Second-order Macroscopic Model Research Articles

A high-resolution meshfree particle method for numerical investigation of second-order macroscopic pedestrian flow models

Recent advancements in empirical observations of human psychological responses have revealed that the governing rules of human behavioral aspects can not only be predicted with adequate deterministic precision but also forged into mathematical formulations. These modeling studies enable crisis managers with reliable simulation tools to gain insights into critical facets of crowd disasters and enhance their decision-making abilities to facilitate safe egress in emergency situations. The macroscopic scale of representation in this context is often invoked to comprehend large-scale collective pattern formation in vast built environments, owing to its computational viability. The associated governing equations are typically constructed in a system of hyperbolic conservation law form, and conducting simulations in real-life complex geometric configurations can pose computational challenges. This article puts forward a high-resolution, shock-capturing meshfree particle method in an Eulerian framework for the numerical approximation of several widely adopted macroscopic pedestrian flow models. It serves as a superior alternative to traditional mesh-based methods, eliminating the need for specific mesh structures and connectivity between them. A classical Generalized Finite Difference Method (GFDM) in conjunction with several consistency conditions on spatial derivative coefficients is adopted to obtain a non-oscillatory and fairly conservative Godunov-type discretization of the governing system. The intercell flux is evaluated using a suitable Riemann solver, and a slope-limited reconstruction procedure of the corresponding Riemann states, adapted to arbitrary point distribution, is employed for improved accuracy. Moreover, the preferred direction of human motion is determined from a local density-dependent eikonal-type equation, which is solved using a generalized version of the Fast Marching Method. A thorough numerical investigation of three well-known second-order macroscopic models, namely Payne-Whitham (PW), Aw-Rascle (AR), and Aw-Rascle-Zhang (ARZ) is carried out through two hypothetical scenarios of crowd evacuation, not only to demonstrate the accuracy and applicability of the proposed approach but also to deepen understanding of their distinctive characteristics. A comparison of the flow flux with the fundamental diagram suggests that the density profile obtained with the ARZ model is equivalent to Hughes' first-order model. In contrast, the PW and AR models have proven to effectively replicate complex, unstable pedestrian phenomena such as stop-and-go waves. Finally, Helbing's counter-intuitive idea that strategically positioned barriers upstream of a bottleneck can essentially expedite evacuation is numerically investigated across various parameter choices.

Stability analysis of stochastic second-order macroscopic continuum models and numerical simulations

Second-order macroscopic continuum models have been constantly improving for decades to reproduce the empirical observations. Recently, a series of experimental studies have suggested that the stochastic factors contribute significantly to destabilizing traffic flow. Nevertheless, the traffic flow stability of the stochastic second-order macroscopic continuum model has not received the attention it deserves in past studies. More importantly, we have found that the destabilizing aspect of stochasticity is still not correctly validated in the existing theoretical stability analysis. In this paper, we analytically study the impact of stochasticity on traffic flow stability for a general stochastic second-order macroscopic model by using the direct Lyapunov method. Numerical simulations have been carried out for different typical stochastic second-order macroscopic models. Our analytical stability analysis has been validated, and our methodology has been proved more efficient. Our study has theoretically revealed that the presence of stochasticity has a destabilizing effect in usual stochastic macroscopic models.

Multilane analysis of a viscous second-order macroscopic traffic flow model

Vehicular flow modeling has received much attention in the past decade due to the consequential effect of the increasing number of vehicles. A notable effect is the congestion on urban and semi-urban roads. Traffic flow models are often the first point of reference in addressing these congestion problems. In that regard, a new viscous second-order macroscopic model is presented to explore some dynamics of multilane traffic. The new model accounts for viscosity and the velocity differentials across infinitely many countable lanes. It is realized that the wave properties of the proposed model are analogous to the driving setting on a Ghanaian highway. This is followed by a mathematical condition to achieving a stable traffic flow. Moreover, the viscous model is recast into its discrete form to address interdependency among unique multiple lanes. A simulation result of an eight-lane infrastructure is presented to explain this conceptualization.

A constrained filtering algorithm for freeway traffic state estimation

ABSTRACTA real-time traffic state estimation algorithm is developed and applied to a freeway. The evolution of the traffic is defined by a second-order macroscopic model which computes, for each section of the freeway, the density, and the mean speed according to several nonlinear equations. Different extensions of the Kalman method were already applied to this model, though none of them considers the natural constraints in the state variables. In this work, a new method that incorporates those natural constraints is applied to the macroscopic model obtaining better results. To validate the proposed method, a simulation over a freeway section was made using two different tools: the macroscopic simulator called METANET and the microscopic simulator called SUMO. Promising results were obtained using both approaches.

