Well-defined Macroscopic Fundamental Diagram Research Articles

Quasi revenue-neutral congestion pricing in cities: Crediting drivers to avoid city centers

This paper introduces a predictive congestion pricing method in cities wherein the tolls alter from region to region. We consider a large urban network is partitioned into multiple regions each with a well-defined Macroscopic Fundamental Diagram (MFD) where multiple routes exist between each origin and destination regions. The proposed cordon pricing method is designed to (i) minimize vehicles’ total time spent in the network and (ii) aim for a revenue-neutral tolling. A controller based on model predictive control (MPC) approach is proposed to determine the (possibly negative) optimal time- and region-varying tolls. The MPC controller comprises a regional MFD-based traffic model with no need of destination information and a long-short term memory neural network (LSTM-NN) to obtain an accurate estimation of inter-region transfer flows. Results of numerical experiments indicate the effectiveness of the proposed congestion pricing method to achieve the two objectives simultaneously, compared with No toll and reactive feedback controllers.

Robust perimeter control with cordon queues and heterogeneous transfer flows

Accumulation-based Macroscopic Fundamental Diagram (MFD) model is widely employed to design perimeter control methods to improve traffic operation in urban networks. While the accumulation-based MFD assumes a low-scatter, non-linear relationship between region production and accumulation, the outflow relationship in formulating dynamics of multi-region networks requires simplifying assumptions. The existing perimeter control methods are grounded on accumulation-based MFD models where the number of transferring vehicles is approximated by the ratio of the instantaneous number of vehicles based on their destinations. Moreover, perimeter control may lead to more vehicles queuing at the region boundary (i.e. cordon queues) which add local impediments on traveling vehicles and impact the accuracy of well-defined MFDs. To address these shortcomings under time-varying conditions, this paper develops a robust perimeter control method based on the Sliding Mode Control to minimize total travel time in the entire network. To test the performance of the proposed control method, a trip-based MFD model is developed that accounts for cordon queues and various trip lengths of individual travelers. In this paper, two-region accumulation-based and trip-based MFD models are compared through numerical experiments. The results pinpoint the proposed robust perimeter control method can effectively alleviate congestion and improve network efficiency during traffic rush hours.

Dynamic System Optimum Analysis of Multi-Region Macroscopic Fundamental Diagram Systems With State-Dependent Time-Varying Delays

This paper investigates the dynamic system optimum (DSO) problem with simultaneous route and departure time assignments for a general traffic network partitioned into multiple regions. Regional traffic congestion is modeled with a well-defined macroscopic fundamental diagram (MFD) mapping the trip completion rate to the vehicular accumulation. To overcome the limitation of inconsistent flow propagation between region boundaries and the corresponding travel time, the state-dependent regional travel time function is explicitly incorporated in the flow propagation of the conventional MFD dynamics. From a systems perspective, the traffic dynamics within a region can be regarded as a dynamic system with an endogenous time-varying delay depending on the system state. Equilibrium condition for the DSO problem is analytically derived through the lens of Pontryagin minimum principle and is compared against the static SO counterpart. The structure of path specific marginal cost is analyzed regarding the path travel cost and early-late penalty function. In contrast to existing analytical methods, the proposed method is applicable for general MFD systems without linearization of the MFD dynamics. Neither approximation of the equilibrium solution nor constant regional delay assumption is required. Numerical examples are conducted to illustrate the characteristics of DSO traffic equilibrium and the corresponding marginal cost together with other dynamic external costs.

Data-Driven Model Free Adaptive Perimeter Control for Multi-Region Urban Traffic Networks With Route Choice

Recent studies have shown that a homogenous urban road network exists a well-defined macroscopic fundamental diagram (MFD), which can be used for perimeter control conveniently. Most of the existing perimeter control strategies are model-based control methods, leading to the result that the control effect may be not good enough if the traffic model is not accurate. In this paper, a novel data-driven strategy called model free adaptive control (MFAC) method is proposed for multi-region perimeter control in order to get rid of the dependency of the aforementioned model-based MFD-based perimeter control methods. Due to the fact that the multi-region urban traffic system (MRUTS) is a complex interconnected system, a decentralized estimation and decentralized MFAC (DED-MFAC) method is utilized to deal with the strong-coupled characteristic of the traffic system. In this framework, MFD is used to determine the desired accumulations in each region and generate the throughput data of the urban traffic system, since the acquisition of trip completion flow is more difficult than accumulations. In addition, route choice is also integrated in the proposed policy to further improve the performance of the urban traffic system. A key advantage of the proposed approach is that it can only use the traffic data instead of the traffic model for real-time perimeter control. The effectiveness of the proposed perimeter control scheme is tested in simulation for a multi-region system, and the results show that it is superior to some other commonly used perimeter control methods.

