Continuous formulations and analytical properties of the link transmission model

The link transmission model (LTM) has great potential for simulating traffic flow in large-scale networks since it is much more efficient and accurate than the Cell Transmission Model (CTM). However, there lack general continuous formulations of LTM, and there has been no systematic study on its analytical properties such as stationary states and stability of network traffic flow. In this study we attempt to fill the gaps. First we apply the Hopf–Lax formula to derive Newell’s simplified kinematic wave model with given boundary cumulative flows and the triangular fundamental diagram. We then apply the Hopf–Lax formula to define link demand and supply functions, as well as link queue and vacancy functions, and present two continuous formulations of LTM, by incorporating boundary demands and supplies as well as invariant macroscopic junction models. With continuous LTM, we define and solve the stationary states in a road network. We also apply LTM to directly derive a Poincaré map to analyze the stability of stationary states in a diverge-merge network. Finally we present an example to show that LTM is not well-defined with non-invariant junction models. We can see that Newell’s model and continuous LTM complement each other and provide an alternative formulation of the network kinematic wave theory. This study paves the way for further extensions, analyses, and applications of LTM in the future.     

Dynamic network loading: A stochastic differentiable model that derives link state distributions

We present a dynamic network loading model that yields queue length distributions, accounts for spillbacks, and maintains a differentiable mapping from the dynamic demand on the dynamic queue lengths. The model also captures the spatial correlation of all queues adjacent to a node, and derives their joint distribution. The approach builds upon an existing stationary queueing network model that is based on finite capacity queueing theory. The original model is specified in terms of a set of differentiable equations, which in the new model are carried over to a set of equally smooth difference equations. The physical correctness of the new model is experimentally confirmed in several congestion regimes. A comparison with results predicted by the kinematic wave model (KWM) shows that the new model correctly represents the dynamic build-up, spillback and dissipation of queues. It goes beyond the KWM in that it captures queue lengths and spillbacks probabilistically, which allows for a richer analysis than the deterministic predictions of the KWM. The new model also generates a plausible fundamental diagram, which demonstrates that it captures well the stationary flow/density relationships in both congested and uncongested conditions.     

Dynamic traffic metering in urban street networks: Formulation and solution algorithm

Traffic metering offers great potential to reduce congestion and enhance network performance in oversaturated urban street networks. This paper presents an optimization program for dynamic traffic metering in urban street networks based on the Cell Transmission Model (CTM). We have formulated the problem as a Mixed-Integer Linear Program (MILP) capable of metering traffic at network gates with given signal timing parameters at signalized intersections. Due to the complexities of the MILP model, we have developed a novel and efficient solution approach that solves the problem by converting the MILP to a linear program and several CTM simulation runs. The solution algorithm is applied to two case studies under different conditions. The proposed solution technique finds solutions that have a maximum gap of 1% of the true optimal solution and guarantee the maximum throughput by keeping some vehicles at network gates and only allowing enough vehicles to enter the network to prevent gridlocks. This is confirmed by comparing the case studies with and without traffic metering. The results in an adapted real-world case study network show that traffic metering can increase network throughput by 4.9–38.9% and enhance network performance.     

Dynamic user equilibrium in public transport networks with passenger congestion and hyperpaths

This paper presents a dynamic user equilibrium for bus networks where recurrent overcrowding results in queues at stops. The route-choice model embedded in the dynamic assignment explicitly considers common lines and strategies with alternative routes. As such, the shortest hyperpath problem is extended to a dynamic scenario with capacity constraints where the diversion probabilities depend on the time at which the stop is reached and on the expected congestion level at that time. In order to reproduce congestion for all the lines sharing a stop, the Bottleneck Queue Model with time-varying exit capacity, introduced in Meschini et al. (2007), is extended. The above is applied to separate queues for each line in order to satisfy the First-In-First-Out principle within every attractive set, while allowing overtaking among passengers with different attractive sets but queuing single file. The application of the proposed model to a small example network clearly reproduces the formation and dispersion of passenger queues due to capacity constraints and thus motivates the implementation of the methodology on a real-size network case as the next step for future research.     

