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In traffic flow, cellular automata=kinematic waves
Abstract
This paper proves that the vehicle trajectories predicted by (i) a simple linear car-following model, CF(L), (ii) the kinematic wave model with a triangular fundamental diagram, KW(T), and (iii) two cellular automata models CA(L) and CA(M) match everywhere to within a tolerance comparable with a single “jam spacing”. Thus, CF(L) = KW(T) = CA(L, M).
... Here, we are interested in continuum models that evolve aggregated quantities like the vehicle density or the mean vehicle velocity in space and time. Those models are also called fluid-dynamic or macroscopic models and a rich literature exists today (e.g., [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]). Among the (inviscid) macroscopic models one typically distinguishes between first-order models given by scalar hyperbolic conservation laws and second-order models comprised of systems of strictly hyperbolic equations. ...
... defined, such that k ( k , v k max ) = k ( k ). Including this parametrization by the Lagrangian marker also in (8) gives rise to the generalized demand and supply functions d 1 ( k , w k ), d 2 ( k , w k ) and s 3 ( k , w k ). The following result motivates the construction of our second-order coupling models in the next section. ...
... A common approach discussed in Garavello and Piccoli [9] is the maximization of the flux in the junction such that the Kirchhoff condition (10) and the demand and supply conditions (7) are fulfilled. Given the traces 1− 0 , 2− 0 and 3+ 0 the demand in the incoming roads and the supply in the outgoing road according to (8) is computed. Only, two cases can occur: In the case ...
The simulation of traffic flow on networks requires knowledge on the behavior across traffic intersections. For macroscopic models based on hyperbolic conservation laws, there exist nowadays many ad‐hoc models describing this behavior. Based on real‐world car trajectory data we propose a new class of data‐driven models with the requirements of being consistent to networked hyperbolic traffic flow models. To this end, the new models combine artificial neural networks with a parametrization of the solution space to the half‐Riemann problem at the junction. A method for deriving density and flux corresponding to the traffic close to the junction for data‐driven models is presented. The model parameters are fitted to obtain suitable boundary conditions for macroscopic first‐ and second‐order traffic flow models. The prediction of various models are compared considering also existing coupling rules at the junction. Numerical results imposing the data‐fitted coupling models on a traffic network are presented exhibiting accurate predictions of the new models.
... Here, we are interested in continuum models that evolve aggregated quantities like the car density or the mean car velocity in space and time. Those models are also called fluid-dynamic or macroscopic models and a rich literature exists today (e.g., [3,5,11,14,15,22,23,37,38,44,46,48,49,54,57,58]). Among the (inviscid) macroscopic models one typically distinguishes between first-order models given by scalar hyperbolic conservation laws and second-order models comprised of systems of strictly hyperbolic equations. ...
... In this variant of model ML1 we allow for adaptations in the fundamental diagrams by taking = max for road = 1, 2, 3 as additional model parameters. Thus, the model assumes marker dependent input fluxes 1 (15) to which the linear model is applied. As a generalization of model ML1 model ML2 also satisfies the Kirchhoff condition (10). ...
... ANNs are highly parameterized functions consisting of successively applied linear affine maps (layers) and component-wise non-linear functions (activation functions), see e.g., [47,51]. Our ANN model for the Riemann solver (9) has 6 input neurons (i.e., the domain of the network as a map is R 6 ) accounting for the vehicle densities and the corresponding fluxes as used for the input of the linear model (15). The input is first normalized and then processed by a hidden layer applying the sigmoid activation function and mapping to 12 neurons. ...
Traffic flow on networks requires knowledge on the behavior across traffic intersections. For macroscopic models based on hyperbolic conservation laws there exist nowadays many ad-hoc models describing this behavior. Based on car trajectory data we propose a novel framework combining data-fitted models with the requirements of consistent coupling conditions for macroscopic models of traffic junctions. A method for deriving density and flux corresponding to the traffic close to the junction for data-driven models is presented. Within the models parameter fitting as well as machine-learning approaches enter to obtain suitable boundary conditions for macroscopic first and second-order traffic flow models. The prediction of various models are compared considering also existing coupling rules at the junction. Numerical results imposing the data-fitted coupling models on a traffic network are presented.
... Vehicles on the network were simulated using the cellular automata model (CAM) proposed by (Daganzo, 2006), which is consistent with kinematic wave theory Whitham, 1955a, 1955b;Richards, 1956). In this framework, each street is broken up into homogeneous discrete cells of length 0.004 miles (equal to average vehicle spacing at jam density), which allows only a single vehicle to occupy any cell at any time period. ...
... The same parameters as the previous section are applied to the cellular automata model (Daganzo, 2006), which is used to simulate the vehicles on the network. In this simulation, vehicle locations on arterials are updated at consistent intervals of 0.36 s while vehicle locations on local roads are updated at consistent intervals of 0.72 s. ...
Unimodal, concave relationships between average network productivity and accumulation or density aggregated across spatially compact regions of urban networks—so called network Macroscopic Fundamental Diagrams (MFDs)—have recently been shown to exist on homogeneous street networks. When present, MFD relationships facilitate the modeling of traffic congestion at a regional level and have led to the development of various regional traffic control strategies. However, real street networks are not homogeneous—they generally have a hierarchical structure where some streets (e.g., arterials) promote higher mobility than others (e.g., local roads). This paper examines how the presence of hierarchical roadway structures may potentially cause non-unimodal patterns in a network's MFD. These are observed using three types of tools: analytical models of simple network structures, simulations of various idealized roadway networks, and empirical data. The impacts of street hierarchy depend on how vehicles use different roadway types to move within the network; i.e., their routing strategy. The findings suggest that the presence of roadway hierarchies may lead to MFDs that have non-unimodal or non-concave patterns on the free-flow branch when vehicles route themselves according to user equilibrium principles, which is closest to what would be observed in realistic situations. Such patterns are contrary to what is traditionally assumed in most MFD-based modeling frameworks. However, the unimodal and concave MFD should be expected under system optimal routing conditions that maximize network productivity for a given traffic state.
... e implementation of BLIP has been studied in recent research. Eichler and Daganzo [2,7] study the feasibility, impacts, and benefits of BLIPs using kinematic wave theory. Although BLIP increases the average traffic density and traffic delay, it would not significantly reduce total road capacity. ...
... CA models have been widely exploited to simulate various traffic flow scenarios and also have been equal to kinematic waves in traffic flow [7]. e CA model is a powerful tool for characterizing macroscopic and microscopic traffic flows. ...
Bus lanes with intermittent priority (BLIPs) are lanes where general traffic is required to give way to approaching buses. BLIPs can improve the reliability of bus services and help maximize the use of road resources. It can be seen as an innovative sharing mobility, such as carsharing, carpooling, and lane sharing. However, implementation of BLIPs has never been feasible until vehicle communications could accommodate the idea. Vehicle-to-vehicle (V2V) communications have broad application prospects in the deployment of BLIPs. This paper develops a two-lane cellular automaton (CA) model to simulate BLIPs and assesses the benefits of connected vehicles for bus operation. In the model, lane-changings are asymmetric with an improved mandatory BLIP lane-changing rule underlying. The effects of BLIPs are explored through numerical simulations, including BLIPs’ impacts on neighboring lanes, travel time saving, fuel consumption, and the punctuality rate of buses. Analysis of traffic flow characteristics of corridors using BLIPs reveals that there is a strong connection among the bus departure interval, clear distance, and road capacity.
... For example, with the cumulative flow as the state variable in Eulerian coordinates, the LWR model is equivalent to a Hamilton-Jacobi equation, which can be solved with the minimum principle in (Newell, 1993) or the variational principle in (Daganzo, 2005). In (Daganzo, 2006), the LWR model was shown to be equivalent to various car-following and cellular automaton models. In (Leclercq et al., 2007), both Hamilton-Jacobi and hyperbolic conservation law formulations of the LWR model were derived and solved in Lagrangian coordinates. ...
We present a second-order formulation of the LWR model based on Phillips' model (Phillips, 1979); but the model is nonstandard with a hyperreal infinitesimal relaxation time. Since the original Phillips model is unstable with three different definitions of stability in both Eulerian and Lagrangian coordinates, we cannot use traditional methods to prove the equivalence between the second-order model, which can be considered the zero-relaxation limit of Phillips' model, and the LWR model, which is the equilibrium counterpart of Phillips' model. Instead, we resort to a nonstandard method based on the equivalence relationship between second-order continuum and car-following models established in (Jin, 2016) and prove that the nonstandard model and the LWR model are equivalent, since they have the same anisotropic car-following model and stability property. We further derive conditions for the nonstandard model to be forward-traveling and collision-free, prove that the collision-free condition is consistent with but more general than the CFL condition (Courant et al., 1928), and demonstrate that only anisotropic and symplectic Euler discretization methods lead to physically meaningful solutions. We numerically solve the lead-vehicle problem and show that the nonstandard second-order model has the same shock and rarefaction wave solutions as the LWR model for both Greenshields and triangular fundamental diagrams; for a non-concave fundamental diagram we show that the collision-free condition, but not the CFL condition, yields physically meaningful results. Finally we present a correction method to eliminate negative speeds and collisions in general second-order models, and verify the method with a numerical example.