Empirical evaluation of influential factors on bifurcation in macroscopic fundamental diagrams

Observations from empirical data in the roadway network showed that the relation between averages of network flow versus density often exhibit hysteresis and bifurcation phenomena, which may obscure the reproducibility of a well-defined macroscopic fundamental diagram (MFD). In this paper, we analyzed large-scale trip data from passenger vehicles in an urban network of South Korea and evaluated the shapes of MFDs over many days. It was found that MFDs have two distinctive, reproducible forms: a well-defined, unique relation on weekends and a bifurcation in high-density regime on weekdays.With regard to the bifurcation, we observed higher network flows in the morning and lower network flows in the evening for the same average network density. This implies that the same set of drivers in the network collectively formed two different trip patterns. Hence, we evaluated possible factors that may have effects on the bifurcation phenomenon. In view of this, four factors – heterogeneity, trip completion rate, detouring ratio and commute trips – were analyzed in this study. The findings showed that travelers’ detours could be a key factor for the occurrence of bifurcation because, by detouring travelers improve neither their own travel times nor network-wide travel times, and thereby degrade the network production.

Dynamic modeling and control of taxi services in large-scale urban networks: A macroscopic approach

Taxis are increasingly becoming a prominent mobility mode in many major cities due to their accessibility and convenience. The growing number of taxi trips and the increasing contribution of taxis to traffic congestion are cause for concern when vacant taxis are not distributed optimally within the city and are unable to find unserved passengers effectively. A way of improving taxi operations is to deploy a taxi dispatch system that matches the vacant taxis and waiting passengers while considering the search friction dynamics. This paper presents a network-scale taxi dispatch model that takes into account the interrelated impact of normal traffic flows and taxi dynamics while optimizing for an effective dispatching system. The proposed model builds on the concept of the macroscopic fundamental diagram (MFD) to represent the dynamic evolution of traffic conditions. The model considers multiple taxi service firms operating in a heterogeneously congested city, where the city is assumed to be partitioned into multiple regions each represented with a well-defined MFD. A model predictive control approach is devised to control the taxi dispatch system. The results show that lack of the taxi dispatching system leads to severe accumulation of unserved taxi passengers and vacant taxis in different regions whereas the dispatch system improves the taxi service performance and reduces traffic congestion by regulating the network towards the undersaturated condition. The proposed framework demonstrates sound potential management schemes for emerging mobility solutions such as fleet of automated vehicles and demand-responsive transit services.

Traffic guidance–perimeter control coupled method for the congestion in a macro network

Macroscopic fundamental diagram (MFD) describes the macro relationship between a network vehicle density and a network space mean flow, without requiring the mastery of complex origin to destination data. Thus, MFD provides an opportunity for the macro control of urban road network. However, most of the existing MFD control methods ignore the active role of traffic guidance in solving congestion problems. This study presents a traffic guidance–perimeter control coupled (TGPCC) method to improve the performance of macroscopic traffic networks. The method considers the optimal cumulative volume of a network as the goal and establishes a programming function according to the network equilibrium rule of traffic flow amongst multiple MFD sub-regions, which regards the minimum delay of network, as the objective. The Logit model for the compliance rate of driver route guidance is established by the stated preference survey. Moreover, the perimeter control (PC) method is proposed for adjusting the phase split of intersections. Finally, three schemes, namely, the TGPCC, PC and the method without PC and guidance are tested on a network with four well-defined MFD sub-regions. Results show that the TGPCC addresses the issue of congestion and decreases the total delay accordingly.

Dynamic modeling and control of taxi services in large-scale urban networks: A macroscopic approach

Abstract Taxis are increasingly becoming a prominent mobility mode in many major cities due to their accessibility and convenience. The growing number of taxi trips and the increasing contribution of taxis to traffic congestion are cause for concern when vacant taxis are not distributed optimally within the city and are unable to find unserved passengers effectively. A way of improving taxi operations is to deploy a taxi dispatch system that matches the vacant taxis and waiting passengers while considering the search friction dynamics. This paper presents a network-scale taxi dispatch model that takes into account the interrelated impact of normal traffic flows and taxi dynamics while optimizing for an effective dispatching system. The proposed model builds on the concept of the macroscopic fundamental diagram (MFD) to represent the dynamic evolution of traffic conditions. The model considers multiple taxi service firms operating in a heterogeneously congested city, where the city is assumed to be partitioned into multiple regions each represented with a well-defined MFD. A model predictive control approach is devised to control the taxi dispatch system. The results show that lack of the taxi dispatching system leads to severe accumulation of unserved taxi passengers and vacant taxis in different regions whereas the dispatch system improves the taxi service performance and reduces traffic congestion by regulating the network towards the undersaturated condition. The proposed framework demonstrates sound potential management schemes for emerging mobility solutions such as fleet of automated vehicles and demand-responsive transit services.

