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We discuss continuous traffic flow network models including traffic lights. A mathematical model for traffic light settings within a macroscopic continuous traffic flow network is presented, and theoretical properties are investigated. The switching of the traffic light states is modeled as a discrete decision and is subject to optimization. A numerical approach for the optimization of switching points as a function of time based upon the macroscopic traffic flow model is proposed. The numerical discussion relies on an equivalent reformulation of the original problem as well as a mixed-integer discretization of the flow dynamics. The large-scale optimization problem is solved using derived heuristics within the optimization process. Numerical experiments are presented for a single intersection as well as for a road network.

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... Practical applications include the modeling, simulation and optimization of traffic models and provide a reliable tool to predict and control traffic behavior. During the last decades an increasing number of articles have been published on road traffic control and urban networks [3,9,11,13,14,15,19,20,24]. In particular, control issues such as ramp metering or traffic light control are claimed to decrease traffic congestions. ...

... Traffic lights are set according to the current traffic conditions on the road, i.e. dynamic stop and release of traffic depending on the density. Similar to [13], we concentrate our investigations on urban intersections where traffic lights for each road are installed. Then, dynamic signals are controlled to adjust their timing and phasing to meet changing traffic conditions. ...

... TRAFFIC LIGHT CONTROL: A CASE STUDY 5 3. Modeling Traffic Lights. There exists a wide variety of literature on traffic light control, see [13] for an overview. The new ingredient in our approach is the traffic light control coupled to the dynamic traffic flow network model presented in Section 2. This approach allows for the adaption of traffic light cycles to the different traffic volume during a day. ...

This article is devoted to traffic flow networks including traffic lights at intersections. Mathematically, we consider a nonlinear dynamical traffic model where traffic lights are modeled as piecewise constant functions for red and green signals. The involved control problem is to find stop and go configurations depending on the current trafic volume. We propose a numerical solution strategy and present computational results.

... The density on the edges is assumed to be constant over the whole edge and there is no interaction between the edges at the junctions, such that the densities are constant over time. This enables us to calculate the travel times on the edges without using (18), since the thief has constant velocity ω = 1 − ρ e . The set of hideout vertices is D = {4, 6, 8}. ...

... Imagine the thief having arrived at a vertex n at time t 0 and having chosen edge e to pass next. Then the edge arrival time a e (t 0 ) for edge e can be determined by solving the IVP (18). From the solution y(t) of this IVP, the arrival time a e (t 0 ) at the end of the edge can be derived. ...

... Step 2. The thief starts his journey at the initial point of a certain road at time t = 0. The thief's position y(t) on a road is determined by solving the ODE (18) with an appropriate solver. When the thief reaches a junction, Definition (3.2) (Decision Criteria) is applied to determine the road to be passed next. ...

In this work we introduce a novel model for the tracking of a thief moving through a road network. The modeling equations are given by a strongly coupled system of scalar conservation laws for the road traffic and ordinary differential equations for the thief evolution. A crucial point is the characterization at intersections, where the thief has to take a routing decision depending on the available local information. We develop a numerical approach to solve the thief tracking problem by combining a time-dependent shortest path algorithm with the numerical solution of the traffic flow equations. Various computational experiments are presented to describe different behavior patterns.

... Our goal in this paper is to devise efficient heuristics to approximately solve the problem of optimal traffic light settings for road networks. We base our work on the model and scenarios presented in [12] and refer the reader to the discussion therein for further references regarding the wide variety of different approaches for traffic light control or related problems such as spillover dissipation [11,23]. ...

... However, our focus is the consideration of a nonlinear flow function that will lead to a more challenging nonlinear mixed-integer problem. A piecewise linear flow function as in [12] will, however, play an important role in section 5.3 for a comparison of the performance of our proposed method with a heuristic based on MILP. ...

