# Stability of Traffic Equilibria in a Day-to-Day Dynamic Model of Route Choice and Adaptive Signal Control

## Abstract

**:**

## 1. Introduction and Background

_{0}control policy [28,29]. He showed that, under the assumption of vertical queuing, the dynamical system is convergent and network capacity is maximized.

## 2. Method

#### 2.1. General Formulation of the Problem

- t represents the generic period (“day”) of the dynamic process;
- ${\mathit{C}}^{t}$ is the vector of mean perceived link costs at the start of day t;
- ${\mathit{F}}^{t}$ is the vector of link flows on day t;
- ${\mathit{G}}^{t}$ is the vector of signal settings on day t;
- $\mathit{K}$ is the vector representing average link costs actually experienced by users as a function of link flows and signal settings;
- $\mathit{S}$ is the vector function representing the relationship between link flows and mean perceived link costs based on an assumed probabilistic route choice model;
- $\mathit{H}$ is the vector function representing the relationship between signal settings and link flows based on an assumed signal control policy;
- $\beta $ ($0<\beta \le 1$) is a parameter quantifying the extent to which the most recent travel experience (“yesterday’s trip”) contributes to the formation of the cost perceived by users at the start of day t;
- $\alpha $ ($0<\alpha \le 1$) is a parameter representing the fraction of travelers who actually reconsider their previous day’s choice.

#### 2.2. Day-to-Day Dynamics of Route Choice and Signal Control in a Two-Link Network

**H**, which is then used in the analysis of Section 2.4.

#### 2.3. A Logit form Signal Control Policy

- i denotes the generic phase of the signal cycle;
- $N$ represents the total number of signal phases;
- ${G}_{i}$ is the green split assigned to phase i;
- ${P}_{i}$ is the traffic pressure in phase i;
- $\gamma \left(0\right)$ is a control parameter.

#### 2.4. A Simple Specification of the Dynamic Process of Route Choice and Signal Control

## 3. Numerical Example #1: Two-Link Network

## 4. Numerical Example #2: Grid Network

- $\overline{d}$ is the average overall vehicular delay (s);
- c is the signal cycle length (s);
- G is the green split;
- $x$ is the flow-to-capacity ratio;
- $\tau $ is the duration of the arrival overflow (h);
- Q is the saturation flow (veh./h).

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Derivation of Expression (45)

## References

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**Figure 2.**Maximum value of $\gamma $ ensuring fixed-point stability as a function of the Logit route choice parameter $\theta $ ($b=2;Q=1$).

**Figure 3.**Minimum value of $\gamma $ ensuring fixed-point stability as a function of the Logit route choice parameter $\theta $ for different values of $\alpha $ and $\beta $ ($b=2;Q=1$).

**Figure 4.**Time evolution of F for $\alpha =0.6;\beta =0.4;\gamma =3;\theta =0.5;b=1.5;Q=1;F\left(0\right)=0.8$.

**Figure 5.**Time evolution of F for $\alpha =0.9;\beta =0.8;\gamma =1.05;\theta =1.5;b=2.5;Q=1;F\left(0\right)=0.2$.

**Figure 6.**Time evolution of F for $\alpha =1;\beta =0.8;\gamma =4.05;\theta =1;b=2;Q=1;F\left(0\right)=0.6$.

**Figure 7.**Time evolution of F for $\alpha =1;\beta =1;\gamma =3.5;\theta =2.5;b=1.5;Q=1;F\left(0\right)=0.1$.

**Figure 8.**Time evolution of F for $\alpha =1;\beta =1;\gamma =3.5;\theta =1;b=2;Q=1;F\left(0\right)=0.49;0.51$.

**Figure 10.**Approximate lower (γ_1) and upper (γ_2) limits of the fixed-point stability region in terms of $\gamma $ as a function of $\theta $ (total trips = 2800 veh./h).

**Figure 11.**Approximate lower (γ_1) and upper (γ_2) limits of the fixed-point stability region in terms of $\gamma $ as a function of $\theta $ (total trips = 3100 veh./h).

**Figure 12.**Approximate lower (γ_1) and upper (γ_2) limits of the fixed-point stability region in terms of $\gamma $ as a function of $\theta $ (total trips = 3400 veh./h).

**Figure 13.**Average intersection delay under Logit (with optimal $\gamma $) and Equisaturation signal control policies as a function of $\theta $ (total trips = 2800 veh./h).

**Figure 14.**Average intersection delay under Logit (with optimal $\gamma $) and Equisaturation signal control policies as a function of $\theta $ (total trips = 3100 veh./h).

**Figure 15.**Average intersection delay under Logit (with optimal $\gamma $) and Equisaturation signal control policies as a function of $\theta $ (total trips = 3400 veh./h).

Link | Free-Flow Travel Time (min.) | Saturation Flow (veh./h) | Capacity (veh./h) |
---|---|---|---|

1 | 5 | – | 1500 |

2 | 5 | – | 1500 |

3 | 5 | 1400 | – |

4 | 12 | 2000 | – |

5 | 5 | 1600 | – |

6 | 5 | 1400 | – |

7 | 12 | 2000 | – |

8 | 5 | 1500 | – |

9 | 5 | – | 1500 |

10 | 5 | – | 1500 |

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**MDPI and ACS Style**

Meneguzzer, C.
Stability of Traffic Equilibria in a Day-to-Day Dynamic Model of Route Choice and Adaptive Signal Control. *Appl. Sci.* **2024**, *14*, 1891.
https://doi.org/10.3390/app14051891

**AMA Style**

Meneguzzer C.
Stability of Traffic Equilibria in a Day-to-Day Dynamic Model of Route Choice and Adaptive Signal Control. *Applied Sciences*. 2024; 14(5):1891.
https://doi.org/10.3390/app14051891

**Chicago/Turabian Style**

Meneguzzer, Claudio.
2024. "Stability of Traffic Equilibria in a Day-to-Day Dynamic Model of Route Choice and Adaptive Signal Control" *Applied Sciences* 14, no. 5: 1891.
https://doi.org/10.3390/app14051891