Hierarchical perimeter control with guaranteed stability for dynamically coupled heterogeneous urban traffic

Perimeter control based on the Macroscopic Fundamental Diagram (MFD) is widely developed for alleviating or postponing congestion in a protected region. Recent studies reveal that traffic conditions might not be improved if the perimeter control strategies are applied to unstable systems where high demand generates heavy and heterogeneously distributed traffic congestion. Therefore, considering stability of the targeted traffic system is essential, for the sake of developing a feasible and then optimal control strategy. This paper sheds light on this direction. It integrates a stability characterization algorithm of MFD system equations into the Model Predictive Control (MPC) scheme, and features respectively an upper and a lower bound of the feasible control inputs, to guarantee system stability. Firstly, the dynamics of traffic heterogeneity and its effect on the MFD are analyzed, using real data from Guangzhou in China. Piecewise affine functions of average flow are proposed to capture traffic heterogeneity in both regional and subregional MFDs. Secondly, stability of a three-state two-region system is investigated via stable equilibrium and surface boundaries analysis. Finally, a three-layer hierarchical control strategy is introduced for the studied two-region heterogeneous urban networks. The first layer of the controller calculates the stable surface boundaries for the given traffic demands and then determines the bounds of control input (split rate). An MPC approach in the second layer is used to solve an optimization problem with two objectives of minimizing total network delay and maximizing network throughput. Heterogeneity among the subregions is minimized in the last layer by implementing simultaneously a subregional perimeter flow control and an internal flow control. The effectiveness and stability of the proposed control approach are verified by comparison with four existing perimeter control strategies.     

A comparative study of Macroscopic Fundamental Diagrams of arterial road networks governed by adaptive traffic signal systems

Using a stochastic cellular automaton model for urban traffic flow, we study and compare Macroscopic Fundamental Diagrams (MFDs) of arterial road networks governed by different types of adaptive traffic signal systems, under various boundary conditions. In particular, we simulate realistic signal systems that include signal linking and adaptive cycle times, and compare their performance against a highly adaptive system of self-organizing traffic signals which is designed to uniformly distribute the network density. We find that for networks with time-independent boundary conditions, well-defined stationary MFDs are observed, whose shape depends on the particular signal system used, and also on the level of heterogeneity in the system. We find that the spatial heterogeneity of both density and flow provide important indicators of network performance. We also study networks with time-dependent boundary conditions, containing morning and afternoon peaks. In this case, intricate hysteresis loops are observed in the MFDs which are strongly correlated with the density heterogeneity. Our results show that the MFD of the self-organizing traffic signals lies above the MFD for the realistic systems, suggesting that by adaptively homogenizing the network density, overall better performance and higher capacity can be achieved.     

Dynamics of heterogeneity in urban networks: aggregated traffic modeling and hierarchical control

Real traffic data and simulation analysis reveal that for some urban networks a well-defined Macroscopic Fundamental Diagram (MFD) exists, which provides a unimodal and low-scatter relationship between the network vehicle density and outflow. Recent studies demonstrate that link density heterogeneity plays a significant role in the shape and scatter level of MFD and can cause hysteresis loops that influence the network performance. Evidently, a more homogeneous network in terms of link density can result in higher network outflow, which implies a network performance improvement. In this article, we introduce two aggregated models, region- and subregion-based MFDs, to study the dynamics of heterogeneity and how they can affect the accuracy scatter and hysteresis of a multi-subregion MFD model. We also introduce a hierarchical perimeter flow control problem by integrating the MFD heterogeneous modeling. The perimeter flow controllers operate on the border between urban regions, and manipulate the percentages of flows that transfer between the regions such that the network delay is minimized and the distribution of congestion is more homogeneous. The first level of the hierarchical control problem can be solved by a model predictive control approach, where the prediction model is the aggregated parsimonious region-based MFD and the plant (reality) is formulated by the subregion-based MFDs, which is a more detailed model. At the lower level, a feedback controller of the hierarchical structure, tries to maximize the outflow of critical regions, by increasing their homogeneity. With inputs that can be observed with existing monitoring techniques and without the need for detailed traffic state information, the proposed framework succeeds to increase network flows and decrease the hysteresis loop of the MFD. Comparison with existing perimeter controllers without considering the more advanced heterogeneity modeling of MFD highlights the importance of such approach for traffic modeling and control.     