Spillback congestion in dynamic traffic assignment: A macroscopic flow model with time-varying bottlenecks

In this paper, we propose a new model for the within-day Dynamic Traffic Assignment (DTA) on road networks where the simulation of queue spillovers is explicitly addressed, and a user equilibrium is expressed as a fixed-point problem in terms of arc flow temporal profiles, i.e., in the infinite dimension space of time’s functions. The model integrates spillback congestion into an existing formulation of the DTA based on continuous-time variables and implicit path enumeration, which is capable of explicitly representing the formation and dispersion of vehicle queues on road links, but allows them to exceed the arc length. The propagation of congestion among adjacent arcs will be achieved through the introduction of time-varying exit and entry capacities that limit the inflow on downstream arcs in such a way that their storage capacities are never exceeded. Determining the temporal profile of these capacity constraints requires solving a system of spatially non-separable macroscopic flow models on the supply side of the DTA based on the theory of kinematic waves, which describe the dynamic of the spillback phenomenon and yield consistent network performances for given arc flows. We also devise a numerical solution algorithm of the proposed continuous-time formulation allowing for “long time intervals” of several minutes, and give an empirical evidence of its convergence. Finally, we carry out a thorough experimentation in order to estimate the relevance of spillback modeling in the context of the DTA, compare the proposed model in terms of effectiveness with the Cell Transmission Model, and assess the efficiency of the proposed algorithm and its applicability to real instances with large networks.     

Guaranteed prediction and estimation of the state of a road network

The paper presents an algorithm for the prediction and estimation of the state of a road network comprising freeways and arterials, described by a Cell Transmission Model (CTM). CTM divides the network into a collection of links. Each link is characterized by its fundamental diagram, which relates link speed to link density. The state of the network is the vector of link densities. The state is observed through measurements of speed and flow on some links. Demand is specified by the volume of vehicles entering the network at some links, and by split ratios according to which vehicles are routed through the network. There is model uncertainty: the parameters of the fundamental diagram are uncertain. There is uncertainty in the demand around the nominal forecast. Lastly, the measurements are uncertain. The uncertainty in each model parameter, demand, and measurement is specified by an interval. Given measurements over a time interval [0, t] and a horizon τ ? 0, the algorithm computes a set of states with the guarantee that the actual state at time (t + τ) will lie in this set, consistent with the given measurements. In standard terminology the algorithm is a state prediction or an estimate accordingly as τ > 0 or =0. The flow exiting a link may be controlled by an open- or closed-loop controller such as a signal or ramp meter. An open-loop controller does not change the algorithm, indeed it may make the system more predictable by tightening density bounds downstream of the controller. In the feedback case, the value of the control depends on the estimated state bounds, and the algorithm is extended to compute the range of possible closed-loop control values. The algorithm is used in a proposed design of a decision support system for the I-80 integrated corridor.     

Quasi-dynamic traffic assignment with residual point queues incorporating a first order node model

Static traffic assignment models are still widely applied for strategic transport planning purposes in spite of the fact that such models produce implausible traffic flows that exceed link capacities and predict incorrect congestion locations. There have been numerous attempts to constrain link flows to capacity. Capacity constrained models with residual queues are often referred to as quasi-dynamic traffic assignment models. After reviewing the literature, we come to the conclusion that an important piece of the puzzle has been missing so far, namely the inclusion of a first order node model. In this paper we propose a novel path-based static traffic assignment model for finding a stochastic user equilibrium in general transportation networks. This model includes a first order (steady-state) node model that yields more realistic turn capacities, which are then used to determine consistent capacity constrained traffic flows, residual point (vertical) queues (upstream bottleneck links), and path travel times consistent with queuing theory. The route choice part of the model is specified as a variational inequality problem, while the network loading part is formulated as a fixed point problem. Both problems are solved using existing techniques to find a solution. We illustrate the model using hypothetical examples, and also demonstrate feasibility on large-scale networks.     

