# Optimization Models of Actuated Control Considering Vehicle Queuing for Sustainable Operation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Vehicle Queuing Model Based on Improved Traffic Wave Model

#### 3.1. Improved Traffic Wave Model

_{j}. The traffic flow behind the wave front was the traveling traffic flow with the speed of critical speed v

_{m}, density of critical density k

_{m}, and headway of the saturated headway h

_{m}.

_{1}, density k

_{1}, and headway h

_{1}. The traffic flow behind the wave front was the congested traffic flow with speed of 0 and the density of congested density k

_{j}.

_{start}represents the starting wave (m/s), and u

_{stop}represents the standing wave (m/s).

_{1}is needed for calculation, which is difficult to obtain directly. In the improved model, shown in Equations (6) and (7), headway h

_{1}was adopted instead of k

_{1}, which could be accurately obtained in real time thanks to the widely used coil detectors in actuated control.

#### 3.2. Vehicle Queuing and Dispersion Process

#### 3.2.1. Vehicle Queuing Process

_{a}

^{(n)}is the average speed of the arriving vehicles (m/s), and h

_{a}

^{(n)}is the average headway (s).

#### 3.2.2. Vehicle Dispersion Process

_{m}.

#### 3.3. Vehicle-Queuing Model

_{rb}

^{(n)}≤ t ≤ t

_{gb}

^{(n)}

_{rb}

^{(n)}is the start time of the red light, t

_{gb}

^{(n)}is the start time of the green light, x

_{0}

^{(n)}is the position of the leading vehicle relative to the stop line (m), x

_{l}

^{(n)}is the position of the trailing vehicle relative to the stop line (m), and L

^{(n)}is the queue length (m).

_{gb}

^{(n)}≤ t ≤ t

_{qd}

^{(n)}

_{qd}

^{(n)}is the moment when the starting wave and the standing wave meet, and is shown in Equation (12).

_{max}

^{(n)}formed at time t

_{qd}

^{(n)}is as follows:

_{qd}

^{(n)}, the vehicles passed through the intersection, as shown in Figure 2c. At this time, there were no queued vehicles.

## 4. Optimization Models of Basic Parameters

#### 4.1. Minimal Green Time Optimization Model

_{1}; and (b) the time required for vehicles gradually leave the intersection, which was recorded as T

_{2}.

_{p}is as follows:

#### 4.2. Maximal Green Time Optimization Model

_{i}is the average vehicle delay in flow direction i (s), q

_{i}is the traffic volume in flow direction i (pcu), D

_{0}is the traffic capacity under current signal timing plan (pcu/h), s

_{i}is the saturated flow in flow direction i (pcu/h), g

_{ei}is the effective green time in flow direction i (s), c is the cycle length (s), $\mu $

_{1}is the weight coefficient of the average vehicle delay, and $\mu $

_{2}is the weight coefficient of the capacity. The traffic engineer can decide the weight coefficient according to the actual operating conditions of the intersection. In this study, $\mu $

_{1}= 2/3 and $\mu $

_{2}= 1/3 were chosen.

_{i}was calculated with the Webster model [30], which is as follows:

_{i}is the flow ratio in flow direction i, which equals to saturated flow dividing q

_{i}[10].

_{max}is the longest queue length calculated in Formula (13). Incorporating Equation (13) into Equation (17) is shown as Equation (18).

_{i}is the green time in flow direction i (s), A is the yellow time (s), l

_{i}is the starting loss time in flowing direction i (s), g

_{i}

^{(min)}is the minimal green time in flow direction i (s), and g

_{i}

^{(max)}is the maximal green time in flow direction i (s).

## 5. Solving Algorithm for the Optimization Model

#### 5.1. Variable Coding

#### 5.2. Fitness Function

#### 5.3. Genetic Manipulation

## 6. Verification

#### 6.1. Verification of Minimal Green Time Calculation Model

#### 6.2. Verification of Maximal Green Time Calculation Model

_{1}, F

_{2}, and F

_{3}, the optimization ratios all exceeded 10%. Under phase scheme F

_{4}, since there were only two green time parameters that could be adjusted, it also had an optimization ratio of 2.10%. Thus, the maximal green time calculation model established in this paper could effectively reduce the average vehicle delay and improve traffic capacity (9.27% average optimization ratio), which verified the effectiveness of the model.