... Besides microscopic and cellular models there has been intense research in continuum models where the temporal and spatial evolution of car densities is prescribed. Based on the level of detail there are gas-kinetic or mesoscopic models (e.g., [32,34,39,46,56,31]) and macroscopic models being fluid-dynamics models (e.g., [8,10,13,21,22,25,26,42,43,47,50,54,55,60,66,68]). Among the (inviscid) macroscopic models one typically distinguishes between first-order models based on scalar hyperbolic equations and second-order models comprised of systems of hyperbolic equations. ...
Based on experimental traffic data obtained from German and US highways, we propose a novel two-dimensional first-order macroscopic traffic flow model. The goal is to reproduce a detailed description of traffic dynamics for the real road geometry. In our approach both the dynamic along the road and across the lanes is continuous. The closure relations, being necessary to complete the hydrodynamic equation, are obtained by regression on fundamental diagram data. Comparison with prediction of one-dimensional models shows the improvement in performance of the novel model.
... Chakroborty and Maurya (2008) [49] conducted comparable analysis of different CA based traffic flow models in terms of microscopic properties (like time headway distribution, acceleration noise, stability, etc.). By focusing on spatio-temporal patterns, phase transitions, and growth patterns of oscillations, cellular automata can capture the intricate behaviors observed in real-world traffic, such as the formation and dissipation of traffic jams, and the transition from free-flow to congested traffic conditions [50], [51]. CA based simulation models are often used for heterogeneous traffic flow analysis, including pedestrian and bicycle flow dynamics after adjusting their parameters to reflect the lower speeds and smaller sizes [52]- [59]. ...
Car-following (CF) algorithms are crucial components of traffic simulations and have been integrated into many production vehicles equipped with Advanced Driving Assistance Systems (ADAS). Insights from the model of car-following behavior help researchers to understand the causes of various macro phenomena that arise from interactions between pairs of vehicles. Car-following Models encompass multiple disciplines, including traffic engineering, physics, dynamic system control, cognitive science, machine learning, deep learning, and reinforcement learning. This paper presents an extensive survey that highlights the differences, complementarities, and overlaps among microscopic traffic flow and control models based on their underlying principles and design logic. It reviews representative algorithms, ranging from theory-based Kinematic Models, Psycho-Physical Models, and Adaptive Cruise Control Models to Learning-based algorithms like Reinforcement Learning (RL) and Imitation Learning (IL). To acknowledge the potential impact on CF models, Large GenAI Models are also included as Knowledge- Driven category. This manuscript discusses the strengths and limitations of these models and explores their applications in different contexts. This review synthesizes existing researches and available datasets across different domains to fill knowledge gaps and offer guidance for future research by identifying the latest trends in car following models and their applications.
... In this section, we conduct microscopic traffic simulation experiments to verify the feasibility of the IBA setting method base on lane multiplexing. As a powerful tool for studying traffic flow in a microenvironment [38], cellular automaton (CA) is adopted to build microscopic traffic models to help us evaluate the operation effect of lane operation strategies. ...
Intermittent bus lanes (IBLs) can alleviate the contradiction between bus priority and the urgent demand of general vehicles for road resources. However, existing IBL strategies seldom pay attention to the setting method of the dynamic bus lanes at intersections, which leads to the still serious delay of buses at intersections in the traffic congestion environment. To tackle this issue, this research explores a novel method of setting the intermittent bus approach (IBA) of intersections for lane sharing and bus priority at intersections. In particular, a time slice division strategy with an intersection signal coordination model is developed to fully and reasonably allocate the idle time of bus lanes at intersections. Besides, considering the lane-changing demands of general vehicles at intersections, the parameters of the IBA lane system are modeled and optimized. For testing and verifying the feasibility of the proposed method, comparative experiments are conducted through microscopic traffic simulation. Results show that the proposed IBA setting method can effectively solve the problem of bus priority failure at intersections. It can maintain the continuity of vehicle running on intersection sections, which better exerts the operational benefits of dynamic bus lanes.
... Gipps' model. For some researchers, further categories, i.e. action-point models (Leutzbach and Wiedemann, 1986) and cellular automata models (Daganzo, 2006) can also be mentioned. For the present paper, to start with, a stopping-distance based approach has been employed, resulting in the author's non-lane-based car following model, also known as a staggered car following model. ...
A more comprehensive work was later published as a journal paper (see the full text provided here).
... Gipps' model. For some researchers, further categories, i.e. action-point models (Leutzbach and Wiedemann, 1986) and cellular automata models (Daganzo, 2006) can also be mentioned. For the present paper, to start with, a stopping-distance based approach has been employed, resulting in the author's non-lane-based car following model, also known as a staggered car following model. ...
A similar work was later published as a journal paper (see the full text provided here).
... For a broader review, refer to others. [7][8][9] Finally, because car-following models are more fundamental to traffic flow theory than hydrodynamic models, which use continuous fluid analogy, it is thought that continuous flow dynamics may not be applicable to non-lanebased flows, since the inclusion of irregular driver behaviours is essential. The overall aim of the paper, therefore, is to treat the problem by simulating car-following behaviour where lane-based driving discipline is weak. ...
A similar work was later published as a journal paper (see the full text provided here).
... (v) We characterize the bus movements in and out of the berths by two parameters (Shen et al., 6 the first-order approximation of the real bus traffic dynamics (see Daganzo, 2006Daganzo, , 2007 178 similar treatments of vehicle traffic). In other words, the time loss incurred by bus 179 acceleration and deceleration is factored into bus dwell times and queueing times, while the 180 detailed acceleration and deceleration processes are simplified. ...
Long bus queues at busy stops plague bus systems in many cities. Since berths are laid-out in tandem, buses' overtaking maneuvers are often prohibited or restricted, which can significantly reduce a bus stop's discharge capacity. When overtaking is allowed, aggressive drivers may perform disruptive oblique insertion maneuvers that would undermine stop capacity and compromise safety. This paper develops parsimonious yet realistic simulation models to examine the impacts of different overtaking policies on bus-stop capacity. Key realistic features are considered, including the oblique insertions resulting from overtaking, impacts of a nearby traffic signal, and bus traffic characteristics (reaction and move-up times). Extensive numerical experiments unveil many new findings. Some are at odds with those reported by previous studies. In addition, we examine two strategies that can improve the stop capacity without incurring disruptive oblique insertions. Practical implications of our findings are discussed, especially on choosing the most productive overtaking policy and means to minimize the capacity lost to buses' mutual blockage at stops. These implications have broad applications to various types of bus stops.
... However, the stochastic nature of the KKW model could be unstable and lead to collisions. Daganzo [18] proved that the CA model matches a triangular fundamental diagram kinematic wave model and the CA model agrees with Newell's "lower order" car-following model. ...
The car-following (CF) model is the core component for traffic simulations and has been built-in in many production vehicles with Advanced Driving Assistance Systems (ADAS). Research of CF behavior allows us to identify the sources of different macro phenomena induced by the basic process of pairwise vehicle interaction. The CF behavior and control model encompasses various fields, such as traffic engineering, physics, cognitive science, machine learning, and reinforcement learning. This paper provides a comprehensive survey highlighting differences, complementarities, and overlaps among various CF models according to their underlying logic and principles. We reviewed representative algorithms, ranging from the theory-based kinematic models, stimulus-response models, and cruise control models to data-driven Behavior Cloning (BC) and Imitation Learning (IL) and outlined their strengths and limitations. This review categorizes CF models that are conceptualized in varying principles and summarize the vast literature with a holistic framework.
... Entry ramps are treated as unsignalized merges in which entering vehicles are assumed to have priority. The cellular automata model (CAM) consistent with kinematic wave theory is used to simulate the behavior of vehicles on the network (36,37). In this framework, the ring is broken up into homogeneous discrete cells of length 0.005 mi (equal to average vehicle spacing at jam density) that allow only a single vehicle to occupy any cell at any time period. ...
