Model predictive control based on urban macroscopic fundamental diagram: A city-scale routing application

Traditional macroscopic fundamental diagram (MFD)-based traffic perimeter metering control strategies rely on full knowledge of vehicle accumulation and inter-regional flow dynamics, assumptions that seldom hold in heterogeneous and highly variable real-world networks. Classical data-driven reinforcement learning methods face similar constraints, often converging slowly and exhibiting low sample efficiency when confronted with such complexities. Motivated by these limitations, this paper proposes a Parameterized Deep Q-Network perimeter control (P-DQNPC) scheme designed for multi-region urban road networks. The framework jointly optimizes discrete actions (regional routing choices) and continuous actions (signal-timing or flow-duration regulation) within a model-free learning structure. The approach is first trained and validated on synthetic MFD data to establish stable and interpretable policy behavior under controlled conditions. It is then transferred and further evaluated using real-world measurements from the Performance Measurement System—San Francisco Bay Area (PeMS-SF), a dataset collected from 18,954 loop detectors across the California State Highway System. PeMS-SF is selected due to its high spatial and temporal resolution, broad network coverage, and strong ability to capture realistic and diverse congestion patterns qualities that support both rigorous validation and generalization to other metropolitan regions. Experimental results show that P-DQNPC consistently outperforms state-of-the-art baselines, including deep deterministic policy gradient, deep Q-network, and No-Control schemes. The proposed method achieves superior regulation of regional accumulations and demonstrates enhanced robustness in large, heterogeneous, and uncertain urban traffic environments.

Urban road networks frequently operate in an oversaturated state during peak hours, where traditional traffic signal control strategies, predominantly grounded in the assumption of fully rational user behavior, fail to capture the bounded rationality inherent in drivers’ route choice decisions under congestion. To address this gap, this paper proposed a novel integrated framework that couples evolutionary game theory (EGT) with dynamic signal control, leveraging the Macroscopic Fundamental Diagram (MFD) for real-time feedback between network-wide traffic states and individual decision-making. Specifically, we model drivers within a control zone as a population choosing between two bounded-rational strategies: “waiting straight” versus “detouring”. A replicator dynamics model governs the evolution of strategy adoption, with payoffs dynamically modulated by the MFD to reflect congestion-dependent travel costs. This behavioral layer is embedded within a receding horizon control (RHC) architecture that optimizes green splits and cycle lengths in real time to minimize total zone-wide delay, solved via Particle Swarm Optimization (PSO). Extensive simulations were conducted on a 6 × 6 grid network in SUMO under high-demand conditions (network saturation, approx. 0.92). Results demonstrate that the proposed method reduces average vehicle delay by 18.7% (from 142.8 s to 116.8 s), decreases queue spillback occurrences by 32.4%, and achieves convergence to an evolutionarily stable state (ESS) within 25 minutes, outperforming fixed-time, adaptive MAXBAND, and multi-agent deep reinforcement learning (MADDPG) baselines. This work establishes a closed-loop paradigm for behavior-aware, state-responsive traffic management in severely congested urban environments.

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.

This paper introduces a methodological framework for assessing the impacts of digital infrastructure disruptions in urban traffic networks, using traffic signal tampering as a case study. A readily quantifiable indicator, the Average Flow Reduction Metric (AFRM), is proposed to capture network-wide flow reduction based on the principles of the Macroscopic Fundamental Diagram (MFD). The framework is applied to a simulated network under various tampering scenarios and routing behaviors, including fixed, flexible, semi, and fully adaptive routing. The results show that AFRM correlates meaningfully with conventional disutility-based network performance metrics and serves as a reliable proxy for network degradation, especially under established or rationally adaptive routing behaviors. Due to its reliance on commonly available traffic measurements, AFRM provides a practical tool for diagnosing and managing digital disruptions in traffic networks. As such, it may support traffic managers in assessing, preparing for, monitoring, and responding to disruptions in real time.

