The macroscopic fundamental diagram explained by a walking experiment in class

This paper applies the Macroscopic Fundamental Diagram (MFD) and the Generalized Macroscopic Fundamental (GMFD) Diagram to estimate traffic characters for an urban area. We used traffic flow data manually counted from intersection traffic video and taxi GPS data to derive MFD and GMFD. The method is suitable to recognize traffic situation. These figures show the property of the road network, which consists of infrastructure and control. Mastering the properties of the network supply side, we can better manage and control traffic demand. In the process of data analysis, we found that the time step for data processing is a determinant for the MFD and GMFD shapes. That means different time step produces different results.

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

Since 1933, the Fundamental Diagram (FD) has been studied and used to represent relationships between (in particular) Density and Flow for vehicles traveling over a particular link. The Macroscopic Fundamental Diagram (MFD) is a relatively new generalisation of FD that is used to represent the relationships between density and flow but for the vehicles traveling over the whole network (either the motorway (highway, freeway) or over all arterial roads in the particular city area). In the current paper, we point on several problems with the construction and the use of FD and MFD that can potentially lead to incorrect results and, hence, jeopardise the use of FD and MFD in practical traffic control systems. These problems should be assessed if we want to avoid mathematical and logical flaws that lead to wrong conclusions.

Part I of the thesis investigates novel urban traffic state estimation methods utilizing probe vehicle data. Chapter 2 proposes a method to integrate the collective effect of dispersed probe data with traffic kinematic wave theory and data mining techniques to model the spatial and temporal dynamics of queue formation and dissipation in arterials. The queue estimation method captures interdependencies in queue evolutions of successive intersections, and moreover, the method is applicable in oversaturated conditions and includes a queue spillover statistical inference procedure. Chapter 3 develops a travel time reliability model to estimate arterial route travel times distribution (TTD) considering spatial and temporal correlations between traffic states in consecutive links. The model uses link-level travel time data and a heuristic grid clustering method to estimate the state structure and transition probabilities of a Markov chain. By applying the Markov chain procedure, the correlation between states of successive links is integrated and the route-level TTD is estimated. The methods in Part I are tested with various probe vehicle penetration rates on case studies with field measurements and simulated data. The methods are straightforward in implementation and have demonstrated promising performance and accuracy through numerous experiments. Part II studies network-level modeling and control of large-scale urban networks. Chapter 4 is the pioneer that introduces the urban perimeter control for two-region urban cities as an elegant control strategy to decrease delays in urban networks. Perimeter controllers operate on the border between the two regions, and manipulate the percentages of transfer flows between the two regions, such that the number of trips reaching their destinations is maximized. The optimal perimeter control problem is solved by the model predictive control (MPC) scheme, where the prediction model and the plant (reality) are formulated by macroscopic fundamental diagrams (MFD). Chapter 5 extends the perimeter control strategy and MFD modeling to mixed urban-freeway networks to provide a holistic approach for large-scale integrated corridor management (ICM). The network consists of two urban regions, each one with a well-defined MFD, and a freeway, modeled by the asymmetric cell transmission model, that is an alternative commuting route which has one on-ramp and one off-ramp within each urban region. The optimal traffic control problem is solved by the MPC approach to minimize total delay in the entire network considering several control policies with different levels of urban-freeway control coordination. Chapter 6 integrates traffic heterogeneity dynamics in large-scale urban modeling and control to develop a hierarchical control strategy for heterogeneously congested cities. Two aggregated models, region- and subregion-based MFDs, are introduced to study the effect of link density heterogeneity on the scatter and hysteresis of MFD. A hierarchical perimeter flow control problem is proposed to minimize the network delay and to homogenize the distribution of congestion. The first level of the hierarchical control problem is solved by the MPC approach, where the prediction model is the aggregated parsimonious region-based MFD and the plant is the subregion-based MFD, which is a more detailed model. At the lower level, a feedback controller tries to maximize the network outflow, by increasing regional homogeneity.