Two key aggregated traffic models are the relationship between average network flow and density (known as the network or flow macroscopic fundamental diagram [flow-MFD]) and the relationship between trip completion and density (known as network exit function or the outflow-MFD [o-FMD]). The flow- and o-MFDs have been shown to be related by average network length and average trip distance under steady-state conditions. However, recent studies have demonstrated that these two relationships might have different patterns when traffic conditions are allowed to vary: the flow-MFD exhibits a clockwise hysteresis loop, while the o-MFD exhibits a counter-clockwise loop. One recent study attributes this behavior to the presence of bottlenecks within the network. The present paper demonstrates that this phenomenon may arise even without bottlenecks present and offers an alternative, but more general, explanation for these findings: a vehicle’s entire trip contributes to a network’s average flow, while only its end contributes to the trip completion rate. This lag can also be exaggerated by trips with different lengths, and it can lead to other patterns in the o-MFD such as figure-eight patterns. A simple arterial example is used to demonstrate this explanation and reveal the expected patterns, and they are also identified in real networks using empirical data. Then, simulations of a congestible ring network are used to unveil features that might increase or diminish the differences between the flow- and o-MFDs. Finally, more realistic simulations are used to confirm that these behaviors arise in real networks.
... At the critical point the characteristic length scale of the system, such as the size S of forest fires or the size of traffic jams as shown here, diverges to infinity, which renders the system scale-free and thus scale invariant. This implies a power-law distribution for the jam size, A microscopic formulation of this theory is the celebrated Newell's car-following model (Newell, 2002), which can in turn be formulated as a cellular automaton (CA) (Daganzo, 2006). The NaSch model, for which critical behavior was first conjectured, is a stochastic version of Daganzo's CA model that introduces bounded acceleration and a probability of slow down for moving vehicles. ...
Self-organized criticality (SOC) is a celebrated paradigm from the 90’s for understanding dynamical systems naturally driven to its critical point, where the power-law dynamics taking place make predictions practically impossible, such as in stock prices, earthquakes, pandemics and many other problems in science related to phase transitions. Shortly thereafter, it was realized that traffic flow might be in the SOC category, implying that conventional traffic management strategies seeking to maximize the local flows can become detrimental. This paper shows that the Kinematic Wave model with triangular fundamental diagram, and many other related traffic models, indeed exhibit SOC, thanks in part to the fractal nature of traffic exposed here on the one hand, and our need to get to our destinations as soon as possible, on the other hand. Important implications for congestion management of traffic near the critical region are discussed, such as: (i) Jam sizes obey a power-law distribution with exponent 1/2, implying that both its mean and variance become ill-defined and therefore impossible to estimate. (ii) Traffic in the critical region is chaotic in the sense that predictions becomes extremely sensitive to initial conditions. (iii) However, aggregate measures of performance such as delays and average speeds are not heavy tailed, and can be characterized exactly by different scalings of the Airy distribution, (iv) Traffic state time-space “heat maps” are self-affine fractals where the basic unit is a triangle, in the shape of the fundamental diagram, containing 3 traffic states: voids, capacity and jams. This fractal nature of traffic flow calls for analysis methods currently not used in our field.
... But because the shape of the triangular diagram is irrelevant thanks to the shear symmetry discussed above, one can choose v max = 1 giving the isosceles triangle in Fig. 3 without loss of generality. This means that the celebrated Newell's car-following model [26], which can be formulated as a cellular automaton [27], is equivalent to elementary cellular automata rule 184 [28], which in turn is equivalent to the totally asymmetric simple exclusion process (TASEP), for which many exact results are known. All experimental results in the remainder of the paper use the NaSch model with v max = 1, and it will be simply referred to as the "traffic model". ...
The analogy between the theory of phase transitions in simple fluids and vehicular traffic flow has long been suspected, promising a new level of understanding of urban congestion by standing on one of the firmer foundations in physics. The obstacle has been the interpretation of the thermal energy of the gas-particle system, which remains unknown. This paper proposes the flow of cars through the network as a viable interpretation, where the fundamental diagram for traffic flow would be analogous to the coexistence curve in gas-liquid phase transitions. Thanks to the power-law form of the coexistence curve, it was possible to formalize that the resulting network traffic model belongs to the Kardar-Parisi-Zhang universality class. The scaling relationships arising in this universality class are found to be consistent with West's scaling theory for cities. It is shown that congestion costs (delays + fuel consumption) scale superlinearly with city population, possibly and worryingly more so than predicted by West's theory. Implications for sustainability and resiliency are discussed.
... En revanche, des expérimentations ont mis en évidence que dans un état de trafic congestionné, cette vitesse est toujours constante (Chiabaut et al., 2009(Chiabaut et al., , 2010. En conséquence, le DF triangulaire est le plus répandu dans la littérature du trafic en raison de sa simplicité et de sa pertinence pour reproduire fidèlement les observations empiriques (Newell, 2002;Leclercq, 2009;Daganzo, 2006). Le DF triangulaire, présenté sur la Fig. 1.4.a, ...
Cette thèse fournit une méthodologie pour l'analyse de la capacité des routes vers l’approche microscopique. Dans un premier temps, un outil de simulation est proposé pour estimer les impacts de la variabilité interindividuelle du comportement de poursuite sur deux variables macroscopiques du trafic routier : distribution de la capacité et valeur de la chute de capacité. Nous étudions le comportement de trois modèles de poursuite existants : Newell simple avec accélération bornée, Gipps et Tampere. En utilisant un scénario à voie unique avec limitation de vitesse sur une zone, une tête de bouchon avec la capacité nominale variable a été créé. Deux méthodes de résolution numérique des modèles de Newell et Tampere sont testées : une méthode classique qui utilise un pas de temps uniforme et une nouvelle méthode proposée qui utilise un pas de temps individualisé. Nous mettons en évidence les effets importants sur les variables macroscopiques induits par la résolution classique lorsque la variabilité interindividuelle est considérée. En utilisant la méthode de résolution proposée, nous choisissons de faire varier les trois paramètres typiques de la poursuite : la distance minimale, le temps de réaction et l'accélération. Nous avons utilisé les modèles Newell et Gipps pour cette tâche. Notre étude a montré que le temps de réaction est le paramètre avec le plus d'impact sur la variation de capacité. Nous avons conclu que la variabilité de ces paramètres n'a pas d'impact significatif sur la chute de capacité (à condition que l'accélération maximale ait une valeur moyenne relativement élevée). De plus, diverses formes de distribution des paramètres (uniforme, gaussienne tronquée et gamma) ont été explorées. On s'est rendu compte que cela n'avait pas d'impact significatif sur la répartition des capacités. Dans un deuxième temps, en utilisant des données empiriques pour les véhicules manuels et automatisés (avec le régulateur de vitesse), nous avons estimé la variabilité expérimentale pour prédire l'impact des véhicules automatisés sur un trafic mixte supposé dans la simulation. Quatre critères de sélection sont proposés pour sélectionner les meilleures trajectoires et garantir un processus de calibration fiable. Une méthode simple testée dans la littérature est utilisée pour la calibration des modèles de Newell et Gipps afin d'estimer la variabilité expérimentale des paramètres. En utilisant les résultats précédents comme données d'entrée dans l'outil de simulation proposé, nous avons prédit la diminution de la capacité avec l'augmentation du taux pénétration des véhicules automatisés. Cela contraste avec les premières prédictions trouvées dans la littérature. De plus, on observe une valeur de chute de capacité significative uniquement avec le modèle Gipps (liée aux faibles valeurs du paramètre d'accélération). La méthodologie proposée améliore les méthodes existantes pour effectuer une étude cohérente sur la variabilité interindividuelle du trafic.
... A ramp connecting two freeway links is considered as a road if its length is more than seven times of average vehicle length (150 ft)(Daganzo, 2006); otherwise, it is counted as a node.H. Yang et al. Transportation Research Part B 167 (2023) [99][100][101][102][103][104][105][106][107][108][109][110][111][112][113][114][115][116][117] ...