Dynamic traffic assignment of cooperative adaptive cruise control

Advances in connected and automated vehicle technologies have resulted in new vehicle applications, such as cooperative adaptive cruise control (CACC). Microsimulation models have shown significant increases in capacity and stability due to CACC, but most previous work has relied on microsimulation. To study the effects of CACC on larger networks and with user equilibrium route choice, we incorporate CACC into the link transmission model (LTM) for dynamic network loading. First, we derive the flow-density relationship from the MIXIC car-following model of CACC (at 100% CACC market penetration). The flow-density relationship has an unusual shape; part of the congested regime has an infinite congested wave speed. However, we verify that the flow predictions match observations from MIXIC modeled in VISSIM. Then, we use the flow-density relationship from MIXIC in LTM. Although the independence of separate links restricts the maximum congested wave speed, for common freeway link lengths the congested wave speed is sufficiently high to fit the observed flows from MIXIC. Results on a freeway and regional networks (with CACC-exclusive lanes) indicate that CACC could reduce freeway congestion, but naïve deployment of CACC-exclusive lanes could cause an increase in total system travel time.     

The representation and implementation of time-dependent inundation in large-scale microscopic evacuation simulations

Multi-agent simulation has increasingly been used for transportation simulation in recent years. With current techniques, it is possible to simulate systems consisting of several million agents. Such multi-agent simulations have been applied to whole cities and even large regions. In this paper it is demonstrated how to adapt an existing multi-agent transportation simulation framework to large-scale pedestrian evacuation simulation. The underlying flow model simulates the traffic-based on a simple queue model where only free speed, bottleneck capacities, and space constraints are taken into account. The queue simulation, albeit simple, captures the most important aspects of evacuations such as the congestion effects of bottlenecks and the time needed to evacuate the endangered area. In the case of an evacuation simulation the network has time-dependent attributes. For instance, large-scale inundations or conflagrations do not cover all the endangered area at once.These time-dependent attributes are modeled as network change events. Network change events are modifying link parameters at predefined points in time. The simulation framework is demonstrated through a case study for the Indonesian city of Padang, which faces a high risk of being inundated by a tsunami.     

Measuring marginal congestion costs of urban transportation: Do networks matter?

In determining the marginal cost of congestion, economists have traditionally relied upon directly measuring traffic congestion on network links, disregarding any “network effects,” since the latter are difficult to estimate. While for simple networks the comparison of the network-based congestion costs with the link-based ones can be done within a theoretical framework, it is important to know whether such network effects in real large-scale networks are quantitatively significant.In this paper we use a strategic transportation planning model (START) to compare marginal congestion costs computed link-by-link with measures taking into account network effects. We find that while in aggregate network effects are not significant, congestion measured on a single link is a poor predictor of total congestion costs imposed by travel on that link. Also, we analyze the congestion proliferation effect on the network to see how congestion is distributed within an urban area.     

Network traffic flow optimization under performance constraints

In this paper, a model-based perimeter control policy for large-scale urban vehicular networks is proposed. Assuming a homogeneously loaded vehicle network and the existence of a well-posed Network Fundamental Diagram (NFD), we describe a protected network throughout its aggregated dynamics including nonlinear exit flow characteristics. Within this framework of constrained optimal boundary flow gating, two main performance metrics are considered: (a) first, connected to the NFD, the concept of average network travel time and delay as a performance metric is defined; (b) second, at boundaries, we take into account additional external network queue dynamics governed by uncontrolled inflow demands. External queue capacities in terms of finite-link lengths are used as the second performance metric. Hence, the corresponding performance requirement is an upper bound of external queues. While external queues represent vehicles waiting to enter the protected network, internal queue describes the protected network’s aggregated behavior.By controlling the number of vehicles joining the internal queue from the external ones, herewith a network traffic flow maximization solution subject to the internal and external dynamics and their performance constraints is developed. The originally non-convex optimization problem is transformed to a numerically efficiently convex one by relaxing the performance constraints into time-dependent state boundaries. The control solution can be interpreted as a mechanism which transforms the unknown arrival process governing the number of vehicles entering the network to a regulated process, such that prescribed performance requirements on travel time in the network and upper bound on the external queue are satisfied. Comparative numerical simulation studies on a microscopic traffic simulator are carried out to show the benefits of the proposed method.     

