# Macroscopic and Multi-Scale Models for Multi-Class Vehicular Dynamics with Uneven Space Occupancy: A Case Study

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## Abstract

**:**

## 1. Introduction

#### 1.1. State of the Art

- Different driving modes (e.g., autonomous vs. classic);
- Different origins and destinations;
- Different lengths (i.e., space occupied);
- Different velocities/flux functions;
- Reserved roads or reserved entry/exit lanes.

#### 1.2. Case Study

#### 1.3. Our Contribution

- The first model is purely macroscopic. Both cars and truck are described by two coupled first-order LWR-based models. Fundamental diagrams are shaped in order to allow cars to move even in the presence of fully congested trucks. Considering that the fundamental diagram of each class is influenced by the presence of the other class, in the case of unstable (rapidly varying) traffic conditions of one class, we observe a scattered behavior in the fundamental diagram of the other class. Numerical results will show that this feature allows the model to catch, at least in part, some second-order (inertial) phenomena in traffic behavior, such as stop and go waves.
- The second model is multi-scale. Cars are described by a first-order LWR-based model, while trucks are described by a second-order microscopic follow-the-leader model. For trucks, we consider the microscopic model used in [3], inspired, in turn, by a model originally proposed in [33] and specifically designed to reproduce stop and go waves. The choice of second-order model for trucks is crucial, since inertia effects are not at all negligible for those vehicles, while they are less important in car dynamics. Finally, note that, since trucks are confined to only one lane and cannot overtake, their dynamics perfectly matches the constituting assumptions of the follow-the-leader model.

**Remark**

**1.**

## 2. Dataset

## 3. Models

#### 3.1. Macroscopic Model

- (L1)
- ${v}_{\mathrm{L}}({\rho}_{\mathrm{L}},{\rho}_{\mathrm{H}})\ge 0$ for all $({\rho}_{\mathrm{L}},{\rho}_{\mathrm{H}})\in \mathcal{D}$ and ${v}_{\mathrm{L}}({\rho}_{\mathrm{L}},{\rho}_{\mathrm{H}})=0$ iff ${\rho}_{\mathrm{L}}={\rho}_{\mathrm{L}}^{*}\left({\rho}_{\mathrm{H}}\right)$, where$${\rho}_{\mathrm{L}}^{*}\left({\rho}_{\mathrm{H}}\right):={\rho}_{\mathrm{L}}^{\mathrm{max}}-{\rho}_{\mathrm{H}}/\beta $$
- (L2)
- ${v}_{\mathrm{L}}({\rho}_{\mathrm{L}},{\rho}_{\mathrm{H}})$ is a decreasing function with respect to ${\rho}_{\mathrm{L}}$ and ${\rho}_{\mathrm{H}}$;
- (L3)
- ${f}_{\mathrm{L}}(0,{\rho}_{\mathrm{H}})=0$ and ${f}_{\mathrm{L}}({\rho}_{\mathrm{L}}^{*}\left({\rho}_{\mathrm{H}}\right),{\rho}_{\mathrm{H}})=0$ for all ${\rho}_{\mathrm{H}}\in [0,{\rho}_{\mathrm{H}}^{\mathrm{max}}]$;
- (L4)
- ${f}_{\mathrm{L}}({\rho}_{\mathrm{L}},{\rho}_{\mathrm{H}})$ is concave with respect to ${\rho}_{\mathrm{L}}$ for any ${\rho}_{\mathrm{H}}$. We define$${\sigma}_{\mathrm{L}}\left({\rho}_{\mathrm{H}}\right):=arg\underset{{\rho}_{\mathrm{L}}}{max}{f}_{\mathrm{L}}({\rho}_{\mathrm{L}},{\rho}_{\mathrm{H}})$$
- (L5)
- ${f}_{\mathrm{L}}({\rho}_{\mathrm{L}},{\rho}_{\mathrm{H}})$ is a decreasing function with respect to ${\rho}_{\mathrm{H}}$ for any ${\rho}_{\mathrm{L}}$.

