# Bi-Level Planning Model of Charging Stations Considering the Coupling Relationship between Charging Stations and Travel Route

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## Abstract

**:**

## 1. Introduction

_{2}emission [1] and one third of the energy consumption [2]. The energy shortage, coupled with the problem of environmental pollution, has brought worldwide attention to electric vehicles (EVs) which brings great environmental and social benefits with zero tailpipe emissions and a lower level of noise pollution. Many countries have already formulated relevant policies to encourage research into EVs and promote their use [3,4]

_{.}For example, countries like France, the Netherlands, and Germany have announced plans to ban the sale of fuel vehicles. Nie et al. [5] proposed a modeling framework to optimize the design of incentive policies. Kontou et al. [6] proposed a framework for minimizing the social cost of replacing gas-powered vehicles with battery electric vehicles (BEVs). Additionally, the result shows that it takes 6–12 years for 80% of conventional vehicles to be replaced with EVs. However, the proportion of EVs in all cars still remains low and the major factors that influence the popularization of EVs are the limited travel range and the lack of charging infrastructure. Surveys show that the majority of customers would expect 300 miles of travel range [7] while, for the time being, EVs can only provide a range of 100 miles or so with a full battery. In order to make up for the limited travel range, to mitigate range anxiety, and to promote the penetration rate of EVs, it is of great importance to plan and build up more fast charging stations, taking into account the faint possibility of increasing the travel range in a short time [8].

- -
- In this study, we consider the coupling relationship between EV users’ travel routes and the location of charging stations. The shortest path would be the best choice, however, when it could cannot meet the traveling demand, we search other routes passing the fast charging stations within a certain deviation so as to analyze the trip feasibility.
- -
- In order to simulate the process of users heading for charging stations to charge, we accord priority consideration to charging when the state of charge (SOC) is less than the range anxiety threshold, and then to charge on relatively sufficient battery. Following this method, we obtain the number of EVs charged at each station, which enables us to achieve a more reliable and practical sizing result.
- -
- Based on the charging times and deviation, we propose the concept of the satisfaction index. The satisfaction index can reflect the convenience that the location of the charging stations would bring to the users.
- -
- In this research, we combine slow charging in the public parking lot and fast charging in the fast charging station, and research the impact of the increase of EV ownership and coverage of slow chargers on the fast charging demand and the travel success ratio.

## 2. Uncertainty Modeling of Travel Pattern

#### 2.1. Daily Trips

_{t}, and the number of destinations is n

_{t}− 1. Based on the probability distribution of daily trips, we use the Monte Carlo simulation (MCS) to obtain the number of daily trips per vehicle.

#### 2.2. Destination of Trip

#### 2.3. Travel Distance

_{1}and σ

_{1}represents the expected value and standard deviation of the lognormal probability density function, respectively, and μ

_{1}= 3.2, σ

_{1}= 0.88.

## 3. Coupling Relationship between Charging Stations and Travel Route

#### 3.1. Impact of Charging Stations on Travel Route

_{1}and D

_{2}(nodes b and d). Through Dijkstra’s algorithm, the shortest path of the trip chain is a-b-d-e-a with a distance of 31. We assume that the EV maximum travel range is 13 which could not sustain b-d. In order to complete the trip, the user needs to charge at nodes b and e to reach D

_{2}before getting back to a. In this case, the travel route would be a-b-e-d-e-a with a distance of 33. Fast charging demands occur at node b and node e. We find that the distribution of charging stations would affect the user’ travel route. When the shortest path cannot satisfy the travel demand, the user will look for other paths. When an EV user passes by a charging station, the necessity of charge is also considered in this paper.

#### 3.2. Potential Trip Chains Considering the Location of Charging Stations

_{i}and D

_{i}

_{+1}and the shortest distance obtained via Dijkstra’s algorithm is d

_{i}

_{,i+1}. With D

_{i}and D

_{i}

_{+1}as the starting and ending points, the potential route passing the charging station C

_{k}comprises of the shortest path from D

_{i}to C

_{k}and from C

_{k}to D

_{i}

_{+1}, and the distances are d

_{i}

_{,k}and d

_{k}

_{,i+1}, respectively. In this case, the deviation to the shortest path d

_{v}is expressed as Equation (3). The definition of the deviation constraint is shown in Equation (4). If the deviation constraint is not satisfied, the route is excluded:

_{1}, C

_{2}, and C

_{3}, respectively, where the dashed lines indicate the process of driving. Figure 7 is a schematic diagram of a set of potential trip chains where the red dots indicate the charging stations passing by.

