Triangular Fundamental Diagram Research Articles

Self-organized criticality of traffic flow: Implications for congestion management technologies

Self-organized criticality (SOC) is a celebrated paradigm from the 90’s for understanding dynamical systems naturally driven to its critical point, where the power-law dynamics taking place make predictions practically impossible, such as in stock prices, earthquakes, pandemics and many other problems in science related to phase transitions. Shortly thereafter, it was realized that traffic flow might be in the SOC category, implying that conventional traffic management strategies seeking to maximize the local flows can become detrimental. This paper shows that the Kinematic Wave model with triangular fundamental diagram, and many other related traffic models, indeed exhibit SOC, thanks in part to the fractal nature of traffic exposed here on the one hand, and our need to get to our destinations as soon as possible, on the other hand.Important implications for congestion management of traffic near the critical region are discussed, such as: (i) Jam sizes obey a power-law distribution with exponent 1/2, implying that both its mean and variance become ill-defined and therefore impossible to estimate. (ii) Traffic in the critical region is chaotic in the sense that predictions becomes extremely sensitive to initial conditions. (iii) However, aggregate measures of performance such as delays and average speeds are not heavy tailed, and can be characterized exactly by different scalings of the Airy distribution, (iv) Traffic state time–space “heat maps” are self-affine fractals where the basic unit is a triangle, in the shape of the fundamental diagram, containing 3 traffic states: voids, capacity and jams. This fractal nature of traffic flow calls for analysis methods currently not used in our field.

Constructing a fundamental diagram for traffic flow with automated vehicles: Methodology and demonstration

Increasingly, commercial vehicles are equipped with automated vehicle (AV) features such as adaptive cruise control systems. The AV feature can automatically control the headway between the current vehicle and the preceding vehicle in an adaptive manner. The automatic control may lead to significantly different car- following motions compared with those of human-driven vehicles, which challenges the applicability of classic traffic flow theory to emerging road traffic with AVs. To investigate the impacts of commercial AVs on traffic flow, this paper proposes a general methodology that combines both empirical experiments and theoretical models to construct a fundamental diagram (FD), i.e., the foundation for traffic flow theory for AV traffic. To demonstrate the empirical experiment settings, we collected high-resolution trajectory data with multiple commercial AVs following one another in a platoon with different headway settings. The field experiment results revealed that the traditional triangular FD structure remains applicable to describe the traffic flow characteristics of AV traffic. Further, by comparing the FDs between AVs and human-driven vehicles, it was found that although the shortest AV headway setting can significantly improve road capacity, other headway settings may decrease road capacity compared with existing human-driven-vehicle traffic. It was also found that headway settings may affect the stability of traffic flow, which has been revealed by theoretical studies but was first verified by empirical AV data. With these findings, mixed traffic flow FDs were derived by incorporating different headway settings and AV penetration rates. The method proposed in this paper, including experiment designs, data collection approaches, traffic flow characteristics analyses, and mixed traffic flow FD construction approaches, can serve as a methodological foundation for studying future mixed traffic flow features with uncertain and evolving AV technologies.

A Fast Lax–Hopf Algorithm to Solve the Lighthill–Whitham–Richards Traffic Flow Model on Networks

Efficient and exact algorithms are important for performing fast and accurate traffic network simulations with macroscopic traffic models. In this paper, we extend the semianalytical Lax–Hopf algorithm in order to compute link inflows and outflows with the Lighthill–Whitham–Richards (LWR) model. Our proposed fast Lax–Hopf algorithm has a very low computational complexity. We demonstrate that some of the original algorithm’s operations (associated with the initial conditions) can be discarded, leading to a faster computation of boundary demands/supplies in network simulation problems for general concave fundamental diagrams (FDs). Moreover, the computational cost can be further reduced for triangular FDs and specific space–time discretizations. The resulting formulation has a performance comparable to the link transmission model and because it solves the original LWR model for a wide range of FD shapes, with any initial configuration, it is suitable to solve a broad range of traffic operations problems. As part of the analysis, we compare the performance of the proposed scheme with that of other well-known computational methods.

