# Joint Optimization of Intersection Control and Trajectory Planning Accounting for Pedestrians in a Connected and Automated Vehicle Environment

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Integrated Optimization Problem

_{info}is defined as the ratio of all equipped vehicles (i.e., CAVs = CVs + AVs) to the total number of vehicles, and the automated level r

_{auto}is defined as the ratio of AVs to all equipped vehicles. These two metrics reflect the penetration rate of CAVs and can be used to represent different implementation stages of new vehicle technologies.

#### 2.1. Model Framework

#### 2.2. Model Formulation

#### 2.2.1. Vehicle Delay

#### 2.2.2. Pedestrian Delay

^{th}W signal start time for the reference approach, ${t}_{\mathrm{FDW}}^{}$ and ${t}_{\mathrm{G}}^{}$ are the start time of FDW and the green signal for vehicles on the reference approach when ${n}_{\mathrm{W}}$ = 0, respectively, $\mathsf{\Delta}{t}_{\mathrm{W}}^{\mathrm{min}}$ is the minimum pedestrian crossing time, and $\mathsf{\Delta}{t}_{\mathrm{PC}}$ is the length of the pedestrian clearance interval. If the triggering pedestrian event happens in approach m = 2, Equation (5) still holds. The explanation of Equation (5) is given below.

_{2}in Figure 3d) rather than the triangular area (i.e., A

_{1}in Figure 3d), which refers to the case of no active repetition. The total pedestrian delay at the intersection is the sum of the expected delays on both sidewalks (n = 1, 2) of each leg, and it is estimated with Equation (7).

#### 2.2.3. Trajectory Design for AVs

_{min}(e.g., we set it to 10 km/h in our simulation) is defined to avoid the vehicle crawling to the intersection for no-stopping purposes. Meanwhile, the design speed cannot exceed the free-flow speed u

_{f}. Overall, the design speed might be equivalent to the optimal speed ${u}_{c,J}^{\mathrm{opt}}$, the minimum speed u

_{min}, or the free-flow speed u

_{f}, and it needs to satisfy ${u}_{\mathrm{min}}\le {u}_{c,J}^{\mathrm{des}}\le {u}_{f}$.

## 3. Solution Algorithm Based on the Ant Colony System

#### 3.1. ACS State Transition Rule

^{-}chooses the node c to move to by applying the rule in Equation (8).

#### 3.2. Local Pheromone Updating Rule

_{0}is the tour length of any feasible (or rough) solution (e.g., a given vehicle departure sequence), and $\left|N\right|$ is the number of nodes (i.e., vehicles in our set). For the purpose of the ACS efficiency, in a decision step, we define L

_{0}by the total time associated with all vehicles discharging in a given departure sequence J, as shown in Equation (12).

#### 3.3. Global Pheromone Updating Rule

## 4. Simulation Settings

^{2}; the vehicle saturation flow rate s

_{m}= 1800 veh/h (m = 1, 2, 3, 4); the free-flow speed u

_{f}= 60 km/h; the length of the zone of interest d = 100 m; the minimum speed for trajectory design u

_{min}= 10 km/h. For the ACS algorithm, after numerical tests (details are provided in Appendix A), we select the parameters as follows: α = 0.1, β = 2, ρ = 0.1, e

_{0}= 0.2, ${n}_{\mathrm{ant}}$ = 10, and ${n}_{\mathrm{iter}}$ = 30. To study the performance of the joint optimization with the ACS algorithm, we set various weights associated to the vehicle delay and the pedestrian delay in Equation (1). In the optimization, a conservative range of the weight ω for pedestrian delay is set to 0 ≤ ω ≤ 0.5, and ω varies with an increment of 0.05.

_{info}) and AV ratios (i.e., r

_{auto}).

_{veh}is the total number of vehicles, and n

_{ped}is the total number of pedestrians. Notice that 1.20 passengers per vehicle is a conservative setting for private cars. Increasing the occupancy of vehicles would only exacerbate the total delay penalties imposed by pedestrians onto vehicle passengers.

