# Multistage and Dynamic Layout Optimization for Electric Vehicle Charging Stations Based on the Behavior Analysis of Travelers

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## Abstract

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## 1. Introduction

- First, multistage CS layout optimization is studied comprehensively with the dynamic EV charging demands of one day. The proposed method cannot only optimize the layout of CSs considering the numbers of EVs at different stages but also ensure the service quality by satisfying the charging demand at different time intervals in a day.
- Second, we designed a two-layer model to avoid the non-linear characteristics caused by the queuing phenomenon, which can improve the computational speed and consider the important queuing phenomenon during EV charging in a large road network.

## 2. Charging Demand Distribution

#### 2.1. Problem Description

#### 2.2. Travel Behavior Analysis

#### 2.3. Charging Behavior Analysis

#### 2.4. Monte Carlo Method

## 3. Mathematical Modeling

#### 3.1. Mixed Integer Linear Programming (MILP) Model

#### 3.1.1. Total Social Cost

#### 3.1.2. Location Constraints

#### 3.1.3. Capacity Constraints

#### 3.1.4. Other Constraints

#### 3.2. M/M/S/K Model of Queuing Theory

## 4. Case Study

#### 4.1. Simulation Results of Charging Demand Distribution

#### 4.2. Layout Optimization Results

#### 4.2.1. Location Optimization Results

#### 4.2.2. Capacity Optimization Results

#### 4.2.3. One-Time Strategy vs. Multistage Strategy

#### 4.3. Sensitivity Analysis

#### 4.3.1. Sensitivity Analysis and the Selection of $\beta $

#### 4.3.2. Sensitivity Analysis of the Relocation Cost

#### 4.3.3. Sensitivity Analysis of Road Network Scales

## 5. Conclusions and Future Work

- The final stage with a relatively large number of EV trajectories has a great influence on the optimization results with both multistage strategy and one-time strategy. TSC with multistage optimization strategy can drop 8.79% from that with one-time optimization strategy in the case study, which can provide some meaningful suggestions for the long-term planning of EV CSs.
- Charging service quality, relocation cost, and road network scales have a significant impact on the optimization results. Surplus supply of charging demand for the charging peak is necessary to decrease the charging demand loss caused by uneven EV arrivals.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Abbreviations | |

CCSs | candidate charging stations |

CSs | charging stations |

EV | electric vehicles |

FCPs | fast charging piles |

MILP | mixed integer linear programming |

PWL | piecewise linear method |

SOC | state of charge |

TSC | total social cost |

UFZ | urban functional zone |

Indices | |

$r$ | trajectory from 1 to $R$ |

$s$ | point on the trajectory from 1 to ${S}_{r}$ |

$i$ | demand point from 1 to $N$ |

$t$ | time interval from 1 to $T$ |

$j$ | candidate charging station from 1 to $M$ |

$k$ | planning stage from 1 to $K$ |

Parameters | |

$R$ | total number of trajectories |

${S}_{r}$ | total number of points on trajectory $r$ |

$N$ | total number of demand points |

$T$ | total number of time intervals |

$M$ | total number of candidate charging stations |

$K$ | total number of planning stages |

${l}_{s}^{r}$ | location of point $s$ on trajectory $r$ |

${t}_{s}^{r}$ | time of point $s$ on trajectory $r$ |

$SO{C}_{s}^{r}$ | $SOC$ of point $s$ on trajectory $r$ |

${P}_{s}^{r}$ | charging probability of point $s$ on trajectory $r$ |

${d}_{s}^{r}$ | shortest path distance from point $s$ to the departure point on trajectory $r$ |

${\delta}_{s}^{r}$ | auxiliary probability of charging probability |

$V$ | average EV speed |

$\gamma $ | driving range per kilowatt-hour |

${C}_{m}$ | rated capacity of EV battery |

$\overline{SOC}$$/\underset{\_}{SOC}$ | upper/lower limit of charging |

$\alpha $ | control coefficient between the detour cost and cost related to CSs |

$\beta $ | charging satisfaction coefficient |

${F}_{i,j,t}$ | monetary cost from $i$ to $j$ in time interval $t$ for an EV |

$F$ | average hourly wage |

${d}_{i,j}$ | shortest path distance from $i$ to $j$ |

${V}_{i,j,t}$ | EV speed from $i$ to $j$ in time interval $t$ |

${J}_{EV,k}$ | detour cost of EV users at stage $k$ |

${J}_{CS,k}$ | cost related to CSs at stage $k$ |

${J}_{CS,j,k}^{cs}$ | construction and relocation cost of CS $j$ at stage $k$ |

${J}_{CS,j,k}^{cp}$ | installation and relocation cost of FCPs in CS $j$ at stage $k$ |

${J}_{CS,j,k}^{cs,r}$$/{J}_{CS,j,k}^{cp,r}$ | operating cost of CS $j$/maintenance cost of FCPs in CS $j$ |

${C}_{c}^{cs}$$/{C}_{o}^{cs}$$/{C}_{r}^{cs}$ | construction/operating/relocation cost of a CS |

${C}_{c}^{cp}$$/{C}_{o}^{cp}$$/{C}_{r}^{cp}$ | installation/maintenance/relocation cost of a FCP |

${P}_{c}$ | charging rated power of FCPs |

$\Delta T$ | a time interval |

${G}_{j,k}$ | upper limit of the number of FCPs caused by the land and the distribution grid |

Variables | |

${h}_{s,i,t}^{r}$ | =1 represents the point $s$ on trajectory $r$ is at demand point $i$ in time interval $t$; 0, otherwise |

${W}_{i,t}$ | traffic flow at demand point $i$ in time interval $t$ |

${D}_{i,t}$ | charging demand at demand point $i$ in time interval $t$ |

${x}_{i,j,k}$ | =1 represents EVs at demand point $i$ choose CCS $j$ for charging; 0, otherwise |

${y}_{j,k}$ | =1 represents CCS $j$ is open at stage $k$; 0, otherwise |

${z}_{j,k}$ | number of FCPs installed in CCS $j$ at stage $k$ |

## Appendix A

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**Figure 1.**The topology of Sioux Falls road network with 24 nodes and 38 edges. Each node (24 in total) represents the road crossing, which corresponds to an urban functional zone (UFZ), and each edge (38 in total), with a number indicating the road length (km), illustrates the bidirectional road. The numbers on the nodes represent the ID of road crossings.

