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Balancing the supply and demand for ride-sourcing companies is a challenging issue, especially with real-time requests and stochastic traffic conditions of large-scale congested road networks. To tackle this challenge, this article proposes a robust and scalable approach that integrates reinforcement learning (RL) and a centralized programming (CP) structure to promote real-time taxi operations. Both real-time order matching decisions and vehicle relocation decisions at the microscopic network scale are integrated within a Markov decision process framework. The RL component learns the decomposed state-value function, which represents the taxi drivers' experience, the off-line historical demand pattern, and the traffic network congestion. The CP component plans nonmyopic decisions for drivers collectively under the prescribed system constraints to explicitly realize cooperation. Furthermore, to circumvent sparse reward and sample imbalance problems over the microscopic road network, this article proposed a temporal-difference learning algorithm with prioritized gradient descent and adaptive exploration techniques. A simulator is built and trained with the Manhattan road network and New York City yellow taxi data to simulate the real-time vehicle dispatching environment. Both centralized and decentralized taxi dispatching policies are examined with the simulator. This case study shows that the proposed approach can further improve taxi drivers' profits while reducing customers' waiting times compared to several existing vehicle dispatching algorithms.

This study studies the tradable credit scheme design problem considering a mixture of local traffic and cross-boundary traffic. The local traffic refers to the travel demand generated by local residents with O-D pairs inside the network, while the cross-boundary traffic is the traffic with either origin or destination or both be outside the network. As the local authority aims to maximize its local social welfare, it determines the quantity of cross-boundary trips by evaluating the revenue of the cross-boundary traffic. Two credit charging schemes are investigated, i.e., a spatially differentiated credit scheme and an anonymous tradable credit scheme. In the first scheme, due to the different charging prices and the selfishness of the local authority, the travel credits are freely tradable within the local travellers only. The cross-boundary travellers have to buy travel credits from the local authority. In the second scheme, the tradable credit scheme is anonymous. The local authority determines the link-specific number of credits to be charged for using that link, while the travel credits are distributed to local travellers only but are allowed for free trading among both local and cross-boundary travellers. Two standard multi-class traffic equilibrium problems are established with side constraints and a credit restriction constraint. The equilibrium link flow patterns under these credit schemes are then demonstrated with local elastic demand. In both tradable credit schemes, the credit price in the trading market is unique under the equilibrium condition.

Connected and automated vehicles (CAVs) and regular human-piloted vehicles (RVs) will coexist in the near future. Research has indicated the feasibility of using CAVs to stabilize traffic flow and ...

This paper develops both adaptive distributed dynamic state feedback control law and adaptive distributed measurement output feedback control law for heterogeneous discrete-time swarm systems with multiple leaders. The convex hull formed by the leaders and the system matrix of leaders is estimated via an adaptive distributed containment observer. Such estimations will feed the followers so that every follower can update the system matrix of the corresponding adaptive distributed containment observer and the system state of their neighbors. The followers cooperate with each other to achieve leader–follower consensus and thus solve the containment control problem over the network. Numerical results demonstrate the effectiveness and computational feasibility of the proposed control laws.

In the literature, networks are usually assumed to operate at their steady states in static traffic assignment models. Given a set of demand management and supply regulation strategies, whether the vehicular traffic on a network would approach or not to a steady state under certain assumptions of behavioral realism is of importance and great interest in practice. Moreover, one primary reason for understanding the long-term traffic evolution dynamics is to control or influence the system trajectory in an efficient manner. To address these problems, we focus on a dynamic feedback control design and its subsequent steady-state analysis for a class of day-to-day (DTD) disequilibrium process so that the network traffic would evolve towards the desired stable traffic equilibrium. We propose an interconnected dynamic system framework for the stability and convergence analysis of DTD traffic adjustment process under the dynamic feedback control. Distinguished from the existing studies, we introduce a notion of output stability as well as the Lyapunov-like conditions to carry out the investigation noting that the steady states of the dynamic tolls that correspond to the Lagrange multipliers of the traffic equilibrium problem can be non-unique. The proposed method gives rise to a new approach to selecting pricing strategy. The proposed method involves only two adjustment parameters with well-identified physical meanings while the requirement on adjustment parameters for convergence and stability is mild. Finally, several numerical examples are presented to illustrate the insightful findings. It is found that, with dynamic feedback control, the network traffic state can converge to the desired equilibrium in a much faster manner than the case without control. The dynamic feedback controller can be implemented as demand management and supply regulation strategies such as road pricing and signal control.