Microstructural Topological Sensitivities of the Second-Order Macroscopic Model for Waves in Periodic Media

We consider scalar waves in periodic media through the lens of a second-order effective i.e. macroscopic description, and we aim to compute the sensitivities of the germane effective parameters due to topological perturbations of a microscopic unit cell. Specifically, our analysis focuses on the tensorial coefficients in the governing mean-field equation – including both the leading order (i.e. quasi-static) terms, and their second-order companions bearing the effects of incipient wave dispersion. The results demonstrate that the sought sensitivities are computable in terms of (i) three unit-cell solutions used to formulate the unperturbed macroscopic model; (ii) two adoint-field solutions driven by the mass density variation inside the unperturbed unit cell; and (iii) the usual polarization tensor, appearing in the related studies of non-periodic media, that synthesizes the geometric and constitutive features of a point-like perturbation. The proposed developments may be useful toward (a) the design of periodic media to manipulate macroscopic waves via the microstructure-generated effects of dispersion and anisotropy, and (b) sub-wavelength sensing of periodic defects or perturbations.

Generalized macroscopic traffic model with time delay

The effect of delay or reaction time on traffic flow dynamics has been investigated widely in the literature using microscopic traffic models. Recent studies using second-order Payne-type models have shown analytically that, on a macroscopic scale, time delay does not contribute to whether traffic instabilities occur. This paper will attempt to show that it all depends on the (macroscopic) model used for the analysis that delay does have effect on traffic instabilities or not. To this end, we will formulate a generalized (linear) stability condition for a second-order macroscopic model with delay and investigate analytically the effect of such delay on traffic instabilities in some specific macroscopic models. It is found that the choice of the equilibrium speed function in a (second order) macroscopic model will determine how delay affects such (linear) stability condition

Optimal control of freeway systems based on a linearized prediction model

AbstractIn this paper a Model Predictive Control scheme is proposed in order to prevent congestion phenomena in freeways by means of variable speed limits. The prediction model to be used in the Model Predictive Control scheme is a linearized version of the first-order dynamic model of traffic flow, and this represents the main novelty of the present work. In particular, we approximate the steady-state speed-density characteristic as a piecewise constant function. The prediction model is then a simplified dynamic model that is obviously less effective to represent the real behaviour of traffic (than the second-order macroscopic model) but its simpler equations makes it suitable to be used on line in real freeway systems. As a matter of fact, in an on-line application of a Model Predictive Control scheme, the finite-horizon optimization problem must be solved in acceptable computational times, allowing a high speed of the regulator computing system; in this work this is achieved thanks to the linearized prediction model. Some numerical results are then reported in the paper, showing the effectiveness of the proposed control scheme.

Asymptotic limits of a discrete Kinetic Theory model of vehicular traffic

The paper proposes a rigorous method to construct the hyperbolic asymptotic limit of the discrete Kinetic Theory model of vehicular traffic proposed in [8]. A second-order macroscopic model of the Payne–Whitham type is derived and the coefficients of the equations are obtained from the detailed description of the microscopic interactions developed in the kinetic model.

Riemann Problem Resolution and Godunov Scheme for the Aw-Rascle-Zhang Model

Recently, Aw and Rascle and Zhang introduced a new second-order macroscopic model, following the theoretical investigations on the Payne-Whitham second-order modelling by Daganzo. The conserved variables in this model are the density and the relative flow. The aim of this paper is to solve the Riemann problem for the Aw-Rascle-Zhang (ARZ) model for all possible initial conditions, using an extended fundamental diagram. The resolution of the Riemann problem provides the user of the model with a set of nontrivial analytical solutions; it is also a prerequisite for the construction of numerical solution schemes. Some examples are given in which analytical solutions of the Riemann problem are discussed and compared to numerical solutions. The ARZ model shows better fit to real data than the embedded Lighthill-Whitham-Richards (LWR) model for the same set of physical parameters.

Parameter-dependent modeling of freeway traffic flow

In the paper a novel non-linear modeling formalism is suggested for freeway traffic flow. A second-order macroscopic model has been transformed into a Linear Parameter Varying (LPV) model. The paper proposes two LPV descriptions for traffic modeling: the first one covers completely the non-linear macroscopic dynamics, while the second one is an approximate description. Real test field topology and measurement data are used for validating the proposed modeling methodology. Simulation examples are given to compare real detector measurements, non-linear and the parameter-dependent responses.

Wave propagation of the traffic flow dynamic model based on wavefront expansion

This paper discusses propagation of perturbations along traffic flow modeled by a modified second-order macroscopic model through the wavefront expansion technique. The coefficients in this expansion satisfy a sequence of transport equations that can be solved analytically. One of these analytic solutions yields information about wavefront shock. Numerical simulations based on a Pade approximation of this expansion were done at the end of this paper and results showed that propagation of perturbations at traffic flow speed conforms to the theoretical analysis results.

Modeling the driver's behavior on second-order macroscopic models of vehicular traffic flow

This paper deals with the modeling and qualitative analysis, with special attention to hyperbolicity analysis, of a large class of second-order hydrodynamic traffic flow models. These models are obtained from conservation equations for mass and momentum properly closed by constitutive relations related to the modeling of the behaviour of drivers in the flow. The first part of the paper deals with defining the general mathematical setting for modeling second-order hydrodynamic equations of traffic flow related to conservation of mass and momentum, and the second part with analyzing the qualitative properties of some specific models. In particular, it is shown how some models known in the literature can be obtained as particular cases of the above framework. Finally, the third part develops a hyperbolicity analysis which is also referred to some specific models.