Macroscopic traffic flow propagation stability for adaptive cruise controlled vehicles

Traffic flow propagation stability is concerned about whether a traffic flow perturbation will propagate and form a traffic shockwave. In this paper, we discuss a general approach to the macroscopic traffic flow propagation stability for adaptive cruise controlled (ACC) vehicles. We present a macroscopic model with velocity saturation for traffic flow in which each individual vehicle is controlled by an adaptive cruise control spacing policy. A nonlinear traffic flow stability criterion is investigated using a wavefront expansion technique. Quantitative relationships between traffic flow stability and model parameters (such as traffic flow and speed, etc.) are derived for a generalized ACC traffic flow model. The newly derived stability results are in agreement with previously derived results that were obtained using both microscopic and macroscopic models with a constant time headway (CTH) policy. Moreover, the stability results derived in this paper provide sufficient and necessary conditions for ACC traffic flow stability and can be used to design other ACC spacing policies.     

Perimeter and boundary flow control in multi-reservoir heterogeneous networks

In this paper, we macroscopically describe the traffic dynamics in heterogeneous transportation urban networks by utilizing the Macroscopic Fundamental Diagram (MFD), a widely observed relation between network-wide space-mean flow and density of vehicles. A generic mathematical model for multi-reservoir networks with well-defined MFDs for each reservoir is presented first. Then, two modeling variations lead to two alternative optimal control methodologies for the design of perimeter and boundary flow control strategies that aim at distributing the accumulation in each reservoir as homogeneously as possible, and maintaining the rate of vehicles that are allowed to enter each reservoir around a desired point, while the system’s throughput is maximized. Based on the two control methodologies, perimeter and boundary control actions may be computed in real-time through a linear multivariable feedback regulator or a linear multivariable integral feedback regulator. Perimeter control occurs at the periphery of the network while boundary control occurs at the inter-transfers between neighborhood reservoirs. To this end, the heterogeneous network of San Francisco is partitioned into three homogeneous reservoirs and the proposed feedback regulators are compared with a pre-timed signal plan and a single-reservoir perimeter control strategy. Finally, the impact of the perimeter and boundary control actions is demonstrated via simulation by the use of the corresponding MFDs and other performance measures. A key advantage of the proposed approach is that it does not require high computational effort and future demand data if the current state of each reservoir can be observed with loop detector data.     

Deriving macroscopic fundamental diagrams from probe data: Issues and proposed solutions

Well-defined relationships between flow and density averaged spatially across urban traffic networks, more commonly known as Macroscopic Fundamental Diagrams (MFDs), have been recently verified to exist in reality. Researchers have proposed using MFDs to monitor the status of urban traffic networks and to inform the design of network-wide traffic control strategies. However, it is also well known that empirical MFDs are not easy to estimate in practice due to difficulties in obtaining the requisite data needed to construct them. Recent works have devised ways to estimate a network’s MFD using limited trajectory data that can be obtained from GPS-equipped mobile probe vehicles. These methods assume that the market penetration level of mobile probe vehicles is uniform across the entire set of OD pairs in the network; however, in reality the probe vehicle market penetration rate varies regionally within a network. When this variation is combined with the imbalance of probe trip lengths and travel times, the compound effects will further complicate the estimation of the MFD.To overcome this deficit, we propose a method to estimate a network’s MFD using mobile probe data when the market penetration rates are not necessarily the same across an entire network. This method relies on the determination of appropriate average probe penetration rates, which are weighted harmonic means using individual probe vehicle travel times and distances as the weights. The accuracy of this method is tested using synthetic data generated in the INTEGRATION micro-simulation environment by comparing the estimated MFDs to the ground truth MFD obtained using a 100% market penetration of probe vehicles. The results show that the weighted harmonic mean probe penetration rates outperform simple (arithmetic) average probe penetration rates, as expected. This especially holds true as the imbalance of demand and penetration level increases. Furthermore, as the probe penetration rates are generally not known, an algorithm to estimate the probe penetration rates of regional OD pairs is proposed. This algorithm links count data from sporadic fixed detectors in the network to information from probe vehicles that pass the detectors. The simulation results indicate that the proposed algorithm is very effective. Since the data needed to apply this algorithm are readily available and easy to collect, the proposed algorithm is practically feasible and offers a better approach for the estimation of the MFD using mobile probe data, which are becoming increasingly available in urban environments.     