_{m}, therefore, the total number of the potential trip chains can be obtained via Equation (5), where n

_{t}is daily trips. Then, we judge the feasibility of each potential trip chain; if the trip is a success, the process will terminate, otherwise, it will continue with the next trip chain until all are analyzed:

#### 3.3. Feasibility Analysis of Trip Chains

- When d ≤ R which indicates that the traveling distance is smaller than the maximum travel range of EVs, charging is not necessary, thus the trip is a success; and
- When d > R which indicates that the traveling distance exceeds the maximum travel range of EVs, charging is necessary and we need to the judge whether the trip is a success and analyze which charging station(s) the user needs to charge at.

_{i}, which satisfies the condition ${d}_{O{D}_{i}}>R$, which means that when the EV cannot reach D

_{i}within its travel range, it shall be charged during the range of OD

_{i}. The principle of simulating the process of users heading for charging stations to charge is as follows:

- If there is no charging station found in the range of (0,R), the user cannot reach the next destination. The trip chain is deemed a failure.
- If there are any charging stations found in the range of (R’,R), we will opt for the charging station closer to the start point. As the battery runs at a lower level, the user will become anxious and choose to charge immediately.
- If there are any charging stations found in the range of (0,R’), we will choose the charging station farthest from the starting point. As the vehicle could still provide sufficient power, the charging demand is not that urgent. The time cost in the charging station would be lower for the user and she/he could charge more if she/he charges at a low-power state, thus, leading to a higher probability of a successful trip.

_{k}within the range of OD

_{i}, then charging station C

_{k}is set as the start point O’ and we continue to judge the feasibility of the remaining route; if that trip chain fails, we continue with the next one until all are judged. If all the potential trip chains cannot meet the demand of the journey, the journey is deemed as a failure, which indicates that the siting of the charging stations cannot satisfy the demand of the EV user.

_{2}, which means that the user needs fast charging in the range of OD

_{2}. Then we search for a charging station and find ${R}^{\prime}\le {d}_{O{C}_{2}}\le R$, which indicates that the user should charge the vehicle at the charging station C

_{2}. Thus, C

_{2}is set as the new starting point O’ as shown in Figure 8b. Repeat the procedures and we find that another fast charging is needed in O’D

_{3}(C

_{2}D

_{3}). Following the simulating principle, the user will charge at C

_{4}, thus, turning C

_{4}into a new starting point O’ as is shown in Figure 8c. At this moment, with ${d}_{{O}^{\prime}{D}_{3}({C}_{4}{D}_{3})}<R$, the trip chain is deemed a success.

## 4. Upper Model Based on the Travel Success Ratio

#### 4.1. Model Formulation

_{q}

_{j}is defined as the success or failure of user j; 1 for success, and 0 for failure. We consider that charging too many times will affect the travel experience. With n

_{c}as the number of fast charging times and n

_{c}

_{max}as the maximum charging times, the mathematical expression for the upper model can be formulated as follows:

_{ev}denotes the number of users whose travel range exceeds the maximum travel range of EVs. d

_{ii’}denotes the shortest distance between the adjacent charging stations. d

_{min}refers to the shortest distance between charging stations. x

_{i}represents the number of charging stations at node i. For each traffic node, up to one charging station is acceptable.

#### 4.2. Solution Method

## 5. Lower Model Based on the Satisfaction Index and Total Social Cost

#### 5.1. Determination of the Optimal Path

_{c}and the deviation d

_{v}. The route that achieves the fewest charging times and the shortest deviation is deemed as the optimal path.

#### 5.2. Sizing Model Based on the Hybrid Method

_{i}by utilizing queuing theory. A greedy algorithm is used to allocate a certain number of chargers at each station based on the relative system density of stations without a prior knowledge of the arrival rate of each station. Therefore, as is shown in Equation (12), the linear function related to the number of EVs n

_{ev,i}served at the charging station i is adopted to describe the arrival rate λ

_{i}of the charging station i. When it comes to the calculation of the capacity, n

_{ev,i}could be used to replace λ

_{i}which would reflect the system density of each charging station correctly:

_{i}of the charging station i can be obtained via Equation (13). We denote the number of chargers at the charging station i as c

_{i}, then the system density ρ

_{i}can be calculated by Equation (14):

_{c,i}is the average charging time at the charging station i which is related to the SOC of EVs when they arrive at the charging station i and it could be obtained via Equation (15). t

_{fast}is the time required for a full fast charge when the EV runs out of power. ${d}_{avc,i}^{\prime}$ denotes the average mileage that has been traveled from the previous charging station to the newly-arrived charging station i, which can be obtained via Equation (16). R is the maximum travel range of EVs. ${d}_{i,j}^{\prime}$ represents the mileage that has been traveled from the previous charging station to the newly-arrived charging station i. n

_{ev,i}is the number of EVs charged at the charging station i.

