# Modeling HDV and CAV Mixed Traffic Flow on a Foggy Two-Lane Highway with Cellular Automata and Game Theory Model

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## Abstract

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## 1. Introduction

- To the best of our knowledge, this study is among the first to address a new problem of modeling mixed traffic flow in foggy weather. Foggy weather would affect the visibility and make the mixed traffic flow more complicated. Thus, we aim to model the mixed traffic flow and analyze its characteristics under different visibility levels.
- We propose four different car-following models within a two-lane highway scenario for mixed traffic flow by considering the limited visibility in foggy weather.
- We develop a game-theoretical approach to lane-changing policies for CAVs, considering the interaction between CAVs and surrounding vehicles (HDV/CAV). Compared with the traditional lane-changing rules, such as the lane-changing rule in STCA, the proposed lane-changing strategy could increase the speed of the mixed traffic flow.

## 2. Literature Review

#### 2.1. Driving Behavior in Foggy Conditions

#### 2.2. Mixed Traffic Flow Models in Highway

#### 2.3. Cellular Automata Model in Traffic Simulation

## 3. Problem Formulation

#### 3.1. The Car-Following Rules

#### 3.1.1. HDV–HDV/CAV

- Acceleration.
- (1)
- When $\Delta {x}_{n}\left(t\right)>{d}_{safe}$, and the preceding vehicle is in the scope of visibility, which means ${d}_{v}>\Delta {x}_{n}\left(t\right)$, if ${v}_{n}\left(t\right)<\Delta {x}_{n}\left(t\right)/t$, then the HDV will accelerate with the probability ${p}_{accelerate}$ according to the following acceleration rule:$${v}_{n}\left(t+1\right)=min\left({v}_{n}\left(t\right)+1,{v}_{max},\Delta {x}_{n}\left(t\right)/t\right)$$
- (2)
- When $\Delta {x}_{n}\left(t\right)>{d}_{safe}$, and the preceding vehicle is not in the scope of visibility, which means ${d}_{v}<\Delta {x}_{n}\left(t\right)$, then the HDV will adjust its speed within the visible distance. The acceleration rule is as follows:$${v}_{n}\left(t+1\right)=min\left({v}_{n}\left(t\right)+1,{d}_{v}/t\right)$$

- Deceleration.
- (1)
- When $\Delta {x}_{n}\left(t\right)<{d}_{safe}$, if ${d}_{v}>\Delta {x}_{n}\left(t\right)$, the HDV will decelerate to ensure safety:$${v}_{n}\left(t+1\right)=min\left({v}_{n}\left(t\right)-1,\Delta {x}_{n}\left(t\right)/t\right)$$
- (2)
- When $\Delta {x}_{n}\left(t\right)<{d}_{safe}$, if ${d}_{v}<\Delta {x}_{n}\left(t\right)$, the HDV cannot judge the distance to the preceding vehicle. In that case, the HDV can adjust its speed within its visible distance to ensure traffic safety; the speed of the vehicle at the next timestep is defined as$${v}_{n}\left(t+1\right)=min\left({v}_{n}\left(t\right)-1,{d}_{v}/t\right)$$

- Randomization.HDV slows down randomly according to Equation (5) with the probability ${p}_{slowdown}$.$${v}_{n}\left(t+1\right)=max\left({v}_{n}\left(t\right)-1,0\right)$$
- Update position.The state of the HDV in this situation from time $t$ to $t+1$ can be described as$${x}_{n}\left(t+1\right)={x}_{n}\left(t\right)+{v}_{n}\left(t\right)$$

#### 3.1.2. CAV–CAV

#### 3.1.3. CAV-HDV

- Acceleration.If $\Delta {x}_{n}>{d}_{n,safe}$$${v}_{n}\left(t+1\right)=min\left({v}_{n}\left(t\right)+1,{v}_{max},\Delta {x}_{n}\left(t\right)/t\right)$$
- Deceleration.If $\Delta {x}_{n}\le {d}_{n,safe}$$${v}_{n}\left(t+1\right)=min\left({v}_{n}\left(t\right)-1,{d}_{n,safe}\right)$$

