Network Macroscopic Fundamental Diagram Research Articles

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.

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.

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

Non-unimodal and non-concave relationships in the network Macroscopic Fundamental Diagram caused by hierarchical streets

Unimodal, concave relationships between average network productivity and accumulation or density aggregated across spatially compact regions of urban networks—so called network Macroscopic Fundamental Diagrams (MFDs)—have recently been shown to exist on homogeneous street networks. When present, MFD relationships facilitate the modeling of traffic congestion at a regional level and have led to the development of various regional traffic control strategies. However, real street networks are not homogeneous—they generally have a hierarchical structure where some streets (e.g., arterials) promote higher mobility than others (e.g., local roads). This paper examines how the presence of hierarchical roadway structures may potentially cause non-unimodal patterns in a network's MFD. These are observed using three types of tools: analytical models of simple network structures, simulations of various idealized roadway networks, and empirical data. The impacts of street hierarchy depend on how vehicles use different roadway types to move within the network; i.e., their routing strategy. The findings suggest that the presence of roadway hierarchies may lead to MFDs that have non-unimodal or non-concave patterns on the free-flow branch when vehicles route themselves according to user equilibrium principles, which is closest to what would be observed in realistic situations. Such patterns are contrary to what is traditionally assumed in most MFD-based modeling frameworks. However, the unimodal and concave MFD should be expected under system optimal routing conditions that maximize network productivity for a given traffic state.

Partitioning of urban networks with polycentric congestion pattern for traffic management policies: Identifying protected networks

AbstractLarge‐scale urban networks are usually loaded heterogeneously with a polycentric congestion pattern, resulting in a highly scattered network macroscopic fundamental diagram (NMFD or MFD). Thus, researchers have tried to partition city networks into homogeneous subzones. In this study, a six‐step partitioning algorithm is proposed. The framework allows the NMFD information to be used. It combines traffic variables and geographic connectivity to obtain partitions. The framework includes graph definition, data preprocessing, feature handling, clusters and partitions identification, and boundaries reshaping. Tests on a simplified grid network and the city of the Melbourne road network demonstrate the suitability of the framework for characterizing the traffic states by the partitions. The framework on the missing data scenarios and monocentric and polycentric traffic concentrations scenarios, as well as applying multiple data types, has been challenged. Thereafter, parameter impact analysis demonstrates that manipulation of the parameters enables users to find the desired partitions. Last but not least, a comparison with an existing method also implies the uniqueness and efficiency of the developed framework.

Mode differentiation in partitioning of mixed bi-modal urban networks

ABSTRACT Empirical studies have demonstrated the existence of a well-defined Network Macroscopic Fundamental Diagram (NMFD). Well-shaped NMFD, however, may not be a universal law because urban networks often display heterogeneous congestion distributions. Therefore, different algorithms have been proposed to partition a heterogeneous network into homogeneous sub-networks. These algorithms mainly consider uni-modal networks, despite the multi-modal nature of urban traffic flows. This paper is one of the first studies dedicated to the partitioning problem of bi-modal networks. First, a three-step partitioning algorithm is developed concerning NMFD-based applications. Second, the role played by mode differentiation in bi-modal network partitioning is investigated through the lens of speed-NMFDs. The results demonstrate that the algorithm performs well. Mode differentiation should be considered in the partitioning process while is defined differently in free-flow and saturated conditions. The links' lengths, bus stop distribution, speed limit zone, and the existence of bus terminals might affect speed-NMFD properties.

Traffic dynamics in bi-modal urban networks: a cross-comparison of outflow 2D-NMFD and 3D-NMFD

Network macroscopic fundamental diagram (NMFD) has provided practical convenience to simulate network-level traffic dynamics using NMFD-based models. Nevertheless, only a few works have been conducted on traffic dynamics in bi-modal urban networks. This paper provides a comparison of the application of outflow 2D-NMFD and 3D-NMFD through simulating different scenarios using accumulation-based model in a single reservoir system. The models are verified using micro-simulation data as ground truth performed on a Manhattan-style grid network and a real network. The results reveal the outflow 2D-NMFD can provide accurate traffic state evolution, while the goodness of fit is the main challenge in the application of outflow 3D-NMFD. Besides, the accumulation-based model needs modifications to construct the bus accumulation profile in the test-bed with a low share of buses. Last but not least, the accuracy of the existing network-level speed models in mixed-traffic condition and its effects on traffic dynamics simulation are validated and discussed.

