# Applicability of the mesoscopic Lattice Boltzmann Method to estimate road capacity depending on lane number decrease

Romain Noël, Alex Coiret

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Romain Noël, Alex Coiret. Applicability of the mesoscopic Lattice Boltzmann Method to estimate road capacity depending on lane number decrease. AIIT 2024 - 4th International Conference, Greening the way forward: sustainable transport infrastructure and systems, AIIT (Italian Association for Traffic and Transport Engineering), Sep 2024, Rome, Italy. 18 p. hal-04710811v2

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Submitted on 26 Sep 2024

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# Application of the lattice Boltzmann method to estimate road capacity decrease depending on lane number and flow density

Noël Romain¹,², Coiret Alex¹

¹ Gustave Eiffel University, SII/Cosys, Nantes, France.

² INRIA, I4S, Rennes, France.

September 19/20, 2024

<figure><img src="image_2.png" /></figure>

Université
Gustave Eiffel
1. Introduction

2. LBM method

3. Results

4. Conclusion
## Why ? How ? What ?

* Predict traffic flows to help public services and road planning.

* Exploring new numerical simulations.

* Try to study the influence of truck concentration, vehicle incoming density on congestions.

## Application

A numerical method to estimate the road capacity at critical points, without modeling each vehicle.
## Meso-scale

* Macro-scale is only looking at density and average speed flow etc.

* Micro-scale is able to track each vehicle.

* The Prigogine theory [4] is focusing on a in-between meso-scale, where vehicles are indiscernible but have statistical effect.

* It has the advantage of being efficient and detailed [1].

## Application to multi-classes

Multi-class traffic flow are addressed through interactions between statistical groups [6]
The LBM is a very efficient and flexible mesoscopic solver.

## LBM building and usage in other field [5]:

* Multi-phase fluid mechanics,

* Biological process,

* Thermal engineering.

<figure><img src="image_3.png" /><figcaption>Figure: Soap bubble</figcaption></figure>

## LBM for road traffic

* The Prigogine theory is a special case of the Boltzmann equation.

* LBM can capture non-linear effects: stop and go effects [2].

* LBM can be extended to multi-class traffic flows [3].
1. Introduction

2. LBM method

3. Results

4. Conclusion
<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>

Rewritten **streaming** step to include number of lane changes:

$$f_{c,i}(x + e_i\delta_t, t + \delta_t) = f_{c,i}(x, t + \delta_t) \frac{n_l(x)}{n_l(x + e_i\delta_t)}, \quad (1)$$
**Statistical interactions:**

$$f_{c,i}(x, t + \delta_t) = f_{c,i}(x, t) + \frac{1}{\tau} (f_{c,i}^{(0)} - f_{c,i}) . \quad (2)$$

Equilibrium distribution $f_{c,i}^{(0)}$ is computed from macroscopic quantities.

**Macroscopic quantities** recovered from statistical moments:

$$\text{density } \rho_c = \sum_{i=0}^{m-1} f_{c,i}(x, t), \quad \text{flow } q_c = \sum_{i=0}^{m-1} e_i f_{c,i}(x, t), \quad (3)$$
Fundamental diagram facing other methods [3]:

<figure><img src="image_6.png" /></figure>

Truck concentration parameter sensitivity:

<figure><img src="image_7.png" /></figure>
1. Introduction

2. LBM method

3. Results

4. Conclusion
<table><caption>Table 1.a Input parameters studied according to different scenarios, see table 2.</caption><thead><tr><th>Parameters</th><th>Values</th><th>Lines</th><th>Scenario</th><th>Situation</th></tr></thead><tbody><tr><td>(ρ<sub>w</sub>, α)</td><td>(0.10,0)</td><td>red dashed on fig. 5</td><td>S1-a</td><td>Free-flow</td></tr><tr><td>(ρ<sub>w</sub>, α)</td><td>(0.105,0)</td><td>blue solid on fig. 5</td><td>S1-b</td><td>Congested</td></tr><tr><td>(ρ<sub>w</sub>, α)</td><td>(0.105,0.115)</td><td>green dot-dashed on fig. 5</td><td>S1-c</td><td>Free-flow</td></tr><tr><td>(ρ<sub>w</sub>, α)</td><td>(0.10,0)</td><td>red dashed on fig. 6</td><td>S2-a</td><td>Free-flow</td></tr><tr><td>(ρ<sub>w</sub>, α)</td><td>(0.105,0)</td><td>blue solid on fig. 6</td><td>S2-b</td><td>Congested</td></tr><tr><td>(ρ<sub>w</sub>, α)</td><td>(0.105,0.115)</td><td>green dot-dashed on fig. 6</td><td>S2-c</td><td>Free-flow</td></tr></tbody></table>