Motorway Tidal Flow Lane Control

The expansion of road infrastructure, in spite of increasing congestion levels, faces severe restrictions from all sorts: economical, environmental, social, or technical. An efficient and, usually, less expensive alternative to improve mobility and the use of available infrastructure is the adoption of traffic management. A particular case of interest occurs when inbound and outbound traffic on a given facility is unbalanced throughout the day. This scenario may benefit of a lane management strategy called tidal flow (or reversible) lane control, in which case the direction of one or more contraflow buffer lanes is reversed according to the needs of each direction. This paper proposes a simple and practical real-time strategy for efficient motorway tidal flow lane control. A state-feedback switching policy based on the triangular fundamental diagram, that requires only aggregated measurements of density, is adopted. A theoretical analysis based on the kinematic wave theory shows that the strategy provides a Pareto-optimal solution. Microsimulations using empirical data from the A38(M) Aston Expressway in Birmingham, UK, are used to demonstrate the operation of the proposed strategy. The robustness of the switching policy to parameter variations is demonstrated by parametric sensitivity analysis. Simulation results confirm an increase of motorway throughput and a smooth operation for the simulated scenarios.

Determining the Breakpoints of Fundamental Diagrams

The breakpoints of fundamental diagrams (FDs) are an important foundation for estimating capacity, identifying traffic conditions, and determining the phase regime of multi-regime FDs. Unfortunately, to our best knowledge, there has been little research on determining breakpoints in a scientific way. This paper applies a Gaussian mixture model (GMM) to describe multi-component traffic conditions, and introduces the GMMs of traffic flow parameters (speed and occupancy). In order to avoid the possible singularities or degeneracies of typical maximum likelihood (ML) estimation, maximum a posteriori (MAP) estimation is introduced for the parameter estimation. The minimum cumulative probability method is then proposed for the determination of critical parameters at breakpoints. Nine datasets collected from the Caltrans Performance Measurement System (PeMS) are used for the estimation and validation of the proposed methods. The results show that the proposed methods can effectively divide up traffic conditions and determine breakpoints. Meanwhile, the effects of mixture weight and vehicle type on breakpoints are investigated and the proposed methods show stable performance in the determination of breakpoints under different conditions. The estimated critical occupancies and speeds are also applied for capacity estimation and calibration of the triangular FD model and three-regime speed-occupancy relationship.

On the value of relative flow data

Traffic flow can be described using three dimensions, i.e., space x, time t and cumulative flow N. This study considers estimating the cumulative flow over space and time, i.e., N(x,t), using relative flow data collected by stationary and moving observers. Stationary observers, e.g., loop-detectors, can observe flow at fixed position over time. Furthermore, automated or other equipped and connected vehicles can serve as moving observers that observe flow relative to their position over time. To present the value of relative flow data, in this paper, we take the perspective of a model-based estimation approach. In this approach, the data is used in two processes: (1) information assimilation of real-time data and models and (2) learning of the models used in information assimilation based on historical data. This paper focuses on traffic state estimation on links. However, we explain that, in absence of stationary observer that are positioned at the link boundaries, it is valuable to consider the information propagation over nodes. Throughout this study a LWR-model with a triangular fundamental diagram (FD) is used to develop the principles that can be used for the two processes. These principles are tested in a simulation (VISSIM) study. This study shows that we can find the traffic flow model parameters and can partially estimate the link boundary conditions based on relative flow data collected by moving observers alone. It also shows that the traffic flow behavior differs partially from the LWR-model with triangular FD, and therefore, we recommend the option to learn and use other traffic flow models in future research. Overall, relative flow data is considered valuable to obtain model learning datasets and to estimate the traffic state.

Nonstandard second-order formulation of the LWR model

ABSTRACTIn this study, we present an equivalent second-order formulation of the LWR model based on Phillips’ model, in which the acceleration rate equals a relaxation term with an infinitesimal relaxation time. We convert the model into a continuum car-following model and demonstrate its equivalence to that of the LWR model. We demonstrate that the nonstandard second-order model is stable, but the original Phillips model is not. We derive conditions for the nonstandard model to be forward-traveling and collision-free, prove that the collision-free condition is consistent with but more general than the CFL condition, and demonstrate that only anisotropic and symplectic Euler discretization methods lead to physically meaningful solutions. Numerically, the nonstandard second-order model has the same shock and rarefaction wave solutions as the LWR model for both Greenshields and triangular fundamental diagrams; for a non-concave fundamental diagram, the collision-free condition, but not the CFL condition, yields physically meaningful results.

