Macroscopic Fundamental Diagram Research Articles

Estimation of a Fundamental Diagram with Heterogeneous Data Sources: Experimentation in the City of Santander

The reduction of urban congestion represents one of the main challenges for increasing sustainability. This implies the necessity to increase our knowledge of urban mobility and traffic. The fundamental diagram (FD) is a possible tool for analyzing the traffic conditions on an urban road link. FD is commonly associated with the links of a transport network, but it has recently been extended to the whole transport network and named the network macroscopic fundamental diagram (NMFD). When used at the link or network level, the FD is important for supporting the simulation, design, planning, and control of the transport system. Recently, floating car data (FCD), which are based on vehicles’ trajectories using GPS, are able to provide the trajectories of a number of vehicles circulating on the network. The objective of this paper is to integrate FCD with traffic data obtained from traditional loop-detector technology for building FDs. Its research contribution concerns the proposal of a methodology for the extraction of speed data from taxi FCD, corresponding to a specific link section, and the calibration of FDs from FCD and loop detector data. The methodology has been applied to a real case in the city of Santander. The first results presented are encouraging, supporting the paper’s thesis that FCD can be integrated with data obtained from loop detectors to build FD.

Risk‐averse perimeter control for alleviating the congestion of an urban traffic network system with uncertainties

AbstractThe perimeter control method is an effective way to alleviate traffic congestion that is based on Macroscopic Fundamental Diagrams (MFDs). However, the strategy may lead to congestion when it ignores the uncertainty of MFDs. To address this problem, this paper presents a risk‐averse perimeter control method. First, the urban traffic network system with uncertainty is modeled using a neural network and scenario tree. Then, this research quantifies the congestion risk caused by uncertainty using an average value‐at‐risk. The next step sees the design of a risk‐averse model predictive control (MPC) controller that takes the multi‐stage risk as the optimization objective and improves robustness by interpolating between the conventional stochastic and worst‐case MPC formulations. Finally, this paper analyzes the risk‐sensitive stability of an urban traffic network system and gives a solvable form of risk‐averse optimal control for this system. Finally, two simulations are conducted to verify the presented method's validity and superiority for an urban traffic network system with uncertainties. The simulation results show that the risk‐averse perimeter control method presented by this paper is superior because it reduces the total travel time by 12.98% compared to Stochastic MPC, by 15.96% compared to bang‐bang control, and by 14.54% compared to proportional‐integral.

Real-time Vulnerability Analysis of Urban Expressway

ABSTRACT This study proposes a new method for evaluating the road network vulnerability per unit time: the real-time vulnerability index (RVI). The practical application of the proposed measure in real traffic scenarios is investigated by using empirical data from Shanghai Expressway.To prevent traffic congestion, a critical threshold of RVI (RVI*) is suggested for traffic control. Furthermore, the hysteresis on macroscopic fundamental diagram (MFD) and RVI is further analyzed in the phase transition of traffic states. The results show that RVI can help derive some representative traffic characteristics in real time and dynamically reflect road network vulnerability: when traffic congestion or accidents occur on the road network, the results observed for the traffic hysteresis on MFD and RVI were consistent, and RVI is easier to observe and more suitable for traffic management. The findings prove that RVI is a reliable indicator to help predict the dynamic operation of the road network.