An efficient and exact event-based algorithm for solving simplified first order dynamic network loading problems in continuous time

In this paper a novel solution algorithm is proposed for exactly solving simplified first order dynamic network loading (DNL) problems for any generalised network. This DNL solution algorithm, termed eLTM (event-based Link Transmission Model), is based on the seminal Lighthill–Witham–Richards (LWR) model, adopts a triangular fundamental diagram and includes a generalised first order node model formulation. Unlike virtually all DNL solution algorithms, eLTM does not rely on time discretisation, but instead adopts an event based approach. The main advantage of this approach is the possibility of yielding exact results. Furthermore, an approximate version of the same algorithm is introduced. The user can configure an a-priori threshold that dictates the approximation error (measurable a-posteriori). Using this approximation the computational effort required decreases significantly, making it especially suitable for large scale applications. The computational complexity is investigated and results are demonstrated via theoretical and real world case studies. Fixed periods of stationary demands are included adopting a matrix demand profile to mimic basic departure time demand fluctuations. Finally, the information loss of the approximate solution is assessed under different configurations.     

Congestion-aware system optimal route choice for shared autonomous vehicles

We study the shared autonomous vehicle (SAV) routing problem while considering congestion. SAVs essentially provide a dial-a-ride service to travelers, but the large number of vehicles involved (tens of thousands of SAVs to replace personal vehicles) results in SAV routing causing significant congestion. We combine the dial-a-ride service constraints with the linear program for system optimal dynamic traffic assignment, resulting in a congestion-aware formulation of the SAV routing problem. Traffic flow is modeled through the link transmission model, an approximate solution to the kinematic wave theory of traffic flow. SAVs interact with travelers at origins and destinations. Due to the large number of vehicles involved, we use a continuous approximation of flow to formulate a linear program. Optimal solutions demonstrate that peak hour demand is likely to have greater waiting and in-vehicle travel times than off-peak demand due to congestion. SAV travel times were only slightly greater than system optimal personal vehicle route choice. In addition, solutions can determine the optimal fleet size to minimize congestion or maximize service.     

临时关闭道路以疏导城市道路交通流的可行性分析

文章基于Braess悖论,以单位车辆的平均出行时间为最优目标,以网络道路系统为约束条件,从逆向思维角度建立了通过临时关闭道路引导交通流以缓解城市道路交通拥挤的Narsh均衡非线性规划模型,并以北京二环道路网络为研究对象进行了实例分析。结果表明,在城市道路网中,关闭部分路段并不能明显缓解交通拥堵现状,但可以在不影响路网整体通行时间的条件下改善路网局部拥堵状况。建议城市道路交通临时疏导方案应该根据路网交通流的分布特点和道路通过能力等因素进行制订。     

System optimal dynamic traffic assignment: Properties and solution procedures in the case of a many-to-one network

Thanks to its high dimensionality and a usually non-convex constraint set, system optimal dynamic traffic assignment remains one of the most challenging problems in transportation research. This paper identifies two fundamental properties of the problem and uses them to design an efficient solution procedure. We first show that the non-convexity of the problem can be circumvented by first solving a relaxed problem and then applying a traffic holding elimination procedure to obtain the solution(s) of the original problem. To efficiently solve the relaxed problem, we explore the relationship between the relaxed problems based on different traffic flow models (PQ, SQ, CTM) and a minimal cost flow (MCF) problem for a special space-expansion network. It is shown that all the four problem formulations produce the same minimal system cost and share one common solution which does not involve inside queues in the network. Efficient solution algorithms such as the network simplex method can be applied to solve the MCF problem and identify such an optimal traffic pattern. Numerical examples are also presented to demonstrate the efficiency of the proposed solution procedure.     