A behavioral microeconomic foundation for car-following models

The objective of this paper is to develop a micro-economic modeling approach for car-following behaviors that may capture different risk-taking tendencies when dealing with different traffic conditions. The proposed framework allows for the consideration of perception subjectivity and judgement errors that may lead to unsafe acceleration driving maneuvers with the possibility of real-end collisions. The modeling approach relies on a generalized utility-based formulation with three specific types of subjective utility functions (SUFs): Prospect Utility (PT) subjective utility function, Constant Relative Risk Aversion (CRRA) subjective utility function, and an Exponential Constant Relative Risk Aversion (ECRA) subjective utility function. The formulation is assessed in terms of its homogeneous macroscopic properties (thus leading to a triangular fundamental diagram) and its non-homogeneous microscopic properties (thus leading to realistic following behavior facing different traffic scenarios).Once tested in terms of feasibility, the modeling approach is calibrated against real-life trajectory data. A Genetic Algorithm (GA) method is adopted to minimize a spacing-mixed-error term while considering inter-driving heterogeneity. The three models (i.e. PT, CRRA and ECRA) produce acceptable error values with the PT model showing the best fit, followed by the ECRA model, followed by the CRRA model. Even though the CRRA and the PT models show similar intensity in the acceleration response to the behavior of a lead vehicle with more disturbance (stochasticity/randomness) seen with the CRRA SUF, the PT and the ECRA models show more realistic wave formation despite their difference in terms of individual acceleration distribution functions. The ECRA model results in an amplified sensitivity to the behavior of the lead vehicle with the acceleration probability distribution function skewed to the left (i.e. towards decelerating rather than accelerating).The calibration exercise is followed by a simulation exercise. The three suggested models produce a homogeneous congestion phase, but a clear transient single/multiple wave formation is seen with the PT and the ECRA models. These latter models are able to reproduce all the congestion regimes observed on real-world surface transportation networks. The ECRA is characterized by a decreased capacity and increased traffic disturbances with additional shockwave formation. Finally, the different models allow the possibility of perception or judgement errors with the explicit incorporation of a collision probability and a collision weight in the suggested formulation approach.

Fundamental diagram estimation by using trajectories of probe vehicles

The fundamental diagram (FD), also known as the flow–density relation, is one of the most fundamental concepts in the traffic flow theory. Conventionally, FDs are estimated by using data collected by detectors. However, detectors’ installation sites are generally limited due to their high cost, making practical implementation of traffic flow theoretical works difficult. On the other hand, probe vehicles can collect spatially continuous data from wide-ranging area, and thus they can be useful sensors for large-scale traffic management. In this study, a novel framework of FD estimation by using probe vehicle data is developed. It determines FD parameters based on trajectories of randomly sampled vehicles and a given jam density that is easily inferred by other data sources. A computational algorithm for estimating a triangular FD based on actual, potentially noisy traffic data obtained by multiple probe vehicles is developed. The algorithm was empirically validated by using real-world probe vehicle data on a highway. The results suggest that the algorithm accurately and robustly estimates the FD parameters.

On the value of relative flow data

Abstract Traffic flow can be described using three dimensions, i.e., space x, time t and cumulative flow N. This study considers estimating the cumulative flow over space and time, i.e., N ( x , t ) , using relative flow data collected by stationary and moving observers. Stationary observers, e.g., loop-detectors, can observe flow at fixed position over time. Furthermore, automated or other equipped and connected vehicles can serve as moving observers that observe flow relative to their position over time. To present the value of relative flow data, in this paper, we take the perspective of a model-based estimation approach. In this approach, the data is used in two processes: (1) information assimilation of real-time data and models and (2) learning of the models used in information assimilation based on historical data. This paper focuses on traffic state estimation on links. However, we explain that, in absence of stationary observer that are positioned at the link boundaries, it is valuable to consider the information propagation over nodes. Throughout this study a LWR-model with a triangular fundamental diagram (FD) is used to develop the principles that can be used for the two processes. These principles are tested in a simulation (VISSIM) study. This study shows that we can find the traffic flow model parameters and can partially estimate the link boundary conditions based on relative flow data collected by moving observers alone. It also shows that the traffic flow behavior differs partially from the LWR-model with triangular FD, and therefore, we recommend the option to learn and use other traffic flow models in future research. Overall, relative flow data is considered valuable to obtain model learning datasets and to estimate the traffic state.

