Macroscopic Fundamental Diagram Research Articles

Effect of the trip-length distribution on network-level traffic dynamics: Exact and statistical results

This paper presents additional results of the generalized bathtub model for urban networks, including a simpler derivation and exact solutions for uniformly distributed trip lengths. It is shown that in steady state this trip-based model is equivalent to the more parsimonious accumulation-based model, and that the trip-length distribution has merely a transient effect on traffic dynamics, which converge to the same point in the macroscopic fundamental diagram (MFD). To understand the statistical properties of the system, a queueing approximation method is proposed to compute the network accumulation variance. It is found that (i) the accumulation variance is much larger than predicted by traditional queueing models, due to the nonlinear dynamics imposed by the MFD, (ii) the trip-length distribution has no effect on the accumulation variance, indicating that a proposed formula for the variance might be universal, and (iii) the system exhibits critical behavior near the capacity state where the accumulation variance diverges. This indicates that the tools from critical phenomena and phase transitions might be useful to understand congestion in cities.

A Neural Network and Generalized Predictive Control Framework for Urban Traffic System Optimal Perimeter Control

Optimal perimeter control is one of the effective control technologies to alleviate urban congestion, which is based on macroscopic fundamental diagrams (MFDs). However, most previous optimal perimeter control methods used a linearization model at the desired point to approximate a nonlinear MFDs system, which is vulnerable to causing a potential mismatch between the linearization model and the environmental dynamics. However, this mismatch often leads to performances degradation. To solve this problem, this paper uses a neural network trained from data to approximate the complex nonlinear urban traffic system globally. To facilitate the design and control, this framework linearizes the nonlinear neural network of the traffic system to an instantaneous linearization model. The optimal perimeter control problem is finally solved by generalized predictive control (GPC) with the instantaneous linearization model. A key advantage of the proposed framework is that the global instantaneous linearization model is more accurate than the linearization model around the desired point. Simulation results show that the proposed framework significantly alleviates congestion and reduces the total travel time spent (TTS) in the traffic system compared with no control, “greedy” feedback control and PID control.

Heterogeneity Aware Emission Macroscopic Fundamental Diagram (e-MFD)

Transportation sector is one of the major producers of greenhouse gases which are responsible for climate change. Finding an appropriate emission estimation tool for large-scale networks is essential for developing efficient emission mitigation strategies. This paper presents an advanced version of the emission macroscopic fundamental diagram (e-MFD) which improves the stability and accuracy of the previous model. A bi-modal function is applied to separate free-flow and congested branches of the e-MFD. The accuracy of the proposed e-MFD is evaluated with both a synthetic grid network and a real-world city-level network. The study also assesses the model’s stability under directional traffic demands and road incidents. A comparison with the original e-MFD also verifies the superiority of the proposed model with higher accuracy. Standard deviation of density used in the proposed model to boost the performance. It is worth mentioning the standard deviation can be recorded with the existing hardware, such as loop detectors, and does not impose a considerable computational complexity. The proposed model can be employed for emission measurement in large-scale networks and hierarchical traffic control systems for more homogeneous congestion distribution and emission control.

From aggregated traffic models to emissions quantification: connecting the missing dots

Traffic models based on the Macroscopic Fundamental Diagram have great potential for future applications of large-scale estimation and monitoring of exhaust emissions. These traffic models offer an aggregated representation of the traffic dynamics in the network, considering mean spatial speeds and average travel distances per zone in the city, which are then used to determine the exhaust emissions using a macroscopic emission model. This aggregated estimation of the travel emissions has an inherent bias when compared to a benchmark scenario consisting of a microscopic estimation of the emissions at the network level. This paper analyzes this discrepancy qualitatively and quantitatively. As our testbed, we consider the traffic data of the 1st of November collected by the project pNEUMA in downtown Athens, Greece. The benchmark scenario consists of the estimation of exhaust emissions using the Vehicle Specific Power model. For the aggregated estimation of travel emissions, we use the aggregated traffic dynamics based on the Macroscopic Fundamental Diagram dynamics and COPERT V emission model. Our results show that the aggregated methodology underestimates CO2 emissions by 30% and NOx emissions by 32%, in agreement with the existing literature. Despite the observed discrepancies between the estimated emissions, the aggregated traffic dynamics approach based on the Macroscopic Fundamental Diagram is computationally lighter than a microscopic one, and therefore more appropriate for large-scale applications.