... In contrast to the usual notation, we use A i (t)γ i (t) in place ofγ i (t) in order to make the usually implicit dependence ofγ i (t) on A i (t) explicit (cf., e.g., [12]) because this reformulation dramatically improves the performance of the solvers for the Stage I problem introduced in section 4. Moreover, we require the Rankine-Hugoniot relation for the conservation of traffic, ...

We consider the problem of computing optimal traffic light programs for urban road intersections using traffic flow conservation laws on networks. Based on a partial outer convexification approach, which has been successfully applied in the area of mixed-integer optimal control for systems of ordinary or differential algebraic equations, we develop a computationally tractable two-stage solution heuristic. The two-stage approach consists of the solution of a (smoothed) nonlinear programming problem with dynamic constraints and a reconstruction mixed-integer linear program without dynamic constraints. The two-stage approach is founded on a discrete approximation lemma for partial outer convexification, whose grid-independence properties for (smoothed) conservation laws are investigated. We use the two-stage approach to compute traffic light programs for two scenarios on different discretizations and demonstrate that the solution candidates cannot be improved in a reasonable amount of time by global state-of-the-art mixed-integer nonlinear programming solvers. The two-stage solution candidates are not only better than results obtained by global optimization of piecewise linearized traffic flow models but also can be computed at a faster rate.

... camera, radar). However, difficulties still exist: (1) extensive field-tests and manual tuning are required to reflect roadway and traffic characteristics; the process is resource-consuming and the outcomes cannot be ported generally; (2) given the complexity of urban traffic systems, it is often not feasible to develop an accurate model-based method without simplified assumptions about operating conditions -which can lead to suboptimal solutions [7], [8]. ...

... Different from model-based control methods where full information of environment is required [7], [8], RL can learn the optimal strategy to control traffic signals via pure interaction with traffic environment. In this paper, we adopt Q-network in our RL approach. ...

Ineffective and inflexible traffic signal control at urban intersections can often lead to bottlenecks in traffic flows and cause congestion, delay, and environmental problems. How to manage traffic smartly by intelligent signal control is a significant challenge in urban traffic management. With recent advances in machine learning, especially reinforcement learning (RL), traffic signal control using advanced machine learning techniques represents a promising solution to tackle this problem. In this paper, we propose a RL approach for traffic signal control at urban intersections. Specifically, we use neural networks as Q-function approximator (a.k.a. Q-network) to deal with the complex traffic signal control problem where the state space is large and the action space can be discrete. The state space is defined based on real-time traffic information, i.e. vehicle position, direction and speed. The action space includes various traffic signal phases which are critical in generating a reasonable and realistic control mechanism, given the prominent spatial-temporal characteristics of urban traffic. In the simulation experiment, we use SUMO, an open source traffic simulator, to construct realistic urban intersection settings. Moreover, we use different traffic patterns, such as major/minor road traffic, through/left-turn lane traffic, tidal traffic, and varying demand traffic, to train a generalized traffic signal control model that can be adapted to various traffic conditions. The simulation results demonstrate the convergence and generalization performance of our RL approach as well as its significant benefits in terms of queue length and wait time over several benchmarking methods in traffic signal control.

... A line of research that applies optimal control to traffic has been developed thanks to the work of Herty et al. [9,81,83,89,94,90,62,88] and Piccoli et al. [34,35,47]. Several approaches have been used to solve the optimal control: adjoint methods, combinatorial methods, mixed-integer methods and instantaneous control. ...

... Une ligne de recherche qui applique le contrôle optimal au trafic routier a été développée grâce au travail de Herty et al. [9,81,83,89,94,90,62,88] et Piccoli et al. [34,35,47]. Plusieurs approches ont été utilisées pour résoudre les problèmes de control optimal: les méthodes de l'adjoint, les méthodes combinatoires, les méthodes "mixted integer" et le contrôle instantané. ...