Continuous-time general link transmission model with simplified fanning, Part II: Event-based algorithm for networks

In this paper a novel solution algorithm is proposed for solving general first order dynamic network loading (DNL) problems in general transport networks. This solution algorithm supports any smooth non-linear two regime concave fundamental diagram and adopts a simplified fanning scheme. It is termed eGLTM (event-based General Link Transmission Model) and is based on a continuous-time formulation of the kinematic wave model that adapts shockwave theory to simplify expansion fans. As the name suggests eGLTM is a generalisation of eLTM, which is a special case that solves the simplified first order model assuming a triangular fundamental diagram. We analyse the impact of modelling delay in the hypocritical branch of the fundamental diagram to assess the differences between the two models. In addition, we propose an additional stream of mixture events to propagate multi-commodity flow in event based macroscopic models, which makes both eLTM and eGLTM suitable for dynamic traffic assignment (DTA) applications. The proposed solution scheme can yield exact solutions as well as approximate solutions at a significantly lesser cost. The efficiency of the model is demonstrated in a number of case studies. Furthermore, different settings for our simplified fanning scheme are investigated as well as an extensive analysis on the effect of including route choice on the algorithms computational cost. Finally, a large scale case study is conducted to investigate the suitability of our newly proposed model in a practical context and assess its efficiency. In this study comparisons between eLTM and eGLTM are included to demonstrate the impact of aforementioned generalisation as well as the multi-commodity extension that is proposed.

Automatic identification of near-stationary traffic states based on the PELT changepoint detection

The existence of stationary states during peak periods has been an underlying assumption of many studies on analysis, operations, control, and management of transportation networks. In Cassidy (1998), a method was proposed to manually identify near-stationary states by visually inspecting transformed curves of cumulative total vehicle counts and occupancies. Such near-stationary states are important for calibrating fundamental diagrams, identifying active bottlenecks and incidents, and quantifying capacity drop magnitudes. To the best of our knowledge, however, there lacks an automatic method that can be applied to efficiently identify near-stationary states from a large amount of data.In this study, we attempt to fill this gap. We start with definitions of steady, stationary, and equilibrium states and discuss their relations. Then we present a novel four-step method for automatically identifying near-stationary states from raw loop-detector data: first, the raw data are pre-processed to obtain healthy datasets, fill in missing values, and normalize averaged vehicle counts and occupancies to the same scale; second, daily time series are partitioned into multiple candidate intervals based on the pruned exact linear time (PELT) changepoint detection method; third, the characteristics of the candidate intervals are calculated; and finally, near-stationary states are selected based on modified Cassidy’s gap and duration criteria. We further close the loop by presenting an algorithm to automatically determine the penalty threshold in the second step and the gap threshold in fourth step to ensure the quantity as well as the quality of identified near-stationary states. In a case study, we apply the proposed method to identify near-stationary states from a large set of 30-s raw loop-detector data at a freeway mainline station. We verify the validity of identified near-stationary states both directly and indirectly. The results show that the identified near-stationary states are valid with high quality and the calibrated triangular fundamental diagram is well-fitted and physically meaningful. We finally conclude by discussing some future improvements and potential applications.

Kinematic wave models of sag and tunnel bottlenecks

Sags and tunnels are critical traffic bottlenecks, as they can cause capacity reduction, capacity drop, and extreme low acceleration rates when vehicles accelerate away from the upstream queue. In this paper, we present a behavioral kinematic wave model to explain the three bottleneck effects of sags and tunnels. Assuming increasing time gaps, we derive location-dependent triangular fundamental diagrams to explain the capacity reduction effect; with a bounded acceleration constraint on the stationary states inside the capacity reduction zone, we demonstrate the occurrence of capacity drop and derive a formula to calculate the dropped capacity from the fundamental diagram, road geometry, and acceleration process; from the structure of continuous standing waves we verify the low acceleration rate out of the upstream queue. We also present a simplified phenomenological model of capacity drop at sag/tunnel bottlenecks and two Cell Transmission Models for numerical simulations. With four stationary trajectories at the Kobotoke tunnel in Japan, we calibrate and validate the behavioral model and find that the theoretical predictions match the observations very well. This study can help to develop better design and control strategies to improve the performance of a sag or tunnel bottleneck.