The information of network-wide future traffic speed distribution and its propagation is beneficial to develop proactive traffic congestion management strategies. However, predicting network-wide traffic speed propagation is non-trivial. This study develops a traffic flow dependency and dynamics based deep learning aided approach (TD²-DL), which predict network-wide high resolution traffic speed propagation by explicitly integrating temporal-spatial flow dependency, traffic flow dynamics with deep learning method techniques. Specifically, we first develop a graph theory-based method to identify the local temporal-spatial traffic dependency of each road among neighboring roads adaptive to the prediction horizon and traffic delay. Then, traffic speed propagation on every road is mathematically described by v-CTM based on traffic initial and boundary conditions. Next, the long short-term memory (LSTM) model is employed to predict boundary conditions factoring the traffic temporal-spatial dependency and historical data predicted by v-CTM. In this way, we well couple the physical models (traffic dependency and v-CTM) with the deep learning approach, and further make them coevolution under this framework. Last, an EKF is used to assimilate predicted traffic speed predicted by v-CTM coupled with the LSTMs and the field traffic data; an FNN is introduced to impute missing and corrupted data for improving the traffic speed prediction accuracy. The numerical experiments indicated that the TD²-DL predicted the network-wide traffic speed propagation in 30 minutes with accuracy varying from 85%-98%. It outperformed the tested models recently developed in literature. The ablation experimental results confirmed the significance of factoring traffic dependency and integrating data imputation and assimilation techniques for improving the prediction accuracy.
... Tere are many trafc models in these categories, such as automated cellular models, car chase models, gas kinetic models, fuid fow models, and hydrodynamics [9][10][11]. ...
For a long time now, traffic equations have been considered, and different modeling has been done for it. In this article, we work on the macroscopic model, especially the most famous light model. Because these models are among the stiff and shocking problems, theoretical methods do not give good answers to these problems. This paper describes a meshless method to solve the traffic flow equation as a stiff equation. In the proposed method, we also use the exponential time differencing (ETD) method and the exponential time differencing fourth-order Runge–Kutta (ETDRK4). The purpose of this new method is to use methods of the moving least squares (MLS) method and a modified exponential time differencing fourth-order Runge–Kutta scheme. To solve these equations, we use the meshless method MLS to approximate the spatial derivatives and then use method ETDRK4 to obtain approximate solutions. In order to improve the possible instabilities of method ETDRK4, approaches have been stated. The MLS method provided good results for these equations due to its high flexibility and high accuracy and has a moving window and obtains the solution at the shock point without any false oscillations. The technique is described in detail, and a number of computational examples are presented.
... But because the shape of the triangular diagram is irrelevant thanks to the shear symmetry discussed above, one can choose v max = 1 giving the isosceles triangle in Fig. 3 without loss of generality. This means that the celebrated Newell's car-following model [26], which can be formulated as a cellular automaton [27], is equivalent to elementary cellular automata rule 184 [28], which in turn is equivalent to the totally asymmetric simple exclusion process (TASEP), for which many exact results are known. All experimental results in the remainder of the paper use the NaSch model with v max = 1, and it will be simply referred to as the "traffic model". ...
The analogy between the theory of phase transitions in simple fluids and vehicular traffic flow has long been suspected, promising a new level
of understanding of urban congestion by standing on one of the firmer foundations in physics. The obstacle has been the interpretation of the
thermal energy of the gas-particle system, which remains unknown. This paper proposes the flow of cars through the network as a viable
interpretation, where the fundamental diagram for traffic flow would be analogous to the coexistence curve in gas-liquid phase transitions.
Thanks to the power-law form of the coexistence curve, it was possible to formalize that the resulting network traffic model belongs to the
Kardar-Parisi-Zhang universality class. The scaling relationships arising in this universality class are found to be consistent with West’s
scaling theory for cities. It is shown that congestion costs (delays + fuel consumption) scale superlinearly with city population, possibly and
worryingly more so than predicted by West’s theory. Implications for sustainability and resiliency are discussed.
... But because the shape of the triangular diagram is irrelevant thanks to the shear symmetry discussed above, one can choose v max = 1 giving the isosceles triangle in Fig. 3 without loss of generality. This means that the celebrated Newell's car-following model [23], which can be formulated as a cellular automaton [24], is equivalent to elementary cellular automata rule 184 [25], which in turn is equivalent to the totally asymmetric simple exclusion process (TASEP), for which many exact results are known. All experimental results in the remainder of the paper use the NaSch model with v max = 1, and it will be simply referred to as the "traffic model". ...
The analogy between the theory of phase transitions in simple fluids and vehicular traffic flow has long been suspected, promising a new level of understanding of urban congestion by standing on one of the firmer foundations in physics. The obstacle has been the interpretation of the thermal energy of the gas-particle system, which remains unknown. This paper proposes the flow of cars through the network as a viable interpretation, where the fundamental diagram for traffic flow would be analogous to the coexistence curve in gas-liquid phase transitions. Thanks to the power-law form of the coexistence curve, it was possible to formalize that the resulting network traffic model belongs to the Kardar-Parisi-Zhang universality class. The scaling relationships arising in this universality class are found to be consistent with West's scaling theory for cities. It is shown that congestion costs (delays + fuel consumption) scale superlinearly with city population, possibly and worryingly more so than predicted by West's theory. Implications for sustainability and resiliency are discussed.
... (v) We characterize the bus movements in and out of the berths by two parameters (Shen et al., 6 the first-order approximation of the real bus traffic dynamics (see Daganzo, 2006Daganzo, , 2007 178 similar treatments of vehicle traffic). In other words, the time loss incurred by bus 179 acceleration and deceleration is factored into bus dwell times and queueing times, while the 180 detailed acceleration and deceleration processes are simplified. ...
... Following another technical route, some researchers attempt to incorporate explainable physics-informed regularizations into traffic prediction algorithms to enhance model adaptability and transferability Li et al., 2021a;Thodi et al., 2021). Even though the corridor-level traffic prediction algorithms and their association with classical traffic flow theory (e.g., kinematic wave theory, continuity equation, and cell transmission model) have been broadly studied (Daganzo, 2006;Celikoglu, 2014;Thodi et al., 2021), the research on incorporating network-scale traffic properties and physical invariants into urban network link-wise traffic prediction methods is still at an early stage (Vlahogianni et al., 2014;Kumar and Raubal, 2021), and empirical experience is limited (Li et al., 2021a). When the scale of traffic prediction changes from a single corridor to the network-scale link-wise level, the physical features and required techniques are vastly different. ...
Network traffic flow prediction on a fine-grained spatio-temporal scale is essential for intelligent transportation systems, and extensive studies have been carried out in this area. However, existing methods are mostly data-driven, with stringent requirements on the amount and quality of data. The collected network-scale traffic data are expected to be complete, sufficient, and representative, containing most traffic flow patterns in the road network. Unfortunately, it is very rare that sufficient and representative traffic data across the whole road network in several consecutive weeks are available for model calibration. In real-world applications, data insufficiency and dataset shift problems are prevalent, resulting in the ‘cold start’ issue in traffic prediction. To deal with the challenges above, this paper develops a two-stage physics-informed transfer learning method for network-scale link-wise traffic flow knowledge transfer under MFD-based physical constraints. In the first stage, the road network is partitioned and similar traffic regions are identified according to the physical invariants and MFD characteristics. In this way, the network-scale link-wise traffic flow pattern transfer between similar regions can be initiated under the assumption that regions with similar aggregated traffic flow patterns are more likely to share comparable link-wise traffic flow features. In the second stage, we propose our knowledge transfer architecture Deep Tensor Adaptation Network (DTAN) to bridge traffic flow knowledge in source and target regions via the parallel Siamese network structure, and further reduce domain discrepancy by imposing two distribution adaptation regularizations. A real-world traffic dataset on the urban expressway network of Beijing is used for numerical tests. The experiment results show that the proposed framework can leverage the trade-off between specific regression task performance in a single region and generalized domain adaptation capacity across multiple regions. The data insufficiency, dataset shift, and heavy computational cost problems are alleviated by improving model transferability. Finally, extensive empirical analysis is carried out to explore traffic flow pattern transferability and its relation to network traffic properties.
... The pioneering work by Gazis et al. (1959) first highlighted that the fundamental diagram (e.g., Greenshields' speed-density relationship) can be linked to the microscopic carfollowing models ( Greenshields et al., 1935 ;Gazis et al., 1961 ). The significant study by Daganzo (2006) proved that, by assuming a triangular flow-density diagram, vehicle trajectories constructed from a simplified kinematic wave model are equivalent to those generated by Newell's simple linear car-following model ( Newell, 2002 ) and two types of cellular automata models within a certain approximation range ( Nagel, 1996 ). A recent effort along this line includes an s-shaped three-parameter (S3) speed-density function by Cheng et al. (2021) with a macro-to-micro consistent car-following model. ...