Two-region Perimeter Control Based on Risk-averse Model Predictive Control

The perimeter control is effective in alleviating traffic congestion. However, previous studies about perimeter control based on macroscopic fundamental diagram didn't consider the influences of the system uncertainty even though this uncertainty may lead to significant congestion. This paper presents a two-region perimeter control method based on a risk-averse model predictive control considering uncertainty. The advantage of this paper is that we design a risk index for uncertainty and design a controller which takes the risk index as an optimization objective. A scenario tree is used to model the uncertainty of the two-region dynamic equations. Average value at risk mapping is employed to calculate the systematic risk due to uncertainty. Then, the optimization objective of a multi-stage risk is used in MPC based on the scenario tree. Finally, the multi-stage risk is formulated as a solvable form. Simulation results show that the proposed perimeter control method reduces the total travel time of the two-region road network with uncertainty and significantly reduces traffic congestion compared to stochastic model predictive control method.

Expected changes in the energy performance of traffic flows as penetrated by electrified vehicles

This paper examines the energy performance of traffic as penetrated by electrified vehicles (BEVs), in comparison with the corresponding behavior of fossil fueled. The evaluation is conducted by combining two approaches that provide macroscopic estimations for the traffic performance and the energy consumption. A Macroscopic Fundamental Diagram (MFD) is used to identify the congestion status of an urban area, under various traffic demands. Speed dependent energy consumption functions able to capture the effect of congestion are fed with the traffic outputs of the MFD to estimate traffic included consumption. Results show that BEVs energy consumption is not sensitive to changes in traffic conditions and the demand, in contrast with the conventional which consumption particularly rises with the traffic demand. A potential rebound effect from an incremental use of BEVs also evaluated.

On the Characterization of Eco-Friendly Paths for Regional Networks

Macroscopic traffic models represent a promising tool to design strategies for ecological routing. To benefit from this tool, we must first characterize the relationship between path emissions and distance traveled or travel time on aggregated networks, i.e., a regional network. This paper investigates this relationship between two toy networks and a real urban network representing the city of Innsbruck (Austria). We utilize an accumulation-based model based on the Macroscopic Fundamental Diagram to mimic the traffic dynamics in the network and utilize the COPERT IV model to estimate the travel emissions, focusing on the carbon dioxide CO2. We show that there is a linear relationship between the total emissions of CO2 and the average travel time of internal paths, i.e. paths that take place completely within a single region. We also show that in some cases, there is a linear relationship between the total emissions and the average travel distance or travel time of paths that cross multiple regions in the network. However, the latter is not always true as traffic dynamics play an important role in path emissions. In other words, eco-friendly paths on regional networks do not necessarily follow the shortest paths in terms of distance or time.

Alpha-fair large-scale urban network control: A perimeter control based on a macroscopic fundamental diagram

