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.

Theoretical arguments and empirical evidence for the emergence of macroscopic-epidemic–type behavior, in the form of Susceptible-Infected-Susceptible (SIS) or Susceptible-Infected-Recovered (SIR) processes in urban traffic congestion from microscopic network flows is given. Moreover, it is shown that the emergence of SIS/SIR implies a relationship between traffic flow and density, which is consistent with observations of the so-called Fundamental Diagram of Traffic (FDT), which is a characteristic signature of vehicle movement phenomena that spans multiple scales. Our results put on more firm grounds recent findings that indicate that traffic congestion at the aggregate level can be modeled by simple contagion dynamics.

Traffic state estimation (TSE) is a core task in intelligent transportation systems (ITSs) that seeks to infer key operational parameters—such as speed, flow, and density—from limited observational data. Existing methods often face challenges in practical deployment, including limited estimation accuracy, insufficient physical consistency, and weak generalization capability. To address these issues, this paper proposes a hybrid estimation framework that integrates multi-agent reinforcement learning (MARL) with the Lighthill–Whitham–Richards (LWR) traffic flow model. In this framework, each roadside detector is modeled as an agent that adaptively learns fundamental diagram (FD) parameters—the free-flow speed and jam density—by fusing local detector measurements with global CAV trajectory sequences via an interactive attention mechanism. The learned parameters are then passed to an LWR solver to perform sequential (rolling) prediction of traffic states across the entire road segment. We design a reward function that jointly penalizes estimation error and violations of physical constraints, enabling the agents to learn accurate and physically consistent dynamic traffic state estimates through interaction with the physics-based LWR environment. Experiments on simulated and real-world datasets demonstrate that the proposed method outperforms existing models in estimation accuracy, real-time performance, and cross-scenario generalization. It faithfully reproduces dynamic traffic phenomena, such as shockwaves and queue waves, demonstrating robustness and practical potential for deployment in complex traffic environments.

Mid-sized Philippine cities commonly rely on fixed-time traffic signal plans that cannot respond to short-term, demand-driven surges, resulting in measurable idle time at stop lines, increased delay, and unnecessary emissions, while adaptive signal control has demonstrated performance benefits, many existing solutions depend on centralized infrastructure and high-bandwidth connectivity, limiting their applicability for resource-constrained local government units (LGUs). This study reports a field deployment of TrafficEZ, a lightweight edge AI signal controller that reallocates green splits locally using traffic-density approximations derived from cabinet-mounted cameras. The controller follows a macroscopic, cycle-level control abstraction consistent with Transportation System Models (TSMs) and does not rely on stationary flow–density–speed (fundamental diagram) assumptions. The system estimates queued demand and discharge efficiency on-device and updates green time each cycle without altering cycle length, intergreen intervals, or pedestrian safety timings. A quasi-experimental pre–post evaluation was conducted at three signalized intersections in El Salvador City using an existing 125 s, three-phase fixed-time plan as the baseline. Observed field results show average per-vehicle delay reductions of 18–32%, with reclaimed effective green translating into approximately 50–200 additional vehicles per hour served at the busiest approaches. Box-occupancy durations shortened, indicating reduced spillback risk, while conservative idle-time estimates imply corresponding CO2 savings during peak periods. Because all decisions run locally within the signal cabinet, operation remained robust during backhaul interruptions and supported incremental, intersection-by-intersection deployment; per-cycle actions were logged to support auditability and governance reporting. These findings demonstrate that density-driven edge AI can deliver practical mobility, reliability, and sustainability gains for LGUs while supporting evidence-based governance and performance reporting.

We develop an operational-level approach to improve pedestrian traffic efficiency and to explain discrepancies among the fundamental diagrams in pedestrian dynamics by experiments and theoretical analysis. Controlled experiments were conducted where participants walked under different acoustic conditions, exploiting the influence of sound on walking behavior. The results demonstrate that (1) regular acoustic stimulation substantially improves pedestrian efficiency: a monotone rhythm slower than the natural pace increases pedestrian flow by up to 36.8%, while periodic tones perform even better; however, flow decreases when pedestrians fail to follow the imposed rhythm. (2) Regular acoustic tones reduce pedestrians’ sensitivity to inter-personal distance and promote step synchronization, thereby suppressing stop-and-go movements and improving the utilization of longitudinal space under high-density conditions. A theoretical framework is developed to link macro-flow with micro-step, and potential engineering applications are discussed. These findings suggest that well designed rhythmic sounds can enhance pedestrian flow in crowd control.