#### 3.2. The Lane-Changing Rules

#### 3.2.1. The Lane-Changing Rules for HDVs

#### 3.2.2. The Lane-Changing Rules for CAVs

## 4. Numerical and Simulation Experiments

#### 4.1. Scenarios and Setting

#### 4.2. Simulation Results

#### 4.3. Comparison with Other Work

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**Speed-density diagrams under different CAV market penetration $p$. (

**a**) dense fog condition (${d}_{v}$ = 50 m), (

**b**) medium fog condition (${d}_{v}$ = 100 m), (

**c**) no fog condition (${d}_{v}$ = 500 m).

**Figure 5.**Flow-density diagrams under different CAV market penetration $p$. (

**a**) Dense fog condition (${d}_{v}$ = 50 m), (

**b**) medium fog condition (${d}_{v}$ = 100 m), (

**c**) no fog condition (${d}_{v}$ = 500 m).

**Figure 6.**Flow–density diagrams under different visibility levels. (

**a**) $p=0\%$, (

**b**) $p=20\%$, (

**c**) $p=40\%$, (

**d**) $p=60\%$, (

**e**) $p=80\%$, (

**f**) $p=100\%$.

Notion | Explanation |
---|---|

${d}_{front}$ | the space of the gap between the subject vehicle and the preceding vehicle in the target lane |

${d}_{back}$ | the space of the gap between the subject vehicle and the succeeding vehicle in the target lane |

${d}_{safe}$ | the space of the gap that needs to be satisfied to be safe |

${d}_{v}$ | the visible distance in foggy environment |

$L$ | the length of the single lane |

$N$ | the total number of vehicles |

$p$ | the CAV proportion in traffic flow |

${p}_{change}$ | the lane-changing probability |

${p}_{accelerate}$ | the acceleration probability |

${p}_{slowdown}$ | the randomization deceleration probability |

$Q$ | the traffic flow |

$t$ | the timestep in the simulation |

$T$ | the simulation time horizon |

${v}_{n}$ | $\mathrm{the}\text{}\mathrm{speed}\text{}\mathrm{of}\text{}\mathrm{the}\text{}\mathrm{subject}\text{}\mathrm{vehicle}\text{}n$ |

${v}_{max}$ | the maximum speed |

${v}_{min}$ | the minimum speed |

$\overline{v}$ | the average speed |

${x}_{n}$ | $\mathrm{the}\text{}\mathrm{position}\text{}\mathrm{of}\text{}\mathrm{the}\text{}\mathrm{subject}\text{}\mathrm{vehicle}\text{}n$ |

${x}_{n+1}$ | $\mathrm{the}\text{}\mathrm{position}\text{}\mathrm{of}\text{}\mathrm{the}\text{}\mathrm{preceding}\text{}\mathrm{vehicle}\text{}n+1$ |

$\Delta {x}_{n}$ | the space gap between the subject vehicle and the preceding vehicle in the current lane |

$\rho $ | the vehicle density |

Decision Making | FV. | ||
---|---|---|---|

Accelerate (n) | Decelerate (1 − n) | ||

SV. | $\mathrm{Change}\text{}\mathrm{lanes}\text{}\left(m\right)$ | $({P}_{11},{Q}_{11}$) | $({P}_{12},{Q}_{12}$) |

$\mathrm{Do}\text{}\mathrm{not}\text{}\mathrm{change}\text{}\mathrm{lanes}\text{}(1-m)$ | $({P}_{21},{Q}_{21}$) | $({P}_{22},{Q}_{22}$) |