This paper introduces a novel α-fair perimeter control (AFPC) scheme for a multi-region urban network based on the macroscopic fundamental diagram (MFD) modeling approach. The goal is to optimize the allocation of available region resources (i.e., available capacity) through perimeter control by metering the flow that transfers between any two neighbor regions while also considering the impacts (i.e., accumulation, average speed and queue) of such decision on road users in the regions. Regional traffic state dynamics are modeled using a multi-region MFD approach while queuing dynamics at a regional perimeter border, caused by control, are modeled using a point queue model. Based on regional average speed, a utility measure for each region is defined. Weights corresponding to regions are assigned dynamically to reflect the prevailing state of regional queues. The problem is then formulated based on these utilities with a novel perimeter control model that identifies the inflows between any two neighboring regions. The unique feature of the developed AFPC model is its ability to find a balanced trade-off between efficiency (i.e., travel time savings) and fairness in the distribution of both resources and impacts. The performance of the proposed model is compared with the no control base scenario as well as four other control scenarios. The results indicate that our proposed approach outperforms other control scenarios in terms of fairness while not degrading efficiency. The sensitivity analysis of a range of α values illustrates the wide spectrum of fair-centered yet efficient solutions that can be successfully achieved. Thus, by manipulating α values and regions’ weights, the AFPC framework can be tweaked to meet the specific fairness and efficiency aspirations of a network while also tailoring the specific regions’ need.

Developing modified congestion index and congestion-based level of service

Traffic flow and travel time-based measures are used to quantify and characterize congestion levels on urban roadway networks. However, under low-density conditions, contradictory results are observed when these two measures are used to identify the prevailing congestion levels. This accentuates the need to develop a new measure of congestion that effectively integrates traffic flow and travel time-based measures. The present study develops a new measure termed modified congestion index (MCI) by integrating traffic flow and travel time-based congestion measures for identifying prevailing congestion levels. As proof of the concept, the application of the methodology was demonstrated using empirical data collected at the corridor and network level and simulated data developed for a real-world urban network. The traffic flow and travel time-based congestion measures were integrated using a generalized least square framework (GLS) to derive MCI. Investigations revealed that traffic states bifurcated based on MCI follow the general philosophy of traffic state bifurcation using the macroscopic fundamental diagram (MFD) and fundamental diagram (FD). MCI, when correlated with existing performance measures, revealed that MCI could characterize both reliability and congestion simultaneously. As an important outcome from a policy and practical application perspective, MCI-based level of service (LOS) thresholds were developed. The application of the LOS thresholds was demonstrated by (a) analysing the effect of traffic signal control using a before-after perspective and (b) ranking and prioritizing segments in an urban network for optimal allocation of funds for developing congestion mitigation strategies. The developed LOS thresholds are expected to facilitate planners and engineers to comprehend the operational level of urban networks in terms of congestion and develop mitigation measures accordingly.

Hierarchical control for stochastic network traffic with reinforcement learning

This study proposes a hierarchical control framework to maximize the throughput of a road network driven by travel demand with uncertainties. In the upper level, a perimeter controller regulates the traffic influx into the core road network. The upper level uses a reinforcement learning algorithm that learns and responds to the traffic dynamics in the core road network without the need for an underlying system model and macroscopic fundamental diagram. The lower level is a local signal control system that regulates the spatial distribution of traffic flow within the core network. The results show that the hierarchical control framework can improve road network throughput by coordinating control actions conducted at the two levels. The improvement in system-wide performance is validated by a range of performance metrics and macroscopic flow-accumulation patterns realized under different control settings. The study contributes to the management of urban road networks with advanced computing technologies.

A Novel Environment Estimation Method of Whole Sample Traffic Flows and Emissions Based on Multifactor MFD

Vehicle emissions seriously affect the air environment and public health. The dynamic estimation method of vehicle emissions changing over time on the road network has always been the bottleneck of air quality simulation. The dynamic traffic volume is one of the important parameters to estimate vehicle emission, which is difficult to obtain effectively. A novel estimation method of whole sample traffic volumes and emissions on the entire road network based on multifactor Macroscopic Fundamental Diagram (MFD) is proposed in this paper. First, the intelligent clustering and recognition methods of traffic flow patterns are constructed based on neural network and deep-learning algorithms. Then, multifactor MFD models are developed considering different road types, traffic flow patterns and weekday peak hours. Finally, the high spatiotemporal resolution estimation method of whole sample traffic volumes and emissions are constructed based on MFD models. The results show that traffic flow patterns are clustered efficiently by the Self-Organizing Maps (SOM) algorithm combined with the direct time-varying speed index, which describe 91.7% traffic flow states of urban roads. The Deep Belief Network (DBN) algorithm precisely recognizes 92.1% of the traffic patterns based on the speeds of peak hours. Multifactor MFD models estimate the whole sample traffic volumes with a high accuracy of 91.6%. The case study shows that the vehicle emissions are evaluated dynamically based on the novel estimation method proposed in this paper, which is conducive to the coordinated treatment of air pollution.