- (H1)
- ${v}_{\mathrm{H}}({\rho}_{\mathrm{L}},{\rho}_{\mathrm{H}})\ge 0$ for all $({\rho}_{\mathrm{L}},{\rho}_{\mathrm{H}})\in \mathcal{D}$ and ${v}_{\mathrm{H}}({\rho}_{\mathrm{L}},{\rho}_{\mathrm{H}})=0$ iff ${\rho}_{\mathrm{H}}={\rho}_{\mathrm{H}}^{*}\left({\rho}_{\mathrm{L}}\right)$, where$${\rho}_{\mathrm{H}}^{*}\left({\rho}_{\mathrm{L}}\right):=min\left\{{\rho}_{\mathrm{H}}^{\mathrm{max}},\beta ({\rho}_{\mathrm{L}}^{\mathrm{max}}-{\rho}_{\mathrm{L}})\right\}$$
- (H2)
- ${v}_{\mathrm{H}}({\rho}_{\mathrm{L}},{\rho}_{\mathrm{H}})$ is a decreasing function with respect to ${\rho}_{\mathrm{L}}$ and ${\rho}_{\mathrm{H}}$;
- (H3)
- ${f}_{\mathrm{H}}({\rho}_{\mathrm{L}},0)=0$ and ${f}_{\mathrm{H}}({\rho}_{\mathrm{L}},{\rho}_{\mathrm{H}}^{*}\left({\rho}_{\mathrm{L}}\right))=0$ for all ${\rho}_{\mathrm{L}}\in [0,{\rho}_{\mathrm{L}}^{\mathrm{max}}]$;
- (H4)
- ${f}_{\mathrm{H}}({\rho}_{\mathrm{L}},{\rho}_{\mathrm{H}})$ is concave with respect to ${\rho}_{\mathrm{H}}$ for any ${\rho}_{\mathrm{L}}$. We define$${\sigma}_{\mathrm{H}}\left({\rho}_{\mathrm{L}}\right):=arg\underset{{\rho}_{\mathrm{H}}}{max}{f}_{\mathrm{H}}({\rho}_{\mathrm{L}},{\rho}_{\mathrm{H}})$$
- (H5)
- ${f}_{\mathrm{H}}({\rho}_{\mathrm{L}},{\rho}_{\mathrm{H}})$ is a decreasing function with respect to ${\rho}_{\mathrm{L}}$ for any ${\rho}_{\mathrm{H}}$.

- The partial coupling phase is in place when$$({\rho}_{\mathrm{L}},{\rho}_{\mathrm{H}})\in {\mathcal{D}}_{1}:=\left\{0\le {\rho}_{\mathrm{H}}\le {\rho}_{\mathrm{H}}^{\mathrm{max}},\phantom{\rule{4pt}{0ex}}0\le {\rho}_{\mathrm{L}}\le {\rho}_{\mathrm{L}}^{\mathrm{max}}-{\rho}_{\mathrm{H}}^{\mathrm{max}}/\beta \right\},$$In this phase, we assume that cars are mainly in the fast lane and do not affect the truck dynamics. Trucks are then independent from cars.For trucks, we choose a triangular fundamental diagram with$${v}_{\mathrm{H}}\left({\rho}_{\mathrm{H}}\right)={V}_{\mathrm{H}}^{\mathrm{max}}\phantom{\rule{1.em}{0ex}}\mathrm{for}\mathrm{all}\phantom{\rule{1.em}{0ex}}{\rho}_{\mathrm{H}}\le {\sigma}_{\mathrm{H}},$$Cars do not interfere with trucks but adapt their dynamics to the presence of them. Moreover, for cars, we choose (a family of) triangular fundamental diagrams, see Figure 6a. Specifically, we set$${v}_{\mathrm{L}}({\rho}_{\mathrm{L}},{\rho}_{\mathrm{H}})=\left\{\begin{array}{ccc}{V}_{\mathrm{L}}^{*}\left({\rho}_{\mathrm{H}}\right)\hfill & \mathrm{if}& {\rho}_{\mathrm{L}}\le {\sigma}_{\mathrm{L}}\left({\rho}_{\mathrm{H}}\right),\hfill \\ {\displaystyle \frac{{V}_{\mathrm{L}}^{*}\left({\rho}_{\mathrm{H}}\right)\phantom{\rule{4pt}{0ex}}{\sigma}_{\mathrm{L}}\left({\rho}_{\mathrm{H}}\right)}{{\rho}_{\mathrm{L}}^{*}\left({\rho}_{\mathrm{H}}\right)-{\sigma}_{\mathrm{L}}\left({\rho}_{\mathrm{H}}\right)}\left(\frac{{\rho}_{\mathrm{L}}^{*}\left({\rho}_{\mathrm{H}}\right)}{{\rho}_{\mathrm{L}}}-1\right)}\hfill & \mathrm{if}& {\sigma}_{\mathrm{L}}\left({\rho}_{\mathrm{H}}\right)<{\rho}_{\mathrm{L}}\le {\rho}_{\mathrm{L}}^{\mathrm{max}}-{\rho}_{\mathrm{H}}^{\mathrm{max}}/\beta ,\hfill \end{array}\right.$$For $({\rho}_{\mathrm{L}},{\rho}_{\mathrm{H}})\in {\mathcal{D}}_{1}$, the model (9) then becomes$$\left\{\begin{array}{c}{\partial}_{t}{\rho}_{\mathrm{L}}+{\partial}_{x}{f}_{\mathrm{L}}({\rho}_{\mathrm{L}},{\rho}_{\mathrm{H}})=\phantom{\rule{0.166667em}{0ex}}0\hfill \\ {\partial}_{t}{\rho}_{\mathrm{H}}+{\partial}_{x}{f}_{\mathrm{H}}\left({\rho}_{\mathrm{H}}\right)=\phantom{\rule{0.166667em}{0ex}}0\hfill \end{array}\right.$$
- The full coupling phase is in place when $({\rho}_{\mathrm{L}},{\rho}_{\mathrm{H}})\in {\mathcal{D}}_{2}:=\mathcal{D}\setminus {\mathcal{D}}_{1}$, see Figure 5. In this case, we assume that there are too many cars to find it convenient to be confined to the fast lane. For this reason, they invade the slow lane, thus influencing the dynamics of trucks. The two equations in system (9) are then fully coupled.As before, we choose for both classes a family of triangular fundamental diagrams which extend, by continuity, those defined in ${\mathcal{D}}_{1}$, as shown in Figure 7.