## 5. Algorithm and Model Analysis

_{info}, r

_{auto}) to represent CAV penetration rates, and we test CS = (1.0, 1.0) and CS = (0.5, 0.5) representing the full AVs and a lower CAV penetration rate, respectively.

## 6. Performance of the Control Algorithm

_{info}(i.e., the ratio of the number of CAVs to the total number of vehicles) from 0.2 to 1, using an automated level r

_{auto}= 0. The r

_{info}begins from 0.2 rather than 0, as we assume that traffic information is collected only from CAVs. By raising r

_{info}from 0.2 to 1 (see Figure 9a), the total person delay declines to 7250 s. This represents a 10% improvement. We then look at the impacts on the performance from different r

_{auto}. The AV penetration ratio is set from 0 to 1 assuming all vehicles are connected, i.e., r

_{info}= 1. As a result, the average person delay decreases consecutively from 7250 s to 6450 s, an 11% improvement (see Figure 9b). This shows the value of the AV trajectory planning. More importantly, the new algorithm outperforms that proposed in [19] in all scenarios.

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Analysis of the Parameters in ACS Algorithm

_{0}, as shown in Figure A1. The smaller value of e

_{0}(between 0 and 0.2) gave better results; thus, we adopted e

_{0}= 0.2.

(a) 2000 veh/h and 3200 ped/h | ||||||||||

α\ρ | 0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 |

0 | 0.7121 | 0.7488 | 0.6797 | 0.7125 | 0.6855 | 0.6500 | 0.6789 | 0.6800 | 0.6798 | 0.7486 |

0.1 | 1.5531 | 0.7176 | 0.6547 | 0.6794 | 0.6883 | 0.7453 | 0.6881 | 0.6598 | 0.7145 | 0.6786 |

0.2 | 2.5234 | 0.8325 | 0.7287 | 0.6820 | 0.6320 | 0.7164 | 0.6851 | 0.6887 | 0.6819 | 0.7159 |

0.3 | 2.2635 | 0.9286 | 0.8403 | 0.8297 | 0.6788 | 0.7220 | 0.6981 | 0.7463 | 0.7496 | 0.6891 |

0.4 | 2.5991 | 1.0304 | 0.6702 | 0.7702 | 0.8116 | 0.7556 | 0.6791 | 0.6464 | 0.6787 | 0.7094 |

0.5 | 3.0222 | 1.7705 | 0.8052 | 0.7335 | 0.6792 | 0.6788 | 0.6792 | 0.7498 | 0.7519 | 0.6834 |

0.6 | 3.0393 | 1.4531 | 1.0529 | 0.8484 | 0.7850 | 0.7225 | 0.6474 | 0.8044 | 0.6823 | 0.6791 |

0.7 | 3.0273 | 1.4790 | 0.9212 | 0.6866 | 0.8921 | 0.6759 | 0.8541 | 0.7831 | 0.6591 | 0.6789 |

0.8 | 3.1171 | 1.6504 | 1.0962 | 0.7501 | 0.7117 | 0.6437 | 0.7553 | 0.6799 | 0.7124 | 0.6848 |

0.9 | 2.7633 | 2.1410 | 1.1633 | 0.8432 | 0.7816 | 0.8833 | 0.7340 | 0.7499 | 0.6807 | 0.7751 |

(b) 3000 veh/h and 3200 ped/h | ||||||||||

α\ρ | 0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 |

0 | 12.9907 | 10.3935 | 11.6400 | 11.9239 | 10.2903 | 10.7171 | 11.9827 | 11.3282 | 13.0788 | 12.0050 |

0.1 | 8.4433 | 5.0415 | 5.9714 | 7.3524 | 7.9087 | 8.5192 | 8.3840 | 8.3412 | 8.9581 | 8.6485 |

0.2 | 9.8332 | 6.1374 | 6.7941 | 6.8732 | 5.8084 | 6.3423 | 7.4827 | 8.5220 | 8.5561 | 8.2074 |