Congested Urban Networks Tend to Be Insensitive to Signal Settings: Implications for Learning-Based Control

This paper highlights several properties of large urban networks that can have an impact on machine learning methods applied to traffic signal control. In particular, we note that the average network flow tends to be independent of the signal control policy as density increases past the critical density. We show that this property, which so far has remained under the radar, implies that no control (i.e. a random policy) can be an effective control strategy for a surprisingly large family of networks, especially for networks with short blocks. We also show that this property makes deep reinforcement learning (DRL) methods ineffective when trained under congested conditions, independently of the particular algorithm used. Accordingly, in contrast to the conventional wisdom around learning-based methods promoting the exploration of all states, we find that for urban networks it is advisable to discard any congested data when training, and that doing so will improve performance under all traffic conditions. Our results apply to all possible grid networks thanks to a parametrization introduced here. The impact of the turning probability was found to be very significant, in particular to explain the loss of symmetry observed in the macroscopic fundamental diagram of the networks, which is not captured by existing theories that rely on corridor approximations without turns. Our findings also suggest that supervised learning methods have enormous potential as they require very little examples to produce excellent policies.

Optimal traffic operation for maximum energy efficiency in signal-free urban networks: A macroscopic analytical approach

The integration of artificial intelligence and wireless communication technologies in communicant autonomous vehicles (CAVs) enables coordinating the movement of CAV platoons at signal-free intersections. The capacity of signal-free intersections can be significantly improved by adjusting traffic variables at a macroscopic scale; however, the resulting improvement in the capacity does not necessarily have a positive impact on the energy consumption of CAVs at the network level. In this research, we develop an analytical model to enhance energy efficiency by optimizing macroscopic traffic variables in signal-free networks. To this end, we adopt a macroscopic modeling approach to estimate the operational capacity by accounting for the stochasticity resulting from the error in synchronizing the arrival and departure of consecutive platoons in crossing directions at intersections. We also develop a macrolevel analytical model to estimate expected energy loss during the acceleration/deceleration maneuver required for resynchronization at intersections as a function of synchronization success probability. We then maximize energy efficiency by minimizing expected energy loss and maximizing expected capacity in a biobjective optimization framework. We solve the energy efficiency problem using an analytical approach to derive a closed-form solution for the optimal traffic speed and the length of the marginal gap between the passage of consecutive platoons in crossing directions through intersections for a (general) normal distribution of the operational error. Having the closed-form solution of the energy efficiency problem, we balance the trade-off between energy loss and operational capacity at a large scale by extending the analytical model to the network level using the Macroscopic Fundamental Diagram (MFD) concept. The results of our two-ring simulation model indicate the accuracy of the proposed analytical model in estimating the macroscopic relationship between the expected energy loss at intersections and the vehicular density in signal-free networks. Our numerical results also show that optimizing the traffic speed and marginal gap length can improve energy efficiency by 31% at the cost of a 16% decrease in maximum capacity.

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.

