# An Optimization Model for the Temporary Locations of Mobile Charging Stations

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## Abstract

**:**

## 1. Introduction

- Mode 1 (L1) is the simplest EV charging solution. In this case, an EV is connected to AC power supply using standard sockets, but it must contain a contractor to disconnect the power supply in case of overload or an electric shock. The EV’s charging is done without communication between it and the station, and the maximum current accepted is 16 A.
- Mode 2 (L2), where the EV is charged from the power supply using standardized single-phase or three-phase sockets and charging conductors containing a box of control integrated into the cable with a pilot function command and a protection system against electrical shocks (RCD). The charge current value should not exceed 32 A.
- Mode 3 (L3), where an EV is connected via a specific scheme to the charging station (EAVE) that has installed control and protection functions. The maximum current is 3 × 63 A.
- Mode 4 (L4), where EVs are charged from stations using direct current.

## 2. Literature Review

## 3. Mathematical Model

- (i)
- Each EV will be assigned to a center at the shortest possible distance from its location.
- (ii)
- Upon arrival at the service center, each EV will stand in a queue with no more than b clients with a probability of at least $\beta $; or in each center will be at least a clients, with a probability of at least $\alpha $; or when arriving at the service center, each EV will be in a queue with no more than b clients, and in each center will be at least a clients, with a probability of at least $\gamma $.
- (iii)
- The battery capacity of the mobile charging station is enough to meet all requests during the considered temporary location period.

#### 3.1. Nonlinear Optimization Problem

#### 3.2. Linear Optimization Problem

**Theorem**

**1.**

**Lemma**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

## 4. Estimating Parameters

- Identity number of the taxi;
- Distance (miles) and trip time (seconds);
- Customer pick-up times and arrival times at destination (day, month, year, hour);
- Pick-up and destination locations of customers (latitude and longitude).

- The taxi has enough energy to make the trip.
- If for intersection i at time t there is no probability of a destination obtained from the NYC data, then all intersections have the same probability of destination.
- Discrete distributions are constructed so that intersections with high probabilities have greater chances of random choice than others. Intersections for which we did not have a destination probability in the data were considered to have the same probability.
- The taxi picks up a customer as soon as it reaches the next intersection.
- The maximum charge capacity of an electric taxi battery is 30 kW, taking the lower and upper limits as 5% and 95%.

## 5. Results and Discussions

## 6. Operational Metrics—Comparative Experiments

## 7. Conclusions

- We formulated a nonlinear optimization problem and an equivalent mixed-linear optimization problem for optimal temporary location of a mobile charging station.
- We obtained new probability-queuing constraints, considering at least a clients in the queue and/or at most b clients in the queue.
- We compared two operational modes for a mobile charging station.

- Different temporary locations are obtained for different probability constraints and days of the week.
- The locations obtained are in areas with no or few fixed charging stations.
- The temporary location operational mode compared to moving operational mode has a smaller mean response time.
- Travel distance is an issue in moving operational mode, increasing operational costs, a problem that does not arise in the temporary location strategy.