This paper proposes a double time-scale model to capture the day-to-day evolution along with the within-day variability of travel time. The proposed model can be used to evaluate short-term to long-term effects of new transport policies and construction of critical infrastructures, and to analyze the resilience of road networks under disruptions. The within-day travel time variability is modeled using the functional data analysis, in which the effects of road traffic congestion and noise of traffic data are considered explicitly. The within-day process is then regarded as the local volatility (or the noise process) to drive the day-to-day process while the latter is described by a modified geometric Brownian motion (GBM). Then, the day-to-day travel time process is obtained by the statistics of the modified GBM. The model probabilistically captures the evolution of day-to-day and within-day travel time processes analytically. Moreover, an efficient method based on the cross-entropy method is developed for calibrating the model parameters. A lasso-like regularization is employed to guarantee a small bias between the model estimations and the measurement counterparts. Finally, two empirical studies are carried out to validate the proposed model at different scales with different traffic scenarios, i.e., a case study on the Guangzhou Airport Expressway (link to path scale) under traffic accident conditions and a case study in New York City (city-scale) to analyze the network resilience under Hurricane Sandy. Both case studies adopted probe vehicle data but with different configurations (fine versus coarse, small versus big data). The empirical studies confirm that the proposed model can accommodate the inherent variability in traffic conditions and data meanwhile being computationally tractable. The numerical results illustrate the applicability of the proposed model as a powerful tool in practice for analyzing the short-term and long-term impacts of disruptions and systematic changes in the performance of road networks.

Macroscopic fundamental diagram (MFD) has been widely used for aggregate modeling of urban traffic network dynamics to tackle the dimensionality problem of microscopic approaches. This paper contributes to the state-of-the-art by proposing a dynamic user equilibrium (DUE) model that enables simultaneous route choice and departure time choice under the MFD framework for various applications such as park-and-ride, vehicle dispatching and relocation. To better capture the traffic flow propagation and to adapt to the fast time-varying demand, the state-dependent travel time function is integrated into the MFD dynamics as an endogenous time-varying delay. The multi-region MFD dynamics with saturated state and inflow constraints is then used as the network loading model to formulate the DUE model through the lens of the differential variational inequality. Necessary conditions for the DUE are analytically derived using the Pontryagin minimum principle. Difficulties raised in handling the dynamic state-dependent nonlinear travel time functions, state and inflow constraints are addressed without model linearization nor enforcing constant delay assumption as conventionally done in the literature. The additional cost induced by inflow capacity and accumulation constraints can capture the hypercongestion represented by the downward sloping part of the MFD without actually activating traffic congestion. Numerical examples solved by using time-discretization solution algorithm illustrate the DUE characteristics and the corresponding dynamic external costs induced by constraints.

The increasing population in cities induces a high travel demand. Unfortunately, due to the limited capacity of urban transport networks, this increasing demand for travel raises various problems a...

In view of the advantages and a promising market prospect of the emerging connected automated vehicles (CAVs), it will be very likely that the roadway is shared by CAVs and RHVs in the near future....