Adaptive Deep Q-Network Algorithm with Exponential Reward Mechanism for Traffic Control in Urban Intersection Networks

The demand for transportation has increased significantly in recent decades in line with the increasing demand for passenger and freight mobility, especially in urban areas. One of the most negative impacts is the increasing level of traffic congestion. A possible short-term solution to solve this problem is to utilize a traffic control system. However, most traffic control systems still use classical control algorithms with the green phase sequence determined, based on a specific strategy. Studies have proven that this approach does not provide the expected congestion solution. In this paper, an adaptive traffic controller was developed that uses a reinforcement learning algorithm called deep Q-network (DQN). Since the DQN performance is determined by reward selection, an exponential reward function, based on the macroscopic fundamental diagram (MFD) of the distribution of vehicle density at intersections was considered. The action taken by the DQN is determining traffic phases, based on various rewards, ranging from pressure to adaptive loading of pressure and queue length. The reinforcement learning algorithm was then applied to the SUMO traffic simulation software to assess the effectiveness of the proposed strategy. The DQN-based control algorithm with the adaptive reward mechanism achieved the best performance with a vehicle throughput of 56,384 vehicles, followed by the classical and conventional control methods, such as Webster (50,366 vehicles), max-pressure (50,541 vehicles) and uniform (46,241 vehicles) traffic control. The significant increase in vehicle throughput achieved by the adaptive DQN-based control algorithm with an exponential reward mechanism means that the proposed traffic control could increase the area productivity, implying that the intersections could accommodate more vehicles so that the possibility of congestion was reduced. The algorithm performed remarkably in preventing congestion in a traffic network model of Central Jakarta as one of the world’s most congested cities. This result indicates that traffic control design using MFD as a performance measure can be a successful future direction in the development of reinforcement learning for traffic control systems.

Urban Road Network Partitioning Based on Bi-Modal Traffic Flows With Multiobjective Optimization

The recent extension of a macroscopic fundamental diagram (MFD) into a bi-modal MFD (or 3D-MFD) provides the relationship among the total network circulating flows and the accumulations of private vehicles and public buses. 3D-MFD reveals the contribution of large occupancy vehicles such as buses in improving urban transportation efficiency. A lot of bi-modal traffic management techniques are introduced based on 3D-MFD to improve the urban traffic efficiency without using detailed origin-destination (OD) information. However, similar to MFD, 3D-MFD is also highly affected by the heterogeneity of a road network. In order to form 3D-MFDs with low scatter to be utilized for further bi-modal traffic management, this paper proposes a partition method to cluster road links into several homogeneous regions for a bi-modal urban network. It is comprised of three layers named as initial partition, merging, and boundary adjusting. At the initial partition layer, Seeded Region Growing (SRG) and the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) are integrated to obtain a number of subregions. A modified Genetic Algorithm (GA) is developed to merge the subregions into larger regions at the merging layer. Then, boundary adjusting is performed by changing the region to which a boundary is clustered to optimize the result. Multi-sensor data collected from Shenzhen in China are utilized to verify the effectiveness of the proposed partition method.

A macroscopic dynamic network loading model using variational theory in a connected and autonomous vehicle environment

Recent studies on macroscopic fundamental diagram (MFD) have shifted towards enhancing MFD dynamics in terms of efficiency and realism. In this paper, a multi-reservoir dynamic network loading (MRDNL) model for a large-scale urban road network is developed. The proposed framework consists of a hyper-link and a hyper-node model. The hyper-link model is constructed using variational theory (VT) by introducing a set of constraints that construct an upper bound for upstream and downstream cumulative accumulations. Using VT, LWR is incorporated into MFD dynamics to enhance traffic propagation at reservoirs. The hyper-link model captures important traffic phenomena, such as kinematic waves, queueing, and congestion, while the hyper-node model functions as a medium for transferring flow to other parts of the multi-reservoir urban network. A numerical scheme is advanced for solving the proposed MRDNL model and its performance is evaluated using a hypothetical traffic network. Compared to previous MFD-based dynamic network loading models, our approach is more computationally efficient since it does not require discretization and offers more realistic solutions by considering multiple kinematic waves and including the dynamics of urban traffic signals. This study, further, incorporated connected and autonomous vehicles (CAVs) into MFD dynamics and evaluated the network-wide effect of introduction of CAVs in urban traffic networks. For a network with 100% CAV market penetration rate, an approximate enhancement of 30% in total network outflow was found.