0.3 | 10.7529 | 7.6624 | 5.5161 | 5.2073 | 5.8491 | 6.5132 | 6.8953 | 6.8258 | 7.5134 | 6.9751 |

0.4 | 11.6295 | 7.1261 | 6.2907 | 6.5372 | 6.0145 | 6.8260 | 6.8493 | 6.5042 | 8.2212 | 7.4090 |

0.5 | 11.9399 | 8.4034 | 6.5555 | 5.7233 | 6.0260 | 6.5581 | 6.1832 | 6.5694 | 7.6405 | 7.7624 |

0.6 | 11.4331 | 9.0894 | 6.8417 | 5.8083 | 6.2012 | 6.2573 | 6.8510 | 7.5058 | 6.6033 | 7.4749 |

0.7 | 11.3156 | 8.3969 | 7.4951 | 7.4255 | 6.7906 | 6.4504 | 5.6615 | 7.1968 | 8.5731 | 9.1214 |

0.8 | 10.1384 | 9.1516 | 5.8145 | 6.6242 | 6.0210 | 6.4746 | 6.2777 | 6.7004 | 6.1851 | 7.1229 |

0.9 | 13.3276 | 9.6023 | 8.7868 | 6.9082 | 6.4102 | 6.1237 | 8.0115 | 6.4153 | 8.4931 | 7.8232 |

_{0}= 0.2, ${n}_{\mathrm{ant}}$ = 10, and ${n}_{\mathrm{iter}}$ = 30.

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**Figure 3.**Illustration of pedestrian delay model. (

**a**–

**c**) All cases to estimate ${t}_{J}^{\mathrm{end}}$, the ending time of the estimation horizon of pedestrian delay for vehicle departure sequence J. In all cases, we consider ${n}_{\mathrm{W}}$, the number of signal changes to “walk” for the reference approach starting from the vehicle or pedestrian event time ${t}_{e}$ and lasting until the last vehicle departure time ${D}_{\left|N\right|,\text{\hspace{0.17em}}J}^{}$. We also consider the times of ${D}_{\left|N\right|,\text{\hspace{0.17em}}J}^{}$. In (

**a**), ${n}_{\mathrm{W}}=0$, and the upper bound of ${D}_{\left|N\right|,\text{\hspace{0.17em}}J}^{}$ is the ending time of “flashing do not walk”. In (

**b**), ${n}_{\mathrm{W}}=0$, and the lower bound of ${D}_{\left|N\right|,\text{\hspace{0.17em}}J}^{}$ is ${t}_{\mathrm{G}}^{}$, the start time of vehicle green signal on the reference approach after ${t}_{e}$. In (

**c**), ${n}_{\mathrm{W}}\ge 1$, and the lower bound of ${D}_{\left|N\right|,\text{\hspace{0.17em}}J}^{}$ is ${t}_{\mathrm{W}=1}^{}$, the start time of W for the reference approach after ${t}_{e}$. (

**d**) An example of the pedestrian delay calculated with the triangular areas from ${t}_{e}$ to ${t}_{J}^{\mathrm{end}}$ on the two conflicting approaches with pedestrian arrival rate $\lambda $.

**Figure 5.**Relationship between the predicted vehicle or pedestrian delay and the actual vehicle or pedestrian delay.

**Figure 6.**Performance of average person delay with different weights for the pedestrian delay in the joint optimization.

**Figure 7.**(

**a**) Average vehicle delay and (

**b**) average pedestrian delay using the proposed approach with ACS.

**Figure 8.**Comparison of the average person delay between the balanced demand and the unbalanced demand.

**Figure 9.**Performance of the proposed algorithm under different connected and automated vehicle (CAV) penetration rates: (

**a**) different r

_{info}when r

_{auto}= 0, and (

**b**) different r

_{auto}when r

_{info}= 1.