- The development of a new strategy in different urban areas to install predictive charging stations;
- Another study in which we intend to use installed charging stations and gather data on the number of cars, the flow of cars in certain time periods and driver behavior given by GPS data;
- A study for increasing the percentage of renewable energy that supplies charging stations; for example, a charging station with PV on the roof;
- Uncertainty analysis with fuzzy intervals;
- Extending the work by considering Markov dependence hypothesis.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Global EV Outlook 2017 Two Million and Counting. Available online: https://webstore.iea.org/global-ev-outlook-2017 (accessed on 3 August 2019).
- European Alternative Fuels. Available online: https://www.eafo.eu/countries/european-union/23640/infrastructure/electricity (accessed on 6 March 2020).
- Electric Vehicles in Europe; European Environment Agency: Copenhagen, Denmark, 2016.
- IEC 61851-1 Standard. Available online: https://webstore.iec.ch/publication/33644 (accessed on 5 August 2019).
- Huang, S.; He, L.; Gu, Y.; Wood, K.; Benjaafar, S. Design of a mobile charging service for electric vehicles in an urban environment. IEEE Trans. Intell. Transp. Syst.
**2014**, 16, 787–798. [Google Scholar] [CrossRef] - Idaho National Laboratory (INL). What Is the Impact of Utility Demand Charges on a DCFC Host? INL/EXT-15-35706. June 2015. Available online: http://avt.inl.gov/pdf/EVProj/EffectOfDemandChargesOnDCFCHosts.pdf (accessed on 23 July 2019).
- Orlando Utilities Commission (OUC). DC Fast Charging Efforts in Orland. Presentation by OUC. 9 December 2014. Available online: http://www.advancedenergy.org/portal/ncpev/blog/news/wp-content/uploads/2014/12/OUC-presentation-NCPEV-14.pdf (accessed on 15 July 2019).
- Gong, D.; Tang, M.; Buchmeister, B.; Zhang, H. Solving Location Problem for Electric Vehicle Charging Stations-A Sharing Charging Model. IEEE Access
**2019**, 7, 138391–138402. [Google Scholar] [CrossRef] - Taxicab Fact Book; NYC Taxi Limousine Commision: New York, NY, USA, 2014. Available online: https://www1.nyc.gov/site/tlc/about/request-data.page (accessed on 12 March 2019).
- Chen, H.; Su, Z.; Hui, Y.; Hui, H. Dynamic charging optimization for mobile charging stations in Internet of Things. IEEE Access
**2018**, 6, 53509–53520. [Google Scholar] [CrossRef] - Cui, S.; Zhao, H.; Zhang, C. Multiple types of plug-in charging facilities’ location-routing problem with time windows for mobile charging vehicles. Sustainability
**2018**, 10, 2855. [Google Scholar] [CrossRef] [Green Version] - Cui, S.; Zhao, H.; Chen, H.; Zhang, C. The Mobile Charging Vehicle Routing Problem with Time Windows and Recharging Services. Comput. Intel. Neurosc.
**2018**. [Google Scholar] [CrossRef] [PubMed] - Chen, H.; Su, Z.; Hui, Y.; Hui, H. Optimal approach to provide electric vehicles with charging service by using mobile charging stations in heterogeneous networks. In Proceedings of the 2016 IEEE 84th Vehicular Technology Conference (VTC-Fall), Montreal, QC, Canada, 18–21 September 2016. [Google Scholar]
- Jia, L.; Zechun, H.; Yonghua, S.; Zhuowei, L. Optimal siting and sizing of electric vehicle charging stations. In Proceedings of the 2012 IEEE International Electric Vehicle Conference, Greenville, SC, USA, 4–8 March 2012. [Google Scholar]
- 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] - Zhu, Z.H.; Gao, Z.Y.; Zheng, J.F.; Du, H.M. Charging station location problem of plug-in electric vehicles. J. Transp. Geogr.
**2016**, 52, 11–22. [Google Scholar] [CrossRef] - Csiszár, C.; Csonka, B.; Földes, D.; Wirth, E.; Lovas, T. Urban public charging station locating method for electric vehicles based on land use approach. J. Transp. Geogr.
**2019**, 74, 173–180. [Google Scholar] [CrossRef] - Xiong, Y.; Gan, J.; An, B.; Miao, C.; Bazzan, A.L. Optimal electric vehicle fast charging station placement based on game theoretical framework. IEEE trans. Intell. Transp. Syst.
**2017**, 19, 2493–2504. [Google Scholar] [CrossRef] - Shukla, A.; Verma, K.; Kumar, R. Planning of Fast Charging Stations in Distribution System Coupled with Transportation Network for Capturing EV flow. In Proceedings of the 2018 8th IEEE India International Conference on Power Electronics (IICPE), Jaipur, India, 13–15 December 2018. [Google Scholar]
- Lu, F.; Hua, G. A location-sizing model for electric vehicle charging station deployment based on queuing theory. In Proceedings of the 2015 International Conference on Logistics, Informatics and Service Sciences (LISS), Barcelona, Spain, 27–29 July 2015. [Google Scholar]
