Macroscopic Fundamental Diagram Research Articles

An Iterative Adaptive Dynamic Programming Approach for Macroscopic Fundamental Diagram-Based Perimeter Control and Route Guidance

Macroscopic fundamental diagrams (MFDs) have been widely adopted to model the traffic flow of large-scale urban networks. Coupling perimeter control and regional route guidance (PCRG) is a promising strategy to decrease congestion heterogeneity and reduce delays in large-scale MFD-based urban networks. For MFD-based PCRG, one needs to distinguish between the dynamics of (a) the plant that represents reality and is used as the simulation tool and (b) the model that contains easier-to-measure states than the plant and is used for devising controllers, that is, the model-plant mismatch should be considered. Traditional model-based methods (e.g., model predictive control (MPC)) require an accurate representation of the plant dynamics as the prediction model. However, because of the inherent network uncertainties, such as uncertain dynamics of heterogeneity and demand disturbance, MFD parameters could be time-varying and uncertain. Conversely, existing data-driven methods (e.g., reinforcement learning) do not consider the model-plant mismatch and the limited access to plant-generated data, for example, subregional OD-specific accumulations. Therefore, we develop an iterative adaptive dynamic programming (IADP)-based method to address the limited data source induced by the model-plant mismatch. An actor-critic neural network structure is developed to circumvent the requirement of complete information on plant dynamics. Performance comparisons with other PCRG schemes under various scenarios are carried out. The numerical results indicate that the IADP controller trained with a limited data source can achieve comparable performance with the “benchmark” MPC approach using perfect measurements from the plant. The results also validate the IADP’s robustness against various uncertainties (e.g., demand noise, MFD error, and trip distance heterogeneity) when minimizing the total time spent in the urban network. These results demonstrate the great potential of the proposed scheme in improving the efficiency of multiregion MFD systems. Funding: The work was jointly funded by the National Natural Science Foundation of China under [Grant 72071214] (R. Zhong), the Dit4Tram project from the European Union’s Horizon 2020 Research and Innovation Programme under [Grant 953783] (N. Geroliminis), and the Research Student Attachment Programme of The Hong Kong Polytechnic University (C. Chen). Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2023.0091 .

New Capacity Model for Malaysian Urban Roads Using Macroscopic Fundamental Diagram and Multiple Linear Regression Approach

New Capacity Model for Malaysian Urban Roads Using Macroscopic Fundamental Diagram and Multiple Linear Regression Approach

Method for the Experimental Identification of Variables and Configurations That Modify the Shape of the Macroscopic Fundamental Diagram

In this paper, we propose a method for establishing if a variable is capable of modifying the Macroscopic Fundamental Diagram (MFD) of a street network. The variables have many different configurations, and a simulation is performed for each one. Then, based on the output data of each simulation, the representative speed, density, and flow of the network are calculated. We use three metrics to establish if a variable affects the MFD: the first establishes a distance between the compared density and speed patterns, the second establishes a distance between capacities, and the third establishes a distance between critical densities. We select four variables to test our method: the precision of driving, the vehicles’ top speeds distribution, the procedure for selecting routes, and the procedure for selecting destinations; we determine whether each of these variables can modify the MFD shape. Additionally, we detect which configurations of a variable are able to reach and exceed the critical density (causing congestion) so we can establish which configurations are sustainable and which are not. The novelties of this work are twofold: (1) we introduce a method to detect if a variable modifies the MFD; (2) we establish if the selected variables modify the MFD.

A comparison of the accumulation-based, trip-based and time delay macroscopic fundamental diagram models