Perimeter control with real-time location-varying cordon

With unbalanced travel demand distribution over time and space, a stationary cordon location hinders the full potential of perimeter flow control based on network Macroscopic Fundamental Diagram (MFD). This paper introduces a perimeter control method wherein the region boundaries alter in real-time to tackle propagation of local pockets of congestion. The nonlinear dynamics of the heterogeneous traffic network are modelled as a switching system. The linearization of the derived switching nonlinear dynamics is conducted considering the accumulation heterogeneity. A Linear Quadratic Regulator (LQR) is employed for gating the flow exchange between regions to minimize network congestion. Several scenarios are examined comparing perimeter control schemes with static and dynamic cordons. Results pinpoint the proposed LQR control with location-varying cordon strategy and a moderate switching interval significantly reduces the vehicles total time spent in the network.

A model predictive perimeter control with real-time partitions

Previous studies through simulation and empirical data have shown that a Network Macroscopic Fundamental Diagram (NMFD) exists and can be used for designing network optimal perimeter control strategies. These control strategies rely on well defined NMFDs, which highly depend on the homogeneity of the traffic condition in the network. However, it is known that traffic dynamics change drastically during the day in different zones in a large-scale network, and different control strategies might lead to heterogeneous traffic distribution across the urban network. One potential direction is re-partitioning the network to maintain the well defined NMFDs. However, re-partitioning the network changes each sub network’s size, such that it makes the well-defined NMFDs unpredictable. This paper provides a model predictive control-based optimization approach for perimeter control using real-time partitioning to avoid this problem and utilize re-partitioning techniques. Results show that the proposed method can be used in a heterogeneous network to improve control performance by redistributing accumulations via re-partitioning over time. Our results, which are compared to no control and the traditional model predictive control, yield that the proposed method is superior to the others.

Impacts of bus stop location and berth number on urban network traffic performance

The effects of operational characteristics of the public transport system on the performance of the urban network traffic flow and the public transport system have been widely investigated at the local level. However, to the best of authors' knowledge, there is no attempt to investigate these characteristics at the network level. This study bridges this gap through the notion of network macroscopic fundamental diagram. In particular, the effects of the bus stop location (i.e. far-side and near-side) and berth number are discussed at the network level through simulating different scenarios in the central business district of the city of Christchurch, New Zealand. In consistent with the local level studies, the outputs show that the far-side bus stops result in better network performance (i.e. larger capacity and critical density range) and a lower median for the network average delay of car traffic. The near-side bus stops instead lead to a lower median for the public transport system. The results reveal that increasing the berth number improves the network capacity and median of the network average delay for both modes. Finally, the impacts of the combination of the far-side and near-side bus stop on network performance have been discussed.

Macroscopic Traffic Dynamics Under Fast-Varying Demand

Aggregated dynamic traffic models based on the network macroscopic fundamental diagram (MFD) have been founded on stationary or steady-state approximations. Theoretically, they are only qualified with slow-varying demand profiles as inputs. In practice, MFD models are often fed with fast-varying demand patterns classically observed during peak hours. This paper investigates the accuracy of existing MFD model outputs in such a case. It also provides a fresh look at hysteresis loops that may arise on MFD shapes and shows how they are connected to the demand profile. First derived are the exact solutions of the Lighthill, Whitham, and Richards (LWR) model for a simple but meaningful test case corresponding to the loading and the recovery of an arterial with a single bottleneck. The spatial integration of these solutions permits to derive the average flow- and outflow-MFD patterns. The average flow-MFD exhibits a clockwise hysteresis loop with higher flows during the network loading than during the recovery. The outflow MFD shows an opposite trend with a counterclockwise hysteresis loop. The detailed analysis of the LWR solution highlights that these patterns are fully driven by the congestion dynamics and different wave propagation during both periods. This complements existing studies that tend to explain hysteresis by heterogeneous network loading due to unbalanced flows at intersections. It should be noted that the size of the production-MFD hysteresis loop is influenced by the demand loading profile, which means that the MFD is more sensitive to the demand than initially expected. The second part of the paper is about the implications of these observations on the MFD simulation. The main conclusion is that the accumulation-based formulation captures properly the aggregate dynamics of the LWR model during saturation but not in free-flow. It should be calibrated with an average flow-MFD that does not account for the hysteresis pattern to properly estimate the outflow. On the contrary, the original formulation of the trip-based model is only valid in free-flow. During saturation, it should be patched to modify the outflow values. The best approach is a hybrid model that combines the trip-based model in free-flow and the accumulation-based model during saturation. Note that in free-flow the trip-based model is better fed by the average flow-MFD with hysteresis as it gives better estimation for the mean speed. Finally, some evidences are presented and tend to confirm that all the conclusions remain valid for more complex urban settings.