ABSTRACT Macroscopic fundamental diagrams (MFDs) for traffic networks have gained theoretical and empirical support with new data collection technologies. However, well-defined MFD curves only exist for specific network topologies and are subject to disturbances, particularly hysteresis phenomena. This study examines hysteresis in MFDs and Network Exit Functions during rush hour conditions. We apply the LWR theory to a highway corridor with a downstream bottleneck and identify a figure-eight hysteresis pattern. We analyze the impact of road topology and demand patterns on hysteresis formation analytically. Empirical data from two bottlenecks provides statistical evidence that continuous bottlenecks cause less hysteresis than discontinuous ones. Our observations confirm counter-clockwise loops in real conditions, attributed to demand asymmetries through theoretical analysis. Numerical experiments using the Cell Transmission Model demonstrate that even slight capacity reductions in homogeneous sections can significantly decrease MFD hysteresis while maintaining downstream outflow, achievable through standard traffic control measures like ramp metering.

The macroscopic fundamental diagram explained by a walking experiment in class

With the continuous advancement of Intelligent Transportation Systems (ITS) in China, the originally open urban regional traffic network is progressively evolving into a virtually controlled and closed system, wherein traffic states can be monitored, measured, and regulated. However, traditional theories on urban traffic supply-demand equilibrium and associated congestion control methods, which are rooted in open-system assumptions, are increasingly inadequate for contemporary applications. Under the ITS environment, it is crucial to redefine the equilibrium principles of urban traffic networks and establish corresponding congestion control theories and methodologies. This paper reviews the developmental status of intelligent transportation management and control systems in Chinese major cities and introduces a theoretical framework for regional congestion control in urban road networks accommodating mixed traffic flows (comprising conventional and new-energy vehicles). By constructing a spatio-temporal virtual closed system through a "data boundary" defined by discrete regional detection points of mixed traffic inflow and outflow within a controlled network area, and integrating Macroscopic Fundamental Diagram model under regional perimeter control strategies, this study proposes a hybrid active-pinning control approach based on regional network controllability. Furthermore, a technical pathway and methodological system are designed to further analyze this theoretical framework.

Traffic state is the foundation of urban transportation system management and operation, requiring a substantial amount of spatiotemporal traffic data. However, because of the limitations of data collection devices and recording technologies, traffic state missing-data problems are inevitable. To address this issue, this study proposes a hybrid framework MFD-TGCN (Macroscopic Fundamental Diagram-Temporal Graph Convolution Network) for traffic state imputation, based on the structure of physics-informed deep learning (PIDL). This framework integrates the strengths of physics-based traffic flow models and data-driven models for spatiotemporal feature dependency. To accommodate the physical features of various traffic state scenarios, three alternative Macroscopic Fundamental Diagram (MFD) models are utilized, and the Weighted Least Square (WLS) algorithm is applied for the initial parameter calibration. Meanwhile, the Graph Convolutional Network (GCN) and Gated Recurrent Unit (GRU) modules are employed effectively to capture the spatiotemporal features. The framework’s performance is evaluated using real-world traffic flow and density data from Chongqing, China, under different missing-data patterns, missing-data ratios, and varying numbers of network detectors, comparing with several categories of baseline methods. The results demonstrate that incorporating the road network information helps capture more spatial features, enhancing the accuracy of traffic network state imputation. The results confirm the superior performances of our framework over state-of-the-art methods in various scenarios, especially in cases with significant and complex missing data. Furthermore, experimental evidence indicates the combined physics and data-driven models framework has great generalizability capability for flexible scenarios in transportation system planning, management and control.

Traffic modeling and control in large-scale urban road networks present significant challenges. The macroscopic fundamental diagram provides a means of formulating dynamical traffic models of such networks, thereby enabling the development of model-based design techniques for state estimation and feedback control. In this article we focus on the computational efficiency of macroscopic fundamental diagram-based nonlinear model predictive control schemes for perimeter control and route guidance actuated networks, which are macroscopic actuation methods involving traffic flow manipulation between adjacent network neighborhoods. A number of economic nonlinear model predictive control schemes, based on direct methods from the numerical optimal control literature, are implemented using a variety of nonlinear programming solvers. The computational efficiency of the schemes is evaluated via computer simulations of congestion control scenarios for macroscopic fundamental diagram-based network models with different numbers of regions using randomly generated traffic demand profiles. The results indicate that the proper pairing of direct methods and solvers yields significant improvements in computational efficiency for macroscopic fundamental diagram-based control schemes, thereby improving the real-time feasibility and, consequently, the field deployment potential of the resulting macroscopic road traffic flow control algorithms.