Macroscopic fundamental diagram (MFD) is widely applied in network modelling and management, such as route guidance and vehicle relocation, which are formulated as generalised dynamic traffic assignment (DTA) problems. MFD can effectively reduce the spatial dimension thus making the generalised DTA problems computationally efficient. In the literature, three MFD models, the accumulation-based model, the trip-based model, and the time delay model, were proposed to capture the traffic flow propagation under different traffic conditions and demand scenarios. However, no consensus has been reached on their computational efficiency and which model should be chosen under certain traffic conditions and demand scenarios. In this paper, we revisit these models regarding two important theoretical properties regarding flow propagation in the DTA, i.e. the first-in-first-out (FIFO) principle and causality. Corresponding dynamic network loading algorithms are designed to compare their numerical accuracy and computational efficiency. Numerical comparisons with Lighthill-Whitham-Richards (LWR) model and a micro simulator confirm that the accumulation-based model is valid in saturation, the trip-based model is valid in free-flow, while the time delay model provides a good approximation in both free-flow and saturation scenarios. On the other hand, violation of strict causality is observed in the accumulation-based and trip-based models, rendering it hard to pursue analytical DTA. This issue is not observed in the time delay model. Overall, the time delay model is a promising alternative for dynamic network loading in large-scale network applications.

Investigating built environment and traffic flow impact on crash frequency in urban road networks

While numerous studies have examined the factors that influence crash occurrence, there remains a gap in understanding the intricate relationship between built environment, traffic flow, and crash occurrences across different spatial units. This study explores how built environment attributes, and dynamic traffic flow characteristics affect crash frequency by focusing on proposed traffic density-based zones (TDZs). Utilizing a comprehensive dataset from Greater Melbourne, Australia, this research emphasizes on the dynamic traffic flow variables and insights from the Macroscopic Fundamental Diagram model, considering parameters such as shockwave velocity and congestion index. The association between the potential influencing factors and crash frequency is examined using a random parameter negative binomial regression model. Results indicate that the data segmentation based on TDZs is instrumental in establishing a more refined crash model compared to traditional planning-based zones, as demonstrated by improved goodness-of-fit measures. Factors including density (e.g., employment density), network design (e.g., road density and highway density), land use diversity (e.g., job-housing balance and land use mixture), and public transit accessibility (e.g., bus route density) are significantly associated with crash occurrence. Furthermore, the unobserved heterogeneity effects of the shockwave velocity and congestion index on crashes are revealed. The study highlights the significance of incorporating dynamic traffic flow variables in understanding crash frequency variations across different spatial units. These findings can inform optimal real-time traffic monitoring, environmental design, and road safety management strategies to mitigate crash risks.

Investigating the Impact of Automated Vehicles on the Safety and Operation of Low-Speed Urban Networks

Automated vehicles (AVs) will appear on the road soon and will influence the properties of road traffic networks such as capacity and safety. While the impact of AV on traffic operations has been discussed extensively, most existing literature in this direction focuses on microscopic and mesoscopic levels. This study examines the effect of AVs on traffic flow operations and crash risks at a network scale. In addition, this study discusses the implications for designing and operating safe, low-speed urban road networks. Simulation experiments were carried out in a grid road network with AVs and human-driven vehicles operating in mixed flow conditions. To assess traffic performance, the macroscopic fundamental diagram (MFD) relationships were estimated, specifically focusing on speed-density correlations. From this analysis, it was possible to extract key traffic performance indices such as capacity and critical speed. Using time-to-collision surrogate safety measures, macroscopic safety diagrams were generated which associate the level of congestion with the potential crash conflicts among vehicles at an aggregated spatial scale. Utilizing this knowledge, a novel multiobjective optimization based on the NSGA-II algorithm was applied to identify the optimal trade-off between efficiency and safety. The presence of AVs was found to have a positive impact on the capacity, critical density, and average speed on a system level, even in low-speed scenarios. Moreover, AVs can result in increased critical density in the network, which suggests that the road system can serve more vehicles at its capacity, thus improving efficiency, while decreasing the number of conflicts. These findings are useful for both traffic planners and operators.

