# Stabilization Strategy of a Novel Car-Following Model with Time Delay and Memory Effect of the Driver

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## Abstract

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## 1. Introduction

- (1)
- Driver memory is considered on the basis of OVM, and it is pointed out that drivers’ memory of previous traffic will have a significant impact on subsequent follow-up behavior.
- (2)
- This study comprehensively designs control term parameters to mitigate delay-induced unstable traffic flow.
- (3)
- The control strategy of this study considers the influence of multiple time delays on the traffic flow, which is closer to the actual traffic situation.
- (4)
- The IDISM and Hopf bifurcation analysis methods are used to determine the stability interval of the system. Thus, an appropriate combination of parameters for a drivers’ memory model with time delay feedback control is determined.

## 2. A Novel Time Delayed Car-Following Model

## 3. Stability Analysis

## 4. The Design of Control Parameters in the Car-Following Model

#### 4.1. Control Strategy for Stability

#### 4.2. Design of Delay Feedback Control Strategy

## 5. Case Studies

#### 5.1. Verification of Control Strategy

#### 5.2. Parameter Calibration

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Eigenvalues of the system for $N=7$ and $\alpha ={\mathrm{s}}^{-1}$. (

**a**) the uncontrolled drivers’ memory model $\kappa =0$; (

**b**) A time delay feedback control item is added, $\kappa =0.7{\mathrm{s}}^{-1},{\tau}_{2}=0.45\mathrm{s}$. (

**a**) the eigenvalue distribution of a drivers’ memory model with no controls. (

**b**) the eigenvalue distribution of a drivers’ memory model with control.

**Figure 3.**The stability diagram of different design time delays for control (

**a**) $\kappa =0.3{\mathrm{s}}^{-1},\alpha =2{\mathrm{s}}^{-1}$; (

**b**) $\kappa =0.6{\mathrm{s}}^{-1},\alpha =2{\mathrm{s}}^{-1}$. Selecting the number of vehicles $N=0,{V}^{\prime}\left(t\right)=1.448$, the controlled driver memory model is stable when $\mathsf{\Omega}=0$, and the system is unstable when $\mathsf{\Omega}\ne 0$.

**Figure 4.**Choose $N=7,h=25\mathrm{m},\alpha =2{\mathrm{s}}^{-1}$ for the speed chart of the vehicle when $\kappa $ and ${\tau}_{1}$ change. (

**a**) ${\tau}_{1}=0.4\mathrm{s},{\tau}_{2}=0.5\mathrm{s}$. (

**b**) ${\tau}_{1}=0.4\mathrm{s},\kappa =0.6{\mathrm{s}}^{-1}$.

**Figure 5.**Two-dimensional stability diagram of a controlled drivers’ memory model with respect to ${\tau}_{2}$, $\kappa $ for $\alpha =2{\mathrm{s}}^{-1}$, $\varpi =0.6$, ${\tau}_{1}=0.5\mathrm{s}$.

**Figure 6.**Vehicle speed diagram with different time delay feedback control parameter combinations. (

**a**) $\kappa =0.1,{\tau}_{2}=0.2\mathrm{s}$ platoon speed diagram (

**b**) $\kappa =0.22,{\tau}_{2}=0.47\mathrm{s}$ platoon speed diagram (

**c**) $\kappa =0.615,{\tau}_{2}=0.2\mathrm{s}$ platoon speed diagram (

**d**) $\kappa =0.465,{\tau}_{2}=0.52\mathrm{s}$ platoon speed diagram (

**e**) $\kappa =0.345,{\tau}_{2}=0.81\mathrm{s}$ platoon speed diagram (

**f**) $\kappa =0.88,{\tau}_{2}=0.955\mathrm{s}$ platoon speed diagram.

**Figure 7.**Vehicle speed with different control strategies. (

**a**) Uncontrolled system. (

**b**) Controlled system.

**Figure 8.**Vehicle trajectory with different control strategies. (

**a**) Uncontrolled system. (

**b**) Controlled system.

**Figure 9.**Controlled drivers’ memory model stability diagram relative to ${\tau}_{{\gamma}_{2}}$ and $\kappa $ for ${\tau}_{{\gamma}_{1}}=0.5\mathrm{s},\alpha =2{\mathrm{s}}^{-1}$.

**Figure 10.**Time–speed diagram of the first vehicle with different combinations of control parameters. (

**a**) $\kappa =0.17,{\tau}_{2}=0.55\mathrm{s}$ platoon speed diagram (

**b**) $\kappa =0.22,{\tau}_{2}=0.47\mathrm{s}$ platoon speed diagram (

**c**) $\kappa =0.645,{\tau}_{2}=0.37\mathrm{s}$ platoon speed diagram (

**d**) $\kappa =0.565,{\tau}_{2}=0.635\mathrm{s}$ platoon speed diagram (

**e**) $\kappa =0.92,{\tau}_{2}=0.56\mathrm{s}$ platoon speed diagram (

**f**) $\kappa =0.785,{\tau}_{2}=0.95\mathrm{s}$ platoon speed diagram.

**Figure 11.**Vehicle trajectory with different control strategies. (

**a**) Uncontrolled system. (

**b**) Controlled system.

**Figure 12.**The hysteresis loop of the controlled driver memory model corresponds to Figure 10.

**Figure 13.**Velocity comparison diagram of controlled and uncontrolled models. (

**a**) Uncontrolled model considering driver memory. (

**b**) Controlled model considering driver memory.

**Figure 14.**Vehicle Speed Change Graph. (

**a**) Uncontrolled model considering driver memory. (

**b**) Controlled model considering driver memory.

Options | Values |
---|---|

Population Size | 100 |

Mutation Rate | 0.1 |

Crossover Rate | 0.95 |

Generations | 1000 |

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

$\alpha \left({\mathrm{s}}^{-1}\right)$ | 0.8334 |

$\varpi $ | 0.9105 |

${h}_{c}\left(\mathrm{m}\right)$ | 21.9235 |

${V}_{\mathrm{max}}\left(\mathrm{m}/\mathrm{s}\right)$ | 32.6472 |

${\gamma}_{1}$ | 0.0746 |

${\gamma}_{2}$ | 0.5983 |

${P}_{error}$ | 0.5007 |

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**MDPI and ACS Style**

Pan, Y.; Wang, Y.; Miao, B.; Cheng, R.
Stabilization Strategy of a Novel Car-Following Model with Time Delay and Memory Effect of the Driver. *Sustainability* **2022**, *14*, 7281.
https://doi.org/10.3390/su14127281

**AMA Style**

Pan Y, Wang Y, Miao B, Cheng R.
Stabilization Strategy of a Novel Car-Following Model with Time Delay and Memory Effect of the Driver. *Sustainability*. 2022; 14(12):7281.
https://doi.org/10.3390/su14127281

**Chicago/Turabian Style**

Pan, Yifan, Yongjiang Wang, Baobin Miao, and Rongjun Cheng.
2022. "Stabilization Strategy of a Novel Car-Following Model with Time Delay and Memory Effect of the Driver" *Sustainability* 14, no. 12: 7281.
https://doi.org/10.3390/su14127281