The Macroscopic Fundamental Diagram (MFD) framework has been widely utilized to describe traffic dynamics in urban networks as well as to design perimeter flow control strategies under stationary (constant) demand and deterministic settings. In real world, both the MFD and demand however suffer from various intrinsic uncertainties while travel demand is of time-varying nature. Hence, robust control for traffic networks with uncertain MFDs and demand is much appealing and of greater interest in practice. In literature, there would be a lack of robust control strategies for the problem. One major hurdle is of requirement on model linearization that is actually a basis of most existing results. The main objective of this paper is to explore a new robust perimeter control framework for dynamic traffic networks with parameter uncertainty (on the MFD) and exogenous disturbance induced by travel demand. The disturbance in question is in general time-varying and stochastic. Our main contribution focuses on developing a control-Lyapunov function (CLF) based approach to establishing a couple of universal control laws, one is almost smooth and the other is Bang-bang like, for different implementation scenarios. Moreover, it is indicated that the almost smooth control is more suited for road pricing while the Bang-bang like control for signal timing. In sharp contrast to existing methods, in which adjusting extensive design parameters are usually needed, the proposed methods can determine the control in an automatic manner. Furthermore, numerical results demonstrate that the control can drive the system dynamics towards a desired equilibrium under various scenarios with uncertain MFDs and travel demand. Both stability and robustness can be substantially observed. As a major consequence, the proposed methods achieve not only global asymptotic stability but also appealing robustness for the closed-loop traffic system.

The current implementation of the spatial-temporal Gaussian process regression has computational complexity O(NM3), where N and M are the number of temporal and spatial data, respectively, and thus can only be applied to data with large N but relatively small M. In this work, we show that by exploring the Kronecker structure in the state-space model realization of the spatial-temporal Gaussian process, we can extend the current implementation with a coordinate transformation and an output transformation (corresponding to data preprocessing), such that the computational complexity is reduced to O(M3 +NM2 +NM) and therefore the proposed implementation can be applied to data with large N and moderately large M. Moreover, the proposed implementation can be parallelized and the computational complexity can be further lowered if parallel computing is adopted.

The aim of this research was to investigate the performance of simulated traffic data for real-time crash prediction when loop detector stations are distant from the actual crash location. Nearly all contemporary real-time crash prediction models use traffic data from physical detector stations; however, the distance between a crash location and its nearest detector station can vary considerably from site to site, creating inconsistency in detector data retrieval and subsequent crash prediction. Moreover, large distances between crash locations and detector stations imply that traffic data from these stations may not truly reflect crash-prone conditions. Crash and noncrash events were identified for a freeway section on I-94 EB in Wisconsin. The cell transmission model (CTM), a macroscopic simulation model, was applied in this study to instrument segments with virtual detector stations when physical stations were not available near the crash location. Traffic data produced from the virtual stations were used to develop crash prediction models. A comparison revealed that the predictive accuracy of models developed with virtual station data was comparable to those developed with physical station data. The finding demonstrates that simulated traffic data are a viable option for real-time crash prediction given distant detector stations. The proposed approach can be used in the real-time crash detection system or in a connected vehicle environment with different settings.

Macroscopic fundamental diagram (MFD), establishing a mapping from the network flow accumulation to the trip completion rate, has been widely used for aggregate modeling of urban traffic network dynamics. Based on the MFD framework, extensive research has been dedicated to devising perimeter control strategies to protect the network from gridlock. Recent research has revealed that the stochasticity and time-varying nature of travel demand can introduce significant scattering in the MFD, thus reducing the definition of the MFD dynamics. However, this type of demand effect on the behavior of the MFD dynamics has not been well studied. In this article, we investigate such effect and propose some appropriate boundary conditions to ensure that the MFD dynamics are well-defined. These boundary conditions can be regarded as travel demand adjustment in traffic rationing. For perimeter control design, a set of sufficient conditions that guarantee the controllability, an important but yet untouched issue, are derived for general multi-region MFD systems. The stability of the network equilibrium and convergence of the network dynamics are then analyzed in the sense of Lyapunov. Both theoretical and numerical results indicate that the network traffic converges to the desired uncongested equilibrium under proper boundary conditions in conjunction with proper control measures. The results are consistent with some existing studies and offer a control systems perspective regarding the demand-oriented behavior analysis of MFD-based network traffic dynamics. A surprising finding is that if the control purpose is to regulate the traffic to a desired level of service, the perimeter control gain can be simply chosen as its desired steady state, that is, the control gain is a constant and can be implemented as proportional control. This property sheds light on the road pricing design based on the MFD framework by minimizing the gap between the actual traffic state and the desired traffic state.

This paper analyses the performance of motorway control strategies subject to real-time flow measurement and modeling uncertainties. The control strategies are derived and tested on the cell transm...