A dynamic multi-region MFD model for ride-sourcing with ridesplitting

Dynamic network-level models directly addressing ride-sourcing services can support the development of efficient strategies for both congestion alleviation and promotion of more sustainable mobility. Recent developments presented models focusing on ride-hailing (solo rides) and traditional ride-sharing, but no work addressed ridesplitting (a type of ride-sourcing service with shared rides) in dynamic contexts. Here, we sought to develop a dynamic aggregated traffic network model capable of representing ride-sourcing services and background traffic in a macroscopic multi-region urban network. We combined the Macroscopic Fundamental Diagram (MFD) with detailed state-space and transition descriptions of background traffic and ride-sourcing vehicles in their activities to formulate mass conservation equations. Accumulation-based MFD models might experience additional errors due to the variation profile of trip lengths, e.g., when vehicles cruise for passengers. We integrate the so-called M-model that utilizes the total remaining distance to capture dynamics of regional and inter-regional flows and accumulations for different vehicle (private or ride-sourcing) states. This aggregated model is capable to reproduce the dynamics of complex systems without using resource-expensive simulations. We also show that the model can accurately forecast the vehicles’ conditions in near-future predictions (e.g., 30 min ahead). Later, a comparison with benchmark models shows lower errors in the proposed model in all states. Finally, we evaluate the model’s robustness to noises in its inputs, and forecast errors remain below 15% even where inputs are 20% off the actual values for ride-sourcing vehicles. The development of this model prepares the path for developing real-time feedback-based management policies such as priority-based perimeter control or repositioning strategies for idle ride-sourcing vehicles and developing regulations over ride-sourcing in congested areas.

Perimeter Control Method of Road Traffic Regions Based on MFD-DDPG.

As urban areas continue to expand, traffic congestion has emerged as a significant challenge impacting urban governance and economic development. Frequent regional traffic congestion has become a primary factor hindering urban economic growth and social activities, necessitating improved regional traffic management. Addressing regional traffic optimization and control methods based on the characteristics of regional congestion has become a crucial and complex issue in the field of traffic management and control research. This paper focuses on the macroscopic fundamental diagram (MFD) and aims to tackle the control problem without relying on traffic determination information. To address this, we introduce the Q-learning (QL) algorithm in reinforcement learning and the Deep Deterministic Policy Gradient (DDPG) algorithm in deep reinforcement learning. Subsequently, we propose the MFD-QL perimeter control model and the MFD-DDPG perimeter control model. We conduct numerical analysis and simulation experiments to verify the effectiveness of the MFD-QL and MFD-DDPG algorithms. The experimental results show that the algorithms converge rapidly to a stable state and achieve superior control effects in optimizing regional perimeter control.

Nonlinear model predictive control of large-scale urban road networks via average speed control

Controlling traffic in large-scale urban road networks is a challenging problem. Aggregated dynamical models, based on the macroscopic fundamental diagram (MFD) of urban traffic, enable model-based control design. As an alternative to perimeter control actuation commonly used in MFD-based control, in this paper, we propose actuation over regional space-mean speeds, which we name average speed control. The method involves manipulation of regional speeds via instrumentation similar to variable speed limits in freeways, or using vehicle-to-infrastructure communication. We develop nonlinear model predictive control schemes considering actuation over average speed and perimeter control. Their performances are compared using simulations on congested scenarios, the results of which suggest potential of the method as an alternative or complementary actuation to perimeter control.

Analysis of superblocks during the transition phase from traditional vehicles to fully automated vehicle environment: A case study of Barcelona City

The planning of vehicle-dependent cities across the world with emerging technologies like automated vehicles must be amended to make them pedestrian-friendly. Therefore, there is a need to reorganise the urban streets to create more spaces for pedestrians, even with the presence of automated vehicles (AVs). This research proposes the concept of a superblock model in the presence of automated vehicles to meet this challenge. This research aims to assess the impacts of AVs on pedestrian-friendly streets using a superblock approach. In the superblock model, cut-through traffic demand is restricted, and traffic generated or attracted to a superblock can have access, allowing for more room for public spaces. This study systematically discusses the superblock model implemented in a network with a grid structure using a genetic algorithm. It addresses the dynamic user equilibrium assignment for demand and topology segmentation. Three scenarios are considered due to the longer period of transition from traditional vehicles to fully AVs. The study evaluates the potential and constraints of implementing the superblock model with different penetration rates of AVs in the system by defining variable traffic demand and the variable number of superblocks. Furthermore, a cost-benefit analysis of the proposed methodology is conducted. The proposed model is implemented on a real-time network of Barcelona city, and the findings of this study are summarised by the macroscopic fundamental diagram, which depicts the improvement in the performance of the network with the inclusion of AVs.