- Said, D.; Cherkaoui, S.; Khoukhi, L. Multi-priority queuing for electric vehicles charging at public supply stations with price variation. Wirel. Commun. Mob. Com.
**2015**, 15, 1049–1065. [Google Scholar] [CrossRef] - Zhu, J.; Li, Y.; Yang, J.; Li, X.; Zeng, S.; Chen, Y. Planning of electric vehicle charging station based on queuing theory. J. Eng.
**2017**, 13, 1867–1871. [Google Scholar] [CrossRef] - Pierre, M.; Jemelin, C.; Louvet, N. Driving an electric vehicle. A sociological analysis on pioneer users. Energy Effic.
**2011**, 4, 511. [Google Scholar] [CrossRef] - Hidrue, M.K.; Parsons, G.R.; Kempton, W.; Gardner, M.P. Willingness to pay for electric vehicles and their attributes. Resour Energy Econ.
**2011**, 33, 686–705. [Google Scholar] [CrossRef] [Green Version] - Procopiou, A.T.; Quirós-Tortós, J.; Ochoa, L.F. HPC-based probabilistic analysis of LV networks with EVs: Impacts and control. IEEE Trans. Smart Grid
**2016**, 8, 1479–1487. [Google Scholar] [CrossRef] - Kattmann, C.; Rudion, K.; Tenbohlen, S. Detailed power quality measurement of electric vehicle charging infrastructure. CIRED-Open Access Proc. J.
**2017**, 2017, 581–584. [Google Scholar] [CrossRef] [Green Version] - Hou, K.; Xu, X.; Jia, H.; Yu, X.; Jiang, T.; Zhang, K.; Shu, B. A reliability assessment approach for integrated transportation and electrical power systems incorporating electric vehicles. IEEE Trans. Smart Grid
**2016**, 9, 88–100. [Google Scholar] [CrossRef] - Gjelaj, M.; Toghroljerdi, S.H.; Andersen, P.B.; Træholt, C. Optimal Infrastructure Planning for EVs Fast Charging Stations based on Prediction of User Behavior. IET Electr. Syst. Transp.
**2019**, 10, 1–12. [Google Scholar] [CrossRef] [Green Version] - Ding, Z.; Lu, Y.; Zhang, L.; Lee, W.J.; Chen, D. A Stochastic Resource-Planning Scheme for PHEV Charging Station Considering Energy Portfolio Optimization and Price-Responsive Demand. IEEE Trans. Ind. Appl.
**2018**, 54, 5590–5598. [Google Scholar] [CrossRef] - He, F.; Wu, D.; Yin, Y.; Guan, Y. Optimal deployment of public charging stations for plug-in hybrid electric vehicles. Transport. Res. B-Meth.
**2013**, 47, 87–101. [Google Scholar] [CrossRef] - Asamer, J.; Reinthaler, M.; Ruthmair, M.; Straub, M.; Puchinger, J. Optimizing charging station locations for urban taxi providers. Transp. Res. Part Policy Pract.
**2016**, 85, 233–246. [Google Scholar] [CrossRef] [Green Version] - Frade, I.; Ribeiro, A.; Gonçalves, G.; Antunes, A.P. Optimal location of charging stations for electric vehicles in a neighborhood in Lisbon, Portugal. Transp. Res. Rec.
**2011**, 2252, 91–98. [Google Scholar] [CrossRef] [Green Version] - He, Y.; Kockelman, K.M.; Perrine, K.A. Optimal locations of US fast charging stations for long-distance trip completion by battery electric vehicles. J. Clean. Prod.
**2019**, 214, 452–461. [Google Scholar] [CrossRef] - Wang, C.; He, F.; Lin, X.; Shen, Z.J.M.; Li, M. Designing locations and capacities for charging stations to support intercity travel of electric vehicles: An expanded network approach. Transp. Res. Part Emerg. Technol.
**2019**, 102, 210–232. [Google Scholar] [CrossRef] - Davidov, S.; Pantoš, M. Planning of electric vehicle infrastructure based on charging reliability and quality of service. Energy
**2017**, 118, 1156–1167. [Google Scholar] [CrossRef] - Hu, D.; Liu, Z.W.; Chi, M. Multiple Periods Location and Capacity Optimization of Charging Stations for Electric Vehicle. In Proceedings of the 2019 China-Qatar International Workshop on Artificial Intelligence and Applications to Intelligent Manufacturing (AIAIM), Doha, Qatar, 1–4 January 2019. [Google Scholar]
- Rahman, I.; Vasant, P.M.; Singh, B.S.M.; Abdullah-Al-Wadud, M.; Adnan, N. Review of recent trends in optimization techniques for plug-in hybrid, and electric vehicle charging infrastructures. Renewable Sustainable Energy Rev.
**2016**, 58, 1039–1047. [Google Scholar] [CrossRef] - Liu, Z.F.; Zhang, W.; Ji, X.; Li, K. Optimal planning of charging station for electric vehicle based on particle swarm optimization. In Proceedings of the IEEE PES Innovative Smart Grid Technologies, Tianjin, China, 21–24 May 2012. [Google Scholar]
- Soares, J.; Sousa, T.; Morais, H.; Vale, Z.; Canizes, B.; Silva, A. Application-specific modified particle swarm optimization for energy resource scheduling considering vehicle-to-grid. Appl. Soft. Comput.
