# Modeling the Dynamic Exclusive Pedestrian Phase Based on Transportation Equity and Cost Analysis

^{*}

## Abstract

**:**

## 1. Introduction

- A transportation equity index (TEI) is proposed to quantify the individual differences under different environment conditions. Additionally, vehicle throughput is converted into passenger throughput to reflect transportation equity between different traffic participants.
- Based on the TEI, a bi-objective optimization model is established to find the best tradeoff between transportation equity and cost. The proposed cost model considers the pedestrian–vehicle interaction, which is more consistent with the actual situation.
- Considering the yielding rate and environment conditions, sensitivity analysis is conducted to determine the application domain of EPP. The results provide operational guidelines for decision-makers to better select the pedestrian phase pattern at signalized intersections.

## 2. Literature Review

#### 2.1. Pedestrian–Vehicle Interaction Research

#### 2.2. Transportation Equity Research

## 3. Model Formulations

#### 3.1. Objective Functions

#### 3.2. Signal Constrains

#### 3.3. Transportation Equity Modeling

#### 3.3.1. Pedestrian Throughput

#### 3.3.2. Vehicle Throughput

#### 3.4. Cost Modeling

#### 3.4.1. Pedestrian–Vehicle Interaction

#### 3.4.2. Safety Cost

## 4. Preliminaries: Data and Methods

#### 4.1. Solution Algorithms

#### 4.2. Data Resource

## 5. Numerical Examples and Sensitivity Analysis

#### 5.1. Metaparameter Analysis

_{H}) was calculated to evaluate the optimization performance of NSGA-II and MOEA. The larger the I

_{H}, the better the algorithm performance will be. The best out of optimization runs with 3000 objective evaluations were selected for each configuration. For NSGA-II, the overall best configuration was P

_{s}/M

_{r}/C

_{r}= 50/0.05/0.3, while the configuration P

_{s}/M

_{r}/C

_{r}= 50/0.05/0.3 was the top performer for MOEA.

_{H}was computed by changing the value of the interest parameter. Figure 3 presents how changes in one parameter affect the optimization performance when fixing the other parameters at their best configurations. It can be concluded that P

_{s}has a stronger influence on I

_{H}than M

_{r}and C

_{r}in the NSGA-II algorithm. NSGA-II does not rely on an external archive; thus, a large population indicates rich initial diversity. In this case, constructing a large enough pool of nondominated solutions was possible. In contrast, MOEA prefers smaller M

_{r}because the neighborhood mechanism in MOEA accelerates the propagation of mutation across the population.

#### 5.2. Optimization Results

_{H}and in the spread of the nondominated front. The reason for the poor performance of MOEA is the inherited replacement strategy that can result in limited population diversity and premature convergence. In contrast, NSGA-II had better global exploration capabilities than MOEA. According to the distribution of nondominated fronts of the two methods, the fronts are approximately convex, with two objectives changing in reverse. It indicates that signal optimization methods lacking cost consideration may lead to transportation inequity between pedestrians and vehicles. The nondominated fronts obtained by the proposed model cover a spectrum of solutions that balance either of the two objectives.

#### 5.3. Sensitivity Analysis

- When the vehicular volume is constant, the EPP setting has a better effect with high pedestrian volumes.
- With the same traffic demand, it is more suitable to set the EPP when the yielding rate is low. The analysis shows that the yielding rate has a significant impact on the EPP setting conditions at intersections.
- Environment factors have effects on the EPP setting condition. With the same traffic demand and yielding rate, the EPP is more suitable under wet conditions. It shows that incorporating the environment condition into the EPP setting criteria is essential.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

EPP | Exclusive Pedestrian Phase |

TWC | Traditional Two-way Control Phase |

TEI | Transportation Equity Index |

MOEA | Multiobjective Evolutionary Algorithm based on Decomposition |

NSGA-II | Nondominated Sorting Genetic Algorithm II |

MINLP | Mixed Integer Nonlinear Programming |

UAV | Unmanned Aerial Vehicle |

## Appendix A

_{1}: equity, f

_{2}: cost) can be calculated by:

