This paper develops a sensitivity analysis framework for the perturbed utility route choice (PURC) model and the accompanying stochastic traffic equilibrium model. We derive analytical sensitivity expressions for the Jacobian of the individual optimal PURC flow and equilibrium link flows with respect to link cost parameters under general assumptions. This allows us to determine the marginal change in link flows following a marginal change in link costs across the network. We show how to implement these results while exploiting the sparsity generated by the PURC model. Numerical examples illustrate the use of our method for estimating equilibrium link flows after link cost shifts, identifying critical design parameters, and quantifying uncertainty in performance predictions. Finally, we demonstrate the method in a large-scale example. The findings have implications for network design, pricing strategies, and policy analysis in transportation planning and economics, providing a bridge between theoretical models and real-world applications.

Urban traffic congestion significantly impacts regional air quality and contributes substantially to pollutant emissions. Suburban freeway corridors are a major source of traffic-related emissions, particularly carbon monoxide (CO), nitrogen oxides (NOx) and hydrocarbons (HC). This paper proposes an emission-aware receding-horizon Variable Speed Limit (VSL) controller with a shadow-penalized one-step update rule for peripheral freeway corridors. Emission rates on freeways exhibit high sensitivity to speed fluctuations and congestion recovery processes. To address this relationship, we generalize the bounded-acceleration LWR (BA-LWR) traffic flow model by allowing the acceleration bound to depend on speed and deriving analytical expressions for flow and travel times. This BA-LWR variant enhances behavioral realism through the incorporation of driver responses to congestion, which is widely recognized as a main cause of the important capacity drop phenomenon. Our approach implements dynamical optimization of a VSL to balance the dual objectives of minimizing both travel time and emissions as quantified by the COPERT V (European Environment Agency, 2023) model. Numerical simulations demonstrate that this framework effectively manages congestion and reduces emissions across various traffic demand scenarios.

Despite strong indirect evidence that physical energy exertion shapes cycling behaviour, it has never been explicitly incorporated into route choice models. This paper introduces a generalisable, individual-specific, physics-based energy model that computes expected energy expenditure (Joules) and energy intensity (Joules per metre) for every link in the network. The model accounts for slope, individual cruising speed preferences, and acceleration losses at intersections using empirically derived, intersection-specific stopping probabilities. Calibration is performed on a large-scale GPS dataset from the Greater Copenhagen Area. Descriptive analysis of observed routes reveals that excess energy is mainly driven by frequent stopping on major roads and dedicated cycleways. We then demonstrate one key application of the model by integrating both total energy (Joules) and energy intensity (Joules per metre) into a series of route choice models (Multinomial Logit, Latent Class, and Mixed Logit). Results confirm statistically significant dispreference for energy attributes and reveal substantial taste heterogeneity: a substantial subset of cyclists exhibits strong dispreference for high-intensity links and is willing to accept considerable detours to avoid them. This study provides a methodological foundation for integrating mechanical energy into route choice models in a way that is individually calibrated, physically grounded, and adaptable to diverse contexts and datasets. By grounding route choice in a quantity that combines slope, stop-and-go dynamics, and cyclist-specific cruising speed into a single measure of effort, the framework both improves the empirical fit of route choice models and offers a more interpretable account of the mechanisms behind observed preferences. The energy model itself has applications beyond route choice, and opens the door to understanding cyclists’ behaviour through more fundamental physical quantities.

The literature suggests that safety is among the most crucial factors influencing decisions regarding cycling. The safety aspect also immediately affects the route choice behaviour of cyclists. Cyclists do not necessarily choose the objectively safest routes; instead, their route choices are influenced by subjective safety. Objective safety, such as crash risk and infrastructure conditions, further shapes route choices by influencing subjective safety.This study estimates several bicycle route choice models for the Copenhagen metropolitan area, where safety is accounted for with explicit indicators: the expected number of crashes and near-crashes on the routes. We employ several utility specifications in terms of the inclusion of the explicit safety indicators to understand how cyclists’ route choice behaviour can be reflected in the models. Although the investigated route choice models already account for multiple network attributes, explicit safety indicators have an additional significant influence. The effect of expected near-crashes on route choice behaviour is less obvious than that of expected crashes, and the former require more in-depth model investigation. The results suggest high inter-respondent heterogeneity in preferences regarding safety.We consolidate the results of route choice models with the computation of elasticity values and scenario analyses. These suggest focusing safety improvements along main corridors is an efficient way to improve overall safety, even though it remains difficult to isolate the impact of specific safety measures from other elements, such as broader road design changes or network-wide effects. Finally, we critically discuss how the safety aspect should be included in route choice models.