<table><caption>Table 2: (a) Characteristics of road sections in scenarios S1 and S2.</caption><thead><tr><th>Scenario name</th><th>Road parameters</th><th>Values</th><th>Space</th><th>Description</th></tr></thead><tbody><tr><td rowspan="9">S1</td><td>n<sub>t</sub></td><td>4</td><td>∀x ∈ [0, 2500]</td><td rowspan="9">lane reduction with 2 sections</td></tr><tr><td>v<sub>m,1</sub></td><td>5</td><td>∀x ∈ [0, 2500]</td></tr><tr><td>v<sub>m,2</sub></td><td>4</td><td>∀x ∈ [0, 2500]</td></tr><tr><td>n<sub>t</sub></td><td>2</td><td>∀x ∈ [2500, 5000]</td></tr><tr><td>v<sub>m,1</sub></td><td>5</td><td>∀x ∈ [2500, 5000]</td></tr><tr><td>v<sub>m,2</sub></td><td>4</td><td>∀x ∈ [2500, 5000]</td></tr><tr><td>n<sub>t</sub></td><td>4</td><td>∀x ∈ [0, 2000]</td></tr><tr><td>v<sub>m,1</sub></td><td>5</td><td>∀x ∈ [0, 2000]</td></tr><tr><td>v<sub>m,2</sub></td><td>4</td><td>∀x ∈ [0, 2000]</td></tr><tr><td rowspan="9">S2</td><td>n<sub>t</sub></td><td>3</td><td>∀x ∈ [2000, 3750]</td><td rowspan="9">lane reduction with 3 sections</td></tr><tr><td>v<sub>m,1</sub></td><td>5</td><td>∀x ∈ [2000, 3750]</td></tr><tr><td>v<sub>m,2</sub></td><td>4</td><td>∀x ∈ [2000, 3750]</td></tr><tr><td>n<sub>t</sub></td><td>2</td><td>∀x ∈ [3750, 5000]</td></tr><tr><td>v<sub>m,1</sub></td><td>5</td><td>∀x ∈ [3750, 5000]</td></tr><tr><td>v<sub>m,2</sub></td><td>4</td><td>∀x ∈ [3750, 5000]</td></tr><tr><td>n<sub>t</sub></td><td>2</td><td>∀x ∈ [3750, 5000]</td></tr><tr><td>v<sub>m,1</sub></td><td>5</td><td>∀x ∈ [3750, 5000]</td></tr><tr><td>v<sub>m,2</sub></td><td>4</td><td>∀x ∈ [3750, 5000]</td></tr></tbody></table>
<figure><img src="image_8.png" /><figcaption>Figure 5. Road change from 4 to 2 lanes: red dashed line corresponds to S1-a, blue solid line corresponds to S1-b, dashed green corresponds to S1-c, see table 1.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Figure 6. Road change from 4 to 3 and 3 to 2 lanes: red dashed line corresponds to S2-a, blue solid line corresponds to S2-b, dashed green corresponds to S2-c, see table 1.</figcaption></figure>
<table><caption>Table 1.b Input parameters studied according to different scenarios, see table 2.</caption><thead><tr><th>Parameters</th><th>Values</th><th>Lines</th><th>Scenario</th><th>Situation</th></tr></thead><tbody><tr><td>(ρ<sub>in</sub>, α)</td><td>(0.169, 0)</td><td>red dashed on fig. 7</td><td>S3-a</td><td>Free-flow</td></tr><tr><td>(ρ<sub>in</sub>, α)</td><td>(0.17, 0)</td><td>blue solid on fig. 7</td><td>S3-b</td><td>Congested</td></tr><tr><td>(ρ<sub>in</sub>, α)</td><td>(0.17, 0.03)</td><td>green dot-dashed on fig. 7</td><td>S3-c</td><td>Free-flow</td></tr><tr><td>(ρ<sub>in</sub>, α)</td><td>(0.169, 0)</td><td>red dashed on fig. 8</td><td>S4-a</td><td>Free-flow</td></tr><tr><td>(ρ<sub>in</sub>, α)</td><td>(0.17, 0)</td><td>blue solid on fig. 8</td><td>S4-b</td><td>Congested</td></tr><tr><td>(ρ<sub>in</sub>, α)</td><td>(0.17, 0.03)</td><td>green dot-dashed on fig. 8</td><td>S4-c</td><td>Free-flow</td></tr></tbody></table>