In order to investigate the stochastic features in urban traffic dynamics, we propose a Stochastic Link Flow Model (SLFM) for signalized traffic networks with demand uncertainties. In the proposed model, the link traffic state is described using four different link state modes, and the probability for each link state mode is determined based on the stochastic link states. The SLFM model is expressed as a finite mixture approximation of the link state probabilities and the dynamic link flow models for all the four link state modes. Using data from microscopic traffic simulator SUMO, we illustrate that the proposed model can provide a reliable estimation of the link traffic states, and as well as good estimations on the link state uncertainties propagating within a signalized traffic network.

To maintain the stability and punctuality of bus systems, an accurate forecast of arrival time is essential to devise control strategies to prevent bus bunching especially under congested traffic conditions. Transit agencies provide travelers with accurate and reliable bus arrival times to downstream stations to improve transit service quality so as to attract more transit riders. Varieties of approaches have been dedicated to providing high prediction accuracy while the measure of the associated uncertainty is ignored. Noting that the quantification of uncertainty is vital for robust performance, this paper proposes data-driven approaches based on the Functional Data Analysis (FDA) and Bayesian Support Vector Regression (BSVR) for short-term bus travel time prediction while anticipating various uncertainties. To capture spatial–temporal dynamic traffic conditions along the route so as to increase the accuracy of the journey time prediction and to capture the skewness in journey time distribution, a probabilistic nested delay operator is adopted. Journey time reliability analysis is then conducted using the skewness of dynamic journey time distribution. An empirical study is carried out by fusing the bus transit date of No. 261 bus route and Floating Car Data (FCD) in Guangzhou. The proposed FDA and BSVR methods applied in conjunction with the probabilistic nested delay operator turn out to be highly competitive when performing forecasts under various traffic conditions. Comparative studies indicate that FDA provides more accurate prediction results and tends to anticipate uncertainties in journey time distribution more effectively.

Travel time is an important index for managers to evaluate the performance of transportation systems and an intuitive measure for travelers to choose routes and departure times. An important part of the literature focuses on predicting instantaneous travel time under recurrent traffic conditions to disseminate traffic information. However, accurate travel time prediction is important for assessing the effects of abnormal traffic conditions and helping travelers make reliable travel decisions under such conditions. This study proposes an online travel time prediction model with emphasis on capturing the effects of anomalies. The model divides a path into short links. A Functional Principal Component Analysis (FPCA) framework is adopted to forecast link travel times based on historical data and real-time measurements. Furthermore, a probabilistic nested delay operator is used to calculate path travel time distributions. To ensure that the algorithm is fast enough for online applications, parallel computation architecture is introduced to overcome the computational burden of the FPCA. Finally, a rolling horizon structure is applied to online travel time prediction. Empirical results for Guangzhou Airport Expressway indicate that the proposed method can capture an abrupt change in traffic state and provide a promising and reliable travel time prediction at both the link and path levels. In the case where the original FPCA is modified for parallelization, accuracy and computational effort are evaluated and compared with those of the sequential algorithm. The proposed algorithm is found to require only a piece rather than a large set of traffic incident records.

This paper presents an adaptive signal controller for managing traffic delays and urban bus service reliability with fully adaptable acyclic timing plans. The signal controller is built upon a reinforcement learning framework that consists of a model-based and a data-driven component. The model-based component is represented by a hybrid kinematic wave traffic model that integrates macroscopic flow-based and microscopic vehicle-based state variables subject to stochastic demands and bus service status. To cope with the high dimensional solution space, the data-driven component is incorporated as a multi-layer artificial neural network and is used to approximate future traffic states and system performances with respect to prevailing control settings. Before the controller can be used, the neural network is to be trained through a series of realised dynamic state transitions via an on-policy temporal difference learning algorithm. The proposed control framework is tested over a real world corridor scenario in London, UK. The proposed controller is able to reduce both traffic delays and bus service variabilities subject to stochastic demands with acyclic timing plans that can be derived in short computational time. This study contributes to the design of adaptive network traffic control for multi-modal networks with connected vehicle technology and advanced learning-based optimisation techniques.