In this thesis we consider two coupled PDE-ODE models. One to model moving bottlenecks and theother one to describe traffic flow at junctions. First, we consider a strongly coupled PDE-ODE systemthat describes the influence of a slow and large vehicle on road traffic. The model consists of a scalarconservation law accounting for the main traffic evolution, while the trajectory of the slower vehicle isgiven by an ODE depending on the downstream traffic density. The moving constraint is expressed byan inequality on the flux, which models the bottleneck created in the road by the presence of the slowerDépôt de thèse – Donnéescomplémentairesvehicle. We prove the existence of solutions to the Cauchy problem for initial data of bounded variation.Moreover, two numerical schemes are proposed. The first one is a finite volume algorithm that uses alocally nonuniform moving mesh. The second one uses a reconstruction technique to display thebehavior of the vehicle. Next, we consider the Lighthill-Whitham-Richards traffic flow model on ajunction composed by one mainline, an onramp and an offramp, which are connected by a node. Theonramp dynamics is modeled using an ordinary differential equation describing the evolution of thequeue length. The definition of the solution of the Riemann problem at the junction is based on anoptimization problem and the use of a right of way parameter. The numerical approximation is carriedout using a Godunov scheme, modified to take into account the effects of the onramp buffer. Aftersuitable modification, the model is used to solve an optimal control problem on roundabouts. Two costfunctionals are numerically optimized with respect to the right of way parameter.

... Another signal control methodology is formulated as a quadratic programming problem to minimize and balance the link queues, thus minimizing the risk of queue spillback [13]. G ottlich et al. [14] presented a numerical approach to the optimization of switching points as a function of time based upon the macroscopic tra c ow model. The numerical discussion relies on an equivalent reformulation of the original problem as well as a mixed-integer discretization of the ow dynamics. ...

... Calculate the distance (blue line in Figure 3) between each solution in the non-dominated sorting (f1 i ; f2 i ) and perpendicular to the regression line passing through point (u i ; v i ), reached in Eq. (14) using the following equation: ...

Traffic jam is a daily problem in nearly all major cities in the world and continues to increase with population and economic growth of urban areas. Traffic lights, as one of the key components at intersections, play an important role in control of traffic flow. Hence, study and research on phase synchronization and time optimization of the traffic lights could be an important step to avoid creating congestion and rejection queues in a urban network. Here, we describe the application of NSGA-II, a multi-objective evolutionary algorithm, to optimize both vehicle and pedestrian delays in an individual intersection. In this paper, we improve NSGA-II algorithm based on the regression line to find a Pareto-optimal solution or a restrictive set of Pareto-optimal solutions based on our solution approaches to the problem, named PDNSGA (Non-dominated Sorting Genetic Algorithm based on Perpendicular Distance). The high speed of the proposed algorithm and its quick convergence makes it desirable for large scheduling with a large number of phases. It is demonstrated that our proposed algorithm (PDNSGA) gives better outputs than those of Moga, NSGA-II, and WBGA in traffic signal optimization problem, statistically .

... and the intersections could be classified into two main groups, 30 i.e., signalized and non-signalized ones. The former equips a 31 set of traffic signals while the latter (also named uncontrolled 32 intersection) is one in which the entrance into the intersection 33 from any of approaches is not controlled by a traffic signal. ...

... AQ:1 = Author: Please confirm or add details for any funding or financial support for the research of this article. AQ:2 = Please confirm the volume no. for ref. and the intersections could be classified into two main groups, 30 i.e., signalized and non-signalized ones. The former equips a 31 set of traffic signals while the latter (also named uncontrolled 32 intersection) is one in which the entrance into the intersection 33 from any of approaches is not controlled by a traffic signal. ...