Hysteresis and the Unobserved Congestion Branch in the Macroscopic Fundamental Diagram: Theoretical Considerations and Modeling

The macroscopic fundamental diagram (MFD) is developed to describe traffic operations aggregated over an area. The MFD is defined by network traffic states as a relationship between the accumulation of vehicles and flow or speed of vehicles. The concept of the MFD has been applied to model traffic dynamics and to design control strategies. For various applications (e.g., routing and departure time choices), the MFD is often assumed to be of a particular shape, consisting of two branches—a free-flow branch and a congestion branch. However, empirical observations show some inconsistencies between the theoretical and the empirical MFD. First, the empirical MFD only presents free-flow branch, i.e., the congestion branch is missing. Second, the MFD presents as hysteresis loop(s). This paper explores these differences, providing insights into urban network traffic dynamics. This work takes the travelers’ departure time choices and user equilibrium (UE) as starting points. We consider demand to be in UE in terms of departure time choices. Using this property of the demand profile, the paper proposes a closed-form expression for average density and outflow. Finally, we show some insights in the urban traffic dynamics: (i) an explanation of the hysteresis phenomenon solely from the perspective of departure time choices and UE and (ii) an explanation of why we hardly observe the MFD congestion branch in real life even in heavily congested networks. Our study shows that, for management purposes, the missing of congestion branch is a result of UE, rather than an indicator of congestion severity.

Network-Wide Implementation of Roundabouts Versus Signalized Intersections on Urban Streets: Analytical and Simulation Comparison

This paper examines the impact of roundabouts implemented at intersections throughout a dense urban network on its operational performance. Metrics considered include the average free-flow speed, flow-moving capacity, trip-serving capacity, and fuel consumption rate. Three intersection strategies are compared: signalized intersections allowing left turns in a permitted manner (TWs), signalized intersections prohibiting left turns (TWLs), and modern roundabouts (RBs). Using the approaches of macroscopic fundamental diagrams and network exit functions, both analytical investigations and microscopic traffic simulations for grid networks were conducted. In general, the results from both analyses agree well. The results reveal that when single-lane roundabouts are applied in networks with a single travel in each direction, the RB network outperforms the TW network for all operational metrics. The RB network also outperforms the TWL network in free-flow speed and flow-moving capacity and has a similar trip-serving capacity as the TWL network. However, when roundabouts with two travel lanes are applied on multi-lane networks, the TWL network exceeds the RB network in both flow-moving and trip-serving capacities. This decrease in the performance of the RB network could possibly come from the complexity imposed on the entering vehicle that wants to use the inner lane. Moreover, because vehicles in the RB network need to accelerate/decelerate more frequently those in the other networks, the RB network generates a higher fuel consumption rate in uncongested and capacity conditions. The findings suggest intersections of roundabouts could be beneficial for networks with a single travel lane in each direction.

A three-dimensional extension control for the dynamic boundary entrance of congested road networks

The dynamics and randomness of traffic demand require changes in the scope and boundary flow of congested traffic areas in a timely manner. Because of this, existing control methods have difficulty accurately detecting the boundary flow and executing an effective control strategy. Combined with the characteristics of extension theory, this paper transforms the boundary flow of congested traffic regions into a fuzzy variable and proposes a variable structure three-dimensional extension control method. Variable structure control consists of two parts. The first part is based on the matter-element model, and the correlation function of extension theory, boundary selection and entrance metering (BSEM) is proposed to evaluate the link density data in the periphery of the congested area and determine its dynamic boundary. According to the results of boundary selection, a proportion integration (PI) controller is designed to regulate the input flow of the congested area. Then, considering the measurement error of road network outflow, an extension lifting feedback (ELF) control is proposed. ELF takes the difference between the actual outflow rate of the road network and the expected outflow rate of the macroscopic fundamental diagram (MFD) curve of the congested region as the characteristic variable, divides the road network data, and adopts different control strategies to improve the input flow adjustment accuracy in the congested area. Finally, a case analysis is performed, and the results show that compared with Bang-Bang control, PI control and BSEM for congested regions without considering outflow fluctuation control, the proposed BSEM-ELF cooperative control can effectively reduce the total delay and average fuel consumption of the road network.