Introduction. This publication provides a comparative analysis of two-phase and three-phase theories of traffic flows. It considers the key differences between these theories and their applicability to real transport systems as well as the phase transitions taken into account by the theories’ models. The main focus is made on empirical data and modeling complex dynamic phenomena on roads. The article highlights the scientific novelty of Kerner’s threephase theory and its advantages in congestion forecasting and managing traffic flows. Materials and methods. The study examines and analyses classical theories of traffic flow, including two-phase models based on the fundamental diagram and the three-phase theory of traffic developed by B. Kerner. Main attention is paid to theoretical aspects, comparative analysis, and interpretation of key provisions of these theories. Study is based on analysis of scientific literature. The main sources of information have been peer-reviewed articles published in leading scientific journals on transport, monographs devoted to the traffic flow theory and its application in traffic management, reports and materials of the international conferences, and other sources covering both classic approaches and current trends in traffic flow modeling. Results . A comparative analysis of the general two-phase theory of traffic flow and Kerner’s three-phase traffic flow theory has been made. In two-phase model, based on the fundamental traffic diagram, the main phases are free flow and dense flows. These phases are characterized by the relationship between density, flow, and vehicle speed. In two-phase model the phase transition occurs when the critical vehicle density is exceeded. The threephase model describes several fundamental properties of phase transitions: from free flow to synchronized flow, from synchronized flow to wide clusters and reverse transitions and their variants. Discussion and conclusion. The main results of the study include a detailed comparison between two theories, allowing us to identify critical aspects and potential directions for further development. Specifically, it has been shown that Kerner’s three-phase model offers greater capabilities for describing metastable states and complex transitions between phases, making it more suitable for analyzing traffic flows in modern megacities.

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 Mixed traffic flow, consisting of human-driven vehicles (HDVs) and connected autonomous vehicles (CAVs), is expected to dominate future roadways. This study develops a heterogeneous traffic flow model in an intelligent connected environment to analyze macroscopic characteristics, including traffic capacity, critical density, and fundamental diagrams, under varying CAV penetration rates. Three car-following modes are considered: HDVs following HDVs, HDVs following CAVs, and CAVs following CAVs, modeled respectively by the Intelligent Driver Model (IDM), Adaptive Cruise Control (ACC), and Cooperative Adaptive Cruise Control (CACC). Model parameters are calibrated using genetic algorithms based on publicly available autonomous driving datasets. Calibration results demonstrate high predictive accuracy, with mean absolute percentage errors of 0.009% (distance) and 2.37% (speed) for IDM, 0.27% and 3.67% for ACC, and near-zero errors for CACC. Analysis of fundamental diagrams shows that increasing CAV penetration significantly enhances traffic flow efficiency, improves stability, and mitigates congestion. The findings provide theoretical guidance for optimizing mixed traffic operations and support the development of intelligent and connected transportation systems.

Mixed traffic flow composed of connected and automated vehicles (CAVs) and human-driven vehicles (HDVs) represents a core characteristic of intelligent transportation systems. However, its operational efficiency is significantly constrained by lane management strategies and CAV cooperative driving behaviors. To investigate this, a cellular automata-based simulation model is developed that integrates multiple car-following rules, a lane-changing strategy, and a platoon coordination mechanism. Through a systematic comparison of 13 lane management strategies in one-way two-lane and three-lane configurations, this study analyzes the influence mechanisms of lane allocation and cooperative driving on traffic flow, considering fundamental diagram characteristics, operating speed, CAV degradation behavior, and maximum platoon size. The results indicate that the performance of different strategies exhibits phased evolution with increasing CAV penetration rates. At low penetration rates, providing relatively independent space for HDVs effectively suppresses random disturbances and improves throughput. At medium to high penetration rates, dedicated CAV lanes—especially those with spatial continuity—enable cooperative platoons to fully leverage their advantages, leading to significant improvements in traffic capacity and operational stability. These findings demonstrate an optimal alignment between cooperative driving mechanisms and lane configurations, offering theoretical support for highway lane management in mixed traffic environments.

Traffic state estimation (TSE) is crucial for intelligent transportation systems, as it provides unobserved parameters for traffic management and control. In this paper, we propose a novel physics-guided graph neural network for TSE that integrates traffic flow theory into an estimation framework. First, we constructed wave-informed anisotropic temporal graphs to capture the time-delayed correlations across the road network, which were then merged with spatial graphs into a unified spatiotemporal structure for subsequent graph convolution operations. Then, we designed a four-layer diffusion graph convolutional network. Each layer is enhanced with squeeze-and-excitation attention mechanism to adaptively capture dynamic directional correlations. Furthermore, we introduced the fundamental diagram equation into the loss function, which guided the model toward physically consistent estimations. Experimental evaluations on a real-world highway dataset demonstrated that the proposed model achieved a higher accuracy than benchmark methods, confirming its effectiveness in capturing complex traffic dynamics.