This paper addresses a traffic signal scheduling (TSS) problem in a heterogeneous traffic network with signalized and non-signalized intersections. The objective is to minimize the total network-wise delay time of all vehicles within a given finite-time window. First, a novel model is proposed to describe a heterogeneous traffic network with signalized and non-signalized intersections. Second, five meta-heuristics are implemented to solve the TSS problem. Based on the problem characteristics, three local search operators and their ensemble are proposed. Then, five meta-heuristics with such an ensemble are proposed to solve the TSS problem. Third, experiments are carried out based on the real traffic data in the Jurong area of Singapore. The performance of the ensemble of local search operators is verified. Ten algorithms, including five meta-heuristics with and without the ensemble, are evaluated by solving 18 cases with different scales. Finally, the algorithm with the best performance is compared against the currently used traffic signal control strategies. The comparisons and discussions show the competitiveness of the proposed model and meta-heuristics.

... The common intersection [17] [22] needs to take into account the impact of its adjacent intersections. Define the common intersection as the signalized intersection so that the coupling degree between it and its adjacent intersections is between 0 and 1. ...

... H. Yang et al. [17] propose an eco-cooperative adaptive cruise control (Eco-CACC) system that receives signal phasing and timing data from downstream signalized intersections via vehicle-toinfrastructure (V2I) communications. A mathematical model of traffic light settings within a macroscopic continuous traffic flow network is presented, and theoretical properties are investigated in [22]. J. Wu et al. [23] propose a new traffic lights control scheme of a simple intersection, taking into account vehicle behavior, integral red and orange phases. ...

This paper proposes a dynamic cooperative traffic control framework for multiple intersections based on virtual-grids to optimize the throughput and ensure fairness among all traffic flows. The traffic flows are divided by virtual grids, we call it Virtual grid based Cooperative Control of Multiple-Intersections (VGCC). The road segment between two intersections has been divided into two parts, which are defined as the reference region and decision region. When vehicle arrives at or goes away from any one part, it registers or deregisters itself to Road Side Unit (RSU). The traffic controller at intersection, called Intersection Control Unit (ICU), collects the traffic information from all road segments and receives traffic messages from adjacent intersections. The proposed signal-scheduling algorithm considers not only the flows at the local intersection with higher passing rates, but also the flows at downstream signalized intersections with higher passing rates. To ensure fairness, the algorithm gives chances to those phases who has a lower passing rate by using the ageing-counter matrix. According to the real time traffic information, this paper makes signal timing for each phase of a signal cycle one by one. Moreover, a Cooperative Collision Avoidance Predictive control (CCAP) algorithm is proposed, which can assist vehicles to pass the next intersection without stopping, by predicting time conflict. The results indicate that the VGCC algorithm significantly decreases the average number of vehicles on a road segment by 30.77%, reduces average queuing length by 28.89%, decreases the average time spent on passing intersection by 26.93% and reduces average waiting time by 35.21% than the intersection of Common Road networks algorithm (CRN).

... In applications, tailored branch-and-bound algorithms have been applied [9]. Recently, in [6] a penalty method was proposed that relies on a combination of tailored basin hopping and interiorpoint method. ...

... Thus, the additional effort of computing the reduced problem is independent of the iteration number of the optimization algorithm, which can grow exponentially with the discretization. Here, the effort depends only linearly on L, since L+1 initial value problems are solved: once for (8) and L times for (9). ...

Mixed-integer optimal control problems governed by PDEs (MIPDECOs) are powerful mod-eling tools but also challenging in terms of theory and computation. We propose a highly efficient state elimination approach for MIPDECOs that are governed by PDEs that have the structure of an abstract ODE in function space. This allows us to avoid repeated calculations of the states for all time steps, and our approach is applied only once before starting the optimization. The presentation of theoretical results is complemented by numerical experiments.

... In applications, tailored branch-and-bound algorithms are applied [7]. Recently, in [5] a penalty method has been proposed that relies on a combination of tailored basin hopping and interior-point method. ...

Mixed-integer optimal control problems governed by PDEs (MIPDECO) are powerful modeling tools but also very challenging in terms of theory and computation. We propose a highly efficient state elimination approach for MIPDECOs that are governed by PDEs that have the structure of an abstract ODE in function space. This allows us to avoid repeated calculations of the states for all time steps and our approach is applied only once before starting the optimization. The presentation of theoretical results are complemented by numerical experiments.