## Appendix B

Parameter | Description |
---|---|

${e}^{v},\text{}{e}^{p}$ | Environmental index of vehicles and pedestrians |

${q}_{i}^{v}$ | Vehicular flow on arm i in current cycle, (pcu/h) |

${q}_{i}^{p}$ | Pedestrian flow on arm i in current cycle, (ped/h) |

${T}_{v}$ | Vehicular throughput in this cycle, (pcu/h) |

${T}_{p}$ | Pedestrian throughput in this cycle, (ped/h) |

${t}_{P}$ | Length of green of pedestrian signal, (s) |

${l}_{1},{l}_{2}$ | Pedestrian crosswalk length, (m) |

${l}_{3}$ | Diagonal crosswalk length, (m) |

$d$ | Crosswalk width, (m) |

${b}_{1}$ | Distance between pedestrians, (m) |

${v}_{p}$ | Speed of crossing pedestrians, (m/s) |

${t}_{2}$ | Lost time due to pedestrian safety concerns at the end of the red light, (s) |

$C$ | Cycle length, (s) |

${g}_{E}$ | Green time of EPP, (s) |

${t}_{R}$ | Time when the right-turning vehicle occupies the sidewalk, (s) |

${l}_{C}$ | Converted vehicle length, (m) |

${l}_{R}$ | Minimum safe distance between pedestrians and right-turning vehicles, (m) |

${v}_{R}$ | Speed of right-turning vehicles, (m/s) |

${t}_{E}$ | EPP period, (s) |

${t}_{P}$ | Length of green for pedestrians to cross the street, (s) |

${t}_{\mathrm{s}}$ | Minimum time for pedestrians to cross the street, (s) |

${t}_{pv}^{p}$, ${t}_{pv}^{v}$ | Pedestrian–vehicle interaction time of pedestrian and vehicle, (s) |

${N}_{1},{N}_{2},{N}_{3}$ | Number of pedestrians passing through the crosswalks in each cycle with the crosswalk length of ${l}_{1},\text{}{l}_{2}$, ${l}_{3}$; |

$n$ | Number of pedestrians in the first row at the initial stage of EPP |

${t}_{\mathrm{gL}},{t}_{\mathrm{gT}}$ | Signal period of left turn and straight in a cycle, respectively |

${t}_{L},{t}_{T,}{t}_{R}$ | Average vehicle headway of left turn, straight, and right turn, respectively |

${y}_{v}$ | Vehicle yielding rate (%) |

${y}_{p}$ | Pedestrian yielding rate (%) |

$N$ | Numbers of pedestrian–vehicle interactions before setting EPP |

${v}_{\mathrm{s}}$ | Speed of vehicles, (m/s) |

${a}_{\mathrm{s}}$ | Acceleration of vehicles, (m/s^{2}) |

${Q}_{i}^{v}$ | Total vehicular demand at corner i, (pcu/h) |

${Q}_{i}^{p}$ | Total pedestrian crossing demand at corner i, (ped/h) |

${B}_{i}$ | the number of potential traffic accidents under TWC control |

$\delta $ | Binary variable representing the pedestrian phase type, $\delta =\{\begin{array}{c}0,EPP\\ 1,TWC\end{array}$ |

${P}_{T}^{2}$ | Average accident number to pedestrian noncompliance ratio |

$\rho $ | Probability of pedestrian noncompliance |

${\alpha}_{ij}$ | Proportion of pedestrian volume from corner i to corner j in total pedestrian demand of corner i |

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**Figure 4.**Nondominated fronts constructed by NSGA-II and MOEA for the corresponding conditions: (

**a**) TWC in wet condition; (

**b**) EPP in wet condition; (

**c**) TWC in dry condition; (

**d**) EPP in dry condition (f

_{1}: equity, f

_{2}: cost).