A hybrid modelling framework for the estimation of dynamic origin–destination flows

The dynamic origin–destination flow estimation (DODE) problem requires scalable methods for large scale traffic networks and consistent techniques for capturing both uncongested and congested traffic conditions. Despite numerous efforts on incorporating multifold data sources and developing manifold mathematical models, the DODE problem remains a challenging problem in terms of both scalability and consistency. To fill this gap, we propose a novel hybrid DODE framework that integrates region-level (macro) and centroid-level (micro) traffic dynamics. The region-level traffic flows are described by the macroscopic fundamental diagram, while the centroid-level traffic flows are represented by the linear mapping of origin–destination flows onto link counts. This hybrid approach enables us to (i) incorporate region-level traffic measures into the problem, addressing scalability issues arising in large scale traffic networks, and (ii) capture non-linear behaviour of traffic in the regional context, enhancing consistency of the estimation results with respect to traffic conditions. The proposed methodology is experimented in a large-scale traffic network, which is benchmarked for DODE problems. The results indicate an outstanding performance of the hybrid DODE particularly in congested traffic conditions and highlight the effectiveness of aggregated (regional) traffic models in enhancing DODE methods with minimal computational burden.

From Corridor to Network Macroscopic Fundamental Diagrams: A Semi-Analytical Approximation Approach

The design of network-wide traffic management schemes or transport policies for urban areas requires computationally efficient traffic models. The macroscopic fundamental diagram (MFD) is a promising tool for such applications. Unfortunately, empirical MFDs are not always available, and semi-analytical estimation methods require a reduction of the network to a corridor that introduces substantial inaccuracies. We propose a semi-analytical methodology to estimate the MFD for realistic urban networks without the information loss induced by the reduction of networks to corridors. The methodology is based on the method of cuts but applies to networks with irregular topologies, accounts for different spatial demand patterns, and determines the upper bound of network flow. Therefore, we consider both flow conservation and the effects of spillbacks at the network level. Our framework decomposes a given network into a set of corridors, creates a hypernetwork, including the impacts of source terms, and then treats the dependencies across corridors (e.g., because of turning flows and spillbacks). Based on this hypernetwork, we derive the free-flow and capacity branch of the MFD. The congested branch is estimated by considering gridlock characteristics and utilizing recent advancements in MFD research. We showcase the applicability of the proposed methodology in a case study with a realistic setting based on the Sioux Falls network. We then compare the results to the original method of cuts and a ground truth derived from the cell transmission model. This comparison reveals that our method is more than five times more accurate than the state of the art in estimating the network-wide capacity and jam density. Moreover, the results clearly indicate the MFD’s dependency on spatial demand patterns. Compared with simulation-based MFD estimation approaches, the potential of the proposed framework lies in the modeling flexibility, explanatory value, and reduced computational cost. Funding: G. Tilg acknowledges support from the German Federal Ministry for Digital and Transport (BMDV) for the funding of the project LSS (capacity increase of urban networks). S. F. A. Batista and M. Menéndez acknowledge support from the NYUAD Center for Interacting Urban Networks (CITIES), funded by Tamkeen under the NYUAD Research Institute Award [CG001]. L. Ambühl acknowledges support from the ETH Research Grant [ETH-27 16-1] under the project name SPEED. L. Leclercq acknowledges funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program [Grant 646592 - MAGnUMproject]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/trsc.2022.0402 .

A new approach to estimating link performance on Indonesian urban roads: deriving the BPR 1964 function

Predictions of traffic volumes on road networks must be precise for transportation planning, and equilibrium trip assignment models based on the Volume Delay Function (VDF) are typically employed. In 1964, the U.S. Bureau of Public Roads (BPR) created a VDF widely used in many countries, including Indonesia. However, using capacity manuals from Western nations in Indonesia did not produce the desired results due to differences in traffic composition and driver behaviour. Using the Indonesian Highway Capacity Manual’s macroscopic fundamental diagram, this paper aims to derive a more accurate VDF for Indonesia. This method enables the development of a VDF tailored to the unique traffic conditions in Indonesia. It can enhance the precision of traffic volume forecasts in the region. The BPR function parameters tends to overestimate travel time delay, making the road network model flow misestimated. This is presumably caused by traffic count survey inability to record the build-up of traffic flow in the urban network during peak traffic periods.