This paper presents an adaptive traffic controller for stochastic road networks with an integrated model-based and data-driven solution framework. The model-based optimisation component operates based upon an underlying kinematic wave model driven by stochastic demand within a prediction horizon. The data-driven optimisation component operates based upon an approximate dynamic programming (ADP) formulation which approximates the state-control interactions over future stages with a parametric approximator. The approximator reduces the computational complexity of the adaptive control problem by parameterising the state and decision space. The parametric approximator is to be iteratively updated with online feeding of realisations of traffic states via a temporal difference (TD) learning process. Our results reveal that incorporation of the model-based component facilitates the training of the ADP-based state approximator, and hence improve the overall performance of the control system. We further develop a decentralised solution approach in which individual intersections are allowed to derive their own control policies in an asynchronous manner. The data-driven ADP approximator would serve as a central agent coordinating the control policies derived at individual intersections in the network. This is shown to be able to improve and stabilise the performance of the overall control system even under congested conditions. This is a significant progress in adaptive control system design with use of decentralised optimisation techniques. The present study contributes to the adaptive network traffic control with uncertainties through use of advanced modelling and optimisation methods.

Recent research indicates that the trip-based models can perform more accurately for capturing network hyper-congestion than the conventional macroscopic fundamental diagram (MFD) dynamics, especially during transient phases. However, due to the complex mathematical structure of the trip-based models, deriving analytical properties of the dynamic user equilibrium (DUE) of departure time choice with the trip-based models is still a challenge. This paper investigates the DUE problem of departure time choice in an isotropic urban network with identical travelers. Traffic dynamics is captured by the basic trip-based model using a speed-MFD that maps the traffic accumulation to the space-mean speed of the network. Necessary conditions for dynamic user equilibrium with and without inflow capacity constraint are derived, respectively. Under dynamic user equilibrium condition, no traveler can reduce her/his travel cost by changing the departure time. The analysis reveals the significant difference between the basic trip-based model and the conventional MFD models regarding the information support involved in the departure time choice. The derivation does not rely on some common assumptions in the literature such as linear travel cost function, no late arrivals, or linear speed-MFD. The numerical example indicates that the inflow capacity constraint can help prevent two peaks in the departure profile and the vehicle accumulation.

This paper presents a set of optimal control formulations for maximising bus service reliability through deriving optimal adjustments on signal timings. The traffic dynamics is captured by an underlying kinematic wave model in Hamilton–Jacobi formulation. With traffic data collected through loop detectors and bus positioning devices, the control actions are carried out through adjusting signal timing plans according to short-term estimations of traffic flows and bus arrivals. We derive the optimality conditions of multi-objective control formulations and present an open loop solution algorithm. The proposed control system is applied to a test arterial developed based upon a real-world scenario in Central London, UK. It is found that the model is capable of regulating bus service reliability through utilising traffic signals while managing delays induced to surrounding traffic. The study generates new insights on managing bus service reliability in busy urban networks.

For nonlinear switched discrete-time systems with input constraints, this paper presents an open-closed-loop iterative learning control (ILC) approach, which includes a feedforward ILC part and a f...

Macroscopic Fundamental Diagram (MFD) model is widely used to describe urban traffic dynamic system. Based on the MFD model, perimeter control methods are developed to ensure the efficiency of the system. However, most existing perimeter control methods would suffer from two shortcomings, i.e., linearization of the MFD function, accurate calibration of MFD and travel demand. These prerequisites would undermine the performance of the system if an accurate calibration cannot be guaranteed. On the other hand, an optimization scheme of network performance without excessive knowledge of state variables but based on traffic data is preferable. In this study, an optimal feedback controller based on the neuro-dynamic that approximates the solution of the Hamilton-Jacobi-Bellman equation (HJB) is introduced. Firstly, the value function is approximated by a neural network. Then the parameters are optimized by the policy iteration method, with the objective of minimizing the cumulative error toward set-point. Furthermore, the optimal control law constrained by a saturated operator is implemented based on real-time observations recursively. The neuro-dynamic controller is tested for the two-regional MFD system. The results confirm that the neuro-dynamic controller can regulate the traffic states converge to the desired uncongested equilibrium.