Effects of traffic control regulation on Network Macroscopic Fundamental Diagram: A statistical analysis of real data

This research aims to verify the influence of traffic control strategies using signal plans at junctions on observed Network Macroscopic Fundamental Diagram (NMFD), as a variable to describe the aggregate traffic conditions of an urban road network.After establishing the theoretical basis of its existence, the research on NMFDs has been finalized for applications connected to traffic congestion control and management. The degree NMFD depends on traffic control strategies has not been fully demonstrated, nor has its relationship with different congestion levels. In the opinion of the authors, experimental NMFDs based on observed traffic data in the presence of different control strategies are key elements in supporting the understanding of this relationship.The paper has been structured into two parts. The first part presents the formulation of an optimization model for the design of signal plans at junctions which incorporates NMFD variables. The more extensive second part examines the effects of traffic control strategies on observed NMFDs, built with real traffic data. The investigation follows a procedure where observed traffic data are analysed to find variables, functions and diagrams at link and network levels (NMFD). The procedure is applied in two adjacent urban zones located in the central area of Santander (Spain). These zones were identified according to traffic characteristics and land uses. The junctions on the road sub-networks in the two zones have different control regulations, but they are used by the same road users (with the same behavioural strategy). Observed traffic data (flows, densities) from loop sensors have been collected and correlated with the signal control plans at different periods of a working day.What emerges from the analysis is that junction regulation affects traffic flow variables at a link and network level. The NMFDs are statistically different when comparing junction regulations at two homogenous adjacent zones. The results apply only to the studied area and should be confirmed in other real transport systems.

Flow exchanges in multi-reservoir systems with spillbacks

Large-scale traffic flow models based on the Network Macroscopic Fundamental Diagram (MFD) are usually grounded on the bathtub analogy and a conservation equation for vehicle accumulation inside a given urban area. Recent studies have proposed a different approach where the MFD defines the spatial mean speed that is shared by all vehicles in a region while their traveling distance is tracked individually. The former approach is also referred to as “accumulation-based” while the latter is usually named “trip-based”. While extensive studies of both model properties have been carried out for the single reservoir case (a unique region), the multi-reservoir setting still requires some research effort in particular to clearly understand how inflow merge at a reservoir entry and outflow diverge at exit should be managed. These two components play a significant role in the evolution of the whole system, when flows are exchanged between multiple reservoirs. One of the crucial questions is to ensure that congestion properly propagates backwards through a succession of reservoirs when oversaturated situations are observed.In this paper, we propose a thorough analysis of how to handle congestion propagation in the accumulation-based framework with several trip lengths or categories, e.g. internal and external trips. This allows to derive a congestion propagation model for the trip-based approach in a multi-reservoir setting. Based on theoretical considerations and simulation studies, we develop a consistent framework to restrict the inflow and adapt to oversaturated traffic conditions in a reservoir including several trip lengths. Two inflow merging schemes are investigated. The first one is inspired from the existing literature and shares the available supply based on the demand flow ratio at the entry boundary. It is called “exogeneous” in contrast to the second “endogenous” scheme, which shares the supply with respect to the internal accumulation ratio on the different routes. At the reservoir exit, a new outflow diverging scheme is also introduced to better reproduce the effect of queuing vehicles that are prevented from exiting the reservoir when congestion spills back from neighboring reservoirs. Compared to the conventional outflow model, our new approach proves to avoid unrealistic gridlocks when the reservoir becomes oversaturated. Both entry and exit flow models are investigated in details considering the accumulation-based and trip-based frameworks. Finally, the most consistent approach is compared with two other existing MFD models for multiple reservoirs. This demonstrates the importance of properly handling entry and exit flows at boundaries.