Vehicle delay model | J | vehicle departure sequence |

N | current vehicle set at each decision step, cars indexed by c (node in ACS algorithm) | |

s_{m} | vehicle saturation flow rate for approach m | |

q_{m} | vehicle flow rate for approach m (veh/h) | |

${u}_{f}$ | free-flow speed | |

${u}_{c,\text{\hspace{0.17em}}J}^{\mathrm{init}}$ | initial speed of vehicle c when entering the intersection within departure sequence J | |

V_{c} | virtual departure time of vehicle c from the downstream end of the intersection | |

D_{c,J} | predicted departure time of vehicle c in departure sequence J from the downstream end of the intersection | |

P_{c,J} | delay penalty, i.e., the time it takes for vehicle c in departure sequence J to cross the intersection zone (based on vehicle’s position within a platoon) | |

C_{c,J} | the start time of the signal phase that discharges vehicle c | |

r_{info} | information level, i.e., the ratio of all equipped vehicles (including connected vehicles and automated vehicles) to the total number of vehicles | |

r_{auto} | automation level, i.e., the ratio of automated vehicles to all equipped vehicles | |

Pedestrian delay model | W | walk |

DW | do not walk | |

FDW | flashing do not walk | |

PC | pedestrian clearance | |

$\omega $ | weights for pedestrian delay, $0\le \omega <1$ | |

${t}_{e}$ | time instant of vehicle or pedestrian event | |

${D}_{\left|N\right|,\text{\hspace{0.17em}}J}^{}$ | last vehicle departure time in departure sequence J, $N\ne \varnothing $ | |

${t}_{J}^{\mathrm{end}}$ | ending time of the pedestrian delay calculation period for departure sequence J | |

${n}_{\mathrm{W}}$ | number of pedestrian signal changes to W for a reference approach during time interval $[{t}_{e},\text{}{D}_{\left|N\right|,\text{\hspace{0.17em}}J}^{}]$ | |

${t}_{\mathrm{W}=k}^{}$ | change time of pedestrian W signal at ${k}^{\mathrm{th}}\text{}(k=\text{}0,\text{}1,\text{}\dots ,\text{}{n}_{\mathrm{W}}$) cycle for a reference approach during time interval $[{t}_{e},\text{}{D}_{\left|N\right|,\text{\hspace{0.17em}}J}^{}]$ | |

$\mathsf{\Delta}{t}_{\mathrm{FDW}}$ | duration of FDW interval | |

$\mathsf{\Delta}{t}_{\mathrm{PC}}$ | duration of pedestrian clearance interval (includes FDW and all-red time) | |

$\mathsf{\Delta}{t}_{\mathrm{W}}^{\mathrm{min}}$ | minimum pedestrian green time (W interval) | |

${t}_{\mathrm{FDW}}$ | start time of FDW interval for pedestrians on a reference approach when ${n}_{\mathrm{W}}$ = 0 | |

${t}_{\mathrm{G}}^{}$ | start time of the green signal for vehicles on a reference Aapproach when ${n}_{\mathrm{W}}$ = 0 | |

λ_{m,n} | average pedestrian flow rate (ped/h) for approach m, sidewalk n | |

${R}_{m,\text{\hspace{0.17em}}r}$ | r-th pedestrian waiting interval in approach m (typically includes FDW and DW) | |

AV trajectory design | ${u}_{c,\text{\hspace{0.17em}}J}^{\mathrm{des}}$ | design speed of vehicle c in departure sequence J |

${u}_{c,\text{\hspace{0.17em}}J}^{\mathrm{opt}}$ | optimal speed of vehicle c in departure sequence J | |

${u}_{c,J}^{\mathrm{init}}$ | initial speed of vehicle c in departure sequence J when entering intersection | |

${u}_{\mathrm{min}}$ | minimum speed for trajectory design | |

u_{f} | free-flow speed |

Parameter | Definition |
---|---|

$\tau ({c}^{-},\text{\hspace{0.17em}}c)$ | pheromone deposited by ant $\widehat{a}$ from the previous node ${c}^{-}$ to the current node c |