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Phase Schemes | Schematic Diagrams |
---|---|

F_{1} | |

F_{2} | |

F_{3} | |

F_{4} |

Phase Schemes | Objective Function and Constraints | |
---|---|---|

F_{1} | Delay | $D=\left({\displaystyle \sum _{i=1}^{8}{d}_{i}\xb7{q}_{i}}\right)\xb7{\left({\displaystyle \sum _{i=1}^{8}{q}_{i}}\right)}^{-1}$ |

Capacity | $CAP={\displaystyle \sum _{i=1}^{8}\frac{{s}_{i}\xb7{g}_{ei}}{c}}$ | |

Constraints | $\left\{\begin{array}{l}{g}_{1}+{g}_{2}-{g}_{5}-{g}_{6}=0\hfill \\ {g}_{3}+{g}_{4}-{g}_{7}-{g}_{8}=0\hfill \\ {g}_{1}+{g}_{2}+{g}_{3}+{g}_{4}+4I=c\hfill \end{array}\right.$ | |

F_{2} | Delay | $D=\left({\displaystyle \sum _{i=1}^{2}{d}_{i}\xb7{q}_{i}}+{\displaystyle \sum _{i=5}^{6}{d}_{i}\xb7{q}_{i}}+{d}^{\prime}\xb7{q}^{\prime}\right)\xb7{\left({\displaystyle \sum _{i=1}^{2}{q}_{i}}+{\displaystyle \sum _{i=5}^{6}{q}_{i}}+{q}^{\prime}\right)}^{-1}$ |

Capacity | $CAP={\displaystyle \sum _{i=1}^{2}\frac{{s}_{i}\xb7{g}_{ei}}{c}}+{\displaystyle \sum _{i=5}^{6}\frac{{s}_{i}\xb7{g}_{ei}}{c}}+\frac{{s}^{\prime}\xb7{g}_{e}{}^{\prime}}{c}$ | |

Constraints | $\left\{\begin{array}{l}{g}_{1}+{g}_{2}-{g}_{5}-{g}_{6}=0\hfill \\ {g}_{1}+{g}_{2}+{g}^{\prime}+3I=c\hfill \end{array}\right.$ | |

F_{3} | Delay | $D=\left({\displaystyle \sum _{i=3}^{4}{d}_{i}\xb7{q}_{i}}+{\displaystyle \sum _{i=7}^{8}{d}_{i}\xb7{q}_{i}}+{d}^{\prime}\xb7{q}^{\prime}\right)\xb7{\left({\displaystyle \sum _{i=3}^{4}{q}_{i}}+{\displaystyle \sum _{i=7}^{8}{q}_{i}}+{q}^{\prime}\right)}^{-1}$ |

Capacity | $CAP={\displaystyle \sum _{i=3}^{4}\frac{{s}_{i}\xb7{g}_{ei}}{c}}+{\displaystyle \sum _{i=7}^{8}\frac{{s}_{i}\xb7{g}_{ei}}{c}}+\frac{{s}^{\prime}\xb7{g}_{e}{}^{\prime}}{c}$ | |

Constraints | $\left\{\begin{array}{l}{g}_{3}+{g}_{4}-{g}_{7}-{g}_{8}=0\hfill \\ {g}_{3}+{g}_{4}+{g}^{\prime}+3I=c\hfill \end{array}\right.$ | |

F_{4} | Delay | $D=\left({d}^{\prime}\xb7{q}^{\prime}+{d}^{\u2033}\xb7{q}^{\u2033}\right)\xb7{\left({q}^{\prime}+{q}^{\u2033}\right)}^{-1}$ |

Capacity | $CAP=\frac{{s}^{\prime}\xb7{g}_{e}{}^{\prime}}{c}+\frac{{s}^{\u2033}\xb7{g}_{e}{}^{\u2033}}{c}$ | |

Constraints | ${g}^{\prime}+{g}^{\u2033}+2I=c$ |

_{e}′, g

_{e}″, g′, g″, q′, q″ represent the saturated flow, average vehicle delay, effective green time, shown green time, and traffic volume in a higher traffic volume direction, respectively, when the phase schemes are F

_{2}, F

_{3}, F

_{4}.

Index | Simulation Time (s) | HCM Model (s) | Relative Error | Optimization Model (s) | Relative Error |
---|---|---|---|---|---|