ABSTRACT Traffic congestion significantly affects speed, and thus energy consumption of heavy goods vehicles (HGVs). One of the ways of correlating traffic state with vehicle speed is fundamental diagrams (FDs). This study develops a methodology to collect national‐level traffic data for England, integrate it with vehicle data, and use the data to construct FDs by type of road in England. Traffic counts and time‐averaged traffic speed are obtained from the National Highways database and Road Traffic dataset, and space‐averaged traffic speed data is obtained from HERE Maps. Missing entries are added using the temporal pattern of traffic flow, and outliers in the count data are filtered using spline‐regression and unsupervised k‐means clustering. Traffic data is classified by road types using information from HERE Maps. FDs are constructed for each type of road and validated using a separate test dataset from the National Highways database. The correlation between macroscopic traffic flow data and microscopic vehicle data is verified by validating the FDs with HGV speed data collected from on‐board telematics systems. The results can be used to predict vehicle speed directly from traffic density using universal HGV FDs for England, that is useful for estimating energy consumption.

Perimeter control is a method used to manage clusters of intersections at regional boundaries, based on the macroscopic fundamental diagram (MFD). However, the MFD‐based model of urban traffic networks often contains parameter uncertainty and system noise, leading to excessive queues at regional boundaries and reducing the effectiveness of perimeter control. To address these challenges, this paper proposes an optimized control framework for urban traffic networks based on the Conditional Value at Risk (CVaR). The framework includes a Markov jump linear system model to simulate the traffic network and uses CVaR to measure congestion risk caused by uncertainty. The CVaR‐based optimization objectives and the regional boundary queue model are established to enhance control strategies. The proposed CVaR‐based optimal perimeter control framework is compared with no control, bang–bang control, and stochastic model predictive control through simulation. Results show that the CVaR‐based framework significantly reduces the number of vehicles and travel time, outperforming the other three control strategies in alleviating congestion. This study demonstrates the potential of CVaR‐based control in improving urban traffic management.

Abstract This study addresses the challenge of estimating traffic states for road links. We propose an innovative approach that leverages partial trajectory data captured by camera-equipped probe vehicles traveling in the opposite lane. The methodology combines state-of-the-art computer vision algorithms for extracting vehicle trajectories from street-view video sequences with a novel estimation technique based on the Cell Transmission Model (CTM) and Genetic Algorithms (GA). Our approach first calibrates Fundamental Diagram (FD) parameters using observed cell densities, then estimates boundary conditions for all space-time diagrams. We validate the method using simulated traffic data from three different types of links and parameter settings. Results show that the proposed methodology can estimate traffic densities in unobserved regions, even with limited data availability. This research contributes to the field by introducing a cost-effective, high-resolution traffic data collection method and a robust estimation technique for comprehensive traffic state information. While the study shows promising results, it also identifies areas for improvement, including refining models, optimizing processes, and testing with real-world data to enhance accuracy and scalability.

The macroscopic fundamental diagram explained by a walking experiment in class

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.

Traffic state estimation (TSE) fundamentally involves solving high-dimensional spatiotemporal partial differential equations (PDEs) governing traffic flow dynamics from limited, noisy measurements. While Physics-Informed Neural Networks (PINNs) enforce PDE constraints point-wise, this paper adopts a physics-informed deep operator network (PI-DeepONet) framework that reformulates TSE as an operator learning problem. Our approach trains a parameterized neural operator that maps sparse input data to the full spatiotemporal traffic state field, governed by the traffic flow conservation law. Crucially, unlike PINNs that enforce PDE constraints point-wise, PI-DeepONet integrates traffic flow conservation model and the fundamental diagram directly into the operator learning process, ensuring physical consistency while capturing congestion propagation, spatial correlations, and temporal evolution. Experiments on the Next Generation Simulation (NGSIM) dataset demonstrate superior performance compared to state-of-the-art baselines. Further analysis reveals insights into optimal function generation strategies and branch network complexity. Additionally, the impact of input function generation methods and the number of functions on model performance is explored, highlighting the robustness and efficacy of proposed framework.

The fundamental diagram of autonomous vehicles: Traffic state estimation and evidence from vehicle trajectories