In this paper, we consider the spatial-temporal data modeling problem with large number of time instants and moderate number of locations. The problem is formulated as a function estimation problem and then handled by the Gaussian process regression method. To lower the computational complexity, we first sample the continuous-time kernel to get a discrete-time kernel, and then we derive its discrete-time state-space model realization, which is free of numerical problems. Then we convert the Gaussian process regression problem to a Kalman filtering and smoothing problem including both the hyper-parameter estimation and prediction. We consider three hyper-parameter estimation methods: the marginal likelihood maximization method, the generalized cross validation method, and the Stein's unbiased risk estimation method. The proposed implementation is tested over a simulated data set and the Colorado weather data set.

Macroscopic fundamental diagram (MFD) has been used for aggregate modeling of urban traffic network dynamics under stationary traffic assumption for dynamic taxi dispatching, vehicle relocation and dynamic pricing schemes to tackle the dimensionality problem of microscopic approaches. A city is assumed to be partitioned into several regions with each admits a well-defined MFD. Integrating state-dependent regional travel time function as an endogenous time-varying delay, the MFD model with time delay, is adopted to describe the traffic dynamics within a region. On the other hand, it is necessary to enable simultaneous route choice and departure time choice under the MFD framework for various applications such as vehicle dispatching and relocation. This paper presents an optimal control framework to model dynamic user equilibria (DUE) with simultaneous route choice behavior and departure time choice for general urban networks. Necessary conditions for the DUE are analytically derived through the lens of Pontryagin minimum principle. In contrast to existing analytical methods, the proposed method is applicable for general MFD systems without approximation schemes of the equilibrium solution. Numerical examples by time discretization are conducted to illustrate the characteristics of DUE and corresponding dynamic external costs.

This paper proposes an integrated freeway traffic flow control framework that aims to minimize the total travel cost, improve greenness and safety for freeway traffic mixed with connected automated vehicles (CAVs) and regular human-piloted vehicles (RHVs). The proposed framework devises an integrated action of several control strategies such as ramp metering, lane changing control (LCC) for CAVs and lane changing recommendation (LCR) for RHVs, variable speed limit control (VSLC) for CAVs and variable speed limit recommendation (VSLR) for RHVs with minimum safety gap control measures for lane changing and merging maneuvers. The CAVs are assumed to follow the system control instructions fully and immediately. In contrast, the RHVs would make decisions in response to the recommendations disseminated and also the behaviors of CAVs. The compliance rate of drivers to the LCR is captured by the underlying traffic flow model. A set of constraints is imposed to restrict VSLC/VSLR and LCC/LCR measures from changing too frequently or too sharply on both temporal and spatial dimensions to avoid excessive nuisance to passengers and traffic flow instability. A reinforcement learning based solution algorithm is proposed. First, a control parameterization technique is adopted to reduce the dimension of the original optimal control problem to increase computational efficiency. Then, a gradient-free Cross-Entropy-Method based algorithm is used to search the optimal parameters to circumvent the non-differentiability of the traffic flow model. The feasibility and effectiveness of the proposed framework are illustrated via numerical examples for a variety of penetration rates of CAVs under various traffic conditions. A sensitivity analysis is conducted to demonstrate the impacts of several important parameters such as the reaction time of the CAVs. It is found that the integrated control strategy can reduce the total travel cost by reducing the lane changing maneuvers and vehicles queuing at the bottleneck meanwhile smooth the traffic flow and suppress the adverse impact of shockwaves. The effect of ramp metering is not significant when the penetration rate of CAVs is high enough. Speed harmonization (with minimum gap control) in conjunction with LCC/LCR would be a better integrated control strategy under high penetration rate of CAVs.