Intelligent cruise control systems and traffic flow stability

There are two kinds of stability associated with traffic flow problems – string stability (or car-following stability) and traffic flow stability. We provide a clear distinction between traffic flow stability and string stability, and such a distinction has not been recognized in the literature, thus far. String stability is stability with respect to intervehicular spacing; intuitively, it ensures the knowledge of the position and velocity of every vehicle in the traffic, within reasonable bounds of error, from the knowledge of the position and velocity of a vehicle in the traffic. String stability is analyzed without adding vehicles to or removing vehicles from the traffic. On the other hand, traffic flow stability deals with the evolution of traffic velocity and density in response to the addition and/or removal of vehicles from the flow. Traffic flow stability can be guaranteed only if the velocity and density solutions of the coupled set of equations is stable, i.e., only if stability with respect to automatic vehicle following and stability with respect to density evolution is guaranteed. Therefore, the flow stability and critical capacity of any section of a highway is dependent not only on the vehicle following control laws and the information used in their synthesis, but also on the spacing policy employed by the control system. Such a dependence has practical consequences in the choice of a spacing policy for adaptive cruise control laws and on the stability of the traffic flow consisting of vehicles equipped with adaptive cruise control features on the existing and future highways. This critical dependence is the subject of investigation here.     

Data fusion algorithm for macroscopic fundamental diagram estimation

A promising framework that describes traffic conditions in urban networks is the macroscopic fundamental diagram (MFD), relating average flow and average density in a relatively homogeneous urban network. It has been shown that the MFD can be used, for example, for traffic access control. However, an implementation requires an accurate estimation of the MFD with the available data sources.Most scientific literature has considered the estimation of MFDs based on either loop detector data (LDD) or floating car data (FCD). In this paper, however, we propose a methodology for estimating the MFD based on both data sources simultaneously. To that end, we have defined a fusion algorithm that separates the urban network into two sub-networks, one with loop detectors and one without. The LDD and the FCD are then fused taking into account the accuracy and network coverage of each data type. Simulations of an abstract grid network and the network of the city of Zurich show that the fusion algorithm always reduces the estimation error significantly with respect to an estimation where only one data source is used. This holds true, even when we account for the fact that the probe penetration rate of FCD needs to be estimated with loop detectors, hence it might also include some errors depending on the number of loop detectors, especially when probe vehicles are not homogeneously distributed within the network.     

A hybrid model predictive control for traffic flow stabilization and pollution reduction of freeways

In this work a control system is developed and analyzed for the suppression of moving jamwaves and the reduction of pollutant concentrations near motorways. The system is based on the second-order macroscopic freeway traffic model METANET, joined by an emission dispersion model, introduced in a previous work of the authors. For the control tasks dedicated controllers are designed, both using the nonlinear model predictive control method. The control objectives require a distinction in the utilized control measures, thus different controllers are designed and used in predefined control modes. The first mode of the controller is responsible for keeping pollutant concentrations below prescribed limits under stable conditions. The second mode of the controller is working in case of a shockwave threat, aiming for traffic stabilization. Between the control modes switching is based on an appropriate rule set that satisfies the stability of the controlled system. The hybrid controller structure is realized by a finite automata. A complex case study is presented for the evaluation of the suggested controller.     

Modelling supported driving as an optimal control cycle: Framework and model characteristics

Driver assistance systems support drivers in operating vehicles in a safe, comfortable and efficient way, and thus may induce changes in traffic flow characteristics. This paper puts forward a receding horizon control framework to model driver assistance and cooperative systems. The accelerations of automated vehicles are controlled to optimise a cost function, assuming other vehicles driving at stationary conditions over a prediction horizon. The flexibility of the framework is demonstrated with controller design of Adaptive Cruise Control (ACC) and Cooperative ACC (C-ACC) systems. The proposed ACC and C-ACC model characteristics are investigated analytically, with focus on equilibrium solutions and stability properties. The proposed ACC model produces plausible human car-following behaviour and is unconditionally locally stable. By careful tuning of parameters, the ACC model generates similar stability characteristics as human driver models. The proposed C-ACC model results in convective downstream and absolute string instability, but not convective upstream string instability observed in human-driven traffic and in the ACC model. The control framework and analytical results provide insights into the influences of ACC and C-ACC systems on traffic flow operations.     