Traffic resilience quantification based on macroscopic fundamental diagrams and analysis using topological attributes

Transportation system disruptions significantly impair transportation efficiency. This paper proposes new indicators derived from the Macroscopic Fundamental Diagram (MFD) dynamics before and after a disruption to evaluate its impact on traffic resilience. Considering that MFD is an intrinsic property of a homogeneously congested transportation network, the resilience losses due to congestion and network disruption are measured separately. The resilience loss is defined as the reduction in trip completion rate, comparing congested cases to uncongested cases or disrupted cases to undisrupted cases. The resilience loss hence also exists for an undisrupted network and is measurable by the proposed method. A Simulation of Urban MObility (SUMO) model is calibrated by real origin–destination patterns, to allow for experiments in scenarios of different demand variations and supply disruptions. Case studies are conducted in Munich, Germany and Kyoto, Japan to test the usefulness of the newly proposed indicators. We furthermore explore the relationship between resilience loss and network topological attributes such as centrality and connectivity from a variety of synthetic disruption experiments in Munich and Kyoto. We find that the resilience loss in a grid-like network as in Kyoto is less dependent on the degradation of network connectivity than in a ring-like network as in Munich.

Bicycle as a traffic mode: From microscopic cycling behavior to macroscopic bicycle flow

Cities allocate dedicated road space to bicycles in favor of active-mode road users. For urban environments with a mass bicycle volume, bicycle traffic congestion is likely to occur. Hence, a thorough understanding of bicycle traffic flow is necessary for the assessment of cycling infrastructure and the development of traffic management strategies considering cycling efficiency. This study aims to investigate bicycle flow characteristics using microscopic traffic simulation. As bicycle flow performance is subject to the non-lane-based movement strategy and the behavioral heterogeneity among cyclists, various scenarios with different microsimulation settings are evaluated. Ultimately, we derive the functional form fundamental diagrams and macroscopic fundamental diagrams using a curve-fitting approach and an analytical method, respectively. Important macroscopic traffic flow parameters, such as capacity, critical speed, critical density, backward wave speed, etc., are estimated. The results show that lane width, overtaking incentive, and desired speed distribution are factors that affect bicycle flow performance. The distinct features of bicycle flow under different traffic states are identified by discussing the simulation outcome and comparing the estimated flow parameters. The findings can be utilized by future research regarding large-scale bicycle traffic flow modeling and control.

Dynamic joint decision of matching parameters and relocation strategies in ride-sourcing systems interacting with traffic congestion

As the ride-sourcing market expands, ride-sourcing fleets have increased urban traffic congestion, and in turn, road congestion is affecting ride-sourcing operations. It is crucial to incorporate this interaction mechanism into the operational decision of ride-sourcing systems. This mechanism can be described in detail by simulation models. Applying simulation-based optimization methods to the real-time operation of ride-sourcing systems is challenging due to the limitations of simulation efficiency. To tackle this issue, this study first proposes a modular simulation model combining time-driven and event-driven mechanisms. The simulation model comprises two layers: the traffic flow module at the bottom, defined by trip-based and multi-region macroscopic fundamental diagrams and cell transmission models, and the ride-sourcing operation module at the top, which considers the interaction between the traffic congestion and ride-sourcing operation and models the random behavior of passengers and drivers. We verify the efficiency and accuracy of the simulation model. The simulation model is utilized as an evaluator of the objective functions and performance measures. Then, we integrate the Bayesian optimization, parallel sampling, and rolling horizon approaches to develop an efficient and effective simulation-based optimization framework for the dynamic joint decision-making of matching radius, matching intervals, and threshold relocation strategies. Six groups of experiments for the traffic scenario in Berlin reveal some interesting findings. (1) The matching rate first increases and then decreases with the matching radius (a concave function), while the closest study does not think so. (2) As the fleet size grows, passenger waiting time reaches its minimum and then rises because of the increased endogenous congestion generated by ride-sourcing fleets. (3) The optimal operation strategy considering endogenous congestion not only leads to an increase in the system revenue and efficiency but also leads to congestion relief, even though the latter was not our initial objective. This benefit is the most pronounced when the fleet size and background traffic are moderate.