#### 3.2. Multi-Scale Model

#### 3.2.1. Microscopic Model for Heavy Vehicles

#### 3.2.2. Full Model

#### 3.3. Extension of the Models to General Road Networks

#### 3.3.1. Any Number of Lanes

#### 3.3.2. Junctions

## 4. Numerical Approximation and Calibration

#### 4.1. Macroscopic Model

#### 4.2. Multi-Scale Model

## 5. Numerical Results

#### 5.1. Macroscopic Model

#### 5.1.1. Test 1A: Creeping

#### 5.1.2. Test 2A: Cars’ Congestion Affects Truck Dynamics

#### 5.1.3. Test 3A: Stop and Go Wave

#### 5.2. Multi-Scale Model

#### 5.2.1. Test 1B: Creeping Effect

#### 5.2.2. Test 2B: Cars’ Congestion Affects Truck Dynamics

#### 5.2.3. Test 3B: Merge

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The Italian motorway A4 Trieste–Venice and its branches to/from Udine, Pordenone and Gorizia, managed by Autovie Venete S.p.A.

**Figure 2.**Typical weekly (from Monday to Sunday) flux data on the A4 motorway of (

**a**) light and (

**b**) heavy vehicles collected on March 2019 near Redipuglia. Smoothed data are plotted in black. Note the flux drop of cars in the middle of the day and of trucks on the weekend.

**Figure 3.**Typical daily (Thursday) flux and velocity data on the A28 motorway of (

**a**–

**c**) light and (

**b**–

**d**) heavy vehicles collected in May 2019 near Sesto al Reghena.

**Figure 4.**Creeping phenomenon registered in May 2019 near Portogruaro: (

**a**) Light vehicles move in the fast lane even if (

**b**) heavy vehicles queue in the slow lane. (

**c**) Light vehicles’ velocity drops from ∼140 to ∼60 km/h and then to ∼20 km/h while (

**d**) heavy vehicles are completely stopped.

**Figure 6.**Fundamental diagrams of the macroscopic model in the partial coupling phase, i.e., $({\rho}_{\mathrm{L}},{\rho}_{\mathrm{H}})\in {\mathcal{D}}_{1}$. (

**a**) Light vehicles, (

**b**) heavy vehicles.

**Figure 7.**Fundamental diagrams of the macroscopic model in the full coupling phase, i.e., $({\rho}_{\mathrm{L}},{\rho}_{\mathrm{H}})\in {\mathcal{D}}_{2}$. (

**a**) Light vehicles, (

**b**) heavy vehicles.

**Figure 9.**(

**a**) Flux–density and (

**b**) velocity–density relationships for cars with real data superimposed.

**Figure 10.**(

**a**) Flux–density and (

**b**) velocity–density relationships for trucks with real data superimposed.

**Figure 11.**Simulated trajectories obtained with real inflow data as left boundary conditions (not all vehicles are plotted for visualization purposes). Horizontal blue lines and vertical black lines indicate, respectively, the position and the duration of the real queue as measured in the field.