Macroscopic relations of urban traffic variables: Bifurcations, multivaluedness and instability

Recent experimental work has shown that the average flow and average density within certain urban networks are related by a unique, reproducible curve known as the Macroscopic Fundamental Diagram (MFD). For networks consisting of a single route this MFD can be predicted analytically; but when the networks consist of multiple overlapping routes experience shows that the flows observed in congestion for a given density are less than those one would predict if the routes were homogeneously congested and did not overlap. These types of networks also tend to jam at densities that are only a fraction of their routes’ average jam density.This paper provides an explanation for these phenomena. It shows that, even for perfectly homogeneous networks with spatially uniform travel patterns, symmetric equilibrium patterns with equal flows and densities across all links are unstable if the average network density is sufficiently high. Instead, the stable equilibrium patterns are asymmetric. For this reason the networks jam at lower densities and exhibit lower flows than one would predict if traffic was evenly distributed.Analysis of small idealized networks that can be treated as simple dynamical systems shows that these networks undergo a bifurcation at a network-specific critical density such that for lower densities the MFDs have predictably high flows and are univalued, and for higher densities the order breaks down. Microsimulations show that this bifurcation also manifests itself in large symmetric networks. In this case though, the bifurcation is more pernicious: once the network density exceeds the critical value, the stable state is one of complete gridlock with zero flow. It is therefore important to ensure in real-world applications that a network’s density never be allowed to approach this critical value.Fortunately, analysis shows that the bifurcation’s critical density increases considerably if some of the drivers choose their routes adaptively in response to traffic conditions. So far, for networks with adaptive drivers, bifurcations have only been observed in simulations, but not (yet) in real life. This could be because real drivers are more adaptive than simulated drivers and/or because the observed real networks were not sufficiently congested.     

Dynamic modeling and control of taxi services in large-scale urban networks: A macroscopic approach

Taxis are increasingly becoming a prominent mobility mode in many major cities due to their accessibility and convenience. The growing number of taxi trips and the increasing contribution of taxis to traffic congestion are cause for concern when vacant taxis are not distributed optimally within the city and are unable to find unserved passengers effectively. A way of improving taxi operations is to deploy a taxi dispatch system that matches the vacant taxis and waiting passengers while considering the search friction dynamics. This paper presents a network-scale taxi dispatch model that takes into account the interrelated impact of normal traffic flows and taxi dynamics while optimizing for an effective dispatching system. The proposed model builds on the concept of the macroscopic fundamental diagram (MFD) to represent the dynamic evolution of traffic conditions. The model considers multiple taxi service firms operating in a heterogeneously congested city, where the city is assumed to be partitioned into multiple regions each represented with a well-defined MFD. A model predictive control approach is devised to control the taxi dispatch system. The results show that lack of the taxi dispatching system leads to severe accumulation of unserved taxi passengers and vacant taxis in different regions whereas the dispatch system improves the taxi service performance and reduces traffic congestion by regulating the network towards the undersaturated condition. The proposed framework demonstrates sound potential management schemes for emerging mobility solutions such as fleet of automated vehicles and demand-responsive transit services.     

Speed-flow relations and cost functions for congested traffic: Theory and empirical analysis

A dynamic ‘car-following’ extension of the conventional economic model of traffic congestion is presented, which predicts the average cost function for trips in stationary states to be significantly different from the conventional average cost function derived from the speed-flow function. When applied to a homogeneous road, the model reproduces the same stationary state equilibria as the conventional model, including the hypercongested ones. However, stability analysis shows that the latter are dynamically unstable. The average cost function for stationary state traffic coincides with the conventional function for non-hypercongested traffic, but rises vertically at the road’s capacity due to queuing, instead of bending backwards. When extending the model to include an upstream road segment, it predicts that such queuing will occur under hypercongested conditions, while the general shape of the average cost function for full trips does not change, implying that hypercongestion will not occur on the downstream road segment. These qualitative predictions are verified empirically using traffic data from a Dutch bottleneck. Finally, it is shown that reduced-form average cost functions, that relate the sum of average travel cost and average schedule delay costs to the number of users in a dynamic equilibrium, certainly need not have the intuitive convex shape, but may very well be concave – despite the fact that the underlying speed-flow function may be convex.     