Exploring the Relationship between Heterogeneity of Vehicle Distribution and the Macroscopic Fundamental Diagram under Segment Disruption Conditions

Abstract: The framework of a well-defined Macroscopic Fundamental Diagram (MFD) relating network-wide flow and density was proposed under the homogeneity assumption that congestion is distributed evenly which is not universal in practice. This study aims to explore deep details of the relation between heterogeneity of vehicle distribution and the MFD under segment disruption conditions. By plotting MFDs and calculating variance, kurtosis and skewness of segment traffic density for different network structures, details of the relation are revealed. Results show that under segment disruption conditions, if the network flow reaches the maximum level at the same density for different network structures, a negative correlation between heterogeneity of vehicle distribution and maximum flow can be established. Similar relation can be observed under other states except free flow state. Nevertheless, heavy congestion may affect this relation because of spillback. Overall, the results are significant for traffic control to maintain networks at a high-efficiency level under segment disruption conditions.

Enhancing model-based feedback perimeter control with data-driven online adaptive optimization

Most feedback perimeter control approaches that are based on the Macroscopic Fundamental Diagram (MFD) and are tested in detailed network structures restrict inflow from the external boundary of the network. Although such a measure is beneficial for the network performance, it creates virtual queues that do not interact with the rest of the traffic and assumes small unrestricted flow (i.e. almost zero disturbance). In reality, these queues can have a negative impact to traffic conditions upstream of the protected network that is not modelled. In this work an adaptive optimization scheme for perimeter control of heterogeneous transportation networks is developed and the aforementioned boundary control limitation is dropped. A nonlinear model is introduced that describes the evolution of the multi-region system over time, assuming the existence of well-defined MFDs. Multiple linear approximations of the model (for different set-points) are used for designing optimal multivariable integral feedback regulators. Since the resulting regulators are derived from approximations of the nonlinear dynamics, they are further enhanced in real-time with online learning/adaptive optimization, according to performance measurements. An iterative data-driven technique is integrated with the model-based design and its objective is to optimize the gain matrices and set-points of the multivariable perimeter controller based on real-time observations. The efficiency of the derived multi-boundary control scheme is tested in microsimulation for a large urban network with more than 1500 roads that is partitioned in multiple regions. The proposed control scheme is demonstrated to achieve a better distribution of congestion (by creating “artificial” inter-regional queues), thus preventing the network degradation and improving total delay and outflow.

Optimal perimeter control synthesis for two urban regions with aggregate boundary queue dynamics

Perimeter control policies for urban regions with Macroscopic Fundamental Diagram (MFD) modeling have been presented in previous works. The control policies might meter the number of transferring vehicles from one region to another, resulting in queueing vehicles at regional boundaries. Concentrated vehicles at boundaries might affect the existence of well-defined MFDs. Most previous works neglect the effect of the boundary concentrated vehicles on the traffic flow dynamics, and do not explicitly consider their effect on the perimeter control policy.This paper introduces a new MFD-based model for two-region networks with aggregate boundary queue dynamics. The dynamic flow characteristics for the two urban regions are modeled by the MFD functions, while aggregate boundary queue dynamics for both regions are modeled by input-output balance differential equations. Maximum lengths are imposed on the aggregate boundary queues, that aim at maintaining the existence of well-defined MFDs and their dynamics.Based on the developed model, the optimal control policy to maximize the total network throughput is found. Analytical solutions for the optimal perimeter control problem, with constrained perimeter control inputs and constrained lengths of aggregate boundary queues, are derived. The optimal synthesis for principal cases are found and verified by numerical tests. The numerical results demonstrate the effect of aggregate boundary queues on the optimal perimeter control policy.

MACROSCOPIC FUNDAMENTAL DIAGRAMS: SIMULATION FINDINGS FOR THESSALONIKI’S ROAD NETWORK

Within the framework of this paper, the existence of a Macroscopic Fundamental Diagram (MFD) for the city of Thessaloniki is discussed, obtained through simulation. Initial findings show that there is a well-defined MFD, which is a property of a network, as it is not influenced by changes in travel demand. The MFD can be used as tool for managing traffic at real-time and for improving accessibility in an area, through pricing strategies and entrance points control based on the accumulation of vehicles.