Road use tolling is an effective way of alleviating congestion. Although many tolling models have been developed, there is gap in the research for a model that: i) is dynamic, ii) accounts for the impacts of tolls on travel demand and departure time choice, iii) accounts for stochasticity in travellers’ route choices, iv) is well-behaved, producing continuous outputs, and v) is computationally feasible to apply to real-life large-scale networks. This paper fills this gap, by developing a tolling model based on the dynamic multi-region Macroscopic Fundamental Diagram (MFD) Stochastic User Equilibrium (SUE) traffic model introduced in Duncan et al. (2025). We begin by extending the model to account for elastic demand and departure time choice. Then, we integrate the model within a toll-price optimisation framework, where the tolling scheme is travel-time-based and the objective function maximises social welfare. We first test the model in a small-scale example multi-region MFD system, and then apply it to estimate an optimal toll-price in a real-life large-scale and detailed case study of Zealand, Denmark. Experiments find that the model is well-behaved and produces smooth objective function surfaces with a unique maximum. Travel behaviour implications of tolling are also realistic, where some travellers opt not to travel by car, some change their departure time, and some change their route. Results suggest that tolling could instigate a positive change in travel behaviour to benefit society.

Micro mobility operators face a significant challenge in optimally determining their daily inventory levels. An analysis of data from station-based and free-floating services in Boston (USA) and Paris (France) reveals that optimal inventory levels are inherently imbalanced, as the majority of stations require no operator intervention, while a minority necessitates substantial adjustments. From a purely data perspective, the fact that some observations are more frequent than others is referred to as Data Imbalance, and it is a well-known problem when working with Machine Learning models. Therefore, this research proposes testing the Balance Cascade (BC), an algorithm designed to handle data imbalance, to solve the inventory problem. It also proposes an enhanced version of the model able to handle multiple outputs, and a new way to compute the penalty factor. The effectiveness of the approach is evaluated using data from Boston's BlueBike station-based bike-sharing system, demonstrating that BC significantly improves prediction accuracy, and that the proposed enhanved version further enhance performances thanks to a more realistic understanding of the penalty factors.

Macroscopic fundamental diagrams (MFDs) and related network traffic dynamics models have received both theoretical support and empirical validation with the emergence of new data collection technologies. However, the existence of well-defined MFD curves can only be expected for traffic networks with specific topologies and is subject to various disturbances, most importantly hysteresis phenomena. This study aims to improve the understanding of hysteresis in Macroscopic Fundamental Diagrams and Network Exit Functions (NEFs) during rush hour conditions. We apply the LWR theory to a highway corridor featuring a location-dependent downstream bottleneck to identify a figure-eight hysteresis pattern, clockwise on the top and counter-clockwise on the bottom. Our empirical observations confirm the occurrence of counter-clockwise loops in real conditions, an effect which we can attribute to demand asymmetries through theoretical analysis. The paper discusses the impact of the road topology and demand patterns on the formation and intensity of hysteresis loops analytically. To substantiate these findings, we analyze empirical MFD data from two bottlenecks and present statistical evidence that, under otherwise identical conditions, a continuous bottleneck causes less hysteresis than a discontinuous one. We conduct numerical experiments using the Cell Transmission Model (CTM) to show that even a slight reduction in the capacity of the homogeneous section can significantly decrease MFD hysteresis while maintaining outflow at the corridor's downstream end. These reductions can be achieved with minimal intervention through standard traffic control measures, such as dynamic speed limits or ramp metering.

This paper analyzes the time-dependent relationship between the mean and variance of travel time on a single corridor under rush hour like congestion patterns. To model this phenomenon, we apply the LWR ((Lighthill Whitham, 1955), (Richards, 1956)) theory on a homogenous freeway with a discontinuous bottleneck at its downstream end, assuming a uni-modal demand profile with a stochastic peak. We establish conditions for typical counterclockwise hysteresis loops under these assumptions. It is demonstrated that shapes of the fundamental diagram which always produce a counterclockwise loop can be interpreted as an indication of aggressive driving behavior, while deviations may occur under defensive driving. This classification enables a detailed explanation of the qualitative physical mechanisms behind this pattern, as well as an analysis of the causes for quantitatively limited deviations. Some of the mathematical properties of the LWR model identified in our analysis have not yet been addressed in the literature and we critically examine the extent to which these reflect actual traffic flow behavior. Our considerations are supported by numerical experiments. The obtained results aim to improve the fundamental understanding of the physical causes of this hysteresis pattern and to facilitate its better estimation in traffic planning and control.

