/10$); (

**h**) Net Profit (c = 5, $\lambda =1/10$).

**Figure 4.**GWO (Grey Wolf Optimization) sensitivity analysis: (

**a**) Effect of α on Net Profit; (

**b**) Effect of α on Number of Selected CSs; (

**c**) Effect of $\u20ac$ on Net Profit; (

**d**) Effect of $\u20ac$ on Number of Selected CSs; (

**e**) Effect of $\mathrm{c}$ on Net Profit; (

**f**) Effect of $c$ on Number of Selected CSs; (

**g**) Effect of $N$ on Net Profit; (

**h**) Effect of $N$ on Number of Selected CSs.

**Table 1.**Experiment parameter settings: (

**a**) GWO parameter settings; (

**b**) Charging Station parameter settings in an urban area; (

**c**) Charging Station parameter settings for Washington D.C.

(a) GWO parameter settings | |

Lower Bound | 0 |

Upper Bound | 1 |

Dimension | Number of possible CSs |

Population Size | 50 |

Number of Generations | 20 |

(b) Charging Station parameter settings in an urban area
| |

N | 10 |

c | 5 |

µ | [1/30, 1/60, 1/120] |

λ | [1/60, 1/10] |

$cu$ ($) | 5 (Level 2), 10 (Level 3) |

$\u20ac$ ($) | 50 (Level 2), 70 (Level 3) |

α | 0.3 |

(c) Charging Station parameter settings for Washington D.C.
| |

N | 10 |

c | 5 |

µ | [1/30, 1/60] |

λ | Proportional to Annual Average Daily Traffic (AADT) in D.C. [36] Low: [0.002–0.08], High: [0.01–0.4] |

Installation & Electricity Cost ($ per min) | 0.0057 (Level 2), 0.018 (Level 3) [37] |

Operating Cost ($ per min) | Proportional to Average Standardized Land Price per Square Foot in D.C. [23] [0.15–0.5] |

$cu$ ($) | Operating Cost + Installation & Electricity Cost |

$\u20ac$ ($) | 15 (Level 2), 18 (Level 3) |

α | 0.3 |

(a) | |||||

Parameter | Measure | Adj SS | Adj MS | F-Value | p-Value (<0.05) |

Entering Probability (α) | Net Profit | 60.76 | 15.19 | 1.37 | 0.247 |

Entering Probability (α) | Number of CS Selected | 253.6 | 63.39 | 2.11 | 0.084 |

Gross Profit per EV ($\u20ac$) | Net Profit | 20,959 | 6986.31 | 489.88 | 0.000 |

Gross Profit per EV ($\u20ac$) | Number of CS Selected | 3685 | 1228.40 | 41.10 | 0.000 |

CS Capacity (N) | Net Profit | 17.75 | 5.916 | 0.72 | 0.545 |

CS Capacity (N) | Number of CS Selected | 105.7 | 35.24 | 1.13 | 0.339 |

(b) | |||||

Measure | CS Level | Mean | StDev | T-Value | p-Value (<0.05) |

Net Profit—High Rate | Level 2 | 46.58 | 2.68 | −6.81 | 0.000 |

Level 3 | 53.50 | 4.32 | |||

Net Profit—Low Rate | Level 2 | 2.42 | 1.10 | −3.98 | 0.000 |

Level 3 | 3.73 | 1.22 | |||

Number of Selected CS—High Rate | Level 2 | 29.60 | 5.35 | 2.06 | 0.045 |

Level 3 | 26.60 | 4.94 | |||

Number of Selected CS—Low Rate | Level 2 | 7.60 | 3.33 | −3.72 | 0.001 |

Level 3 | 11.56 | 4.15 | |||

Net Profit—High Rate | Level 3 | 53.5 | 4.32 | −1.71 | 0.100 |

Level 3 SOC | 61 | 21.6 | |||

Net Profit—Low Rate | Level 3 | 3.73 | 1.22 | −0.19 | 0.850 |

Level 3 SOC | 3.8 | 1.25 | |||

Number of Selected CS—High Rate | Level 3 | 26.6 | 4.94 | 0.03 | 0.977 |

Level 3 SOC | 26.56 | 4.93 | |||

Number of Selected CS—Low Rate | Level 3 | 11.56 | 4.15 | 0.51 | 0.610 |

Level 3 SOC | 10.96 | 4.12 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Shabbar, R.; Kasasbeh, A.; Ahmed, M.M.
Charging Station Allocation for Electric Vehicle Network Using Stochastic Modeling and Grey Wolf Optimization. *Sustainability* **2021**, *13*, 3314.
https://doi.org/10.3390/su13063314

**AMA Style**

Shabbar R, Kasasbeh A, Ahmed MM.
Charging Station Allocation for Electric Vehicle Network Using Stochastic Modeling and Grey Wolf Optimization. *Sustainability*. 2021; 13(6):3314.
https://doi.org/10.3390/su13063314

**Chicago/Turabian Style**

Shabbar, Rawan, Anemone Kasasbeh, and Mohamed M. Ahmed.
2021. "Charging Station Allocation for Electric Vehicle Network Using Stochastic Modeling and Grey Wolf Optimization" *Sustainability* 13, no. 6: 3314.
https://doi.org/10.3390/su13063314