**Figure 2.**Flow chart of simulating spatial-temporal distributions of traffic flow and charging demand based on the Monte Carlo method.

**Figure 3.**A two-layer layout optimization model of EV charging stations. TSC-$\eta $-$\beta $ relationship can be acquired by changing the presumed parameter $\beta $ in the MILP model in different loops, which can help determine the desirable value of $\beta $ and corresponding optimal layout results.

**Figure 4.**The spatial-temporal distribution of traffic flow and charging demand with 10,000 trajectories in a day. The data are the average of the demand points with the same kind of UFZs. It can be found that the traffic flow and charging demand fluctuate in one day. (

**a**) The distribution of traffic flow; (

**b**) the distribution of charging demand.

**Figure 5.**The locations of optimal CSs at different stages based on the Sioux Falls road network. The chosen CSs at each stage are labeled by the number of FCPs in them. The size of each circular node represents the relative amount of EV charging demand at each demand point in a day. EVs at the demand points with the same color will go to the CS in the corresponding color area for charging. (

**a**) Stage 1; (

**b**) Stage 2; (

**c**) Stage 3; (

**d**) Stage 4; (

**e**) Stage 5; (

**f**) layout optimization results with a one-time strategy.

**Figure 7.**Optimal results with different charging satisfaction coefficient β. Maximum here stands for the maximum charging demand loss rate for a CS in different time intervals on a day, and average is for all CSs at all stages.

Parameters | Symbols | Value |
---|---|---|

Construction cost of a new CS | ${C}_{c}^{cs}$ | USD 163,000 |

Installation cost of a FCP | ${C}_{c}^{cp}$ | USD 23,500 |

Relocation cost of a CS ^{1} | ${C}_{r}^{cs}$ | USD 26,000 |

Relocation cost of a FCP | ${C}_{r}^{cp}$ | USD 500 |

Average hourly wage | $F$ | USD 8.2 |

EV battery capacity ^{2} | ${C}_{m}$ | 40 kW·h |

Driving range per kilowatt-hour ^{2} | $\gamma $ | 6.075 km/kW·h |

Rated charging power of FPCs ^{2} | ${P}_{c}$ | 80 kW |

Upper/lower limit of charging | $\overline{SOC}$$,\underset{\xaf}{SOC}$ | 0.80, 0.20 |

**Table 2.**Comparison of the optimization results of the one-time optimization strategy and the multistage optimization strategy.

Strategy | Total Number of CSs | Total Number of FCPs | Detour Cost (M$) | Cost Related to CSs (M$) | Total Social Cost (M$) | η |
---|---|---|---|---|---|---|

One-time | 8 | 77 | 1.9899 | 4.6703 | 6.6602 | 8.097% |

Multistage | 8 | 77 | 2.0456 | 4.0295 | 6.0751 | 0.012% |

Variation | 0 | 0 | +2.80% | −13.72% | −8.79% | +8.085% |

**Table 3.**Computational efficiency of the MILP model for different network scales. All numerical examples have the same parameter settings as the case based on Sioux Falls.

Name | Number of Nodes | Number of Lines | Average Road Length (km) | Total Social Cost (M$) | Optimization Time (s) | Gap |
---|---|---|---|---|---|---|

Sioux Falls ^{1} | 24 | 38 | 4.13 | 6.0751 | 1.20 | 0 |

Mumford 0 ^{2} | 30 | 90 | 4.47 | 5.8642 | 1.44 | 0 |

Mumford 1 ^{2} | 70 | 210 | 4.58 | 7.7376 | 34.22 | 0 |

Mumford 2 ^{2} | 110 | 385 | 4.70 | 8.5771 | 164.01 | 0 |

Mumford 3 ^{2} | 127 | 425 | 4.62 | 8.8478 | 128.29 | 0 |

Berlin Friedrichshain ^{1} | 224 | 523 | 4.60 | 11.217 | 138.28 | <0.5% |

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**MDPI and ACS Style**

Chen, F.; Feng, M.; Han, B.; Lu, S.
Multistage and Dynamic Layout Optimization for Electric Vehicle Charging Stations Based on the Behavior Analysis of Travelers. *World Electr. Veh. J.* **2021**, *12*, 243.
https://doi.org/10.3390/wevj12040243

**AMA Style**

Chen F, Feng M, Han B, Lu S.
Multistage and Dynamic Layout Optimization for Electric Vehicle Charging Stations Based on the Behavior Analysis of Travelers. *World Electric Vehicle Journal*. 2021; 12(4):243.
https://doi.org/10.3390/wevj12040243

**Chicago/Turabian Style**

Chen, Feng, Minling Feng, Bing Han, and Shaofeng Lu.
2021. "Multistage and Dynamic Layout Optimization for Electric Vehicle Charging Stations Based on the Behavior Analysis of Travelers" *World Electric Vehicle Journal* 12, no. 4: 243.
https://doi.org/10.3390/wevj12040243