$\eta ({c}^{-},\text{\hspace{0.17em}}c)$ | heuristic information used by ant $\widehat{a}$ from the previous node ${c}^{-}$ to the current node c |

${p}_{\widehat{a}}({c}^{-},\text{\hspace{0.17em}}c)$ | probability for ant $\widehat{a}$ to select the next visit node c based on biased exploration |

${\mathsf{\Omega}}_{\widehat{a}}({c}^{-})$ | set of nodes that remain to be visited by ant $\widehat{a}$ positioned on the previous node ${c}^{-}$ |

$\rho $ | pheromone decay parameter in local pheromone updating rule $(0<\rho <1$) |

$\alpha \text{\hspace{0.17em}}$ | pheromone decay parameter in global pheromone updating rule $(0<\alpha <1$) |

$\beta $ | parameter for heuristic information $(\beta >0$) |

${e}_{0}\text{\hspace{0.17em}}$ | parameter for the exploitation of next vehicle c $(0\le {e}_{0}\le 1$) |

L_{0} | tour length (or discharge time of all vehicles) of any feasible solution for the initialization in local pheromone updating rule |

L_{global} | tour length (or discharge time of all vehicles) of so-far best path from the beginning of the iteration in global pheromone updating rule |

${n}_{\mathrm{ant}}$ | number of ants in ACS algorithm |

${n}_{\mathrm{iter}}$ | number of iterations in ACS algorithm |

**Table 3.**Comparison between ACS algorithm and an optimal control method (based on full enumeration) for different ACS parameter sets.

Method | Accuracy | Computation Time ^{1} (s) | |||||
---|---|---|---|---|---|---|---|

Ant colony system (ACS) | $({n}_{\mathrm{ant}}$$,\text{}{n}_{\mathrm{iter}}$) | CS = (1.0, 1.0) | CS = (0.5, 0.5) | CS = (1.0, 1.0) | CS = (0.5, 0.5) | ||

(10, 10) | 86.73% | 96.91% | 1.9 | (2.5) | 1.5 | (1.6) | |

(10, 20) | 95.67% | 98.75% | 3.5 | (4.3) | 2.8 | (2.3) | |

(10, 30) | 97.97% | 99.79% | 4.1 | (5.8) | 3.2 | (3.5) | |

(15, 10) | 92.65% | 98.80% | 3.5 | (5.0) | 2.1 | (3.1) | |

(15, 20) | 96.72% | 99.99% | 4.3 | (6.7) | 2.9 | (4.6) | |

(15, 30) | 98.52% | 100% | 6.1 | (8.1) | 3.9 | (6.3) | |

Enumeration method | - | 100% | 100% | 338.9 | (-) | 145.1 | (-) |

^{1}The time without parentheses refers to the case when we use an upper limit of 20 vehicles in the optimization, and the time within parentheses refers to the case when all vehicles are considered in the optimization. The maximum time is chosen from 20 tested samples.

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**MDPI and ACS Style**

Yin, B.; Menendez, M.; Yang, K.
Joint Optimization of Intersection Control and Trajectory Planning Accounting for Pedestrians in a Connected and Automated Vehicle Environment. *Sustainability* **2021**, *13*, 1135.
https://doi.org/10.3390/su13031135

**AMA Style**

Yin B, Menendez M, Yang K.
Joint Optimization of Intersection Control and Trajectory Planning Accounting for Pedestrians in a Connected and Automated Vehicle Environment. *Sustainability*. 2021; 13(3):1135.
https://doi.org/10.3390/su13031135

**Chicago/Turabian Style**

Yin, Biao, Monica Menendez, and Kaidi Yang.
2021. "Joint Optimization of Intersection Control and Trajectory Planning Accounting for Pedestrians in a Connected and Automated Vehicle Environment" *Sustainability* 13, no. 3: 1135.
https://doi.org/10.3390/su13031135