Network-based real option models

Building on earlier work to incorporate real option methodologies into network modeling, two models are proposed. The first is the network option design problem, which maximizes the expanded net present value of a network investment as a function of network design variables with the option to defer the committed design investment. The problem is shown to be a generalized version of the network design problem and the multi-period network design problem. A heuristic based on radial basis functions is used to solve the problem for continuous link expansion with congestion effects. The second model is a link investment deferral option set, which decomposes the network investment deferral option into individual, interacting link or project investments. This model is a project selection problem under uncertainty, where each link or project can be deferred such that the expanded net present value is maximized. The option is defined in such a way that a lower bound can be solved using an exact method based on multi-option least squares Monte Carlo simulation. Numerical tests are conducted with the classical Sioux Falls network and compared to earlier published results.     

Comparing whole-link travel time models

In a model commonly used in dynamic traffic assignment the link travel time for a vehicle entering a link at time t is taken as a function of the number of vehicles on the link at time t. In an alternative recently introduced model, the travel time for a vehicle entering a link at time t is taken as a function of an estimate of the flow in the immediate neighbourhood of the vehicle, averaged over the time the vehicle is traversing the link. Here we compare the solutions obtained from these two models when applied to various inflow profiles. We also divide the link into segments, apply each model sequentially to the segments and again compare the results. As the number of segments is increased, the discretisation refined to the continuous limit, the solutions from the two models converge to the same solution, which is the solution of the Lighthill, Whitham, Richards (LWR) model for traffic flow. We illustrate the results for different travel time functions and patterns of inflows to the link. In the numerical examples the solutions from the second of the two models are closer to the limit solutions. We also show that the models converge even when the link segments are not homogeneous, and introduce a correction scheme in the second model to compensate for an approximation error, hence improving the approximation to the LWR model.     

An instantaneous kinematic wave theory of diverging traffic

Diverging junctions are an important type of bottlenecks, which can reduce capacities and initiate and propagate traffic congestion in a road network. In this paper, we propose a kinematic wave theory for modeling dynamics of non-cooperative diverging traffic, in which traffic dynamics of vehicles to one direction are assumed to be independent of those to other directions instantaneously. During a short time interval, the kinematic wave model of diverging traffic is decoupled into a number of nonlinear resonant systems. From analytical solutions to the Riemann problem of a decoupled system, a new definition of partial traffic demand is introduced, so that diverging flows can be easily computed with the supply–demand method. Then a Cell Transmission Model is proposed to solve the kinematic wave model of diverging traffic by taking into account of the interactions among different traffic streams. Simulation results demonstrate that vehicles follow the First-In-First-Out principle in the long run, and the model converges when we decrease the cell and time-step sizes. In addition, it is shown that traffic streams to different directions segregate in a selfish manner, and the total throughput of a diverging junction is not maximized as in existing diverge models. In the future, more theoretical and empirical studies are needed for a better understanding of this and other diverge models.     

Robust optimization for emergency logistics planning: Risk mitigation in humanitarian relief supply chains

This paper proposes a methodology to generate a robust logistics plan that can mitigate demand uncertainty in humanitarian relief supply chains. More specifically, we apply robust optimization (RO) for dynamically assigning emergency response and evacuation traffic flow problems with time dependent demand uncertainty. This paper studies a Cell Transmission Model (CTM) based system optimum dynamic traffic assignment model. We adopt a min–max criterion and apply an extension of the RO method adjusted to dynamic optimization problems, an affinely adjustable robust counterpart (AARC) approach. Simulation experiments show that the AARC solution provides excellent results when compared to deterministic solution and sampling based stochastic programming solution. General insights of RO and transportation that may have wider applicability in humanitarian relief supply chains are provided.