This paper addresses the underrepresentation of traffic activity in Life Cycle Assessment (LCA) practice despite its critical influence on the energy and environmental footprint of both electrified and conventional vehicles. To bridge this gap, the paper proposes a new framework that enhances the integration of traffic dynamics into fleet LCAs while maintaining computational simplicity. The approach combines Macroscopic Fundamental Diagrams (MFDs), which estimate network-level traffic performance, with an average-speed-based emissions model to evaluate on-road energy use and emissions performance of traffic. This quantification is further extended by applying life cycle inventory emission factors to account for upstream and downstream impacts, including energy production, vehicle manufacturing, and end-of-life treatment. The framework is demonstrated through a case study involving urban traffic networks in Zurich and Thessaloniki. Results illustrate the method’s capacity to evaluate multiple vehicles within realistic flow scenarios and adaptability to variable traffic conditions, offering a practical and scalable tool for improved energy and environmental assessment of road transport fleets.

Scaling methods for estimating macroscopic fundamental diagrams in urban networks with sparse stationary sensor coverage

In contemporary perimeter control strategies, reliance on simplified accumulation-based macroscopic fundamental diagram (MFD) models with minimal scatter in macroscopic variables is common. Yet, the inherent state-dependent stochasticity driving road network evolution challenges the efficacy of such deterministic approaches, potentially undermining desired objectives when supply and demand variabilities are ignored. To fill the gap, our study integrates an analytical state-dependent stochastic dynamics model into the MFD framework, incorporating Brownian noise to capture the stochastic essence of traffic evolution. This advancement enables the modeling of stochastic gridlock processes. We formulate a stochastic optimal control problem for perimeter regulation based on this novel MFD model, targeting both delay minimization and gridlock prevention. Utilizing the stochastic Pontryagin maximum principle (SPMP), we express the optimal control conditions as a coupled forward-backward stochastic differential equation (FBSDE). To solve this FBSDE, we reformulate it as an optimization problem and develop neural network-assisted solution methodologies. Our numerical studies validate the convergence of the proposed solution method and demonstrate significant performance enhancements: not only can gridlocks be effectively mitigated by accounting for the model’s stochastic nature and associated objectives, but also the total cumulative trip completion rate improves by up to 25% when optimizing for both delay reduction and gridlock avoidance. These findings underscore the potential of integrating stochastic dynamics into perimeter control strategies for more resilient and efficient traffic management systems.

Given the computational cost of simulations, many studies test Traffic Signal Controllers (TSCs) in limited scenarios, raising concerns about their generalizability. In this study, robustness refers to a Decentralized TSC's (DTSC) ability to maintain performance across varying traffic levels. Some algorithms excel in free-flow conditions but struggle under high congestion. To address this, we propose a standardised testing platform based on the two-bin model–an abstraction of a grid network–to assess DTSC robustness systematically. We introduce two metrics to quantify robustness by measuring the gap between optimal network flow and actual TSC performance under diverse conditions. The known upper bound of the Macroscopic Fundamental Diagram (MFD) for the two-bin model enables these calculations. We evaluate DTSCs inspired by proportional and Back-Pressure (BP) policies and validate insights using microsimulation in PTV Vissim. This work lays the foundation for standardised TSC evaluations and advancing traffic signal control research.