Convexification approaches for regional route guidance and demand management with generalized MFDs

Traffic congestion is one of the main concerns in big cities with many adverse socioeconomic effects. A promising solution is to simultaneously regulate the admission of vehicles (i.e., demand management) and redistribute traffic flows within the network (i.e., route guidance). In this work, we integrate demand management with route guidance within a Model Predictive Control framework using regional traffic dynamics with generalized Macroscopic Fundamental Diagram (MFD) shapes. Dealing with generalized MFD shapes is challenging due to the resulting nonlinear and non-convex optimization formulation. To tackle this challenge, we develop two real-time solution approaches: (i) a successive convexification approach that constructs convex bounding sets for all nonlinear terms, and (ii) a linear approximation approach that solves the problem using triangular macroscopic fundamental diagram approximations. The proposed approaches offer a trade-off between execution speed and solution quality, as the linear approximation approach runs faster while the successive convexification approach yields better quality and accuracy solutions. Macroscopic simulation results illustrate the efficiency of the successive convexification and linear approximation approaches yielding an optimality gap of less than 3.5% and 10% in all considered cases, respectively. Furthermore, both approaches outperform a state-of-practice nonlinear solver in terms of solution quality and execution time. Finally, substantial gains are also obtained regarding travel time and traffic flow efficiency in a realistic microsimulation environment.

Effects of loop detector position on the macroscopic fundamental diagram

Loop detectors are probably the widest-used technology for traffic state estimation. Previous research has shown that loop detector positions within the link significantly affect the estimation of the macroscopic fundamental diagram (MFD) of a given network. This paper examines the biases produced by the positioning of loop detectors on the MFD, using both analytical and simulation methods, as well as empirical data from UTD19. We confirm earlier results that a uniform distribution of loop detector positions reduces the bias. We discovered that: (i) subsets of the MFD determined by the loop detector position can help estimate whether the loop detector MFD will have a bias; (ii) non-uniform distribution of loop detectors is more likely to cause a discrepancy in the position subsets of the MFD, particularly if detectors in the network are positioned more downstream with a greater variation; and (iii) a lower ratio of link length to green signal time increases the possibility of bias in loop detector MFD, while the impact of the aggregation interval was found to be negligible. This research opens the possibility for the bias of MFD induced by the loop detector data to be corrected by only using itself.

Modeling and Control of Network Macroscopic Fundamental Diagram during Holidays: A Case Study of Qingming Festival in Tianjin

Modeling and Control of Network Macroscopic Fundamental Diagram during Holidays: A Case Study of Qingming Festival in Tianjin

Train a central traffic prediction model using local data: A spatio-temporal network based on federated learning

With the growing complexity of onboard sensors and the widespread deployment of road sensors, deep learning enables fine-grained traffic prediction using massive amounts of raw traffic data, which facilitates accurate analysis of traffic information in the Internet of Vehicles (IoV). However, most existing studies focus on using all the local data to jointly build a prediction model, facing severe challenges of data security and privacy concerns as well as substantial communication overhead. To address these challenges, in this paper, we propose the Spatial-Temporal Traffic Prediction Network based on federated learning (F-STTP-Net), which only updates the model parameters to the centralized server without any private data. Firstly, we design a sub-area division method, which divides the road network into sub-areas with different macroscopic fundamental diagram properties. Then, we propose a local training model for each sub-area, which uses the graph attention network (GAT) and the long short-term memory (LSTM) to capture the spatio-temporal dependence of the road network. The model uses the branch structure to predict the traffic volume of each intersection in the sub-area. Finally, the local models are aggregated based on federated learning to form a powerful central model, which bridges the constraints on global data sharing and privacy guarantee. We conduct experiments on the real-life dataset in Xuchang Lotus Lake 5G automated vehicles demonstration area to demonstrate that F-STTP-Net can achieve excellent prediction performance without the interaction of sub-area raw data. In addition, the proposed model has a strong generalization ability and can be quickly transferred to a new sub-area.