<table><caption>Table 2: (b) Characteristics of road sections in scenarios S3 and S4.</caption><thead><tr><th>Scenario name</th><th>Road parameters</th><th>Values</th><th>Space</th><th>Description</th></tr></thead><tbody><tr><td rowspan="10">S3</td><td>n<sub>1</sub></td><td>3</td><td>∀x ∈ [0, 2500]</td><td rowspan="10">speed reduction with 2 sections</td></tr><tr><td>v<sub>m,1</sub></td><td>5</td><td>∀x ∈ [0, 2500]</td></tr><tr><td>v<sub>m,2</sub></td><td>4</td><td>∀x ∈ [0, 2500]</td></tr><tr><td>n<sub>1</sub></td><td>3</td><td>∀x ∈ [2500, 5000]</td></tr><tr><td>v<sub>m,1</sub></td><td>3</td><td>∀x ∈ [2500, 5000]</td></tr><tr><td>v<sub>m,2</sub></td><td>3</td><td>∀x ∈ [2500, 5000]</td></tr><tr><td>n<sub>1</sub></td><td>3</td><td>∀x ∈ [0, 2000]</td></tr><tr><td>v<sub>m,1</sub></td><td>5</td><td>∀x ∈ [0, 2000]</td></tr><tr><td>v<sub>m,2</sub></td><td>4</td><td>∀x ∈ [0, 2000]</td></tr><tr><td rowspan="10">S4</td><td>n<sub>1</sub></td><td>3</td><td>∀x ∈ [2000, 3750]</td><td rowspan="10">speed reduction with 3 sections</td></tr><tr><td>v<sub>m,1</sub></td><td>4</td><td>∀x ∈ [2000, 3750]</td></tr><tr><td>v<sub>m,2</sub></td><td>4</td><td>∀x ∈ [2000, 3750]</td></tr><tr><td>n<sub>1</sub></td><td>3</td><td>∀x ∈ [3750, 5000]</td></tr><tr><td>v<sub>m,1</sub></td><td>3</td><td>∀x ∈ [3750, 5000]</td></tr><tr><td>v<sub>m,2</sub></td><td>3</td><td>∀x ∈ [3750, 5000]</td></tr><tr><td>n<sub>1</sub></td><td>3</td><td>∀x ∈ [3750, 5000]</td></tr><tr><td>v<sub>m,1</sub></td><td>3</td><td>∀x ∈ [3750, 5000]</td></tr><tr><td>v<sub>m,2</sub></td><td>3</td><td>∀x ∈ [3750, 5000]</td></tr></tbody></table>
<figure><img src="image_10.png" /><figcaption>Figure 7. Road change from 5 to 3 maximum speed: red dashed line corresponds to S3-a, blue solid line corresponds to S3-b, dashed green corresponds to S3-c, see table 1.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Figure 8. Road change from 5 to 4 and 4 to 3 maximum speed: red dashed line corresponds to S4-a, blue solid line corresponds to S4-b. dashed green corresponds to S4-c, see table 1.</figcaption></figure>
1. Introduction

2. LBM method

3. Results

4. Conclusion
* LBM is a fast and efficient method,

* it yields results comparable at macro-scale,

* the introduced mesoscopic fluctuation could help to capture more realistic behaviour,

* it has the advantage to be more computational efficient over the microscopic simulations

* In perspectives, this work should be extended to a large set of incoming vehicle density to better characterize the critical flows
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Lattice Boltzmann model for traffic flow.

*Physical Review E*, 77(3):036108, March 2008.

[3] Romain Noël, Laurent Navarro, and Guy Courbebaisse.

Lattice boltzmann method for heterogeneous multi-class traffic flow.

*Journal of Computational and Theoretical Transport*, 50(1):27–51, 2021.
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[5] S. Succi.
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