Equilibrium analysis and route guidance in large-scale networks with MFD dynamics

Recent studies have demonstrated that Macroscopic Fundamental Diagram (MFD), which provides an aggregated model of urban traffic dynamics linking network production and density, offers a new generation of real-time traffic management strategies to improve the network performance. However, the effect of route choice behavior on MFD modeling in case of heterogeneous urban networks is still unexplored. The paper advances in this direction by firstly extending two MFD-based traffic models with different granularity of vehicle accumulation state and route choice behavior aggregation. This configuration enables us to address limited traffic state observability and to scrutinize implications of drivers’ route choice in MFD modeling. We consider a city that is partitioned in a small number of large-size regions (aggregated model) where each region consists of medium-size sub-regions (more detailed model) exhibiting a well-defined MFD. This paper proposes a route guidance advisory control system based on the aggregated model as a large-scale traffic management strategy that utilizes aggregated traffic states while sub-regional information is partially known. In addition, we investigate the effect of equilibrium conditions (i.e. user equilibrium and system optimum) on the overall network performance, in particular MFD functions.

Dynamics of heterogeneity in urban networks: aggregated traffic modeling and hierarchical control

Real traffic data and simulation analysis reveal that for some urban networks a well-defined Macroscopic Fundamental Diagram (MFD) exists, which provides a unimodal and low-scatter relationship between the network vehicle density and outflow. Recent studies demonstrate that link density heterogeneity plays a significant role in the shape and scatter level of MFD and can cause hysteresis loops that influence the network performance. Evidently, a more homogeneous network in terms of link density can result in higher network outflow, which implies a network performance improvement. In this article, we introduce two aggregated models, region- and subregion-based MFDs, to study the dynamics of heterogeneity and how they can affect the accuracy scatter and hysteresis of a multi-subregion MFD model. We also introduce a hierarchical perimeter flow control problem by integrating the MFD heterogeneous modeling. The perimeter flow controllers operate on the border between urban regions, and manipulate the percentages of flows that transfer between the regions such that the network delay is minimized and the distribution of congestion is more homogeneous. The first level of the hierarchical control problem can be solved by a model predictive control approach, where the prediction model is the aggregated parsimonious region-based MFD and the plant (reality) is formulated by the subregion-based MFDs, which is a more detailed model. At the lower level, a feedback controller of the hierarchical structure, tries to maximize the outflow of critical regions, by increasing their homogeneity. With inputs that can be observed with existing monitoring techniques and without the need for detailed traffic state information, the proposed framework succeeds to increase network flows and decrease the hysteresis loop of the MFD. Comparison with existing perimeter controllers without considering the more advanced heterogeneity modeling of MFD highlights the importance of such approach for traffic modeling and control.

Equilibrium Analysis and Route Guidance in Large-scale Networks with MFD Dynamics

Recent studies have demonstrated that Macroscopic Fundamental Diagram (MFD), which provides an aggregated model of urban traffic dynamics linking network production and density, offers a new generation of real-time traffic management strategies to improve the network performance. However, the effect of route choice behavior on MFD modeling in case of heterogeneous urban networks is still unexplored. The paper advances in this direction by firstly extending two MFD-based traffic models with different granularity of vehicle accumulation state and route choice behavior aggregation. This configuration enables us to address limited traffic state observability and to scrutinize implications of drivers’ route choice in MFD modeling. We consider a city that is partitioned in a small number of large-size regions (aggregated model) where each region consists of medium-size sub-regions (more detailed model) exhibiting a well-defined MFD. This paper proposes a route guidance advisory control system based on the aggregated model as a large-scale traffic management strategy that utilizes aggregated traffic states while sub-regional information is partially known. In addition, we investigate the effect of equilibrium conditions (i.e. user equilibrium and system optimum) on the overall network performance, in particular MFD functions.

Perimeter and boundary flow control in multi-reservoir heterogeneous networks

In this paper, we macroscopically describe the traffic dynamics in heterogeneous transportation urban networks by utilizing the Macroscopic Fundamental Diagram (MFD), a widely observed relation between network-wide space-mean flow and density of vehicles. A generic mathematical model for multi-reservoir networks with well-defined MFDs for each reservoir is presented first. Then, two modeling variations lead to two alternative optimal control methodologies for the design of perimeter and boundary flow control strategies that aim at distributing the accumulation in each reservoir as homogeneously as possible, and maintaining the rate of vehicles that are allowed to enter each reservoir around a desired point, while the system’s throughput is maximized. Based on the two control methodologies, perimeter and boundary control actions may be computed in real-time through a linear multivariable feedback regulator or a linear multivariable integral feedback regulator. Perimeter control occurs at the periphery of the network while boundary control occurs at the inter-transfers between neighborhood reservoirs. To this end, the heterogeneous network of San Francisco is partitioned into three homogeneous reservoirs and the proposed feedback regulators are compared with a pre-timed signal plan and a single-reservoir perimeter control strategy. Finally, the impact of the perimeter and boundary control actions is demonstrated via simulation by the use of the corresponding MFDs and other performance measures. A key advantage of the proposed approach is that it does not require high computational effort and future demand data if the current state of each reservoir can be observed with loop detector data.