**Figure 6.**Impacts of pedestrian and vehicle demand: (

**a**) Transportation equity comparison; (

**b**) transportation cost comparison.

**Figure 7.**Phase setting and traffic demand under different vehicle yielding rates in the first condition: (

**a**) vehicle yielding rate, 20%; (

**b**) vehicle yielding rate, 40%; (

**c**) vehicle yielding rate, 60%; (

**d**) vehicle yielding rate, 80%.

**Figure 8.**Phase setting and traffic demand under different vehicle yielding rates in the second condition: (

**a**) vehicle yielding rate, 20%; (

**b**) vehicle yielding rate, 40%; (

**c**) vehicle yielding rate, 60%; (

**d**) vehicle yielding rate, 80%.

Notation | Definition | Value |
---|---|---|

Monetary Parameters | ||

${C}_{v}$ | Unit average delay cost of one vehicle per hour ($/h) | 6 |

${C}_{p}$ | Unit average delay cost of one pedestrian per hour ($/h) | 4 |

${C}_{A}$ | The average cost of an accident ($/accident) | 65,000 |

${P}_{T}^{1}$, ${P}_{T}^{2}$ | Average accident number to pedestrian noncompliance ratio | 0.00286 |

${K}_{p}^{\prime}$ | Reduction coefficient of the exclusive right-turn lane on pedestrians under TWC | 0.6 |

$\rho $ | Probability of pedestrian noncompliance | 0.25 |

Crossing Parameters | ||

$l$ | Pedestrian crosswalk length, (m) | 15 |

${l}_{3}$ | Diagonal crosswalk length, (m) | 28 |

$d$ | Crosswalk width, (m) | 5 |

${b}_{1}$ | Distance between pedestrians, (m) | 0.75 |

${l}_{C}$ | Converted vehicle length, (m) | 6 |

${l}_{R}$ | Minimum safe distance between pedestrians and right-turn vehicles, (m) | 0.8 |

Signal Parameters | ||

${g}_{i,min}^{v}$ | Minimal green time for vehicles (s) | 10 |

${C}_{min}$ | Minimum cycle length, (s) | 34 |

${C}_{maz}$ | Maximum cycle length, (s) | 200 |

${t}_{E}$ | EPP period, (s) | 26 |

${t}_{2}$ | Lost time due to pedestrian safety concerns at the end of the red light, (s) | 2 |

$t$ | Minimum length of the acceptable gap for crossing (s) | 5 |

Vehicle & Pedestrian Parameters | ||

${v}_{p}$ | Speed of crossing pedestrians, (m/s) | 1.2 |

${v}_{R}$ | Speed of right-turn vehicles, (m/s) | 2.78 |

${\mu}_{i}$ | Average flow rate of turning vehicles I, (pcu/s) | 0.14 |

${q}_{i}^{p}$ | Pedestrian flow on arm i in the current cycle, (ped/h) | 2000 |

${q}_{i}^{v}$ | Vehicle flow on arm i in the current cycle, (pcu/h) | 1000 |

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## Share and Cite

**MDPI and ACS Style**

Lu, Y.; Wang, T.; Wang, Z.; Li, C.; Zhang, Y.
Modeling the Dynamic Exclusive Pedestrian Phase Based on Transportation Equity and Cost Analysis. *Int. J. Environ. Res. Public Health* **2022**, *19*, 8176.
https://doi.org/10.3390/ijerph19138176

**AMA Style**

Lu Y, Wang T, Wang Z, Li C, Zhang Y.
Modeling the Dynamic Exclusive Pedestrian Phase Based on Transportation Equity and Cost Analysis. *International Journal of Environmental Research and Public Health*. 2022; 19(13):8176.
https://doi.org/10.3390/ijerph19138176

**Chicago/Turabian Style**

Lu, Yining, Tao Wang, Zhuangzhuang Wang, Chaoyang Li, and Yi Zhang.
2022. "Modeling the Dynamic Exclusive Pedestrian Phase Based on Transportation Equity and Cost Analysis" *International Journal of Environmental Research and Public Health* 19, no. 13: 8176.
https://doi.org/10.3390/ijerph19138176