Recent advances in traffic flow theory at the network level, namely the Macroscopic Fundamental Diagram (MFD), reveals the existence of well-defined laws of congestion dynamics at aggregated levels. The same knowledge for multimodal networks however is limited. It is critical to understand how urban space can be allocated and managed for multimodality. The objective is to develop aggregated modeling and optimization approaches, which will contribute on the knowledge of congestion dynamics in cities of different structures and mode usages, and ultimately facilitate the design of efficient and equitable urban transport policies. Building on the knowledge of the single-mode MFD theory, a bi-modal MFD model considering the effect of mode conflict is proposed for mixed networks of buses and cars. A system-level model is developed for multiple-region city network. The flow dynamics among regions are described by a regional level flow conservation law. A non-linear optimization framework is performed to optimize space allocation, minimizing the total passenger cost, given certain demand, city structure and road facility. Then, parking limitation is integrated in the proposed multi-modal system model, where vehicles cruising for parking are also integrated. The extra delay of cruising is captured by a geometric distribution related to the time-dependent parking availability and estimated at the aggregated level. The delay cost to other users is also estimated via the bi-modal MFD, and it shows the effect of cruising on all travelers who do not require parking. Optimal parking pricing policies for on-street and garage parking are obtained through the optimization framework, as well. The existence of a three-dimensional MFD (3D-MFD) for mixed bi-modal networks is investigated and analyzed via micro-traffic simulation studies. A 3D-MFD relates vehicular production of a network (flow, travel distance) to the density of cars and buses, where the impact of each mode on network performance can be directly observed. To further compare the modal impact on performance, the Bus-Car Unit equivalent value is estimated, indicating that this value is state- and mode-composition dependent rather than deterministic. In addition to the conventional vehicle-flow-based analysis, a passenger 3D-MFD is derived which provides a different perspective of the flow characteristics in bi-modal networks. Simulation study on 3D-MFD based perimeter-control shows promising performance in real-time control. The final part of the thesis concerns the MFD-controlled congestion pricing. Feedback-type control mechanisms are proposed to determine and adjust the time-dependent tolls, based on congestion level as expressed by the MFD. One pricing scheme also considers userâ s adaptation to the toll cost, allowing a great flexibility in toll adjustment, and deals with the promotion of public transport usage. The performance of the pricing schemes is investigated in an existing agent-based model where the complex travel behavior in real-life is reasonably reproduced. Results demonstrate that the pricing schemes are effective in congestion reduction. Remarkably, smooth behavioral equilibrium in long-term operation is found under such pricing schemes. Furthermore, user heterogeneity with respect to value-of-time is introduced in the agent-based model. By realizing and treating this heterogeneity, pricing strategies can achieve even higher efficiency and equitable benefit.

This paper explores the concept of the Network Fundamental Diagram for two-dimensional pedestrian networks. In doing so, we investigate if the average performance of the network can be described as a function of the average density or accumulation of the network, in line with recent findings on discrete vehicular traffic networks. We show that this is indeed the case, by considering data from walking experiments and from simulation using a microscopic pedestrian model. It turns out that the shape of the diagram is determined by a number of factors, such as the shape and size of the area, its use and its function, and the composition of the pedestrian flow. Finally, we propose several applications of the diagram for pedestrian networks.

Macroscopic Fundamental Diagram (MFD) is widely used in traffic state evaluation due to its description of the macro level of urban road network. This study focuses on the discrimination and short-term prediction of macro traffic states in urban road networks, using MFD combined with FCM clustering for state partitioning to characterize different macro states of the road network. To predict the MFD state, this paper builds two LSTMs to perform short-term predictions on two important parameters in MFD: road network weighted flow qw and road network weighted density kw. The parameters of the first three statistical intervals and the predicted time period are used as inputs to output the MFD parameters for the predicted time period. To ensure that the two LSTM structures and hyper-parameters settings can achieve the best prediction performance for MFD, the parameter optimization process of both should be included in the same search framework for hyper-parameters search. Therefore, this paper uses GA algorithm combined with multi-objective particle swarm optimization algorithm as the solving algorithm, with the accuracy of solving two MFD parameters and the accuracy of MFD point positioning as the hyper-parameters solving objectives. The study was validated using actual road network data from Hong Kong, and the results showed that the method proposed in this paper has an MRE prediction error of less than 7.8% for the two parameters of MFD, and can predict the future temporal trend of the two parameters, demonstrating the feasibility of MFD related predictions. The model's prediction of the overall shape and change trajectory trend of MFD is consistent with reality, and some test sets in MFD state prediction show high accuracy, the overall accuracy is 81.45%. To verify the effectiveness of the multi-objective search algorithm, typical LSTM models and RNN models were used for comparison. The experiment proved that the model used in this study performed better in error control and state prediction. This study explores and practices a short-term prediction method for road network MFD parameters, MFD status, and their changing trends. Provided path reference for urban road network prediction, traffic control status, and MFD related research.