Macroscopic Fundamental Diagram for pedestrian networks: Theory and applications

The Network or Macroscopic Fundamental diagram (MFD) has been a topic receiving a lot of attention in the past decade. Both from a theoretical angle and from a more application-oriented perspective, the MFD has proven to be a powerful concept in understanding and managing vehicular network dynamics.In particular, the application in traffic management has inspired the research presented in this contribution, where we explore the existence and the characteristics of the pedestrian Macroscopic Fundamental Diagram (p-MFD). This is first of all done from a theoretical perspective, which results in the main contribution of this research showing how we can derive the p-MFD from assumed local fundamental diagrams (FDs). In doing so, we show that we can relate the average (out-)flow from a pedestrian network as a function of the average spatial density ρ¯ and the density spatial variation σ2. We show that the latter is essential to provide a reasonable description of the overall network conditions. For simple relations between density and speed (i.e. Greenshields and Underwood fundamental diagrams), we derive analytical results; for more commonly used FDs in pedestrian flow theory, such as the triangular FD of Newell or the FD of Weidmann, we show the resulting relation by proposing a straightforward simulation approach.As a secondary contribution of the paper, we show how the p-MFD can be constructed from pedestrian trajectory data stemming from either microsimulation or from experimental studies. We argue that the results found are in line with the theoretical results, providing further evidence for the validity of the p-MFD concept. We furthermore discuss concepts of hysteresis, also observed in vehicular network dynamics, due to the differences in the queue build up and recuperation phases.We finally present some applications of the presented concepts in crowd management, network level-of-service determination, and coarse-scale modelling.

Incorporating within link dynamics in an agent-based computationally faster and scalable queue model

ABSTRACTThe growing pace of urbanization increases the need of simulation models to handle large-scale scenarios in reasonable time. The present study proposes a fast spatial queue model, which is anchored to an agent-based travel demand simulation framework. The existing queue model is extended to produce more realistic flow dynamics by introducing backward travelling holes to mixed traffic conditions. In this approach, the space freed by a leading vehicle is not immediately available to the following vehicle. The resulting dynamics resembles Newell's simplified kinematic wave model. The space freed corresponding to each leaving vehicle is named as ‘hole’ and, as following vehicles occupy the space freed by leading vehicles, the hole travels backward. This results in triangular fundamental diagrams for traffic flow. The robustness of the model is tested with flow density and average bike passing rate contours. Spatio-temporal trajectories are presented to differentiate the queuing patterns. Finally, a comparison of the computational performance of the different link and traffic dynamics of the queue model is made.

Validation of an extended discrete first-order model with variable speed limits

This paper validates the prediction model embedded in a model predictive controller (MPC) of variable speed limits (VSLs). The MPC controller was designed based on an extended discrete first-order model with a triangular fundamental diagram. In our previous work, the extended discrete first-order model was designed to reproduce the capacity drop and the propagation of jam waves, and it was validated with reasonable accuracy without the presence of VSLs. As VSLs influence traffic dynamics, the dynamics including VSLs needs to be validated, before it can be applied as a prediction model in MPC. For conceptual illustrations, we use two synthetic examples to show how the model reproduces the key mechanisms of VSLs that are applied by existing VSL control approaches. Furthermore, the model is calibrated by use of real traffic data from Dutch freeway A12, where the field test of a speed limit control algorithm (SPECIALIST) was conducted. In the calibration, the original model is extended by using a quadrangular fundamental diagram which keeps the linear feature of the model and represents traffic states at the under-critical branch more accurately. The resulting model is validated using various traffic data sets. The accuracy of the model is compared with a second-order traffic flow model. The performance of two models is comparable: both models reproduce accurate results matching with real data. Flow errors of the calibration and validation are around 10%. The extended discrete first-order model-based MPC controller has been demonstrated to resolve freeway jam waves efficiently by synthetic cases. It has a higher computation speed comparing to the second-order model-based MPC.