**2013**, 13, 4264–4280. [Google Scholar] [CrossRef] [Green Version] - Awasthi, A.; Venkitusamy, K.; Padmanaban, S.; Selvamuthukumaran, R.; Blaabjerg, F.; Singh, A.K. Optimal planning of electric vehicle charging station at the distribution system using hybrid optimization algorithm. Energy
**2017**, 133, 70–78. [Google Scholar] [CrossRef] - Xu, S.; Feng, D.; Yan, Z.; Zhang, L.; Li, N.; Jing, L.; Wang, J. Ant-based swarm algorithm for charging coordination of electric vehicles. Int. J. Distrib. Sens. Netw.
**2013**. [Google Scholar] [CrossRef] - Fazelpour, F.; Vafaeipour, M.; Rahbari, O.; Rosen, M.A. Intelligent optimization to integrate a plug-in hybrid electric vehicle smart parking lot with renewable energy resources and enhance grid characteristics. Energy Convers. Manag.
**2014**, 77, 250–261. [Google Scholar] [CrossRef] - Yan, X.; Duan, C.; Chen, X.; Duan, Z. Planning of Electric Vehicle charging station based on hierarchic genetic algorithm. In Proceedings of the 2014 IEEE Conference and Expo Transportation Electrification Asia-Pacific (ITEC Asia-Pacific), Beijing, China, 31 August–3 September 2014. [Google Scholar]
- Marianov, V.; Serra, D. Location–allocation of multiple-server service centers with constrained queues or waiting times. Ann. Oper. Res.
**2002**, 111, 35–50. [Google Scholar] [CrossRef] - Marianov, V.; Serra, D. Probabilistic, maximal covering location—allocation models forcongested systems. J. Reg. Sci.
**1998**, 38, 401–424. [Google Scholar] [CrossRef] - Sayarshad, H.R.; Chow, J.Y. Non-myopic relocation of idle mobility-on-demand vehicles as a dynamic location-allocation-queueing problem. Transport. Res. E-Log.
**2017**, 106, 60–77. [Google Scholar] [CrossRef] - Wolff, R.W. Stochastic Modeling and the Theory of Queues; Prentice Hall: Englewood Cliffs, NJ, USA, 1989; Volume 14. [Google Scholar]
- Electric Vehicle Charging Stations in New York; New York Government: New York, NY, USA, 2017. Available online: https://data.ny.gov/Energy-Environment/Electric-Vehicle-Charging-Stations-in-New-York/7rrd-248n (accessed on 16 June 2019).
- TomTom Statistics. June 2019. Available online: https://www.tomtom.com/engb/traffic-index/new-york-traffic (accessed on 16 June 2019).
- Tseng, C.M.; Chau, S.C.K.; Liu, X. Improving viability of electric taxis by taxi service strategy optimization: A big data study of New York city. IEEE Trans. Intell. Transp. Syst.
**2018**, 99, 1–13. [Google Scholar] [CrossRef] [Green Version] - Tseng, C.M.; Chau, C.K. Personalized prediction of vehicle energy consumption based on participatory sensing. IEEE Trans. Intell. Transp. Syst.
**2017**, 18, 3103–3113. [Google Scholar] [CrossRef] [Green Version] - Müllner, D. fastcluster: Fast Hierarchical, Agglomerative Clustering Routines for R and Python. J. Stat. Softw.
**2013**, 53, 1–18. [Google Scholar] [CrossRef] [Green Version] - R Core Team. R: A Language and Environment for Statistical Computing; R Foundation for Statistical Computing: Vienna, Austria, 2018; Available online: https://www.R-project.org/ (accessed on 12 February 2019).
- Helmus, J.; Van Den Hoed, R. Unravelling user type characteristics: Towards a taxonomy for charging infrastructure. World Electr. Veh. J.
**2015**, 7, 589–604. [Google Scholar] [CrossRef] [Green Version] - Philipsen, R.; Schmidt, T.; Van Heek, J.; Ziefle, M. Fast-charging station here, please! User criteria for electric vehicle fast-charging locations. Transp. Res. Part F Traffic Psychol. Behav.
**2016**, 40, 119–129. [Google Scholar] [CrossRef] - Bizon, N. Energy efficiency for the multiport power converters architectures of series and parallel hybrid power source type used in plug-in/V2G fuel cell vehicles. Appl. Energy
**2013**, 102, 726–734. [Google Scholar] [CrossRef] - Bizon, N. Energy Efficiency of Multiport Power Converters used in Plug-In/V2G Fuel Cell Vehicles. Appl. Energy
**2012**, 96, 431–443. [Google Scholar] [CrossRef] - Bizon, N. Nonlinear control of fuel cell hybrid power sources: Part II –Current control. Appl. Energy
**2011**, 88, 2574–2591. [Google Scholar] [CrossRef] - Bizon, N. Nonlinear control of fuel cell hybrid power sources: Part I –Voltage control. Appl. Energy
**2011**, 88, 2559–2573. [Google Scholar] [CrossRef]