Functional form selection and calibration of macroscopic fundamental diagrams

Macroscopic fundamental diagram (MFD) is widely applied in network-level traffic control and management with most applications necessitating a well-calibrated MFD. With various data sources, more and more empirical MFDs are documented, while the MFD functional form is predetermined by traffic engineers based on their prior experiences. To our best, no generally accepted functional form has been identified. An automatic functional form selection method is yet to be devised. To meet this, a two-step MFD calibration framework is proposed to enable both the functional form selection and the estimation of parameters in this paper. A math program problem is first developed to identify a proper functional form from a set of candidate functions via random sampling of the measurement data. A mean-field variational Bayesian (MFVB) algorithm is then proposed to estimate the parameters of the selected MFD functions using the full measurement dataset. Both calibrations with and without the MFD dynamics are evaluated. The comparison between these calibration results highlights that the calibration considering the MFD dynamics can better characterize network traffic dynamics subject to dynamic travel demand and traffic control measures. Leveraging functional form selection and the computational advantages of the MFVB method, the two-step framework can significantly reduce the computational burden. Results using simulated data and empirical data validate the effectiveness and efficiency of the two-step framework. Furthermore, different functional forms are identified for different cities, highlighting the importance of functional form selection in the MFD calibration.

Macroscopic modeling of mixed bi-modal urban networks: A hybrid model of accumulation- and trip-based principles

Network-level traffic flow models either assume steady-state urban flows (i.e. accumulation-based models) or track the movement of all vehicles (i.e. trip-based models). The steady-state assumption present in the accumulation-based models may pose a challenge in light of the multi-modal nature of urban flows. It might be indeed a rough assumption for the flow of some transportation modes like buses, cruising-for-parking vehicles, taxis, and on-demand vehicles. Trip-based models address this concern, however, they need significant parameter calibration effort and are not computationally efficient, which substantially reduces the practicality of these models in real-world applications. Nevertheless, despite the critical importance of developing multi-modal traffic flow models, few attempts have been made to investigate these models in network macroscopic fundamental diagram (NMFD)-related literature. This paper bridges this gap by developing a hybrid network-level traffic flow model for mixed bi-modal (i.e. car and bus) networks. The present hybrid model reproduces the dynamics of car flows via accumulation-based model principles while tracking the movement of buses using the trip-based model. This effort also includes the development of a new FIFO-based entrance function to ensure different modes experience the same delay under saturated traffic conditions. Different numerical experiments are conducted to study the hybrid model performance and to compare it with that of accumulation-based and trip-based models in both steady-state and transition periods under different traffic conditions. Our observations reveal that the hybrid model simulates the dynamics of cars and buses by closely following the behavior of its components under free-flow conditions. The model also outperforms the accumulation-based model under saturated traffic conditions while being considerably less demanding than the trip-based model. A further investigation of the model performance is performed for networks with different bus shares in both free-flow and saturated traffic conditions, confirming the results of the initial numerical experiments. The hybrid model’s computational efficiency is demonstrated. The potential real-world applications of the hybrid model in development of bi-modal network-level simulation models, NMFD-based control strategies along with bus space allocation policies, public transport operation problems, modeling of cruising-for-parking vehicles, taxis, and on-demand vehicles, and modeling and application of autonomous modular vehicles are discussed and future research directions are highlighted.

Evaluating the Effectiveness and Transferability of a Data-Driven Two-Region Perimeter Control Method Using Microsimulation

Regional perimeter control based on the existence of macroscopic fundamental diagrams has been widely studied as an effective tool to regulate traffic and prevent oversaturation in dense urban areas. Significant research efforts have been performed concerning the modeling aspects of perimeter control. More recently, data-driven techniques for perimeter control have shown remarkable promise; however, few studies have examined the transferability of these techniques. While it is surely of the highest priority to devise effective perimeter control methods, the ability of such methods to transfer the learned knowledge and quickly adapt control policies to a new setting is critical, particularly in real-life situations where training a method from scratch is intractable. This work seeks to bridge this research gap by comprehensively examining the effectiveness and transferability of a reinforcement-learning-based perimeter control method for a two-region urban network in a microsimulation setting. The results suggest: 1) the presented data-driven method demonstrates promising control effectiveness in comparison with no perimeter control and an extended greedy controller and 2) the method can readily transfer its learned knowledge and adapt its control policy with newly collected data to simulation settings with different traffic demands, driving behaviors, or both.