Extending the Link Transmission Model with non-triangular fundamental diagrams and capacity drops

The original Link Transmission Model as formulated by Yperman et al. (2006) simulates traffic according to Lighthill–Whitham–Richards theory with a very small numerical error, yet only supports triangular fundamental diagrams. This paper relaxes that restriction in two steps. Firstly, we extend the model to handle any continuous concave fundamental diagram, and prove that this extension is still consistent with Lighthill–Whitham–Richards theory. Secondly, we extend the theory and model to handle a capacity drop, explicitly looking into the handling of both the onset and release of congestion. The final model is still first-order and suitable for general networks. Numerical examples show that it qualitatively improves on the original model due to uniquely featuring complex traffic patterns including stop-and-go waves, with crisp shockwaves between traffic states, as well as acceleration fans.

Calibration of fundamental diagram using trajectories of probe vehicles: Basic formulation and heuristic algorithm

Abstract: A fundamental diagram (FD), also known as flow–density relation, is one of the most important principles in traffic flow theory. It would be valuable if an FD could be calibrated by using GPS-equipped probe vehicles; since they can continuously collect data from wide spatiotemporal area, compared to traditional fixed sensors. This paper proposes methods for calibrating an FD from trajectories of sampled vehicles. We formulate a method that identifies values of a free-flow speed and a critical density of a tri-angular FD, while it relies on exogenous assumptions on FD's functional form and a value of its jam density. Then, a heuristic algorithm for FD calibration in actual traffic environment is developed based on the proposed method. It was validated using traffic data generated by microscopic traffic simulator. The results suggested that the proposed methods can calibrate an FD precisely and robustly. It implies that FDs in road sections on which congestion happens frequently can be calibrated using probe vehicles, if probe vehicle data were collected for a long period. Therefore, the proposed methods would contribute to significant improvement of applicability of probe vehicle-based traffic management methods.

Automatic fitting procedure for the fundamental diagram

ABSTRACTThe fundamental diagram of a road, including free-flow capacity and queue discharge rate, is very important for traffic engineering purposes. In the real word, most traffic measurements come from stationary loop detectors. This paper proposes a method to fit Wu's fundamental diagram to loop detector data. Wu's fundamental diagram is characterised by five parameters, being free-flow speed, wave speed, free-flow capacity, queue discharge rate and jam density. The proposed method entails fixing the wave speed and the free-flow speed. The method consists of two steps. We first use a triangular fundamental diagram to separate the congested branch from the free-flow branch. Then, the remaining three parameters of Wu's fundamental diagram are fitted on each branch using a least-square fit. This method is shown to be robust for cases tested in real life, and hence very noisy, data.

Analysis of moving bottlenecks considering a triangular fundamental diagram

Abstract A significant number of research efforts have studied and analyzed the case in which a vehicle is moving slower than the traffic stream. This phenomenon, known as a moving bottleneck, results in a disruption of traffic flow and may significantly impact the traffic stream behavior upstream, downstream and abreast the slow moving vehicle. In this paper, a macroscopic approach for modeling moving bottlenecks is developed using microscopically derived data considering a triangular fundamental diagram. The passing flow rates of different moving bottleneck scenarios are determined using a previously developed microscopic model based on simulated data derived from the INTEGRATION software. Using the simulation results, an explicit expression of the bottleneck diagram, a flow-density relationship that defines the phenomenon macroscopically is proposed and the behavior of the traffic stream downstream and abreast the moving obstruction is depicted. It is demonstrated that the behavior of the traffic stream downstream of the slow vehicle as well as the acceleration behavior while passing is governed by the demand level. Such a result is coherent and consistent, to a significant extent, with two decades of research related to modeling moving bottlenecks and constitutes a potential feasible and more detailed description of the phenomenon in the case of a triangular fundamental diagram. Finally, it is noteworthy that the research subject of this paper could be considered as a first step in developing a numerical and practitioner-friendly framework for the analysis of moving bottlenecks that does not involve approaching the problem from its theoretical perspective.