An identification model of critical control sub-regions based on macroscopic fundamental diagram theory

As traffic congestion intensifies, identification and control of critical areas to balance road network traffic flow are of paramount importance to mitigate and prevent traffic congestion, based on the road network traffic realities. Most previous traffic sub-region division methods of static characteristics are not suitable for dynamic traffic management. This article proposes the division model of traffic sub-regions and identification method of critical control areas based on macroscopic fundamental diagram (MFD) theory closely integrated with dynamic characteristics of road network traffic. To be specific, the existence of small-scale road network MFD is confirmed at first through the traffic data transmitted from Jiangbei district in Chongqing, China. The initial control sub-regions of the road network are formulated with each small-scale road network as a control area. Then, the traffic state value of sub-regions is calculated by means of regression analysis. A set of rules are designed to aggregate the initial sub-regions, which are associated with the traffic state and geographically adjacent, and the adjusted traffic sub-regions are obtained by means of correlation coefficient analysis. Finally, the identification method of traffic critical control sub-regions is proposed based on clustering analysis. In addition, the identification model of critical control sub-regions is applied and analyzed in the experimental road network of Jiangbei district in Chongqing, China. The results demonstrate that the proposed model is flexible and efficient enough to improve the control over road networks, promoting application of traffic sub-region division based on MFD in dynamic traffic control over actual road networks.

Minimal parameter formulations of the dynamic user equilibrium using macroscopic urban models: Freeway vs city streets revisited

This paper investigates the dynamic user equilibrium (DUE) on a single origin-destination pair with two alternative routes, a freeway with a fixed capacity and the surrounding city-streets network, modeled with a network macroscopic fundamental diagram (NMFD). We find using suitable transformations that only a single network parameter is required to characterize the DUE solution, the freeway to NMFD capacity ratio. We also show that the stability and convergence properties of this system are captured by the constant demand case, which corresponds to an autonomous dynamical system that admits analytical solutions. This solution is characterized by two critical accumulation values that determine if the steady state is in free-flow or gridlock, depending on the initial accumulation. Additionally, we also propose a continuum approximation to account for the spatial evolution of congestion, by including variable trip length and variable NMFD coverage area in the model. It is found that gridlock cannot happen and that the steady-state solution is independent of surface network parameters. These parameters do affect the rate of convergence to the steady-state solution, but convergence rates appear virtually identical when time is expressed in units of the NMFD free-flow travel time.

Effects of iterative learning based signal control strategies on macroscopic fundamental diagrams of urban road networks

Urban traffic flows are inherently repeated on a daily or weekly basis. This repeatability can help improve the traffic conditions if it is used properly by the control system. In this paper, we propose a novel iterative learning control (ILC) strategy for traffic signals of urban road networks using the repeatability feature of traffic flow. To improve the control robustness, the ILC strategy is further integrated with an error feedback control law in a complementary manner. Theoretical analysis indicates that the ILC-based traffic signal control methods can guarantee the asymptotic learning convergence, despite the presence of modeling uncertainties and exogenous disturbances. Finally, the impacts of the ILC-based signal control strategies on the network macroscopic fundamental diagram (MFD) are examined. The results show that the proposed ILC-based control strategies can homogenously distribute the network accumulation by controlling the vehicle numbers in each link to the desired levels under different traffic demands, which can result in the network with high capacity and mobility.

Clustering route choices methodology for network performance analysis

ABSTRACTThe Network Macroscopic Fundamental Diagram (NMFD) provides a quick overlook of a network performance. Recent studies show that inhomogeneous density spatial distributions significantly reduce the network efficiency. However, few studies investigate the triggers of such distributions. In this paper, we investigate how route choices influence the level of performance of a network and we propose a complete methodology to analyse such a relationship. To simplify the exploration, route choice alternatives are clustered in homogeneous groups with respect to the percentage of overlapping and a large set of flow distribution among routes is tested. The investigations are conducted using both analytical formulations and simulation techniques based on a mesoscopic traffic flow simulator. Analytical formulations define the boundary between fluid and congested network loading. The sensitivity to route choices is then assessed. Then, route choice effects on the network level of performance is quantified using the NMFD. The different evaluated scenarios confirm that the network performance is highly affected by route choice, both by paths' selection and flow distribution. Moreover, the study of hysteresis loops emerging in the NMFD shows that route choice impacts significantly the spatial distribution of congestion.

Assessing the Impact of Rainfall on Traffic Operation of Urban Road Network

The research reported in this paper quantifies the impact of rainfall on traffic operation of urban road network. The macroscopic analysis method of the network is the Macroscopic Fundamental Diagram (MFD), which can be used to describe and estimate the level-of-service of road network and evaluate the network-wide traffic state. Quantitative mastering the impact of rainfall on MFD of urban network is not only important for the further understanding of the MFD stability, but also important for the transportation planning and traffic management under rainy conditions. The results of the empirical analysis indicate that rainfall has an obviously diminishing effect on traffic variables of the network's MFD. The average reductions in production, accumulation and weighted speed are 0.2%, 7.8% and 5.4%, respectively. Moreover, rain has greater negative impact on network's MFD in the evening peak.