Corrigendum to “Estimating MFDs in Simple Networks with Route Choice”

Estimating MFDs in Simple Networks with Route Choice

Estimating MFDs in simple networks with route choice

Accurate estimation of macroscopic fundamental diagram (MFD) is the precondition of MFD’s application. At present, there are two traditional estimation methods of road network’s MFD, such as the loop detector data (LDD) estimation method and the floating car data (FCD) estimation method, but there are limitations in both traditional estimation methods. In order to improve the accuracy of road network MFD estimation, a few scholars have studied the fusion method of road network MFD estimation, but there are still some shortcomings on the whole. However, based on the research of adaptive weighted averaging (AWA) fusion method for MFD estimation of road network, I propose to use the MFD’s two parameters of road network obtained by LDD estimation method and FCD estimation method, and establish a back-propagation neural network data fusion model for MFD parameters of road network (BPNN estimation fusion method), and then the micro-traffic simulation model of connected-vehicle network based on Vissim software is established by taking the intersection group of the core road network in Tianhe District of Guangzhou as the simulation experimental area, finally, compared and analyzed two MFD estimation fusion methods of road network, in order to determine the best MFD estimation fusion method of road network. The results show that the mean absolute percent error (MAPE) of the parameters of road network’s MFD and the average absolute values of difference values of the state ratio of road network’s MFD are both the smallest after BPNN estimation fusion, which is the closest to the standard MFD of road network. It can be seen that the result of BPNN estimation fusion method is better than that of AWA estimation fusion method, which can improve the accuracy of road network MFD estimation effectively.

Road network traffic management and control are the key mechanisms to alleviate urban traffic congestion. With this study, we aimed to characterize the traffic flow state of urban road networks using the Macroscopic Fundamental Diagram (MFD) to support area traffic control. The core property of an MFD is that the network flow is maximized when network traffic stays at an optimal accumulation state. The property can be used to optimize the temporal and spatial distribution of traffic flow with applications such as gating control. MFD construction is the basis of these MFD-based applications. Although many studies have been conducted to construct MFDs, few studies are dedicated to improving the accuracy considering the reliability of different sources of data. To this end, we propose an MFD construction method using multi-source data based on Dempster–Shafer evidence (DS evidence) theory considering the reliability of different data sources. First, the MFD was constructed using VTD and CSD, separately. Then, the fused MFD was derived by quantifying the reliability of different sources of data for each MFD parameter based on DS evidence theory. The results under real data and simulated data show that the accuracy of the constructed MFDs was greatly improved considering the reliability of different data sources (the maximum MFD estimation error was reduced by 22.3%). The proposed method has the potential to support the evaluation of traffic operations and the optimization of signal control schemes for urban traffic networks.

Macroscopic Fundamental Diagram for pedestrian networks: Theory and applications

The Macroscopic Fundamental diagram (MFD) has proven to be a powerful concept in understanding and managing vehicular network dynamics, both from a theoretical angle and from a more application-oriented perspective. In this contribution, we explore the existence and the characteristics of the pedestrian Macroscopic Fundamental Diagram (p-MFD). From a theoretical perspective, the main contribution of this research shows how we can derive the p-MFD from assumed local fundamental diagrams (FDs). We show that we can relate the average (out-) flow from a pedestrian network as a function of the average spatial density ρ and the density spatial variation σ2. We show that the latter is essential to provide a reasonable description of the overall network conditions. For simple linear relations between density and speed, we derive analytical results; for more commonly used FDs in pedestrian flow theory we show the resulting relation using a straightforward simulation approach. As a secondary contribution of the paper, we show how the p-MFD can be constructed from pedestrian trajectory data stemming from either microsimulation or from experimental studies. The results found are in line with the theoretical result, providing further evidence for the validity of the p-MFD concept. We furthermore discuss concepts of hysteresis, due to the differences in the queue build up and recuperation phases. We end with applications of the presented concepts, e.g. in crowd management.

This study proposes a novel GPS-based methodology for Macroscopic Fundamental Diagram (MFD) estimation to overcome limitations of fixed detectors and inaccurate penetration rate assumptions. The approach dynamically identifies stop-line positions using spatiotemporal floating car data, calculates maximum queue lengths per signal cycle by combining floating car positions with estimated arriving vehicle lengths, and establishes a speed-based nonlinear model to determine queuing vehicle counts. A dynamic scaling coefficient derived from maximum queue lengths enables assumption-free estimation of total regional vehicles when applied to the floating car population. Validation using Chengdu data demonstrates significant improvements: unary cubic curves achieve optimal fitting for MFD relationships (R2 up to 0.9157); the HMM-CRF hybrid map-matching algorithm reduces average position error by 29% and intersection mismatch rate by approximately 40%; simulation results show queue length estimation accuracy of RMSE 22.8m and MAPE 18.5%, while MFD estimation error for maximum network flow drops from −17.5% to −3.5%, representing an 80% relative accuracy improvement. The proposed methodology provides robust technical support for urban road network assessment and management by enabling high-precision acquisition of MFDs from floating car data, effectively addressing critical challenges in macroscopic traffic modeling and monitoring. This advancement presents potential value for perimeter control applications and other MFD-based traffic management strategies.