Demand and state estimation for perimeter control in large-scale urban networks

State observability and demand estimation are two main issues in large-scale traffic networks which hinder real-world application of real-time control strategies. This study proposes a novel combined estimation and control framework (CECF) to develop perimeter control strategies based on macroscopic fundamental diagram (MFD). The proposed CECF is designed to operate with limited real-time traffic data and capture discrepancies in a priori demand estimates. The CECF is developed with a moving horizon estimator (MHE) that estimates traffic states, route choices and demand flows considering region accumulations and boundary flows observed from the network. The estimated traffic states are incorporated into a model predictive controller (MPC) scheme to derive the control decisions, which are then executed in the urban network. A novel accumulation-based MFD model is developed in this study to address the observability problem, which is incorporated into MHE and MPC schemes. The proposed CECF is implemented in a large-scale traffic network where several demand scenarios are tested. The results confirm the success of the CECF in overcoming observability issues and improving the performance of perimeter control strategies.

Characterizing the Impact of Autonomous Vehicles on Macroscopic Fundamental Diagrams

With the rapid development of autonomous driving, Autonomous Vehicles (AVs) have started to appear on public roads, which has inevitably affected current traffic conditions and the operations of Manual Vehicles (MVs). Current research on AVs’ influence has mainly been conducted at individual level of driving behaviors, while few studies have focused on the overall network level to consider the traffic flow pattern due to mixed traffic. In this work, considering varying signal control schemes and demand loading patterns, we conducted simulation experiments based on a grid network and a real-world network in Beijing using SUMO. Traffic flow with the mixture of MVs, low-level AVs (LAVs), and high-level AVs (HAVs) were emulated so to investigate how the network performs at various levels of mixed traffic. Driving behaviors between the three types of vehicles were calibrated using driving data drawn from OpenACC dataset, and Waymo Open Dataset. The capacity and critical accumulation of the Macroscopic Fundamental Diagram (MFD) were chosen as the key indicators of network performance. We found that AVs positively boost network capacity (up to 19.0% increase) but a negative influence on critical accumulation was also observed (up to 9.0% decrease). However, the positive impact of OpenACC and Waymo’s AVs on macroscopic traffic is still far from ideal since they may be too conservative. AVs can boost flow when traffic is in unsaturated or saturated states. However, when traffic flow is oversaturated, AVs can instead cause flow and average speed to drop faster than that in the MV-only scenario.

Competition in Ride-Hailing Service Operations: Impacts on Travel Distances and Service Performance

Competition in the ride-hailing market can influence traffic congestion and deteriorate the quality of service. A customer can request to be matched with a ride-hailing vehicle from their preferred company, which might not be the nearest vacant vehicle. This can increase both the customer’s matching and pick-up waiting time and the vehicle’s travel distance to the customer, and contribute to traffic congestion. Recent studies focus on the long-term competition effect by considering network equilibrium. In this work, we target a shorter timeframe and investigate how competition influences the passenger–driver matching process, the consequent vehicle travel to the customer, and more globally the system at the operational level. To this end, we propose a modeling and simulation framework based on the Macroscopic Fundamental Diagram (MFD). We apply the so-called M-model, a continuum approximation of the trip-based MFD. Compared with the accumulation-based approach, it explicitly monitors the remaining travel distance of all vehicles. We extend the mathematical M-model decomposition and focus on accurate dynamic estimation of trip lengths for the different vehicle states based on the immediate system state. For this, we suggest creating an additional proxy simulation framework replicating the demand requests and the service vehicle movements. We propose calibrating the matching function by sampling observations on a proxy grid network. Finally, we assess and compare different matching processes that define diverse competition scenarios: competition, cooperation, and competition with partial cooperation (coopetition). The cooperation scenario shows the best results in service performance.