... In these contexts, a crucial role is played by the design, analysis and numerical implementation of controls acting at the nodes of the network. The investigation of optimal control properties of time varying parameters corresponding to junction distribution coefficients have been considered for example in [25,26,27,49,54,57,58,79] while inflow controls have been analyzed in [32,41,53,74]. ...

The paper proposes a general framework to analyze control problems for conservation law models on a network. Namely we consider a general class of junction distribution controls and inflow controls and we establish the compactness in $L^1$ of a class of flux-traces of solutions. We then derive the existence of solutions for two optimization problems: (I) the maximization of an integral functional depending on the flux-traces of solutions evaluated at points of the incoming and outgoing edges; (II) the minimization of the total variation of the optimal solutions of problem (I). Finally we provide an equivalent variational formulation of the min-max problem (II) and we discuss some numerical simulations for a junction with two incoming and two outgoing edges.

... If they are available, such methods can provide global optimal solutions. This approach is followed for example in [9] for gas networks or in [11] for traffic flow. ...

We extend the convergence analysis for methods solving PDE-constrained
optimal control problems containing both discrete and continuous control
decisions based on relaxation and rounding strategies to the class of first
order semilinear hyperbolic systems in one space dimension. The results are
obtained by novel a-priori estimates for the size of the relaxation gap based
on the characteristic flow, fixed-point arguments and particular regularity
theory for such mixed-integer control problems. As an application we consider a
relaxation model for optimal flux switching control in conservation laws
motivated by traffic flow problems.

... Step 3. Traffic signal control system starts to optimize the signal priority strategy for the transit vehicle based on the signal status, transit status, and traffic flow in the network. The traffic signal parameters optimum functions, formulas, and its restrained conditions employ methods and algorithms in the HCM 2000 (Highway Capacity Manual 2000) [26][27][28][29], such as determining delay, progression adjustment factor, and signal cycle length. Priority strategy selection and adjustment Discrete Dynamics in Nature and Society 7 of signal parameters are based on the performance indices of the optimization results from HCM 2000. ...

This study presents methods of transit signal priority without transit-only lanes for a transit-based emergency evacuation in a sudden-onset disaster. Arterial priority signal coordination is optimized when a traffic signal control system provides priority signals for transit vehicles along an evacuation route. Transit signal priority is determined by “transit vehicle arrival time estimation,” “queuing vehicle dissipation time estimation,” “traffic signal status estimation,” “transit signal optimization,” and “arterial traffic signal coordination for transit vehicle in evacuation route.” It takes advantage of the large capacities of transit vehicles, reduces the evacuation time, and evacuates as many evacuees as possible. The proposed methods were tested on a simulation platform with Paramics V6.0. To evaluate and compare the performance of transit signal priority, three scenarios were simulated in the simulator. The results indicate that the methods of this study can reduce the travel times of transit vehicles along an evacuation route by 13% and 10%, improve the standard deviation of travel time by 16% and 46%, and decrease the average person delay at a signalized intersection by 22% and 17% when the traffic flow saturation along an evacuation route is
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... Notable strategies proposed in the last few decades include, e.g., MAXBAND [11,12], TRANSYT [11,12], SCOOT [13], OPAC [14], PRODYN [15], CRONOS [16], and RHODES [17]. To solve the traffic light control problem, many researchers proposed various optimization approaches, e.g., particle swarm optimization [18], distributed coordination of exploration and exploitation [19] and Mixed-integer programming [20] and so on. Among the various approaches, meta-heuristics are very flexible and robust to the problem scale and random variables, have significant advantages in computational efficiency in terms of CPU time for large-scale real-lift applications, and become the new trend for solving reallife traffic light control problems. ...