Coordinated perimeter control of urban road network based on traffic carrying capacity model

The perimeter traffic control of urban road network is of great significance for alleviating traffic congestion. Most of the existing perimeter control methods are design schemes based on macroscopic fundamental diagram (MFD), accompanied by effective algorithmic techniques. However, when applying MFD-based models for perimeter traffic control, it is always restricted by the heterogeneity of road network. In response to this problem, it is urgent to find a new model, which can not only get rid of the influence of heterogeneity, but also describe the evolution of traffic flow to guide the perimeter control. Research displays that the road network traffic carrying capacity model, a model designed to calculate the maximum number of vehicles in the road network under a certain service level, can meet these requirements. First, through the theoretical demonstration of the mathematical model, it is proved that the network heterogeneity neither affects the existence of traffic carrying capacity nor changes its parameter value. This further clarifies that there is no need to divide the homogeneous sub-regions in the preliminary steps of the proposed perimeter control. Second, an innovative road network perimeter coordination control framework is developed, which has a simple operation process and high computational efficiency. The main components of the structural framework include data collector, sub-region coordination controller, and perimeter signal controller. In particular, the core idea of coordinated control is to adjust and model the distribution mode of vehicles in the road network through the transfer of traffic flow between sub-regions, thereby achieving the optimal operation efficiency of the transportation system. Finally, the SUMO simulation software is employed to verify and evaluate the proposed model. The results show that compared with the no perimeter control and the MFD-based perimeter control, the proposed perimeter control strategy is demonstrated to have impressive performance. A key advantage is that the application of the proposed approach does not need to consider the road network heterogeneity, and has great results in both non-congested and congested conditions. Findings from this study can provide a novel direction for the application of urban regional traffic control.

Safety and Sustainable Development of Automated Driving in Mixed-Traffic Urban Areas—Considering Vulnerable Road Users and Network Efficiency

Next to environmental aspects, establishing areas for safe and economically viable automated driving in mixed-traffic settings is one major challenge for sustainable development of Autonomous Vehicles (AVs). This work investigates safety in the interactions between AVs, human-driven vehicles, and vulnerable road users, including cyclists and pedestrians, within a simulated urban environment in the Dutch city of Rotterdam. New junction and pedestrian models are introduced, and virtual AVs with an occlusion-aware driving system are deployed to deliver cargo autonomously. The safety of applying this autonomous cargo delivery service is assessed using a large set of Surrogate Safety Indicators (SSIs). Furthermore, Macroscopic Fundamental Diagrams (MFDs) and travel time loss are incorporated to evaluate the network efficiency. By assessing the impact of various measures involving Vehicle-to-Vehicle (V2V), Vehicle-to-Infrastructure (V2I), Vehicle-to-Everything (V2X) communications, infrastructure modifications, and driving behavior, we show that traffic safety and network efficiency can be achieved in a living lab setting for the considered case. Our findings further suggest that V2X gets implemented, new buildings are not placed close to intersections, and the speed limit of non-arterial roads is lowered.

Adaptive Coordinated Variable Speed Limit between Highway Mainline and On-Ramp with Deep Reinforcement Learning

Highway merging bottleneck is challenged with serious traffic conflicts between on-ramp and mainline vehicles, causing significant capacity drop and drastic speed changes. The paper proposes an adaptive coordinated variable speed limit model to manage highway speed of on-ramp and mainline continuous sections without priority to mainline. That helps to remove the speed difference between the vehicles from on-ramp and mainline flooding into the merging zone, and to sustain actual traffic density close to critical density to counteract capacity drop as indicated with macroscopic fundamental diagram. The method of deep reinforcement learning based on deep deterministic policy gradient is employed to solve the proposed model with a row of continuous control variables. Simulation platform with VISSIM 5.3 is established, and the proposed method can enhance traffic flow through the merging zone by around 10% and 19% under static and dynamic demand, respectively, in addition to reduced density and speed variation by around 30%. This research provides insights into the management of highway capacity so as to secure traffic efficiency and reliability for the merging zone.