**Figure 2.**Charging stations in Brooklyn, New York [48].

**Figure 4.**Location of fixed charging stations obtained through computation optimization, Brooklyn, New York.

Region | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 |
---|---|---|---|---|---|---|

Africa/Middle East | <15 | <20 | <25 | <40 | <50 | ∼50 |

Asia Pacific | ∼200 | ∼350 | ∼650 | ∼800 | ∼950 | ∼1000 |

Europe | ∼75 | ∼200 | ∼240 | ∼350 | ∼450 | ∼500 |

Latin America | <5 | <5 | <10 | <15 | <25 | <25 |

North America | ∼70 | ∼200 | ∼260 | ∼350 | ∼400 | ∼450 |

Charging Time | Type of Source (kW) | Tension (V) | Maximum Current (A) | Mode |
---|---|---|---|---|

6–9 h | mono-phase 3.3 kW | 230 V AC | 16 A | Mode 1 |

2–3 h | triple-phase 10 kW | 400 V AC | 16 A | Mode 1 |

3–4 h | mono-phase 7 kW | 230 V AC | 32 A | Mode 2 |

20–30 min | triple-phase 43 kW | 400 V AC | 63 A | Mode 3 |

20–30 min | voltage continues 50 kW | 400–500 V DC | 100–125 A | Mode 4 |

1–2 h | triple-phase 24 kW | 400 V AC | 32 A | Mode 2 |

Without Probability Constraints | b = 1, $\mathit{\beta}$ = 0.90 | a = 1, $\mathit{\alpha}$ = 0.90 | a = 1, b = 1, $\mathit{\gamma}$ = 0.90 | |
---|---|---|---|---|

Monday | ||||

Solution | 569.51 | 593.92 | 572.41 | 572.41 |

B | 2 | 2 | 2 | 2 |

SoC${}_{max}$ | 119 | 132 | 132 | 132 |

SoC${}_{min}$ | 15 | 31 | 10 | 10 |

Location | ||||

mobile charging station | 12, 15 | 5, 12 | 18, 20 | 18, 20 |

Location | ||||

fixed charging station | 28, 55, 60, 61, 64 | 22, 55, 64 | 28, 52, 60, 61 | 28, 52, 60, 61 |