Parameter | Scenario 1 | Scenario 2 | Scenario 3 |
---|---|---|---|

${d}_{v}\left(\mathrm{m}\right)$ | 50 | 100 | 500 |

${v}_{max}\left(\mathrm{km}/\mathrm{h}\right)$ | 40 | 60 | 100 |

${p}_{change-HDV}$ | 0.2 | 0.3 | 0.4 |

${p}_{slowdown}$ | 0.4 | 0.2 | 0.1 |

$\mathit{\rho}$ | $\mathit{p}$ | ${\mathit{d}}_{\mathit{v}}=50\mathbf{m}$ | ${\mathit{d}}_{\mathit{v}}=500\mathbf{m}$ | ||||
---|---|---|---|---|---|---|---|

$\overline{\mathit{v}}$ in STCA- | $\overline{\mathit{v}}$ in GT-Based | Increase | $\overline{\mathit{v}}$ in STCA- | $\overline{\mathit{v}}$ in GT-Based | Increase | ||

(veh/km/h) | LC (km/h) | LC (km/h) | (%) | LC (km/h) | LC (km/h) | (%) | |

$\rho =15\text{}$ | 0% | 29.61 | 31.94 | 7.87 | 71.26 | 73.67 | 3.38 |

20% | 30.97 | 33.49 | 8.13 | 72.09 | 75.85 | 5.22 | |

40% | 31.01 | 34.62 | 8.15 | 79.77 | 80.17 | 5.01 | |

60% | 32.49 | 35.44 | 8.4 | 81.89 | 85.91 | 4.91 | |

80% | 34.03 | 36.76 | 8.02 | 85.09 | 89.99 | 5.76 | |

100% | 35.18 | 37.77 | 7.36 | 87.66 | 92.91 | 5.99 | |

$\rho =75$ | 0% | 4.36 | 7.2 | 65.14 | 11.44 | 21.42 | 87.23 |

20% | 5.29 | 8.91 | 68.43 | 13.39 | 25.86 | 93.13 | |

40% | 6.13 | 10.81 | 76.35 | 16.15 | 30.15 | 86.69 | |

60% | 7.28 | 12.93 | 77.61 | 17.74 | 34.73 | 95.77 | |

80% | 8.49 | 15.26 | 79.74 | 20.73 | 38.2 | 84.27 | |

100% | 9.36 | 17.07 | 82.37 | 24.53 | 42.86 | 74.72 | |

$\rho =135$ | 0% | 1.42 | 1.52 | 7.04 | 3.16 | 4.97 | 57.28 |

20% | 1.85 | 2.12 | 14.59 | 4.47 | 6.68 | 49.44 | |

40% | 2.47 | 2.92 | 18.21 | 4.92 | 9.08 | 84.55 | |

60% | 3.35 | 3.92 | 17.01 | 6.29 | 11.79 | 87.44 | |

80% | 4.56 | 5.32 | 16.67 | 7.42 | 13.62 | 83.56 | |

100% | 5.49 | 6.57 | 19.67 | 8.03 | 15.59 | 94.15 |

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**MDPI and ACS Style**

Gong, B.; Wang, F.; Lin, C.; Wu, D.
Modeling HDV and CAV Mixed Traffic Flow on a Foggy Two-Lane Highway with Cellular Automata and Game Theory Model. *Sustainability* **2022**, *14*, 5899.
https://doi.org/10.3390/su14105899

**AMA Style**

Gong B, Wang F, Lin C, Wu D.
Modeling HDV and CAV Mixed Traffic Flow on a Foggy Two-Lane Highway with Cellular Automata and Game Theory Model. *Sustainability*. 2022; 14(10):5899.
https://doi.org/10.3390/su14105899

**Chicago/Turabian Style**

Gong, Bowen, Fanting Wang, Ciyun Lin, and Dayong Wu.
2022. "Modeling HDV and CAV Mixed Traffic Flow on a Foggy Two-Lane Highway with Cellular Automata and Game Theory Model" *Sustainability* 14, no. 10: 5899.
https://doi.org/10.3390/su14105899