... BS will communicate with these sensors via cellular, Wi-Fi, WiMAX or DSRC, and make decision on GLPT for road bound. These architectures are shown in Figure 5. [11] Fig. 5. Infrastructure-based architectures [11] B. Existing Traffic Light System SCOOT and SCATS are the two widely implemented dynamic cycle TLS in the world [12]. Both TLSs use inductive loop detector to monitor real-time traffic flow for road intersections. ...

Traffic congestion is a major issue that happens across urban cities around the world. In 2015, annual economic losses of Malaysia caused by traffic congestion is estimated about RM20 billion. Fixed cycle traffic light system (TLS) is first introduced on road intersection to solve traffic congestion. However, fixed cycle TLS in unable to cope with dramatically increase of registered vehicles. In this paper, development of Internet of Things (IoT) device for traffic management system is proposed. An Intel Edison collects real-time traffic flow and communicate with Microsoft Azure IoT cloud server. The cloud server assign priorities to each road bound based on their current traffic volume. Green light phase time (GLPT) is then calculated utilizing a dynamic algorithm. Simulation results showed that dynamic cycle TLS reduces queue length and waiting time on the road intersection by 68% and 67% respectively. Additionally, a monitoring application is designed to ease traffic officer in monitoring real-time traffic flow.

... Timing control generally uses the Webster timing method, which chooses the optimal cycle time by applying the minimum traffic delay and makes the green time ratio proportionally distributed by the maximum flow ratio of each phase. Induction control measures the traffic flow by presetting coils at the entrance of each lane and meets the traffic demand by adjusting the green time ratio of the cycle [4]. Besides, control methods such as fuzzy control [5], queuing theorybased method [6], and model-based method [7,8] are also used in traffic lights control. ...

With rapid development of the urbanization, how to improve the traffic lights efficiency has become an urgent issue. The traditional traffic light control is a method that calculates a series of corresponding timing parameters by optimizing the cycle length. However, fixing sequence and duration of traffic lights is inefficient for dynamic traffic flow regulation. In order to solve the above problem, this study proposes a traffic light timing optimization scheme based on deep reinforcement learning (DRL). In this scheme, the traffic lights can output an appropriate phase according to the traffic flow state of each direction at the intersection and dynamically adjust the phase length. Specifically, we first adopt Proximal Policy Optimization (PPO) to improve the convergence speed of the model. Then, we elaborate the design of state, action, and reward, with the vehicle state defined by Discrete Traffic State Encoding (DTSE) method. Finally, we conduct experiments on real traffic data via the traffic simulation platform SUMO. The results show that, compared to the traditional timing control, the proposed scheme can effectively reduce the waiting time of vehicles and queue length in various traffic flow modes.

... Due to the large scale of an ordinary traffic network, which usually consists of hundreds of intersections and thousands of road links, the high computational complexity in optimization becomes the major hurdle for the real-time scheduling strategy. To solve the traffic signal control problem, many researchers proposed various optimization approaches, e.g., particle swarm optimization [16], distributed coordination of exploration and exploitation [17] and mixed-integer programming [18] and so on. Among these different approaches, meta-heuristics are attractive method for the LUTSCP. ...

... Mixed-integer optimal control is a field of increasing importance as practical applications often include discrete decisions in addition to continuousvalued control variables. Examples of such problems can be found, e.g., in [1,2,4,5,8,14,15,18]. One way to approach mixed-integer optimal control problems is by solving necessary optimality conditions provided by the well-known maximum principle. ...

The article discusses a numerical approach to solve optimal control problems in discrete time that involve continuous and discrete controls. Special attention is drawn to the modeling and treatment of dwell time constraints. For the solution of the optimal control problem in discrete time, a dynamic programming approach is employed. A numerical example is included that illustrates the impact of dwell time constraints in mixed integer optimal control.

... In the literature [20], a continuous traffic flow network model is discussed, in which the switching of 72 the traffic light states is modelled as a discrete decision. The numerical discussion relied on the 73 equivalent reformulation of the original problem and a mixed-integer discretization of the flow 74 dynamics. ...