Saturday | ||||

Solution | 729.5 | 760.83 | 733.39 | 733.39 |

B | 2 | 2 | 2 | 2 |

SoC${}_{max}$ | 142 | 149 | 142 | 142 |

SoC${}_{min}$ | 26 | 26 | 26 | 26 |

Location | ||||

mobile charging station | 12, 18 | 5, 12 | 18, 20 | 18, 20 |

Location | ||||

fixed charging station | 26, 52, 55, 61, 64 | 22, 49, 55, 62, 64 | 26, 52, 54, 61 | 26, 52, 54, 61 |

Without Probability Constraints | b = 1, $\mathit{\beta}$ = 0.90 | a = 1, $\mathit{\alpha}$ = 0.90 | a = 1, b = 1, $\mathit{\gamma}$ = 0.90 | |
---|---|---|---|---|

Monday | ||||

Solution | 460.38 | 482.12 | 541.62 | 541.62 |

B | 1 | 1 | 2 | 2 |

SoC${}_{max}$ | 150 | 150 | 97 | 97 |

SoC${}_{min}$ | 9 | 39 | 9 | 9 |

mean(SoC) | 52 | 77 | 52 | 52 |

Location | ||||

mobile charging station | 9 | 5 | 15, 20 | 15, 20 |

Location | ||||

fixed charging station | 48, 55, 60, 61, 64 | 51, 55, 64 | 42, 50, 60, 61 | 42, 50, 60, 61 |

Saturday | ||||

Solution | 356.57 | 368.31 | 356.98 | 356.98 |

B | 1 | 1 | 1 | 1 |

SoC${}_{max}$ | 142 | 142 | 142 | 142 |

SoC${}_{min}$ | 21 | 21 | 25 | 25 |

mean(SoC) | 55 | 55 | 73 | 73 |

Location | ||||

mobile charging station | 13 | 12 | 13 | 13 |

Location | ||||

fixed charging station | 55, 57, 59 | 49, 55, 62 | 57, 61 | 57, 61 |

Service Area | Response Time | (minutes) | Miss Ratio | Queuing Time (minutes) | Traveled Distance (km) | |
---|---|---|---|---|---|---|

Operational mode | Moving | Location | Moving | Location | Location | Moving |

${\lambda}_{1}$ = 0.00005 | ||||||

180 km${}^{2}$ | 179 | 105 | 0% | 0% | 15 | 864 |

100 km${}^{2}$ | 91 | 51 | 20% | 20% | 15 | 578 |

50 km${}^{2}$ | 37 | 34 | 0% | 0% | 15 | 192 |

${\lambda}_{2}$ = 0.0001 | ||||||

180 km${}^{2}$ | 181 | 84 | 6.25% | 6.25% | 24 | 1108 |

100 km${}^{2}$ | 74 | 54 | 0% | 0% | 17 | 533 |

50 km${}^{2}$ | 41 | 33 | 0% | 0% | 15 | 222 |

${\lambda}_{3}$ = 0.0002 | ||||||

180 km${}^{2}$ | 319 | 94 | 19% | 0% | 24 | 1276 |

100 km${}^{2}$ | 106 | 61 | 23% | 23% | 17 | 776 |

50 km${}^{2}$ | 60 | 36 | 18.75% | 18.75% | 17 | 509 |

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**MDPI and ACS Style**

Răboacă, M.-S.; Băncescu, I.; Preda, V.; Bizon, N.
An Optimization Model for the Temporary Locations of Mobile Charging Stations. *Mathematics* **2020**, *8*, 453.
https://doi.org/10.3390/math8030453

**AMA Style**

Răboacă M-S, Băncescu I, Preda V, Bizon N.
An Optimization Model for the Temporary Locations of Mobile Charging Stations. *Mathematics*. 2020; 8(3):453.
https://doi.org/10.3390/math8030453

**Chicago/Turabian Style**

Răboacă, Maria-Simona, Irina Băncescu, Vasile Preda, and Nicu Bizon.
2020. "An Optimization Model for the Temporary Locations of Mobile Charging Stations" *Mathematics* 8, no. 3: 453.
https://doi.org/10.3390/math8030453