1 | 22.40 | 22.80 | 1.79% | 21.60 | 3.57% |

2 | 20.80 | 22.80 | 9.62% | 21.60 | 3.85% |

3 | 23.80 | 25.40 | 6.72% | 25.20 | 5.88% |

4 | 19.79 | 22.80 | 15.21% | 21.60 | 9.15% |

5 | 20.00 | 22.80 | 14.00% | 21.60 | 8.00% |

6 | 26.34 | 29.40 | 11.62% | 28.60 | 8.58% |

7 | 21.20 | 23.00 | 8.49% | 22.00 | 3.77% |

8 | 22.65 | 24.60 | 8.61% | 23.40 | 3.31% |

9 | 21.30 | 22.80 | 7.04% | 21.60 | 1.41% |

10 | 21.56 | 22.80 | 5.75% | 21.60 | 0.19% |

11 | 21.30 | 22.80 | 7.04% | 21.60 | 1.41% |

12 | 22.62 | 24.60 | 8.75% | 23.40 | 3.45% |

13 | 21.70 | 22.80 | 5.07% | 21.60 | 0.46% |

14 | 24.38 | 22.80 | 6.48% | 21.60 | 11.40% |

15 | 21.79 | 22.80 | 4.64% | 21.60 | 0.87% |

16 | 21.90 | 23.40 | 6.85% | 23.80 | 8.68% |

17 | 20.52 | 22.80 | 11.11% | 21.60 | 5.26% |

18 | 22.40 | 24.60 | 9.82% | 23.40 | 4.46% |

19 | 18.36 | 21.00 | 14.38% | 19.80 | 7.84% |

20 | 19.51 | 21.00 | 7.64% | 19.80 | 1.49% |

21 | 21.00 | 22.80 | 8.57% | 21.60 | 2.86% |

22 | 26.10 | 28.30 | 8.43% | 27.60 | 5.75% |

23 | 20.33 | 21.00 | 3.30% | 19.80 | 2.61% |

24 | 16.90 | 19.20 | 13.61% | 18.00 | 6.51% |

25 | 24.13 | 27.20 | 12.72% | 26.40 | 9.41% |

26 | 24.89 | 27.20 | 9.28% | 26.40 | 6.07% |

27 | 21.12 | 22.80 | 7.95% | 21.60 | 2.27% |

28 | 18.57 | 20.60 | 10.93% | 19.80 | 6.62% |

29 | 20.80 | 21.00 | 0.96% | 19.80 | 4.81% |

30 | 26.58 | 28.20 | 6.09% | 28.00 | 5.34% |

Queue Length (m) | Simulation Time (s) | HCM Model | Optimization Model | ||
---|---|---|---|---|---|

Time (s) | Relative Error | Time (s) | Relative Error | ||

25 | 9.92 | 11.56 | 18.13% | 10.06 | 3.90% |

35 | 12.01 | 13.20 | 10.41% | 12.13 | 3.39% |

45 | 18.41 | 19.73 | 7.29% | 19.13 | 3.83% |

55 | 21.82 | 23.54 | 8.42% | 22.53 | 4.84% |

65 | 24.54 | 26.17 | 7.94% | 25.22 | 4.91% |

Average relative error | 10.44% | 4.18% |

Traffic Flow Direction | Traffic Volume (pcu/h) | Saturated Flow (pcu/h) | Ratio | |
---|---|---|---|---|

SB | TH | 912 | 2750 | 0.33 |

LT | 356 | 1550 | 0.23 | |

NB | TH | 832 | 2750 | 0.30 |

LT | 144 | 1550 | 0.09 | |

WB | TH | 288 | 2450 | 0.08 |

LT | 428 | 2150 | 0.20 | |

EB | TH | 248 | 2750 | 0.09 |

LT | 216 | 1550 | 0.14 |

Phase Schemes | Index | WB LT (s) | B TH (s) | NB LT (s) | SB TH (s) | EB LT (s) | WB TH (s) | SB LT (s) | NB TH (s) |
---|---|---|---|---|---|---|---|---|---|

F_{1} | ① | 32 | 15 | 37 | 54 | 32 | 15 | 37 | 54 |

② | 18 | 22 | 20 | 53 | 17 | 23 | 20 | 53 | |

F_{2} | ① | 24 | 10 | 39 | 39 | 24 | 10 | 39 | 39 |

② | 25 | 15 | 36 | 36 | 14 | 26 | 36 | 36 | |

F_{3} | ① | 16 | 16 | 18 | 24 | 16 | 16 | 18 | 24 |

② | 13 | 13 | 12 | 36 | 13 | 13 | 22 | 26 | |

F_{4} | ① | 12 | 12 | 20 | 20 | 12 | 12 | 20 | 20 |

② | 10 | 10 | 22 | 22 | 10 | 10 | 22 | 22 |

Phase Schemes | Index | Average Vehicle Delay (s) | Traffic Capacity (pcu/h) | Optimization Ratio |
---|---|---|---|---|

F_{1} | ① | 59.62 | 4279 | / |

② | 42.69 | 4472 | 12.79% | |

F_{2} | ① | 24.75 | 3185 | / |

② | 24.15 | 3742 | 10.75% | |

F_{3} | ① | 26.07 | 3328 | / |

② | 22.57 | 3741 | 11.45% | |

F_{4} | ① | 7.84 | 7733 | / |

② | 7.57 | 7668 | 2.10% | |

Average optimization ratio | 9.27% |

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**MDPI and ACS Style**

Wang, X.; Wu, X.; Liu, J.
Optimization Models of Actuated Control Considering Vehicle Queuing for Sustainable Operation. *Sustainability* **2022**, *14*, 8998.
https://doi.org/10.3390/su14158998

**AMA Style**

Wang X, Wu X, Liu J.
Optimization Models of Actuated Control Considering Vehicle Queuing for Sustainable Operation. *Sustainability*. 2022; 14(15):8998.
https://doi.org/10.3390/su14158998

**Chicago/Turabian Style**

Wang, Xinyue, Xianyu Wu, and Jiarui Liu.
2022. "Optimization Models of Actuated Control Considering Vehicle Queuing for Sustainable Operation" *Sustainability* 14, no. 15: 8998.
https://doi.org/10.3390/su14158998