# A Distributed and Hierarchical Optimal Control Method for Intelligent Connected Vehicles in Multi-Intersection Road Networks

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## Abstract

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## 1. Introduction

## 2. Distributed and Hierarchical Optimal Control Architecture

## 3. Methodology

#### 3.1. Controller Design for Multi-Intersection Road Networks at the Cloud Decision Layer

- Speed constraints: $0\le {V}_{i}{}^{p}(k|t)\le {V}_{\mathrm{max}}$
- Traffic flow density constraints: $0\le {p}_{i}{}^{p}(k|t)\le {P}_{\mathrm{max}}$
- where $k=0,1,2,\dots ,{N}_{p}$,${N}_{p}$ is the prediction step, ${V}_{\mathrm{max}}$ represents the maximum road speed, and ${P}_{\mathrm{max}}$ represents the maximum road density.

#### 3.2. Controller Design for ICVs at the Vehicle Control Layer

- Speed constraints: ${v}_{\mathrm{min}}\le {v}_{s}{}^{p}(k|t)\le {v}_{\mathrm{max}}$,
- Acceleration constraints: ${a}_{\mathrm{min}}\le \Delta {v}_{s}{}^{p}(k|t)\le {a}_{\mathrm{max}}$,
- Torque constraints: ${T}_{\mathrm{min}}\le {u}_{i}(k|t)\le {T}_{\mathrm{max}}$,
- Speed terminal constraint: ${v}_{1}{}^{p}({N}_{p}|t)={v}_{i}{}^{*}({N}_{p}|t)$,
- Position terminal constraint: ${S}_{1}{}^{p}({N}_{p}|t)={s}_{i}{}^{*}({N}_{p}|t)$,
- Torque terminal constraint: ${T}_{q,1}{}^{p}({N}_{p}|t)={h}_{1}({v}_{i}{}^{*}({N}_{p}|t))$,
- where ${v}_{\mathrm{min}}$ and ${v}_{\mathrm{max}}$, respectively, represent the minimum and maximum speed of the vehicles; ${a}_{\mathrm{min}}$ and ${a}_{\mathrm{max}}$ respectively, represent the minimum and maximum acceleration of the vehicles; ${T}_{\mathrm{min}}$ and ${T}_{\mathrm{max}}$, respectively, represent the minimum and maximum torque of the vehicles.

- Speed constraints: ${v}_{\mathrm{min}}\le {v}_{s}{}^{p}(k|t)\le {v}_{\mathrm{max}}$
- Acceleration constraints: ${a}_{\mathrm{min}}\le \Delta {v}_{s}{}^{p}(k|t)\le {a}_{\mathrm{max}}$
- Torque constraints: ${T}_{\mathrm{min}}\le {u}_{i}(k|t)\le {T}_{\mathrm{max}}$
- Speed terminal constraint: ${v}_{s}{}^{p}({N}_{p}|t)={v}_{js}{}^{a}({N}_{p}|t)$
- Position terminal constraint with the preceding vehicle: ${S}_{s}{}^{p}({N}_{p}|t)={S}_{1}{}^{a}({v}_{1}{}^{a}({N}_{p}|t))-d$
- Position terminal constraint with the pilot vehicle: ${S}_{s}{}^{p}({N}_{p}|t)={S}_{js}{}^{a}({N}_{p}|t)-(s-1){d}_{des}$
- Torque terminal constraint: ${T}_{q,s}{}^{p}({N}_{p}|t)={h}_{s}({v}_{js}^{a}({N}_{p}|t))$

## 4. Numerical Experiments

#### 4.1. Scenario Settings and Parameters

#### 4.2. Results

#### 4.2.1. Comparison Tests of Traffic Efficiency and Energy-Saving under Different Traffic Flows

#### 4.2.2. Comparison Tests of Traffic Efficiency and Energy Saving under Sensitivity Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 7.**Comparison results of three methods under different flows: (

**a**) comparison results of delay; (

**b**) comparison results of consumption.

**Figure 8.**Comparison results of three methods at intersection subregions under different flows: (

**a**) test results in the intersection subregion N5; (

**b**) test results in the intersection subregion N6; (

**c**) test results in the intersection subregion N7; (

**d**) test results in the intersection subregion N8.

**Figure 9.**Test results with two methods under different traffic flow conditions in multi-intersection road networks: (

**a**) comparison results of traffic delay; (

**b**) comparison results of fuel consumption.

**Figure 10.**Test results with two methods under different traffic flow conditions in the intersection subregions: (

**a**) test results in the intersection subregion N5; (

**b**) test results in the intersection subregion N6, (

**c**) Test results in the intersection subregion N7, (

**d**) Test results in the intersection subregion N8.

Physical Meaning/Variables | Numerical Value/Unit |
---|---|

$\mathrm{Vehicle}\mathrm{mass}/m$ | 1700/kg |

$\mathrm{Maximum}\mathrm{road}\mathrm{density}/{P}_{\mathrm{max}}$ | 110/veh/lane/km |

$\mathrm{Maximum}\mathrm{road}\mathrm{velocity}/{V}_{\mathrm{max}}$ | 70/km/h |

$\mathrm{Free}\text{-}\mathrm{flow}\mathrm{velocity}/{V}_{free}$ | 50/km/h |

$\mathrm{Maximum}\mathrm{speed}/{v}_{\mathrm{max}}$ | 70/km/h |

$\mathrm{Minimum}\mathrm{speed}/{v}_{\mathrm{min}}$ | 20/km/h |

$\mathrm{Maximum}\mathrm{acceleration}/{a}_{\mathrm{max}}$ | 2/m/s^{2} |

$\mathrm{Minimum}\mathrm{acceleration}/{a}_{\mathrm{min}}$ | −2/m/s^{2} |

$\mathrm{Maximum}\mathrm{Torque}/{T}_{\mathrm{max}}$ | 600/N·m |

$\mathrm{Minimum}\mathrm{Torque}/{T}_{\mathrm{min}}$ | −600/N·m |

$\mathrm{Time}\mathrm{headway}/{\tau}_{h}$ | 0.65 |

$\mathrm{minimum}\mathrm{car}\text{-}\mathrm{following}\mathrm{distance}/{d}_{\mathrm{min}}$ | 1/m |

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**MDPI and ACS Style**

Yu, J.; Jiang, F.; Kong, W.; Luo, Y.
A Distributed and Hierarchical Optimal Control Method for Intelligent Connected Vehicles in Multi-Intersection Road Networks. *World Electr. Veh. J.* **2022**, *13*, 34.
https://doi.org/10.3390/wevj13020034

**AMA Style**

Yu J, Jiang F, Kong W, Luo Y.
A Distributed and Hierarchical Optimal Control Method for Intelligent Connected Vehicles in Multi-Intersection Road Networks. *World Electric Vehicle Journal*. 2022; 13(2):34.
https://doi.org/10.3390/wevj13020034

**Chicago/Turabian Style**

Yu, Jie, Fachao Jiang, Weiwei Kong, and Yugong Luo.
2022. "A Distributed and Hierarchical Optimal Control Method for Intelligent Connected Vehicles in Multi-Intersection Road Networks" *World Electric Vehicle Journal* 13, no. 2: 34.
https://doi.org/10.3390/wevj13020034