Cooperative traffic control of a mixed network with two urban regions and a freeway

Currently most optimization methods for urban transport networks (i) are suited for networks with simplified dynamics that are far from real-sized networks or (ii) apply decentralized control, which is not appropriate for heterogeneously loaded networks or (iii) investigate good-quality solutions through micro-simulation models and scenario analysis, which make the problem intractable in real time. In principle, traffic management decisions for different sub-systems of a transport network (urban, freeway) are controlled by operational rules that are network specific and independent from one traffic authority to another. In this paper, the macroscopic traffic modeling and control of a large-scale mixed transportation network consisting of a freeway and an urban network is tackled. The urban network is partitioned into two regions, each one with a well-defined Macroscopic Fundamental Diagram (MFD), i.e. a unimodal and low-scatter relationship between region density and outflow. The freeway is regarded as one alternative commuting route which has one on-ramp and one off-ramp within each urban region. The urban and freeway flow dynamics are formulated with the tool of MFD and asymmetric cell transmission model, respectively. Perimeter controllers on the border of the urban regions operating to manipulate the perimeter interflow between the two regions, and controllers at the on-ramps for ramp metering are considered to control the flow distribution in the mixed network. The optimal traffic control problem is solved by a Model Predictive Control (MPC) approach in order to minimize total delay in the entire network. Several control policies with different levels of urban-freeway control coordination are introduced and tested to scrutinize the characteristics of the proposed controllers. Numerical results demonstrate how different levels of coordination improve the performance once compared with independent control for freeway and urban network. The approach presented in this paper can be extended to implement efficient real-world control strategies for large-scale mixed traffic networks.     

Stability of transportation networks under adaptive routing policies

Growing concerns regarding urban congestion, and the recent explosion of mobile devices able to provide real-time information to traffic users have motivated increasing reliance on real-time route guidance for the online management of traffic networks. However, while the theory of traffic equilibria is very well-known, fewer results exist on the stability of such equilibria, especially in the context of adaptive routing policy. In this work, we consider the problem of characterizing the stability properties of traffic equilibria in the context of online adaptive route choice induced by GPS-based decision making. We first extend the recent framework of “Markovian Traffic Equilibria” (MTE), in which users update their route choice at each intersection of the road network based on traffic conditions, to the case of non-equilibrium conditions, while preserving consistency with known existence and uniqueness results on MTE. We then exhibit sufficient conditions on the network topology and the latency functions for those MTEs to be stable in the sense of Lyapunov for a single destination problem. For various more restricted classes of network topologies motivated by the observed properties of travel patterns in the Singapore network, under certain assumptions we prove local exponential stability of the MTE, and derive analytical results on the sensitivity of the characteristic time of convergence to network and traffic parameters. The results proposed in this work are illustrated and validated on synthetic toy problems as well as on the Singapore road network with real demand and traffic data.     

Highway voting system: Embracing a possible paradigm shift in traffic data acquisition

The integration of internet and mobile phones has opened the door to a new wave of utilizing private vehicles as probes not only for performance evaluation but for traffic control as well, gradually replacing the role of traffic surveillance systems as the dominant source of traffic data. To prepare for such a paradigm shift, one needs to overcome some key institutional barriers, in particular, the privacy issue. A Highway Voting System (HVS) is proposed to address this issue in which drivers provide link- and/or path-based vehicle data to the traffic management system in the form of “votes” in order to receive favorable service from traffic control. The proposed HVS offers a platform that links data from individual vehicles directly with traffic control. In the system, traffic control responds to voting vehicles in a way similar to the current system responding to prioritized vehicles and providing the requested services accordingly. We show in the paper that the proposed “voting” system can effectively resolve the privacy issue which often hampers traffic engineers from getting detailed data from drivers. Strategies to entice drivers into “voting” so as to increase the market penetration level under all traffic conditions are discussed. Though the focus of the paper is on addressing the institutional issues associated with data acquisition from individual vehicles, other research topics associated with the proposed system are identified. Two examples are given to demonstrate the impact of the proposed system on algorithm development and traffic control.     