The Bounded Choice Model with Local Detour Threshold (BCM-LDT) route choice model (Rasmussen et al. 2024) proposes that local detouredness (the extent to which a route detours on its subparts) is an influential factor upon route choice probability and choice set formation. The current paper first extends the BCM-LDT to capture correlations between overlapping used routes, formulating a Bounded Path Size (BPS) LDT model. It then develops a heuristic solution method for computing BPS-LDT probabilities on large-scale networks. The method involves pre-processing necessary sub-route information from representative universal choice sets. Solution tricks are proposed which are shown to considerably improve computation times. A modified maximum likelihood estimation procedure is developed for estimating the BPS-LDT model. Empirical evidence from a simulation study and real-life large-scale case study show that parameter estimates can be identified, are statistically significant and unique, and that local detouredness is an influential factor upon route choice probability.

Urban traffic congestion significantly impacts air quality and contributes substantially to pollutant emissions. Effective traffic management strategies, therefore, require models that accurately capture both traffic dynamics and associated emissions. This paper proposes a Model Predictive Control (MPC) framework aimed at emission reduction in integrated urban-freeway networks. The traffic network is partitioned into an urban core, modeled via a Macroscopic Fundamental Diagram (MFD), and a peripheral freeway, where traffic flow is captured using a Bounded Acceleration extension of the classical Lighthill-Whitham-Richards (BA-LWR) ([1]–[3]) model. The BA-LWR model enhances behavioral realism by incorporating driver responses to congestion, notably the phenomenon of capacity drop. Control measures include variable speed limits (VSL) and ramp metering, which are dynamically optimized through MPC to balance traffic efficiency and emissions minimization. Emissions are quantified using the COPERT III [4] model, while routechoice behavior between urban and freeway routes is modeled probabilistically based on real-time travel conditions. Simulations illustrate the framework's capability to manage congestion and emissions effectively under variable traffic demand scenarios.

We present a risk-aware perimeter-style controller that couples safety and efficiency targets in large, heterogeneous urban traffic networks. The network is compressed into two interacting "reservoirs" whose dynamics follow the Generalized Bathtub Model, while accidents are described by a self-exciting (Hawkes) counting process whose intensity depends on vehicle exposure, speed dispersion between reservoirs and accident clustering. Accident occurrences feed back into operations through an analytically simple degradation factor that lowers speed and discharge capacity in proportion to the live accident load. A receding-horizon policy minimizes a mixed delay-safety objective that includes a variance penalty capturing risk aversion; the resulting open-loop problem is shown to possess a bang-bang optimum whose gates switch only at accident times. This structure enables an event-triggered MPC that only re-optimizes when new accidents occur, reducing on-line computation significantly. Parameters are calibrated using OpenStreetMap data for metropolitan Copenhagen to analyze traffic dynamics during morning peak commuter demand. Monte-Carlo simulations demonstrate delay savings of up to 30

Urban traffic congestion significantly impacts regional air quality and contributes substantially to pollutant emissions. Suburban freeway corridors are a major source of traffic-related emissions, particularly nitrogen oxides (NOx) and carbon dioxide (CO2). This paper proposes a Model Predictive Control (MPC) framework aimed at emission reduction on peripheral freeway corridors. Emission rates on freeways exhibit high sensitivity to speed fluctuations and congestion recovery processes. To address this relationship, we develop and analyze a bounded-acceleration continuum traffic flow model. By introducing an upper limit on vehicle acceleration capabilities, we enhance behavioral realism through the incorporation of driver responses to congestion, which is widely recognized as a main cause of the important capacity drop phenomenon. Our approach implements dynamically optimized variable speed limits (VSLs) at strategic corridor locations, balancing the dual objectives of minimizing both travel time and emissions as quantified by the COPERT V [1] model. Numerical simulations demonstrate that this framework effectively manages congestion and reduces emissions across various traffic demand scenarios.