This paper studies a large-scale urban traffic light scheduling problem (LUTLSP). A centralized model is developed to describe the LUTLSP, where each outgoing flow rate is described as a nonlinear mixed logical switching function over the source link’s density, the destination link’s density and capacity, and the driver’s potential psychological response to the past traffic light signals. The objective is to minimize the total network-wise delay time of all vehicles in a time window. Three metaheuristic optimization algorithms, named as Jaya algorithm, water cycle algorithm (WCA), and harmony search (HS) algorithm are implemented to solve the LUTLSP. Since we adopt a discrete-time formulation of LUTLSP, we firstly develop a discrete version of Jaya and WCA. Secondly, some improvement strategies are proposed to speed up the convergence of applied optimizers. Thirdly, a feature based search operator is utilized to improve the search performance of reported optimization methods. Finally, experiments are carried out based on the real traffic data in Singapore. The HS, WCA, Jaya, and their variants are evaluated by solving 11 cases of traffic networks. The comparisons and discussions verify that the considered metaheuristic optimization methods can effectively solve the LUTLSP considerably surpassing the existing traffic light control strategy.

... Also, for the cycle program of traffic lights, Particle Swarm Optimization (PSO) based approach is proposed and used together with a microscopic traffic simulator [20], and a case study in the reallife traffic network validated the significant profits of the proposed approach [4]. In the literature [21], a continuous traffic flow network model is discussed, in which the switching of the traffic light states is modelled as a discrete decision. The numerical discussion relied on the equivalent reformulation of the original problem and a mixed-integer discretization of the flow dynamics. ...

... Even so, the effectiveness of the measures that are being taken is in question and, to help analyze and design actuation techniques, numerous numerical simulation and optimal control works have been developed. The use of models based on partial differential equations is common, both in the study of traffic flow in an urban network [8,9,15,13,12], and in the analysis of atmospheric pollution [1,10,23,25]. However, the studies that combine both aspects are not so common, and those who do it (usually only interested in simulating the contamination caused by the traffic, as can be seen, for example, in [23] or [4]), they usually start from a previously known vehicular flow, which makes difficult the search for actuations on the road that are optimal in terms of contamination. ...

Air pollution is one of the most important environmental problems nowadays. In large metropolitan areas, the main source of pollution is vehicular traffic. Consequently, the search for traffic measures that help to improve pollution levels has become a hot topic today. In this article, combining a 1D model to simulate the traffic flow over a road network with a 2D model for pollutant dispersion, we present a tool to search for traffic operations that are optimal in terms of pollution. The utility of this tool is illustrated by formulating the problem of the expansion of a road network as a problem of optimal control of partial differential equations. We propose a complete algorithm to solve the problem, and present some numerical results obtained in a realistic situation posed in the Guadalajara Metropolitan Area (GMA), Mexico.

... The application of partial differential equations models is usual, both in the analysis of traffic flow in urban networks [20,8,13,12,17,16] and in the study of air pollution [1,14,25,26]. Nevertheless, the number of works combining both issues is much more restricted (see, for instance, [24], [18], [5], [15] or [6]), and usually assume a previously known vehicular flow, which restricts the design of a road network optimal in terms of pollution and travel times. ...

Within the framework of numerical simulation and optimal control of partial differential equations, in this work we deal with the mathematical modelling and optimal management of urban road networks. In particular, we are interested in finding the optimal management of the network intersections in order to reduce traffic congestion and atmospheric pollution. So, we consider two different multi-objective control problems (the former from a cooperative viewpoint, the latter within a hierarchical paradigm), propose a complete numerical algorithm to solve them, and, finally, present several numerical tests for a realistic case posed in the Guadalajara Metropolitan Area (Mexico), where the possibilities of our methodology are shown.