**Figure 12.**Test 1A: (

**a**) Density and (

**b**) velocity of light and heavy vehicles as a function of space at final time. (

**c**) Density of light and (

**d**) heavy vehicles in space–time.

**Figure 13.**Test 2A: (

**a**) Density and (

**b**) velocity of light and heavy vehicles as a function of space at final time. (

**c**) Density of light and (

**d**) heavy vehicles in space–time.

**Figure 14.**Test 3A: (

**a**) Density and (

**b**) velocity of light and heavy vehicles as a function of space at $t=\Delta t$ (i.e., just after the initial time). (

**c**) Density of light and (

**d**) heavy vehicles in space–time. The evolution of the initial perturbation in the truck density starting at 9 km is perfectly visible, which creates, in turn, a perturbation in the car density.

**Figure 15.**Test 1B: (

**a**) Trajectories of trucks in space–time (for visualization purposes, not all trucks are actually plotted). When the first truck stops, a queue is formed behind. (

**b**) Car density, car velocity and car maximal density given the number of trucks at final time. Creeping is visible between 7 and 9 km.

**Figure 16.**Test 2B: (

**a**) Trajectories of trucks in space–time (for visualization purposes, not all trucks are actually plotted). They stop for a while and then accelerate. (

**b**) Car density, car velocity and car maximal density given the number of trucks at final time.

**Figure 17.**Test 3B: (

**a**–

**c**) Trajectories of trucks in space–time on the first incoming road, second incoming road and outgoing road, respectively (for visualization purposes, not all trucks are actually plotted). (

**d**,

**e**) Car density on second incoming road and outgoing road, respectively.

Light Vehicles | Heavy Vehicles | |
---|---|---|

veh. length + safety dist. (km) | $7.5\times {10}^{-3}$ | $18\times {10}^{-3}$ |

max max density (veh/km) | ${\rho}_{\mathrm{L}}^{*}\left(0\right)=267$ | ${\rho}_{\mathrm{H}}^{*}\left(0\right)=56$ |

min max density (veh/km) | ${\rho}_{\mathrm{L}}^{*}\left({\rho}_{\mathrm{H}}^{\mathrm{max}}\right)=133$ | ${\rho}_{\mathrm{H}}^{*}\left({\rho}_{\mathrm{L}}^{\mathrm{max}}\right)=0$ |

max max speed (km/h) | ${v}_{\mathrm{L}}(0,0)=130$ | ${v}_{\mathrm{H}}(0,0)=90$ |

min max speed (km/h) | ${v}_{\mathrm{L}}(0,{\rho}_{\mathrm{H}}^{\mathrm{max}})=65$ | ${v}_{\mathrm{H}}({\rho}_{\mathrm{L}}^{\mathrm{max}},0)=0$ |

max max flux (veh/h) | ${f}_{\mathrm{L}}({\sigma}_{\mathrm{L}}\left(0\right),0)=4200$ | ${f}_{\mathrm{H}}(0,{\sigma}_{\mathrm{H}}\left(0\right))=1500$ |

min max flux (veh/h) | ${f}_{\mathrm{L}}({\sigma}_{\mathrm{L}}\left({\rho}_{\mathrm{H}}^{\mathrm{max}}\right),{\rho}_{\mathrm{H}}^{\mathrm{max}})=1200$ | ${f}_{\mathrm{H}}({\rho}_{\mathrm{L}}^{\mathrm{max}},{\sigma}_{\mathrm{H}}\left({\rho}_{\mathrm{L}}^{\mathrm{max}}\right))=0$ |

$\delta $ | $50\times {10}^{-3}$ | km |

${\Delta}_{\mathrm{close}}$ | $25\times {10}^{-3}$ | km |

${\Delta}_{\mathrm{far}}$ | $50\times {10}^{-3}$ | km |

${V}_{\mathrm{H}}^{\mathrm{max}}$ | 90 | km/h |

${\tau}_{\mathrm{dec}}$ | $2\times {10}^{-4}$ | h |

${\tau}_{\mathrm{acc}}$ | $1.4\times {10}^{-2}$ | h |

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**MDPI and ACS Style**

Briani, M.; Cristiani, E.; Ranut, P. Macroscopic and Multi-Scale Models for Multi-Class Vehicular Dynamics with Uneven Space Occupancy: A Case Study. *Axioms* **2021**, *10*, 102.
https://doi.org/10.3390/axioms10020102

**AMA Style**

Briani M, Cristiani E, Ranut P. Macroscopic and Multi-Scale Models for Multi-Class Vehicular Dynamics with Uneven Space Occupancy: A Case Study. *Axioms*. 2021; 10(2):102.
https://doi.org/10.3390/axioms10020102

**Chicago/Turabian Style**

Briani, Maya, Emiliano Cristiani, and Paolo Ranut. 2021. "Macroscopic and Multi-Scale Models for Multi-Class Vehicular Dynamics with Uneven Space Occupancy: A Case Study" *Axioms* 10, no. 2: 102.
https://doi.org/10.3390/axioms10020102