Multi-region Macroscopic Fundamental Diagram (MFD) traffic equilibrium models have been developed as a more easily calibratable, maintainable, and computationally efficient alternative to traditional link-network traffic assignment models with full disaggregate network representation. There are four gaps in the research into these models that we highlight: i) the lack of stochasticity accounted for in the modelling of regional path choice, ii) the estimation of parameters of regional path choice models within the traffic equilibrium, iii) regional path choices being based on region travel times actually experienced (rather than instantaneous travel times), and iv) the paucity of real-life case studies. Motivated by these gaps, this paper presents a new dynamic multi-region MFD Stochastic User Equilibrium (SUE) model, and applies it in a real-life case study. The traffic dynamics are described by a new traffic propagation model utilising features of a space-time graph. Regional path choices can be based on region travel times actually experienced. The model produces continuous equilibrated regional path choice probability outputs, thereby facilitating the development of a rigorous statistical estimation procedure for calibrating parameters from tracked regional path choice data. This estimation procedure is operationalised in a large-scale and detailed multi-region MFD system, with 39 underlying rural and urban regions and 96 directional, superimposed motorway regions, 135 regions in total. Results provide empirical evidence to support hypotheses that regional path choice modelling should consider stochasticity, regional path overlap, multiple attributes, and experienced region travel times. Numerical experiments also demonstrate continuity, differences between the instantaneous and experienced dynamic models, relative insensitivity to the time-slice grain, and realism of the model.

One-stage (implicit) choice set formation models offer a computationally efficient way to model how individuals consider alternatives. Among these, the Bounded Choice Model (BCM) stands out for its consistent, utility-based cutoffs. However, the BCM is non-differentiable, which limits its usefulness: key outputs such as elasticities and standard errors cannot be computed analytically. To overcome this, we introduce the Smooth Bounded Choice Model (SBCM). This model assumes a new smooth truncated logistic distribution for the error terms and applies a smooth approximation to the maximum function used in defining the reference utility. As a result, the SBCM is infinitely differentiable, while preserving core features of the BCM, such as bounding, continuity, and the ability to collapse to the Multinomial Logit (MNL) model under specific conditions. Importantly, the SBCM is not just a smoother version of the BCM. Its more flexible distributional assumptions can better capture actual choice behaviour and allow for meaningful differences in predicted probabilities. We derive closed-form expressions for choice probabilities, gradients, Hessians, elasticities, and standard errors, and present a practical estimation method. The SBCM is tested in three case studies: one mode choice and two route choice settings (bicycle and public transport). In all cases, it outperforms both the BCM and MNL in terms of model fit and interpretability. While the BCM has so far been limited to car route choice, we show that the SBCM is widely applicable across various discrete choice contexts.

This paper presents a hierarchical longitudinal control architecture for autonomous truck platoons that jointly addresses safety, string stability, and economic efficiency. The framework integrates a high-rate safety projection filter, a spacing-regulation layer based on a lag-aware proportional-integral-derivative (PID) controller, and a slow-timescale economic optimizer balancing fuel consumption and travel time. The safety layer guarantees collision avoidance under bounded actuation delays by enforcing forward invariance of a velocity-aware headway constraint through a high-order control barrier function. The regulation layer shapes the spacing-error dynamics into a second-order form with interpretable parameters for damping and natural frequency while explicitly accounting for actuator lag. At the macroscopic level, fuel use is modeled by a tractive-power relation that captures aerodynamic benefits of close spacing, enabling a long-term optimization of speed trajectories subject to comfort and energy trade-offs. We show that the closed-loop dynamics converge to the Optimal Velocity Model with Relative Velocity (OVRV) under undisturbed conditions and derive worst-case upper bounds for platoon stabilization time. Numerical case studies demonstrate the superiority of the proposed design over an canonical baseline controllers in both transient behavior and long-term energy efficiency.