The Peak Stability Analysis through Hysteresis Phenomenon on Heterogeneous Networks

The macroscopic fundamental diagram (MFD) is a nonuniversal changing process over network traffic status which indicates different shapes in different networks. Hysteresis is observed in the MFD of some urban networks. It is a unique phenomenon when the network remains at low stability level and usually appears around the congestion period. This paper analyzed network peak stability through focusing on hysteresis. The formation mechanism of hysteresis is deduced from the mathematical method based on previous research studies. The precondition of hysteresis and the changing process of network state can be figured by mathematical deduction. It indicates that hysteresis only occurs conditionally in the period of macroscopic congestion and is not a universal phenomenon. Heterogeneity is an important factor leading to network instability. The hysteresis patterns of different peaks in MFD are different due to the variation of network flow. Real data are collected from Atlanta’s urban network to verify the analysis of hysteresis. To discuss the changing process of hysteresis in different peaks, a three-stage division is proposed and time series is presented as a third dimension in MFD. It is worth noticing that the existence and form of hysteresis in morning and evening peaks are different. Although there is a higher peak flow in the morning peak, the stability of the evening peak performs better when hysteresis occurs in the network. The different fluctuations in the morning and evening peaks are exhibited through the 3D version of MFD. The otherness of hysteresis in different peaks is explained through a 3D coordinate system with cross-compared corresponding indexes.

Structure of road networks and the shape of the macroscopic fundamental diagram.

The macroscopic fundamental diagram (MFD) is a large-scale description of the traffic in an urban area and relates the average car flow to the average car density. This MFD has been observed empirically in several cities but how its properties are related to the structure of the road network has remained unclear so far. The MFD displays in general a maximum flow q^{*} for an optimal car density k^{*} which are crucial quantities for practical applications. Here, using numerical modeling and dimensional arguments, we propose scaling laws for these quantities q^{*} and k^{*} in terms of the road density, the intersection density, the average car size and the maximum velocity. This framework is able to explain the scaling observed empirically for several cities in the world, such as the scaling of k^{*} with the road density, the relation between q^{*} and k^{*} and the impact of buses on the overall capacity q^{*}. This work opens the way to a better understanding of the traffic on a road network at a large urban scale.

A practical method for assessing sustainable traffic levels of roundabouts in urban areas using the macroscopic fundamental diagram

In traffic engineering, queue lengths, waiting times, and delay values are denoted as measures of effectiveness (MOEs). In the case of roundabouts, MOEs are generally calculated starting from the gap-acceptance theory in which the driver behaviour needs to be properly modelled. This research presents an unconventional technique, based on Macroscopic Fundamental Diagrams (MFDs), to describe sustainable traffic levels and (MOEs) of roundabouts, in terms of capacity, speed, density and delay. The deduction of MFDs is based on relatively simplified theoretical hypotheses, but the outcomes could be of interest in several practical applications. MFDs are estimated in the cases of single and double-lane roundabouts, modelled by microscopic simulations starting from a real-world case study in Italy. MFDs are obtained for a wider sample size of origin-destination traffic matrices taking into consideration time intervals of 5 min and 1 h. The results of this research demonstrate that the traffic distribution between arms can influence the shape of a roundabout-MFD. However, the proposed procedure allows us to evaluate the performance of a roundabout as a whole, taking into account numerous MOEs including the total capacity, the critical density and the Level of Service (LOS). Therefore, the use of MFDs could represent a complementary procedure for evaluating the performance of roundabouts.

Load balancing of multi-AGV road network based on improved Q-learning algorithm and macroscopic fundamental diagram

To address the challenges of traffic congestion and suboptimal operational efficiency in the context of large-scale applications like production plants and warehouses that utilize multiple automatic guided vehicles (multi-AGVs), this article proposed using an Improved Q-learning (IQL) algorithm and Macroscopic Fundamental Diagram (MFD) for the purposes of load balancing and congestion discrimination on road networks. Traditional Q-learning converges slowly, which is why we have proposed the use of an updated Q value of the previous iteration step as the maximum Q value of the next state to reduce the number of Q value comparisons and improve the algorithm’s convergence speed. When calculating the cost of AGV operation, the traditional Q-learning algorithm only considers the evaluation function of a single distance and introduces an improved reward and punishment mechanism to combine the operating distance of AGV and the road network load, which finally equalizes the road network load. MFD is the basic property of road networks and is based on MFD, which is combined with the Markov Chain (MC) model. Road network traffic congestion state discrimination method was proposed to classify the congestion state according to the detected number of vehicles on the road network. The MC model accurately discriminated the range near the critical point. Finally, the scale of the road network and the load factor were changed for several simulations. The findings indicated that the improved algorithm showed a notable ability to achieve equilibrium in the load distribution of the road network. This led to a substantial enhancement in AGV operational efficiency.