In this paper, a novel mesoscopic multilane model is proposed to enable simultaneous simulation of mandatory and discretionary lane-changing behaviors to realistically capture multilane traffic dynamics. The model considers lane specific fundamental diagrams to simulate dynamic heterogeneous lane flow distributions on expressways. Moreover, different priority levels are identified according to different lane-changing motivations and the corresponding levels of urgency. Then, an algorithm is proposed to estimate the dynamic mandatory and discretionary lane-changing demands. Finally, the lane flow propagation is defined by the reaction law of the demand–supply functions, which can be regarded as an extension of the Incremental-Transfer and/or Priority Incremental-Transfer principles. The proposed mesoscopic multilane cell transmission model is calibrated and validated on a complex weaving section of the State Route 241 freeway in Orange County, California, showing both the positive and negative impact of lane changing maneuvers, e.g., balancing effect and capacity drop, respectively. Moreover, the empirical study verifies that the model requires no additional data other than the cell transmission model does. Thus, the proposed model can be deployed as a simple simulation tool for accessing dynamic mesoscopic multilane traffic state from data available to most management centers, and also the potential application in predicting the impact of traffic incident or lane control strategy.

The modified cell transmission model (MCTM) is formulated as a linear complementarity system (LCS) in this paper. The LCS formulation presented here consists of a discrete time linear system and a set of complementarity conditions. The discrete time linear system corresponds to the flow conservation equations while the complementarity conditions govern the sending and receiving functions defined by a series of ‘min’ operations in the MCTM. Technical difficulties encountered in application of the CTM and its extensions such as the hard nonlinearity caused by the ‘min’ operator can be avoided by the proposed LCS model. Several basic properties of the proposed LCS formulation, for example, existence and uniqueness of solution, are analysed based on the theory of linear complementarity problem. By this formulation, the theory of LCS developed in control and mathematical programming communities can be applied to the qualitative analysis of the CTM/MCTM. It is shown that the CTM/MCTM is equivalent to a convex programme which can be converted into a constrained linear quadratic control problem. It is found that these results are irrelevant to the cell partition, that is, different cell partitions will not change the uniqueness and convexity of solution. This property is essential for stability analysis and control synthesis. The proposed LCS formulation makes the CTM/MCTM convenient for the design of traffic state estimators, ramp metering controllers.

The macroscopic fundamental diagram (MFD) can effectively reduce the spatial dimension involved in dynamic optimization of traffic performance for large-scale networks. Solving the Hamilton-Jacobi-Bellman (HJB) equation takes center stage in yielding solutions to the optimal control problem. At the core of solving the HJB equation is the value function that represents choosing a sequence of actions to optimize the system performance. However, this problem generally becomes intractable for possible discontinuities in the solution and the curse of dimensionality for systems with all but modest dimension. To address these challenges, a neural network is used to approximate the value function to obtain the optimal controls through policy iteration. Furthermore, a saturated operator is embedded in the neural network approximator to handle the difficulty caused by the control and state constraints. This policy iteration can be implemented as an iterative data-driven technique that integrates with the model-based optimal design based on real-time observations. Numerical experiments are conducted to show that the neuro-dynamic programming approach can achieve optimization goals while stabilizing the system by regulating the traffic state to the desired uncongested equilibrium.

Summary Despite its importance in macroscopic traffic flow modeling, comprehensive method for the calibration of fundamental diagram is very limited. Conventional empirical methods adopt a steady state analysis of the aggregate traffic data collected from measurement devices installed on a particular site without considering the traffic dynamics, which renders the simulation may not be adaptive to the variability of data. Nonetheless, determining the fundamental diagram for each detection site is often infeasible. To remedy these, this study presents an automatic calibration method to estimate the parameters of a fundamental diagram through a dynamic approach. Simulated flow from the cell transmission model is compared against the measured flow wherein an optimization merit is conducted to minimize the discrepancy between model-generated data and real data. The empirical results prove that the proposed automatic calibration algorithm can significantly improve the accuracy of traffic state estimation by adapting to the variability of traffic data when compared with several existing methods under both recurrent and abnormal traffic conditions. Results also highlight the robustness of the proposed algorithm. The automatic calibration algorithm provides a powerful tool for model calibration when freeways are equipped with sparse detectors, new traffic surveillance systems lack of comprehensive traffic data, or the case that lots of detectors lose their effectiveness for aging systems. Furthermore, the proposed method is useful for off-line model calibration under abnormal traffic conditions, for example, incident scenarios. Copyright © 2015 John Wiley & Sons, Ltd.