The Multinomial Logit (MNL) model is widely used in route choice modelling due to its simple closed-form choice probability function. However, MNL assumes that the error terms are independently and identically distributed with infinite support. As a result, it imposes homoscedasticity, meaning that long and short trips share the same error variance, disregards correlations between overlapping routes, and assigns non-zero choice probabilities to all available routes, regardless of their cost. This paper addresses these limitations by developing a closed-form route choice model. We introduce the Bounded q-Product Logit (BqPL) model, which incorporates heteroscedastic error terms with bounded support. The parameter q controls the rate at which error term variance increases with trip cost, and routes that violate cost bounds receive zero choice probabilities, implicitly defining the route choice set. Furthermore, we extend the BqPL model to account for correlations between overlapping routes by integrating path size correction terms within the choice probability function, resulting in the Bounded Path Size q-Product Logit (BPSqPL) model. We illustrate the properties of the BPSqPL model on small-scale networks, contrasting it with a range of existing choice models into which it can collapse. We then present a method to estimate the model parameters and standard errors, using bootstrapping. Finally, we estimate the model using a large-scale bicycle route choice case study, comparing its goodness-of-fit, interpretability, and forecasting ability with relevant collapsing models. We also test the impact of the choice set size on the estimated parameters. The results underscore the importance of addressing the three key limitations of the MNL model and demonstrate the effectiveness of the BPSqPL model in doing so.

This paper analyzes the use of variable speed limits to optimize travel time reliability for commuters. The investigation focuses on a traffic corridor with a bottleneck subject to the capacity drop phenomenon. The optimization criterion is a linear combination of the expected value and standard deviation of average travel time, with traffic flow dynamics following the kinematic wave model (Lighthill, 1955; Richards, 1956). We develop two complementary models to optimally set variable speed limits: In the first model, daily peak traffic demand is conceptualized as a stochastic variable, and the resulting model is solved through a three-stage optimization algorithm. The second model is based on deterministic demand, instead modeling bottleneck capacity as a stochastic process using a stochastic differential equation (SDE). The practical applicability of both approaches is demonstrated through numerical examples with empirically calibrated data.

Most choice models, e.g. Multinomial Logit (MNL), rely on random utility theory, which assumes that a compensatory utility maximization decision rule explains an individual's choice behaviour. Research has shown, however, that behaviour is sometimes better explained by non-compensatory decision rules. While some research has used Latent Class Choice Models (LCCMs) to account for multiple decision rules, many of them - such as the disjunctive rule - have yet to be explored. This paper formulates, estimates, and evaluates a LCCM that combines the MNL with a Generalised Random Disjunctive Model (GRDM), a new choice model we develop. Addressing deficiencies of existing disjunctive choice models, the GRDM allows for relative importance between attributes and is insensitive to irrelevant attributes. Unlike most non-compensatory models, it is tractable and incorporates random error terms for capturing unobserved heterogeneity across choice situations. The GRDM can be expressed as a Universal Logit (UL) model, which helps derive welfare metrics such as Marginal Rates of Substitution and elasticities and makes it possible to estimate the model with traditional software packages. The LCCM combining the GRDM and the MNL is estimated in two large-scale case studies: cyclists' route choice and public transport route choice. Results are compared with other relevant LCCM specifications and the individual choice models, where it is found that the MNL + GRDM LCCM provides the best fit to the data. We also interpret the fitted parameters and calculate the Marginal Rates of Substitution, which align with behavioural expectations.

This study introduces the novel concept of local detouredness, i.e. detours on subsections of a route, as a new phenomenon for understanding and modelling route choice. Traditionally, Stochastic User Equilibrium (SUE) traffic assignment models have been concerned with judging the attractiveness of a route by its total route cost. However, through empirical analysis we show that considering solely the global properties of a route is insufficient. We find that it is important to consider local detouredness both when determining realistic and tractable route choice sets and when determining route choice probabilities. For example, analysis of observed route choice data shows that route usage tends to decay with local detouredness, and that there is an apparent limit on the amount of local detouredness seen as acceptable. No existing models can account for this systematically and consistently, which is the motivation for the new route choice model proposed in this paper: the Bounded Choice Model with Local Detour Threshold (BCM-LDT). The BCM-LDT model incorporates the effect of local detouredness on route choice probability, and has an in-built mechanism that assigns zero probabilities to routes violating a bound on total route costs and/or a threshold on local detouredness. Thereby, the model consistently predicts which routes are used and unused. Moreover, the probability expression is closed-form and continuous. SUE conditions for the BCM-LDT are given, and solution existence is proven. Exploiting the special structure of the problem, a novel solution algorithm is proposed where flow averaging is integrated with a modified branch-and-bound method that iteratively column-generates all routes satisfying local and global bounds. Numerical experiments are conducted on small-scale and large-scale networks, establishing that equilibrated solutions can be found and demonstrating the influence of the BCM-LDT parameters on choice set size and flow allocation.