Learning and managing stochastic network traffic dynamics: an iterative and interactive approach

ABSTRACT This study examines the potential of an iterative and interactive approach to learn network traffic dynamics and optimise tolling strategies considering time-varying stochastic traffic. A tractable ‘truth model’ based on the stochastic Macroscopic Fundamental Diagram is developed to represent the transportation system to be learned and managed. A ‘twin model’ that mirrors the truth model is formulated and calibrated for testing and optimising tolling adjustment strategies with the help of reinforcement learning. The optimised prices are then put into the ‘truth model’ to evaluate network efficiency improvement. The above procedure is iterative and interactive, which can be applied for congestion management in the period-to-period tolling adjustment fashion. Numerical studies show that the proposed iterative and interactive pricing strategy is able to enhance network efficiency even under limited information and/or inaccurate learning of the system. This illustrates the great potential of utilising iterative and interactive frameworks for congestion management.

Optimal Perimeter Control for Multi-Region Traffic Networks Based on Macroscopic Fundamental Diagram

Optimal Perimeter Control for Multi-Region Traffic Networks Based on Macroscopic Fundamental Diagram

Partially observable Markov decision process for perimeter control based on a stochastic macroscopic fundamental diagram

The use of a macroscopic fundamental diagram (MFD) for perimeter control is an effective strategy for managing traffic flow on regional road networks. However, existing methods either assume a MFD with low scattering or assume complete availability of information, which is not realistic. In order to address these limitations, we propose a novel approach that incorporates a stochastic evolution dynamic into the MFD and introduces a partially observable Markov decision process for perimeter control. Our approach assumes that the MFD is randomly distributed between upper and lower boundaries, characterized by a physically interpretable parameter, and that real-time observations represent cumulative data in specific regions. We focus on three sub-objectives: minimizing travel time, minimizing capacity loss due to hysteresis, and preventing gridlock. By utilizing the Bellman equation, we solve the problem and demonstrate that our proposed control method enhances trip completion rates by 7.2% compared to the benchmark case without perimeter control. Furthermore, the inclusion of stochasticity leads to an additional 1.2% improvement in trip completion rates. Through extensive numerical tests, we establish the effectiveness of our approach in optimizing traffic flow regulation on regional road networks.

SPATIAL PARTITION FOR HETEROGENEOUS CITY NETWORKS COMPOSED OF FACTORS THAT INFLUENCE THE DISTRIBUTION OF THE MACROSCOPIC FUNDAMENTAL DIAGRAM

Using a Macroscopic Fundamental Diagram (MFD) to implement partition control is effective in improving mobility in heterogeneous city networks. As one of the most complex issues in partition control, accurate sub-region partition is critical for control effectiveness. Current partition methods focus on the link density and precondition of existing MFD and disregard the factors that influence MFD distribution. To overcome this drawback, this study uses the characteristic value of the link and the intersection connected to the link as the analysis object and proposes an MFD sub-region partitioning method for large-scale networks. Firstly, the influences of road state parameters on MFD distribution are classified into traffic flow parameters, network physical properties, network operation mechanisms and emergencies. Simulation experiments are conducted to determine the degree to which these classifications affect MFD distribution. Secondly, a partition method combined with the link density and influence parameters of MFD is developed. The method is used for a preliminarily division of a road network through Minimum Spanning Tree (MST) and depth partition by the Normalised Cut (Ncut) algorithm. Finally, a case study is conducted in an actual city centre network, and results show that the developed method is superior to the single method based simply on link density.