_{av}is fixed and often much greater than the number of chargers needed. The chargers are allocated according to the system density of each charging station, that is, we fix one charger at the charging station with the highest system density until all the chargers are set. The number of charging stations is recorded as p and the flowchart of sizing with the hybrid method is shown in Figure 11.

#### 5.3. Grid Constraints

_{m}is voltage at bus m. ${V}_{m}^{\mathrm{min}}$ and ${V}_{m}^{\mathrm{max}}\text{}$ are the minimum and maximum voltage level, respectively. n

_{bus}is the number of nodes in the distribution system.

_{i}is the capacity of charging station i, which is related to the number of chargers. ${S}_{i}^{\mathrm{max}}$ is the maximum allowable capacity of the charging station and p is the number of charging stations.

_{mn}is the current in the feeder between node m and node n. I

_{mn}

_{max}is the maximum allowable current in each feeder.

#### 5.4. Satisfaction Index

_{j}represents the satisfaction index of user j and it could be obtained via Equation (21). The charging times and the deviation are combined and normalized in Equation (21), and charging times has more influence on the satisfaction index. It is noted that a smaller satisfaction index indicates a better planning scheme:

_{v,j}denotes the deviation of the user j and λR is the maximum deviation. n

_{c,j}is the charging times of user j and n

_{c}

_{max}is the maximum charging times.

#### 5.5. Total Social Cost

_{init,i}represents the fixed investment cost for charging station i. We classify the charging stations according to the number of fast chargers for each charging station and the fixed investment cost is greatly depending on the level of the charging station [35]. s denotes the area occupied by a fast charger including the parking area. C

_{lan,i}represents the land rental cost for five years, which is related to the type of the area. c

_{i}is the number of chargers at the charging station i. P

_{ch}is the charger rated power while C

_{con}refers to the construction cost of the fast charger. c

_{w,i}is the time cost per hour for each user which is related to the location of the charging station i. t

_{c,i}is the average charging time at the charging station i. T

_{fy}is the days in five years. n

_{ev,i}is the number of EVs charged at the charging station i. p is the number of charging stations.

_{av}is much greater than the number of chargers needed in the sizing model, we assume that users can recharge immediately at charging stations and the calculation of C

_{2}is simplified so that the average charging time is assumed as the waiting time. The total social cost C is utilized as an indicator to evaluate the optimal solutions, which equals the sum of the construction and investment costs of the charging stations and the waiting cost of the users.

## 6. Feasibility Analysis Considering Slow Charging at Destinations

_{slow}is the slow charger rated power. t

_{slow}is the slow charging time and μ

_{3}denotes the expected slow charging time. σ

_{3}is the standard deviation, μ

_{3}= 3.5, σ

_{3}= 0.8.

_{slow,i}and the added travel range d

_{ei}is shown in Equation (25). The maximum travel range of EVs will be updated according to Equation (26) and we continue with the trip feasibility analysis described in Section 3.3.

_{oi}indicates the mileage that has been traveled from the previous charging station to the newly arrived destination. t

_{slow,i}P

_{slow,i}represents the energy recharged at destination i through slow charging and Equation (25) turns the energy recharged into the added driving range.

_{2}is t

_{slow}. The added travel range d

_{e}after charging and the maximum travel range R could be obtained via Equations (25) and (26). With the optimization method and analysis approach proposed in Section 3, we know that the user could successfully reach destination D

_{3}and the trip would be a success. While in the situation Figure 12a without considering slow charging, the user cannot reach destination D

_{3}directly and would have to charge at charging station C

_{4}to continue with the journey.

## 7. Case Study

#### 7.1. Planning Area

^{2}, as shown in Figure 13. The planning area comprises of three parts: commercial area, industrial area and residential area, and includes 81 traffic nodes. For the commercial area, 20 nodes; the industrial area, 31 nodes; and 30 nodes for the residential area. It would be better to set the charging stations around the traffic nodes than on the road [22]. Therefore, in this study, all the traffic nodes are the candidate locations of the charging stations. For the actual construction, the location could be adjusted around the traffic node. The IEEE 123-node distribution test system is used as the corresponding power grid, which could be found in [45]. The maximum allowable voltage drop is 7% and over-voltage up to 5%.