Although the macroscopic volume-delay function (VDF) has been widely used in static traffic assignment for transportation planning, the planning community has long recognized its deficiencies as a static function in capturing traffic flow dynamics and queue evolution process. In the existing literature, many queueing-based and simulation-based dynamic traffic assignment (DTA) models involving various traffic flow parameters have been proposed to capture traffic system dynamics on different spatial scales; however, how to calibrate these DTA models could still be a challenging task in its own right, especially for real-world congested networks with complex traffic dynamics. By extending the fluid-based polynomial arrival queue (PAQ) model with quadratic inflow rates proposed by Newell (1982) and cubic inflow rates by Cheng et al. (2022), this paper attempts to propose a cross-resolution Queueing-based Volume-Delay Function (QVDF) to explicitly establish a coherent connection between (a) the macroscopic average travel delay performance function in a long-term planning horizon and (b) the mesoscopic dynamic queuing model during a single oversaturated period. By introducing two types of elasticity functional forms, this paper develops a relationship from the macroscopic inflow demand-to-capacity (D/C) ratio to the congestion duration of a bottleneck, from the congestion duration to the magnitude of speed reduction. The QVDF can be directly utilized to provide closed-form expressions for both average travel delay performance and the time-dependent speed profiles. The proposed cross-resolution QVDF provides a numerically reliable and theoretically rigorous performance function to characterize oversaturated bottlenecks at both macroscopic and mesoscopic scales.
... The triangular FD introduces distinct properties to the CTM model. Daganzo (2006) showed that CTM is equivalent to some elementary cellular automata models, and also to the simplified car-following model by Newell (2002) in which vehicles trajectories in car-following scenarios are shifted in time and space. This is because, in a triangular FD, characteristic speeds are constant in both of the free-flow and congested regimes, and are thereby insensitive to density variations ( = 0 ⁄ ). ...
Congestions and rear-end crashes are two undesirable phenomena of freeway traffic flows, which are interrelated and highly affected by human psychological factors. Congestions on freeways increase rear-end crash risk, and rear-end crashes can initiate or aggravate congestions. Since congestion are everyday problems, and crashes are rare events, congestion management and crash risk prevention strategies are often implemented through separate research directions. As a result, traffic control studies focusing on increasing efficiency may increase the risk of rear-end crashes, whereas those focusing on improving traffic safety may not necessarily be desirable from efficiency perspective. Both freeway traffic flow and safety management will be more challenging in the era of connected driving. In connected environments, the role of human psychological factors on the traffic flow dynamics and traffic safety will be much more pronounced.
The motivation behind this Ph.D. research is to pave the way for traffic management scenarios that result in more efficient and safer freeway traffic in the era of connected driving. As such, the research aim is to develop a robust understanding of freeway traffic flow dynamics and their safety implications with respect to human psychological factors. This research selects the continuum framework to understand traffic flow dynamics and proactive safety assessment framework to understand the safety implications of traffic flow dynamics.
A comprehensive and critical literature review is conducted to understand the state-of-the-art of continuum traffic flow models. The review effort aims to obtain a robust knowledge about existing discussions and debates over continuum models’ analytical properties and real-world performances. A major part of the review explores the research gaps in continuum models for the era of connected and automated vehicles (CAVs). It is found that none of the existing continuum models can describe the role of complex human psychological factors (e.g., risk perception) on traffic flow dynamics. As well, the critical review revisits the properties and issues with continuum models for both conventional and CAV traffic flows. The review aims to take a close look on a wide range of theoretical, practical, and behavioral issues that must be kept in mind when developing new continuum models.
Next, this research conducts a comprehensive benchmarking study on single-pipe continuum models by using traffic data from the German A5 autobahn. Model families are examined based on the review effort, and suitable representative models are selected within the families. A set of benchmarking criteria is designed, ranging from the operational measures (e.g., delay and travel time) as well as the complex traffic phenomena (e.g., scattering, oscillations, capacity drop, and hysteresis phenomena). Suitable real-world traffic scenarios are carefully selected concerning the benchmarking criteria, and the selected models are comprehensively assessed for real-world scenarios.
Based on the understanding obtained from the review and benchmarking efforts, a novel behavioural continuum model (Non-Equilibrium Traffic model based on Risk Allostasis Theory, i.e., NET-RAT) is developed. NET-RAT fills a huge gap in the literature, that is, lack of behavioural continuum models with respect to a well-established human factor theory. Perceived risk and preferred response time are selected as the major human psychological factors of the conventional and connected environment. Perceived risk is defined in terms of the proportion of stopping distance which is a car-following safety measure. The selected human psychological factors are incorporated into the full velocity difference car-following model (FVDM). A novel continuum model is derived from the extended FVDM, and the interplay between preferred response time and perceived risk is formulated within the well-established risk allostasis theory. The analytical properties of NET-RAT are explored in detail, and its relationship with the notable existing continuum models are discussed. NET-RAT’s performance against real-world traffic is investigated comprehensively and compared with the models studied in the benchmarking effort.
The results demonstrates that NET-RAT outperforms the existing continuum models in the benchmarking effort regarding some aspects of real-world traffic, e.g., travel time estimation and qualitative propagation of jam front. It is shown that drivers that have higher perception of risk can aggravate congestions by increasing shockwave speeds. Such drivers, however, can stabilize traffic flow regardless of traffic conditions due to responding quickly to initial perturbations. On the other hand, those with lower perceptions of risk can reduce congestions, but can initiate traffic instabilities in the intermediately congested states due to increased response time. This research also explores the implications of NET-RAT for the environments, where drivers are provided with information about traffic and their risk perception may be affected.
Next, this dissertation proposes a hybrid methodological framework combining probabilistic and machine learning models to develop the relationships between safety and macroscopic state variables within a flexible conflict-based safety assessment framework. Time spent in conflict is introduced as the total time spent by all vehicles in rear-end conflicts, where the conflict instances are defined based on the proportion of stopping distance and a flexible threshold. The proposed hybrid framework can assess the time spent in conflict for all underlying car-following interactions using only macroscopic state variables, and thus, overcoming the need for trajectory data. Besides, it provides an endogenous safety dimension to the fundamental relations of freeway traffic flows that can be utilized to balance freeway traffic flow efficiency and safety. For instance, control studies can utilize the proposed framework to minimize total travel time while also minimizing total time spent in conflict for crash-prone situations such as shockwaves and traffic oscillations.
Finally, the proposed safety assessment model is utilized in conjunction with NET-RAT to investigate the safety implications of various driving behaviours in relation to risk perception. It is shown that drivers with low perception of risk spend more time in conflict rear-end conflicts during critical situations from safety perspective, e.g., when entering shockwaves and when undergoing stop-and-go waves.
The proposed methodology in this Ph.D. is a pathway for unifying freeway traffic flow modelling and rear-end crash prevention in the era of mixed traffic, when human factors play significant role on both congestion and rear-end crashes.
... In accordance with the literature, the approach was also introduced in order to deal with the lane changing problem in which microscopic modelling is suitable for realistic acceleration reproduction but cannot be applied for lane changes [33] therefore further investigations may found in Daganzo [34] and Laval and Daganzo [35]. They proposed a model based on a Kinematic Wave (KW; [36] model for traffic stream simulation and a micro model for the slower vehicles' representation. ...
Background
This paper compares a hybrid traffic flow model with benchmark macroscopic and microscopic models. The proposed hybrid traffic flow model may be applied considering a mixed traffic flow and is based on the combination of the macroscopic cell transmission model and the microscopic cellular automata.
Modelled variables
The hybrid model is compared against three microscopic models, namely the Krauß model, the intelligent driver model and the cellular automata, and against two macroscopic models, the Cell Transmission Model and the Cell Transmission Model with dispersion, respectively. To this end, three main applications were considered: (i) a link with a signalised junction at the end, (ii) a signalised artery, and (iii) a grid network with signalised junctions.
Results
The numerical simulations show that the model provides acceptable results. Especially in terms of travel times, it has similar behaviour to the microscopic model. By contrast, it produces lower values of queue propagation than microscopic models (intrinsically dominated by stochastic phenomena), which are closer to the values shown by the enhanced macroscopic cell transmission model and the cell transmission model with dispersion. The validation of the model regards the analysis of the wave propagation at the boundary region.
... The triangular FD introduces distinct properties to the CTM model. Daganzo (2006) showed that CTM is equivalent to some elementary cellular automata models and also to the simplified car-following model by Newell (2002) in which vehicles trajectories in car-following scenarios are shifted in time and space. This is because, in a triangular FD, characteristic speeds are linearly degenerate (LD) in both free-flow and congested regimes, meaning that they are constant and insensitive to density variations ( = 0 ⁄ ). ...