Cooperative traffic control of a mixed network with two urban regions and a freeway

Currently most optimization methods for urban transport networks (i) are suited for networks with simplified dynamics that are far from real-sized networks or (ii) apply decentralized control, which is not appropriate for heterogeneously loaded networks or (iii) investigate good-quality solutions through micro-simulation models and scenario analysis, which make the problem intractable in real time. In principle, traffic management decisions for different sub-systems of a transport network (urban, freeway) are controlled by operational rules that are network specific and independent from one traffic authority to another. In this paper, the macroscopic traffic modeling and control of a large-scale mixed transportation network consisting of a freeway and an urban network is tackled. The urban network is partitioned into two regions, each one with a well-defined Macroscopic Fundamental Diagram (MFD), i.e. a unimodal and low-scatter relationship between region density and outflow. The freeway is regarded as one alternative commuting route which has one on-ramp and one off-ramp within each urban region. The urban and freeway flow dynamics are formulated with the tool of MFD and asymmetric cell transmission model, respectively. Perimeter controllers on the border of the urban regions operating to manipulate the perimeter interflow between the two regions, and controllers at the on-ramps for ramp metering are considered to control the flow distribution in the mixed network. The optimal traffic control problem is solved by a Model Predictive Control (MPC) approach in order to minimize total delay in the entire network. Several control policies with different levels of urban-freeway control coordination are introduced and tested to scrutinize the characteristics of the proposed controllers. Numerical results demonstrate how different levels of coordination improve the performance once compared with independent control for freeway and urban network. The approach presented in this paper can be extended to implement efficient real-world control strategies for large-scale mixed traffic networks.

Macroscopic Traffic Control of a Mixed Urban and Freeway Network

In this paper, the macroscopic traffic control of a large-scale mixed transportation network consisting of freeway and urban network is tackled. The urban network is partitioned in two regions, each one with a well-defined macroscopic fundamental diagram (MFD), i.e. a unimodal and low-scatter relationship between network density and outflow. The freeway is regarded as one alternative commuting route which has one on-ramp and one off-ramp within each urban region. The urban and freeway flow dynamics are formulated with the tool of MFD and asymmetric cell transmission models, respectively. Four controllers are considered to control the flow distribution between urban regions and freeway: (i) two on the border of urban regions operating to manipulate the perimeter interflow rates between the two regions, and (ii) two other controllers on the on-ramps for ramp metering to control the flow rates from urban roads to the freeway. The optimal traffic control problem for the mixed network is solved by a receding horizon approach in order to maximize the number of trips that reach their destinations. The results of this paper can be extended to develop efficient control strategies for large-scale mixed traffic networks.

Properties of a well-defined macroscopic fundamental diagram for urban traffic

A field experiment in Yokohama (Japan) revealed that a macroscopic fundamental diagram (MFD) linking space-mean flow, density and speed exists on a large urban area. It was observed that when the highly scattered plots of flow vs. density from individual fixed detectors were aggregated the scatter nearly disappeared and points grouped along a well defined curve. Despite these and other recent findings for the existence of well-defined MFDs for urban areas, these MFDs should not be universally expected. In this paper we investigate what are the properties that a network should satisfy, so that an MFD with low scatter exists. We show that the spatial distribution of vehicle density in the network is one of the key components that affect the scatter of an MFD and its shape. We also propose an analytical derivation of the spatial distribution of congestion that considers correlation between adjacent links. We investigate the scatter of an MFD in terms of errors in the probability density function of spatial link occupancy and errors of individual links’ fundamental diagram (FD). Later, using real data from detectors for an urban arterial and a freeway network we validate the proposed derivations and we show that an MFD is not well defined in freeway networks as hysteresis effects are present. The datasets in this paper consist of flow and occupancy measures from 500 fixed sensors in the Yokohama downtown area in Japan and 600 loop detectors in the Twin Cities Metropolitan Area Freeway network in Minnesota, USA.