#### 7.2. Simulation Results Considering Only Fast Charging

#### 7.3. Simulation Results Using FCLM

#### 7.4. Simulation Results Considering Slow Charging at Destinations

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 8.**The feasibility analysis of the trip chain; (

**a**) the first analysis; (

**b**) the second analysis; (

**c**) the third analysis.

**Figure 12.**The feasibility analysis of the trip chain considering slow charge; (

**a**) analyze the feasibility of the trip chain without considering slow charge; (

**b**) analyze the feasibility of the trip chain considering slow charge.

**Figure 19.**The traffic flow of each traffic node before and after the planning of the fast charging stations; (

**a**) the traffic flow before the planning of the fast charging stations; (

**b**) the traffic flow after the planning of the fast charging stations

**Table 1.**The proportion of tour types; H-H: from the residential area to the residential area; H-O: from the residential area to the commercial area; H-W: from the residential area to the industrial area; O-H: from the commercial area to the residential area; O-O: from the commercial area to the commercial area; O-W: from the commercial area to the industrial area; W-H: from the industrial area to the residential area; W-O: from the industrial area to the commercial area; W-W: from the industrial area to the industrial area.

Tour Type | H-H | H-O | H-W | O-H | O-O | O-W | W-H | W-O | W-W |
---|---|---|---|---|---|---|---|---|---|

Proportion/% | 11.80 | 25.93 | 10.08 | 26.58 | 11.27 | 1.53 | 8.89 | 2.62 | 1.30 |

Parameter | Value | Unit |
---|---|---|

p | 4 | - |

n_{cmax} | 2 | - |

P_{ev} | 15 [25] | kWh |

P_{slow} | 3.5 [23] | kW |

ω | 15 [23] | kWh/100 km |

η | 0.9 [46] | - |

R | 100 | km |

λ | 0.1 | - |

d_{min} | 10 | km |

P_{ch} | 96 [47] | kW |

s | 30 [48] | m^{2} |

C_{av} | 100 | - |

C_{con} | 208.33 [17] | $/kW |

Station Level | Fixed Investment Cost/×10^{3} $ | Minimum Number of Chargers |
---|---|---|

1 | 1061 | 45 |

2 | 800 | 30 |

3 | 477 | 15 |

4 | 323 | 8 |

Area Type | Residential Area | Industrial Area | Commercial Area |
---|---|---|---|

Land rental cost ($/m^{2}) [40] | 330 | 109 | 1070 |

Time cost ($/h) | 5.76 [49] | 5.25 | 3.82 |

Feasible Solutions | Siting Results | Number of Chargers | Rank of Charging Stations |
---|---|---|---|

1 | 25, 33, 41, 57 | 29, 23, 24, 24 | 3, 3, 3, 3 |

2 | 25, 42, 41, 57 | 34, 18, 22, 26 | 2, 3, 3, 3 |

3 | 48, 25, 58, 70 | 36, 32, 24, 8 | 2, 2, 3, 4 |

4 | 48, 25, 58, 69 | 36, 32, 24, 8 | 2, 2, 3, 4 |

5 | 48, 24, 35, 58 | 39, 29, 13, 19 | 2, 3, 4, 3 |

6 | 48, 24, 71, 58 | 35, 35, 9, 21 | 2, 2, 4, 3 |

7 | 48, 25, 69, 58 | 38, 34, 8, 20 | 2, 2, 4, 3 |

8 | 41, 57, 35, 39 | 37, 32, 23, 8 | 2, 2, 3, 4 |

9 | 25, 50, 42, 65 | 38, 26, 13, 23 | 2, 3, 4, 3 |

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Zang, H.; Fu, Y.; Chen, M.; Shen, H.; Miao, L.; Zhang, S.; Wei, Z.; Sun, G.
Bi-Level Planning Model of Charging Stations Considering the Coupling Relationship between Charging Stations and Travel Route. *Appl. Sci.* **2018**, *8*, 1130.
https://doi.org/10.3390/app8071130

**AMA Style**

Zang H, Fu Y, Chen M, Shen H, Miao L, Zhang S, Wei Z, Sun G.
Bi-Level Planning Model of Charging Stations Considering the Coupling Relationship between Charging Stations and Travel Route. *Applied Sciences*. 2018; 8(7):1130.
https://doi.org/10.3390/app8071130

**Chicago/Turabian Style**

Zang, Haixiang, Yuting Fu, Ming Chen, Haiping Shen, Liheng Miao, Side Zhang, Zhinong Wei, and Guoqiang Sun.
2018. "Bi-Level Planning Model of Charging Stations Considering the Coupling Relationship between Charging Stations and Travel Route" *Applied Sciences* 8, no. 7: 1130.
https://doi.org/10.3390/app8071130