This paper provides a comprehensive review of continuum traffic flow models. A comprehensive review of models for conventional traffic is presented that classifies the models into various families regarding their derivation bases. Previous discussions and debates over the performance of models developed for conventional traffic are covered in detail, and wherever applicable, new insights are provided on the properties and interpretations of the existing models. A review of the recent attempts at incorporating connected and automated vehicles (CAVs) traffic flow into the continuum framework is also conducted. The paper also analyses the strengths, limitations, and properties of the existing model families for CAV traffic flow. Research gaps and the issues inherent to CAVs are highlighted, and future directions in the era of connected and automated vehicles are discussed.
... In addition, based on NS model, Ge et al. (2005) introduced the variable safety distance to analyze the mixed traffic flow, the results show that the implementation of fast and slow traffic diversion can effectively expand the traffic flow and reduce the generation of congestion. Within a tolerance comparable with a single jam spacing, Daganzo (2006) proved that the vehicle trajectories predicted by a simple linear car-following model, the kinematic wave model with a triangular fundamental diagram, and two cellular automata models CA (carfollowing model) and CA (kinematic wave model) match everywhere. TT model was firstly proposed by Takayasu and Takayasu (1993), which improved NS model by considering density of vehicles, revealed the phase conversion between the blocking phase and the non-blocking phase. ...
The vulnerability of road network for dangerous goods transportation (RNDGT) under cascading failure considering intentional attack is analyzed. We introduce the time characteristics of load distribution and node recovery ability into previous cascading failure model, subdivide the state of failed node into normal state, partial failure state and complete failure state. Six traffic load distribution strategies including Average Distribution (AD), Betweenness Distribution (BD), Capacity Distribution (CD), Degree Distribution (DD), Tightness Distribution (TD) and Surplus Load Distribution (SLD) are selected to study the load re-distribution of failed nodes. In addition, three kinds of intentional node attack strategies including Degree Attack (DA), Betweenness Attack (BA) and Tightness Attack (TA) are selected to study the impact on the vulnerability. By referring the application of cellular automata applied in epidemic spreading field, we establish a new cascading failure model of RNDGT. The improved maximum connectivity and node failure rate based on node degree are applied to analyze the vulnerability. A case study is conducted by using the RNDGT of Dalian as the background. The previous Motter-Lai model (M-L) is applied as the comparison approach. TA strategy has the least impact on increasing network vulnerability, SLD strategy is the best to reduce network vulnerability.
... It has already been shown that variants of the NaSch model deliver precisely the same results for vehicle trajectories as kinematic wave models and linear vehicle-following models [33]. ...
Dealing with traffic congestion is one of the most pressing challenges for cities. Transport authorities have implemented several strategies to reduce traffic jams with varying degrees of success. The use of reversible lanes is a common approach to improve traffic congestion during rush hours. A reversible lane can change its direction during a time interval to the more congested direction. This strategy can improve traffic congestion in specific scenarios. Most reversible lanes in urban roads are fixed in time and number; however, traffic patterns in cities are highly variable and unpredictable due to this phenomenon’s complex nature. Therefore, reversible lanes may not improve traffic flow under certain circumstances; moreover, they could worsen it because of traffic fluctuations. In this paper, we use cellular automata to model adaptive reversible lanes(aka dynamic reversible lanes). Adaptive reversible lanes can change their direction using real-time information to respond to traffic demand fluctuations. Using real traffic data, our model shows that adaptive reversible lanes can improve traffic flow up to 40% compared to conventional reversible lanes. Our results show that there are significant fluctuations in traffic flow even during rush hours, and thus cities would benefit from implementing adaptive reversible lanes.
... Microscopic models follow the dynamics of each vehicle and are described by ordinary differential equations (ODEs), see e. g. [2,16,19,33,34,38]. Macroscopic models, based on fluid dynamics, consider aggregated quantities such as the density of vehicles and are governed by partial differential equations (PDEs), see e. g. [3,7,12,17,32,41,46]. Kinetic models [24,28,33,36,39,40] are between the previous two classes since they can be derived by microscopic models while macroscopic models can be derived by kinetic descriptions. ...
... Congestion observed in the transportation systems aggregates various amounts of uncertainty at different levels, making it challenging to achieve a clear understanding regarding empirically observed traffic instabilities. In the literature, lane changing has been frequently indicated as an important factor of oscillations under high-density flows (Ahn and Cassidy, 2007;Daganzo, 2006;Laval and Leclercq, 2010;Zheng, 2014;Zheng et al., 2013). Moreover, oscillations appear even on one-lane roads, when a small fluctuation caused by a perturbation, grows larger and then the homogeneous movement with constant inter-vehicle spacing between following and leading vehicles cannot be maintained, leading eventually to a jam (Stern et al., 2018;Sugiyama et al., 2008). ...
Road traffic congestion is the result of various phenomena often of random nature and not directly observable with empirical experiments. This makes it difficult to clearly understand the empirically observed traffic instabilities. The vehicles’ acceleration/deceleration patterns are known to trigger instabilities in the traffic flow under congestion. It has been empirically observed that free-flow pockets or voids may arise when there is a difference in the speeds and the spacing between the follower and the leader increases. During these moments, the trajectory is dictated mainly by the characteristics of the vehicle and the behaviour of the driver and not by the interactions with the leader. Voids have been identified as triggers for instabilities in both macro and micro level, which influence traffic externalities such as fuel consumption and emissions. In the literature, such behaviour is usually reproduced by injecting noise to the results of car-following models in order to create fluctuations in the instantaneous vehicles’ acceleration.
This paper proposes a novel car-following approach that takes as input the driver and the vehicle characteristics and explicitly reproduces the impact of the vehicle dynamics and the driver’s behaviour by adopting the Microsimulation Free-flow aCceleration (MFC) model. The congested part of the model corresponds to the Lagrangian discretization of the LWR model and guarantees a full consistency at the macroscopic scale with congested waves propagating accordingly to the first-order traffic flow theory.
By introducing naturalistic variation in the driving styles (timid and aggressive drivers) and the vehicle characteristics (specification from different vehicle models), the proposed model can reproduce realistic traffic flow oscillations, similar to those observed empirically. An advantage of the proposed model is that it does not require the injection of any noise in the instantaneous vehicle accelerations.
The proposed methodology has been tested by studying a) the traffic flow oscillations produced by the model in a one-lane road uphill simulation scenario, b) the ability of the model to reproduce car-following instabilities observed in three car-following trajectory datasets and c) the ability of the model to produce realistic fuel consumption estimates. The results prove the robustness of the proposed model and the ability to describe traffic flow oscillations as a consequence of the combination of driving style and vehicle’s technical specifications.
... This is the simplest model able to predict the main features of traffic flow, and it has become the standard analysis tool and traffic flow theory. It turns out that there are several models in the literature that are equivalent to the kinematic wave model: the celebrated Newell's car-following model [63] is the car-following version, which can be formulated as a cellular automaton (CA) model [64]. But [65] showed that the shape of the triangular fundamental diagram is irrelevant thanks to a symmetry in the kinematic wave model, allowing us to use an isosceles fundamental diagram, which has many useful properties in practice; see [65] for the details. ...
This paper highlights several properties of large urban networks that can have an impact on machine learning methods applied to traffic signal control. In particular, we show that the average network flow tends to be independent of the signal control policy as density increases. This property, which so far has remained under the radar, implies that deep reinforcement learning (DRL) methods becomes ineffective when trained under congested conditions, and might explain DRL's limited success for traffic signal control. Our results apply to all possible grid networks thanks to a parametrization based on two network parameters: the ratio of the expected distance between consecutive traffic lights to the expected green time, and the turning probability at intersections. Networks with different parameters exhibit very different responses to traffic signal control. Notably, we found that no control (i.e. random policy) can be an effective control strategy for a surprisingly large family of networks. The impact of the turning probability turned out to be very significant both for baseline and for DRL policies. It also explains the loss of symmetry observed for these policies, which is not captured by existing theories that rely on corridor approximations without turns. Our findings also suggest that supervised learning methods have enormous potential as they require very little examples to produce excellent policies.
We propose a simple and novel method to actively resolve the temporary congestion caused by a freeway mainline vehicle’s facilitating maneuver of creating a gap for on-ramp merging vehicles, under under-critical mainline conditions. We first present an analytical finding derived from the kinematic wave model with a triangular fundamental diagram. That is, when the prevailing mainline traffic is under-critical, the total delay of the mainline vehicles affected by the gap creation does not depend on the choice of speed by which the gap is created, namely the facilitating speed. This is because the recovery wave speed is constantly equal to the characteristic wave speed of the congestion regime,
. In light of this observation, to improve mainline traffic efficiency, we propose the strategy of active congestion resolving, which features a recovery wave of a speed higher than
. Characterizing CAVs’ car-following behaviors by Newell’s simplified car-following model, we analytically show that such a recovery wave can be achieved by properly modifying the values of the affected CAVs’ car-following characteristic parameters and adopting the modified values in a proper way during congestion resolving. Then, in the presence of this active congestion resolving strategy, an optimization program is formulated to seek an optimal facilitating speed that can balance between traffic efficiency and speed variation. Simulation experiments are conducted to validate the effectiveness of the proposed method.