The joint impact of road density and width on road network performances —a simulation approach based on macroscopic fundamental diagram

Using high-frequency spatial data in the traffic field, this paper explores the macroscopic fundamental diagram (MFD) and the traffic capacity of the regional road network in Guiyang City using the averaging method of the functional spatial autoregressive (FSAR) model. It reveals the traffic conditions and characteristics of Guiyang City’s regional road network. Initially, high-frequency traffic speed data are represented using functional principal component bases, transforming the parameter estimation of the FSAR model into a problem of estimating the coefficients of these basis functions. The maximum likelihood estimation method is subsequently employed to estimate the model parameters. Furthermore, the Mallows model averaging (MMA) approach is used to assign weights to the models, allowing for the computation of a weighted traffic density. Using the derived weighted traffic density and flow, the MFD of the road network is constructed, and the corresponding traffic capacity is estimated. The proposed averaging method of the FSAR model is validated through numerical simulations, demonstrating enhanced accuracy under limited sample conditions and varying spatial dependencies. The methodology is then applied to the regional road network of Guiyang City to assess traffic congestion. Results indicate that Guanshanhu District, located in the central urban area, exhibits the highest traffic capacity within the region.

Accurate network macroscopic fundamental diagram (NMFD) estimation requires accurate spatial mean speed estimates that are hard to get from loop detector devices (LDD) as they capture local speeds only. This study introduces a correction method able to reconstruct the mean speed from loop data leveraging floating car devices (FCD) during the training phase. This significantly improves LDD-based NMFD estimation. Unlike previous studies focusing on local speed corrections, our approach integrates LDD and FCD using machine learning techniques to enhance link-average speed estimation. We compare the performance of Ordinary Least-Squares Regression (OLSR), Random Forest (RF), Multilayer perceptron (MLP), and XGBoost models, using a comprehensive dataset to evaluate their effectiveness in aligning LDD speeds with FCD as ground truth average speed. The RF model showed the highest accuracy, with a 37.44% improvement over original LDD speeds and a Root-Mean-Squared Error (RMSE) of 9.54 km/h. Additionally, we investigated the impact of LDD positions and demand by separating loading and unloading periods. The RF model’s adjustments were especially effective under these conditions, resulting in an improved accuracy of 44.9%. Our analysis of NMFDs further validated the RF model’s robustness and accurate estimation of NMFDs using corrected LDD. We validate our methodology across two urban networks, Athens and Lyon, demonstrating its spatial and temporal transferability. Results show that bias correction significantly improves LDD speed accuracy, leading to more reliable NMFD estimation. This work advances data fusion methods for urban traffic monitoring and provides a robust framework for addressing LDD positional bias in large-scale traffic studies.

Macroscopic fundamental diagram (MFD) is susceptible to traffic flow heterogeneity under incidents, which makes it hard to obtain an accurate estimation of the perimeter control threshold for effective traffic incident management. This study thus proposed a real-time MFD estimation method considering the impact of incidents. By incorporating newly defined variables, the road vulnerability index and incident parameters, an extended MFD model was developed to capture and quantify the impact of traffic incidents on traffic flow dynamics. Simulation experiments were conducted, and an estimation accuracy of more than 97% could be obtained for threshold estimation, with obvious superiority as compared to estimation methods without regard to the impact of incidents. The efficiency of using the proposed method for perimeter control was also validated, showing an improvement of 9% and 11.6% in average delay and number of stops respectively, as compared with perimeter control without considering the impact of traffic incidents.

Abstract The partitioning of a heterogeneously loaded road network into homogeneous, compact subregions is a fundamental prerequisite for the implementation of network‐level traffic management and control based on the network macroscopic fundamental diagram. This study proposes a flexible road network partitioning framework that leverages the powerful feature extraction capabilities of self‐supervised graph neural networks and employs a multi‐objective optimization approach to balance regional homogeneity and compactness. A graph contrastive learning model is proposed to extract meaningful node embeddings that incorporate topology and attribute similarity information. Based on the learned node embeddings, the partition is determined by a parameter‐free hierarchical clustering method and a subregion identification algorithm. Boundary tuning is then modeled as a bi‐objective optimization problem to maximize regional homogeneity and compactness. A Pareto local search algorithm is developed to approximate the Pareto front. This study further demonstrates the extension of the proposed methods to scenarios with missing data. Finally, the methods are validated on real road networks with automatic license plate recognition data.

Integrating passenger transportation costs into service network design: A bilevel optimal control approach using macroscopic fundamental diagram