Continuum signalized junction model for dynamic traffic networks: Offset,spillback, and multiple signal phases

This paper extends the continuum signalized intersection model exhaustively studied in Han et al. (2014) to more accurately account for three realistic complications: signal offsets, queue spillbacks, and complex signal phasing schemes. The model extensions are derived theoretically based on signal cycle, green split, and offset, and are shown to approximate well traffic operations at signalized intersections treated using the traditional (and more realistic) on-and-off model. We propose a generalized continuum signal model, which explicitly handles complex vehicle spillback patterns on signalized networks with provable error estimates. Under mild conditions, the errors are small and bounded by fixed values that do not grow with time. Overall, this represents a significant improvement over the original continuum model, which had errors that grew quickly with time in the presence of any queue spillbacks and for which errors were not explicitly derived for different offset cases. Thus, the new model is able to more accurately approximate traffic dynamics in large networks with multiple signals under more realistic conditions. We also qualitatively describe how this new model can be applied to several realistic intersection configurations that might be encountered in typical urban networks. These include intersections with multiple entry and exit links, complex signal phasing, all-red times, and the presence of dedicated turning lanes. Numerical tests of the models show remarkable consistency with the on-and-off model, as expected from the theory, with the added benefit of significant computational savings and higher signal control resolution when using the continuum model.     

A multi-class model-based control scheme for reducing congestion and emissions in freeway networks by combining ramp metering and route guidance

The paper proposes a multi-class control scheme for freeway traffic networks. This control scheme combines two control strategies, i.e. ramp metering and route guidance, in order to reduce the total time spent and the total emissions in a balanced way. In particular, the ramp metering and route guidance controllers are feedback predictive controllers, i.e. they compute the control actions not only on the basis of the measured system state, but also on the basis of the prediction of the system evolution, in terms of traffic conditions and traffic emissions. Another important feature of the controllers is that they have a multi-class nature: different classes of vehicles are considered and specific control actions are computed for each class. Since the controllers are based on a set of parameters that need to be tuned, the overall control framework also includes a module to properly determine the gains of the controllers. The simulation analysis reported in the paper shows the effectiveness of the proposed control framework and, in particular, the possibility of implementing control policies that are specific for each vehicle type.     

Evidence of convective instability in congested traffic flow: A systematic empirical and theoretical investigation

An extended open system such as traffic flow is said to be convectively unstable if perturbations of the stationary state grow but propagate in only one direction, so they eventually leave the system. By means of data analysis, simulations, and analytical calculations, we give evidence that this concept is relevant for instabilities of congested traffic flow. We analyze detector data from several hundred traffic jams and propose estimates for the linear growth rate, the wavelength, the propagation velocity, and the severity of the associated bottleneck that can be evaluated semi-automatically. Scatter plots of these quantities reveal systematic dependencies. On the theoretical side, we derive, for a wide class of microscopic and macroscopic traffic models, analytical criteria for convective and absolute linear instabilities. Based on the relative positions of the stability limits in the fundamental diagram, we divide these models into five stability classes which uniquely determine the set of possible elementary spatiotemporal patterns in open systems with a bottleneck. Only two classes, both dominated by convective instabilities, are compatible with observations. By means of approximate solutions of convectively unstable systems with sustained localized noise, we show that the observed spatiotemporal phenomena can also be described analytically. The parameters of the analytical expressions can be inferred from observations, and also (analytically) derived from the model equations.     

On the distribution of urban road space for multimodal congested networks

Transport systems in real cities are complex with many modes of transport sharing and competing for limited road space. This work intends to understand how space distributions for modes and interactions among modes affect network traffic performance. While the connection between performance of transport systems and general land allocation is the subject of extensive research, space allocation for interacting modes of transport is an open research question. Quantifying the impact of road space distribution on the performance of a congested multimodal transport system with a dynamic aggregated model remains a challenge. In this paper, a multimodal macroscopic fundamental diagram (MFD) is developed to represent the traffic dynamics of a multimodal transport system. Optimization is performed with the objective of minimizing the total passenger hours traveled (PHT) to serve the total demand by redistributing road space among modes. Pricing strategies are also investigated to provide a higher demand shift to more efficient modes. We find by an application to a bi-modal two-region city that (i) the proposed model captures the operational characteristics of each mode, and (ii) optimal dynamic space distribution strategies can be developed. In practice, the approach can serve as a physical dynamic model to inform space distribution strategies for policy makers with different goals of mobility.