The analogy between the theory of phase transitions in simple fluids and vehicular traffic flow has long been suspected, promising a new level of understanding of urban congestion by standing on one of the firmer foundations in physics. The obstacle has been the interpretation of the thermal energy of the particle system, which remains unknown. This paper proposes the flow of cars through the network as a viable interpretation, where the fundamental diagram of traffic flow would be analogous to the coexistence curve in gas-liquid phase transitions. Through the power-law form of the coexistence curve, it was possible to formalize that the resulting network traffic model belongs to the Kardar–Parisi–Zhang universality class. The scaling relationships arising in this universality class were found to be consistent with West’s scaling theory for cities. It was shown that congestion costs (delays + fuel consumption) scaled superlinearly with city populations, possibly, and worryingly, more so than predicted by West’s theory. Implications for sustainability and resiliency are discussed.
Vehicular traffic flow is a fascinating real-world system in which interactions at the vehicle scale, occurring on the scale of seconds, shape urban-scale flow patterns that develop over the scale of hours. A fundamental goal of traffic modeling is the development of mathematical equations that reproduce the fundamental laws that govern the flow behavior, without requiring detailed information about the human drivers or their vehicles. Starting from such models, it is shown how an additional scale can arise that lies between the microscopic vehicle scale and the macroscopic urban scale, namely: the mesoscopic scale of traffic waves. In certain flow regimes, uniform traffic flow is dynamically unstable, and small perturbations grow into traveling waves. Suitable models can quantify how this “phantom traffic jam” phenomenon lead to significant energy waste and reduction in flow efficiency. Moreover, via models and simulations, it is demonstrated how the introduction of a small number of automated vehicles can serve to recover the efficient uniform flow state.
Researchers widely use the two-fluid model (TFM) to evaluate the performance of
urban networks, primarily because of its ability to compare various links in a network at similar traffic-density levels. However, the TFM is deterministic and does not capture the stochastic relation between speed and density. The present study develops a modified two-fluid model (MTFM). The variance function or the distribution of speed or travel time for a given density is incorporated using a percentile-based indicator, travel time uncertainty (TTU). The percentile-based indicators for speed distribution are more robust than the variance or other moment-based indicators. The effect of TTU is incorporated using two parameters. The applicability of the proposed MTFM is demonstrated using empirical data collected at the corridor and network levels. The TFM and MTFM were calibrated by formulating a nonlinear optimization problem. Based on the investigation using the corridor and network-level data, it was concluded that the MTFM showed better performance than the existing model. Therefore, the MTFM can better capture the heterogeneity of speed-density relation than the TFM
This paper highlights several properties of large urban networks that can have an impact on machine learning methods applied to traffic signal control. In particular, we note that the average network flow tends to be independent of the signal control policy as density increases past the critical density. We show that this property, which so far has remained under the radar, implies that no control (i.e. a random policy) can be an effective control strategy for a surprisingly large family of networks, especially for networks with short blocks. We also show that this property makes deep reinforcement learning (DRL) methods ineffective when trained under congested conditions, independently of the particular algorithm used. Accordingly, in contrast to the conventional wisdom around learning-based methods promoting the exploration of all states, we find that for urban networks it is advisable to discard any congested data when training, and that doing so will improve performance under all traffic conditions. Our results apply to all possible grid networks thanks to a parametrization introduced here. The impact of the turning probability was found to be very significant, in particular to explain the loss of symmetry observed in the macroscopic fundamental diagram of the networks, which is not captured by existing theories that rely on corridor approximations without turns. Our findings also suggest that supervised learning methods have enormous potential as they require very little examples to produce excellent policies.
Before a car-following model can be applied in practice, it must first be validated against real data in a process known as calibration. This paper discusses the formulation of calibration as an optimization problem and compares different algorithms for its solution. The optimization consists of an arbitrary car following model, posed as either an ordinary or delay differential equation, being calibrated to an arbitrary source of trajectory data that may include lane changes. Typically, the calibration problem is solved using gradient free optimization. In this work, the gradient of the optimization problem is derived analytically using the adjoint method. The computational cost of the adjoint method does not scale with the number of model parameters, which makes it more efficient than evaluating the gradient numerically using finite differences. Numerical results are presented that show that quasi-Newton algorithms using the adjoint method are significantly faster than a genetic algorithm and also achieve slightly better accuracy of the calibrated model.
Network macroscopic fundamental diagrams (MFDs) have recently been shown to exist in real-world urban traffic networks. The existence of an MFD facilitates the modeling of urban traffic network dynamics at a regional level, which can be used to identify and refine large-scale network-wide control strategies. To be useful, MFD-based modeling frameworks require an estimate of the functional form of a network’s MFD. Analytical methods have been proposed to estimate a network’s MFD by abstracting the network as a single ring-road or corridor and modeling the flow–density relationship on that simplified element. However, these existing methods cannot account for the impact of turning traffic, as only a single corridor is considered. This paper proposes a method to estimate a network’s MFD when vehicles are allowed to turn into or out of a corridor. A two-ring abstraction is first used to analyze how turning will affect vehicle travel in a more general network, and then the model is further approximated using a single ring-road or corridor. This approximation is useful as it facilitates the application of existing variational theory-based methods (the stochastic method of cuts) to estimate the flow–density relationship on the corridor, while accounting for the stochastic nature of turning. Results of the approximation compared with a more realistic simulation that includes features that cannot be captured using variational theory—such as internal origins and destinations—suggest that this approximation works to estimate a network’s MFD when turning traffic is present.
We introduce a stochastic discrete automaton model to freeway traffic. Monte-Carlo simulations of the model show a transition from laminar traffic flow to start-stop-waves with increasing vehicle density, as is observed in real freeway traffic. For special cases analytical results can be obtained.
A multi-lane traffic flow model realistically captures the disruptive effects of lane- changing vehicles by recognizing their limited ability to accelerate. While they accelerate, these vehicles create voids in the traffic stream that affect its character. Bounded acceleration explains two features of freeway traffic streams: the capacity drop of freeway bottlenecks, and the quantitative relation between the discharge rate of moving bottlenecks and bottleneck speed. The model com- bines a multilane kinematic wave module for the traffic stream, with a detailed constrained-motion model to describe the lane-changing maneuvers, and a behavioral demand model to trigger them. The behavioral demand model has only one parameter. It was held constant in all experiments.
In this paper and in part II, we give the theory of a distinctive type of wave motion, which arises in any one-dimensional flow problem when there is an approximate functional relation at each point between the flow q (quantity passing a given point in unit time) and concentration k (quantity per unit distance). The wave property then follows directly from the equation of continuity satisfied by q and k. In view of this, these waves are described as 'kinematic', as distinct from the classical wave motions, which depend also on Newton's second law of motion and are therefore called 'dynamic'. Kinematic waves travel with the velocity partial q/partial k, and the flow q remains constant on each kinematic wave. Since the velocity of propagation of each wave depends upon the value of q carried by it, successive waves may coalesce to form 'kinematic shock waves'. From the point of view of kinematic wave theory, there is a discontinuous increase in q at a shock, but in reality a shock wave is a relatively narrow region in which (owing to the rapid increase of q) terms neglected by the flow-concentration relation become important. The general properties of kinematic waves and shock waves are discussed in detail in section 1. One example included in section 1 is the interpretation of the group-velocity phenomenon in a dispersive medium as a particular case of the kinematic wave phenomenon. The remainder of part I is devoted to a detailed treatment of flood movement in long rivers, a problem in which kinematic waves play the leading role although dynamic waves (in this case, the long gravity waves) also appear. First (section 2), we consider the variety of factors which can influence the approximate flow-concentration relation, and survey the various formulae which have been used in attempts to describe it. Then follows a more mathematical section (section 3) in which the role of the dynamic waves is clarified. From the full equations of motion for an idealized problem it is shown that at the 'Froude numbers' appropriate to flood waves, the dynamic waves are rapidly attenuated and the main disturbance is carried downstream by the kinematic waves; some account is then given of the behaviour of the flow at higher Froude numbers. Also in section 3, the full equations of motion are used to investigate the structure of the kinematic shock; for this problem, the shock is the 'monoclinal flood wave' which is well known in the literature of this subject. The final sections (section section 4 and 5) contain the application of the theory of kinematic waves to the determination of flood movement. In section 4 it is shown how the waves (including shock waves) travelling downstream from an observation point may be deduced from a knowledge of the variation with time of the flow at the observation point; this section then concludes with a brief account of the effect on the waves of tributaries and run-off. In section 5, the modifications (similar to diffusion effects) which arise due to the slight dependence of the flow-concentration curve on the rate of change of flow or concentration, are described and methods for their inclusion in the theory are given.
