# Generalization Second Order Macroscopic Traffic Models via Relative Velocity of the Congestion Propagation

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## Abstract

**:**

## 1. Introduction

## 2. Generalized Model

#### 2.1. Relative Velocity of the Congestion Propagation

#### 2.2. Model Equations

**Theorem**

**1.**

**Remark**

**1.**

- First, we can define the relative velocity of the congestion propagation by using equilibrium velocity, as was carried out in ([8]), $c\left(\rho \right)=\rho \frac{\partial V\left(\rho \right)}{\partial \rho}$. The function for equilibrium velocity $V\left(\rho \right)$ (fundamental diagram) can be empirically set using traffic detector data for a long period of time for each segment of the road.
- Second, we can do this without using equilibrium velocity or any form of a fundamental diagram. We approximate the value of the relative velocity of the congestion propagation using traffic density and velocity measured at the current time instant:$$c\left(\rho \right)=\frac{{\rho}_{in}+{\rho}_{out}}{2}\left(\frac{{v}_{out}-{v}_{in}}{{\rho}_{out}-{\rho}_{in}}\right)$$

## 3. Computational Method

## 4. Numerical Results

- Free flow: $0\le \rho <{\rho}_{1}$
- Synchronized flow: ${\rho}_{1}\le \rho <{\rho}_{2}$
- Wide moving jam: ${\rho}_{2}\le \rho \le {\rho}_{max}$

- Free flow: $\left\{\begin{array}{c}V\left(\rho \right)={\alpha}_{2}\rho +{\alpha}_{1}\hfill \\ c\left(\rho \right)={\alpha}_{2}\rho \hfill \end{array}\right.0\le \rho <{\rho}_{1}$
- Synchronized flow: $\left\{\begin{array}{cc}& V\left(\rho \right)={\beta}_{2}\rho +{\beta}_{1}+\frac{{\beta}_{0}}{\rho}\hfill \\ & c\left(\rho \right)={\beta}_{2}\rho -\frac{{\beta}_{0}}{\rho}\hfill \end{array}\right.,{\rho}_{1}\le \rho <{\rho}_{2}$
- Wide moving jam: $\left\{\begin{array}{cc}& V\left(\rho \right)={c}_{*}\left(\frac{{\rho}_{max}}{\rho}-1\right)\hfill \\ & c\left(\rho \right)=-\frac{{c}_{*}{\rho}_{max}}{\rho}\hfill \end{array}\right.,{\rho}_{2}\le \rho \le {\rho}_{max}$

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Traffic detectors $\#1$ and $\#2$ chosen in the segment of the I-580 freeway in California, USA. Data from detector $\#1$ were used as a left boundary condition for modeling, and data from the downstream detector $\#2$ were used for the verification of modeling results.

**Figure 3.**Equilibrium speed $V\left(\rho \right)$ (solid green line) and relative velocity of the congestion propagation $c\left(\rho \right)=\rho \frac{\partial V\left(\rho \right)}{\partial \rho}$ (dashed blue line) as the functions of density for the first lane of I-580 together with the historic data for the one-year period. Left: detector $\#1$, right: detector $\#2$.

**Figure 4.**Equilibrium speed $V\left(\rho \right)$ (solid green line) and relative velocity of the congestion propagation $c\left(\rho \right)=\rho \frac{\partial V\left(\rho \right)}{\partial \rho}$ (dashed blue line) as functions of density aggregated over four lanes of I-580, together with the historic data for the one-year period. Left: detector $\#1$, right: detector $\#2$.

**Figure 5.**The comparison of calculated flows (top) and velocities (bottom) from the first lane of I-580 and observed values from detector $\#2$ (dashed grey lines). On the right, flow (top) and velocity (bottom) relative errors in logarithmic scale. The simulation results obtained with a different expression for the relative velocity of the congestion propagation, $c\left(\rho \right)$, are shown in different colors: green, model (with $c\left(\rho \right)=\rho \frac{\partial v\left(\rho \right)}{\partial \rho}$ from (29); blue, the same model (15) with $c\left(\rho \right)=\rho \frac{\partial V\left(\rho \right)}{\partial \rho}$ as an empirical function of density.

**Figure 6.**The comparison of calculated flows (top) and velocities (bottom) aggregated over four lanes of I-580 and observed values from detector $\#2$ (dashed grey lines). On the right, flow (top) and velocity (bottom) relative errors in logarithmic scale. The simulation results obtained with a different expression for the relative velocity of the congestion propagation, $c\left(\rho \right)$, are shown in different colors: green, model (with $c\left(\rho \right)=\rho \frac{\partial v\left(\rho \right)}{\partial \rho}$ from (29); blue, the same model (15) with $c\left(\rho \right)=\rho \frac{\partial V\left(\rho \right)}{\partial \rho}$ as an empirical function of density.

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**MDPI and ACS Style**

Kholodov, Y.; Alekseenko, A.; Kazorin, V.; Kurzhanskiy, A.
Generalization Second Order Macroscopic Traffic Models via Relative Velocity of the Congestion Propagation. *Mathematics* **2021**, *9*, 2001.
https://doi.org/10.3390/math9162001

**AMA Style**

Kholodov Y, Alekseenko A, Kazorin V, Kurzhanskiy A.
Generalization Second Order Macroscopic Traffic Models via Relative Velocity of the Congestion Propagation. *Mathematics*. 2021; 9(16):2001.
https://doi.org/10.3390/math9162001

**Chicago/Turabian Style**

Kholodov, Yaroslav, Andrey Alekseenko, Viktor Kazorin, and Alexander Kurzhanskiy.
2021. "Generalization Second Order Macroscopic Traffic Models via Relative Velocity of the Congestion Propagation" *Mathematics* 9, no. 16: 2001.
https://doi.org/10.3390/math9162001