The manner in which vehicles follow each other on a highway without passing and the propagation of disturbances down a line of vehicles has been investigated further. Criteria are derived for both local and asymptotic stability in a chain of vehicles. The influence of next nearest neighbors as well as a statistical theory of stability is discussed. “Acceleration noise” is proposed as a parameter that might be employed to characterize the driver-car-road complex under various conditions. Some preliminary experimental measurements of acceleration noise are discussed.
We investigate critical properties of a class of number-conserving cellular automata (CAs) which can be interpreted as deterministic models of traffic flow with anticipatory driving. These rules are among the only known CA rules for which the shape of the fundamental diagram has been rigorously derived. In addition, their fundamental diagrams contain nonlinear segments, as opposed to the majority of number-conserving CAs which exhibit piecewise-linear diagrams. We found that the nature of singularities in the fundamental diagram of these rules is the same as for rules with piecewise-linear diagrams. The current converges toward its equilibrium value as t-1/2, and the critical exponent bgr is equal to unity. This supports the conjecture of universal behaviour at singularities in number-conserving rules. We discuss properties of phase transitions occurring at singularities as well as properties of the intermediate phase.
This paper uses the method of kinematic waves, developed in part I, but may be read independently. A functional relationship between flow and concentration for traffic on crowded arterial roads has been postulated for some time, and has experimental backing (§2). From this a theory of the propagation of changes in traffic distribution along these roads may be deduced (§§2, 3). The theory is applied (§4) to the problem of estimating how a ‘hump’, or region of increased concentration, will move along a crowded main road. It is suggested that it will move slightly slower than the mean vehicle speed, and that vehicles passing through it will have to reduce speed rather suddenly (at a ‘shock wave’) on entering it, but can increase speed again only very gradually as they leave it. The hump gradually spreads out along the road, and the time scale of this process is estimated. The behaviour of such a hump on entering a bottleneck, which is too narrow to admit the increased flow, is studied (§5), and methods are obtained for estimating the extent and duration of the resulting hold-up. The theory is applicable principally to traffic behaviour over a long stretch of road, but the paper concludes (§6) with a discussion of its relevance to problems of flow near junctions, including a discussion of the starting flow at a controlled junction. In the introductory sections 1 and 2, we have included some elementary material on the quantitative study of traffic flow for the benefit of scientific readers unfamiliar with the subject.
It is postulated that lane-changing vehicles create voids in traffic streams, and that these voids reduce flow. This mechanism is described with a model that tracks lane changers precisely, as particles endowed with realistic mechanical properties. The model has four easy-to-measure parameters and reproduces without re-calibration two bottleneck phenomena previously thought to be unrelated: (i) the drop in the discharge rate of freeway bottlenecks when congestion begins, and (ii) the relation between the speed of a moving bottleneck and its capacity.
This paper proves that a class of first order partial differential equations, which include scalar conservation laws with concave (or convex) equations of state as special cases, can be formulated as calculus of variations problems. Every well-posed problem of this type, no matter how complicated, even in multi-dimensions, is reduced to the determination of a tree of shortest paths in a relevant region of space-time where "cost" is predefined. Thus, problems of this type can be practically solved with fast network algorithms. The new formulation automatically identifies the unique, single-valued function, which is stable to perturbations in the input data. Therefore, an auxiliary "entropy" condition does not have to be introduced for the conservation law. In traffic flow applications, where one-dimensional conservation laws are relevant, constraints to flow such as sets of moving bottlenecks can now be modeled as shortcuts in space-time. These shortcuts become an integral part of the network and affect the nature of the solution but not the complexity of the solution process. Boundary conditions can be naturally handled in the new formulation as constraints/shortcuts of this type.
This paper proves that kinematic wave (KW) problems with concave (or convex) equations of state can be formulated as calculus of variations problems. Every well-posed problem of this type, no matter how complicated, is reduced to the determination of a shortest tree in a relevant region of spacetime where cost is predefined. A duality between KW theory and /least cost networks is thus unveiled. In the new formulation space-time curves that constrain flow, such as sets of moving bottlenecks, become space-time shortcuts. These shortcuts become part of the network and affect the nature of the solution but not the speed with which it can be obtained. Complex boundary conditions are naturally handled in the new formulation as constraints/shortcuts of this type.
This paper shows how particle hopping models fit into the context of traffic flow theory, that is, it shows connections between fluid-dynamical traffic flow models, which derive from the Navier-Stokes-equation, and particle hopping models. In some cases, these connections are exact and have long been established, but have never been viewed in the context of traffic theory. In other cases, critical behavior of traffic jam clusters can be compared to instabilities in the partial differential equations. Finally, it is shown how all this leads to a consistent picture of traffic jam dynamics.---In consequence, this paper starts building a foundation of a comprehensive dynamic traffic theory, where strengths and weaknesses of different models (fluid-dynamical, car- following, particle hopping) can be compared, and thus allowing to systematically choose the appropriate model for a given question.
This paper proves that the solution of every well-posed kinematic wave (KW) traffic problem with a concave flow-density relation is a set of least-cost (shortest) paths in space-time with a special metric. The equi-cost contours are the vehicle trajectories. If the flow-density relation is strictly concave the set of shortest paths is unique and matches the set of waves. Shocks, if they arise, are curves in the solution region where the shortest paths end. The new formulation extends the range of applications of kinematic wave theory and simplifies it considerably. For example, moving restrictions such as slow buses, which cannot be treated easily with existing methods, can be modeled as shortcuts in space-time. These shortcuts affect the nature of the solution but not the complexity of the solution process. Hybrid models of traffic flow where discrete vehicles (e.g., trucks) interact with a continuum KW stream can now be easily implemented.
A very simple “car-following” rule is proposed wherein, if an nth vehicle is following an (n−1)th vehicle on a homogeneous highway, the time-space trajectory of the nth vehicle is essentially the same as the (n−1)th vehicle except for a translation in space and in time. It seems that such a rule is at least as accurate as any of the more elaborate rules of car-following that have been proposed over the last 50 years or so. Actually, the proposed model could be interpreted as a special case of existing models but with fewer parameters and a different logic. At least this should form a reasonable starting point for investigating other phenomena.
For a freeway having various entrance and exit ramps, the methods described in Part I are used to relate the cumulative flow curve at any junction to the net cumulative entrance flow at this junction, and the cumulative flow curves for the freeway at the next upstream junction and/ or the next downstream junction. If the type of flow-density relations typical of freeway traffic are idealized by a triangular shaped curve with only two wave speeds, one for free-flowing traffic (positive) and the other for congested traffic (negative), then the relationship is easy to evaluate. The cumulative flow curve at the junction is simply the lower envelope of a translation of the cumulative curve from upstream and a different translation of the cumulative curve from downstream. This relationship is the basic building block for a freeway flow prediction model described in Part III.
In the theory of "kinematic waves," as described originally by Lighthill and Whitham in 1955, the evaluation of the shock path is typically rather tedious. Instead of using this theory to evaluate flows or densities, one can use it to evaluate the cumulative flow A(x, t) past any point x by time t. It is shown here how a formal solution for A(x, t) can be evaluated directly from boundary or initial conditions without evaluation at intermediate times and positions. If there are shocks, however, this solution will be multiple-valued. The correct solution, which is the lower envelope of all such formal solutions, will automatically have discontinuities in slope describing the passage of a shock. To evaluate A(x, t) at any particular location x, it is not necessary to follow the actual path of the shock. The solution can be evaluated directly in terms of the boundary data by either graphical or numerical techniques.
On the multi-lane equivalence of certain models of traffic flow
- J Laval
- C Daganzo
J A Laval and C F Daganzo. On the multi-lane equivalence of certain models of traffic flow.
Technical Report (in preparation), Inst. Trans. Studies, Univ. of California, Berkeley, CA,
2004.
Multi-lane hybrid flow traffic model: quantifying the impacts of lane-changing maneuvers on traffic flow. Institute of Transportation Studies Working Paper UCB-ITS-WP-2004-01
- J A Laval
- C F Daganzo
A simplified theory of kinematic waves in highway traffic: (i) general theory; (ii) queuing at freeway bottlenecks; (iii) multi-dimensional flows
- Newell