... Within this context, partial differential equations models are usually employed both in the analysis of urban traffic flow in road networks (Coclite et al. 2017;Garavello and Piccoli 2009;Garavello et al. 2016;Goatin et al. 2016;Gottlich et al. 2015;Holden and Risebro 1995) and in the investigation of atmospheric pollution (Alvarez-Vázquez et al. 2015a;García-Chan et al. 2014;Orun et al. 2018;Skiba and Parra-Guevara 2013;Stockie 2011). Nevertheless, the compounding of both topics has been much less addressed (we can mention, for instance, Berrone et al. 2012;Canic et al. 2015;García-Chan et al. 2017;Gottlich et al. 2011;Parra-Guevara and Skiba 2003), and is usually based on the assumption of previous knowledge of the vehicular flow, which is not adapted to analyze the management of a road network that may be optimal to travel times and contamination levels. ...

Air contamination and road congestion are two major problems in modern cities. Both are closely related and present the same source: traffic flow. To deal with these problems, governments impose traffic restrictions preventing the entry of vehicles into sensitive areas, with the final goal of decreasing pollution levels. Unfortunately, these restrictions force drivers to look for alternative routes that usually generate traffic congestions, resulting in longer travel times and higher levels of contamination. In this work, blending computational modelling and optimal control of partial differential equations, we formulate and analyse a bilevel optimal control problem with air pollution and drivers’ travel time as objectives and look for optimal solutions in the sense of Stackelberg. In this setting, the leader (local government) implements traffic restrictions meanwhile the follower (drivers set) acts choosing travel preferences against leader constraints. We discretize the problem and propose a numerical algorithm to solve it, combining genetic-elitist algorithms and interior-point methods. Finally, computational results for a realistic case posed in the Guadalajara Metropolitan Area (Mexico) are shown.

... Notable strategies proposed in the last few decades include, e.g., MAXBAND [11,12], TRANSYT [11,12], SCOOT [13], OPAC [14], PRODYN [15], CRONOS [16], and RHODES [17]. To solve the traffic light control problem, many researchers proposed various optimization approaches, e.g., particle swarm optimization [18], distributed coordination of exploration and exploitation [19] and Mixed-integer programming [20] and so on. Among the various approaches, meta-heuristics are very flexible and robust to the problem scale and random variables, have significant advantages in computational efficiency in terms of CPU time for large-scale real-lift applications, and become the new trend for solving reallife traffic light control problems. ...

... In this context, partial differential equations models are usually employed both in the analysis of urban traffic flow in road networks [4][5][6][7][8][9] and in the investigation of atmospheric pollution [10][11][12][13][14]. Nevertheless, the compounding of both topics has been much less addressed (we can mention, for instance, [15][16][17][18][19]), and is usually based on the assumption of a previous knowledge of the vehicular flow, which is not adapted to analyze the management of a road network that may be optimal with respect to travel times and contamination levels. ...

Two major problems in modern cities are air contamination and road congestion. They are closely related and present a similar origin: traffic flow. To face these problems, local governments impose traffic restrictions to prevent the entry of vehicles into sensitive areas, with the final aim of dropping down air pollution levels. However, these restrictions force drivers to look for alternative routes that usually generate congestions, implying both longer travel times and higher levels of air pollution. In this work, combining optimal control of partial differential equations and computational modelling, we formulate a multi-objective control problem with air pollution and drivers' travel time as objectives and look for its optimal solutions in the sense of Stackelberg. In this problem, local government (the leader) implements traffic restrictions meanwhile the set of drivers (the follower) acts choosing travel preferences against leader constraints. Numerically, the discretized problem is solved by combining genetic-elitist algorithms and interior-point methods, and computational results for a realistic case posed in the Guadalajara Metropolitan Area (Mexico) are shown.

... and energy crisis [2], and how to effectively reduce the energy consumption of vehicles has become a hot research area [3], [4]. In addition to decreasing the fuel consumption by optimizing traffic light signals [5], [6], various eco-driving algorithms also have been proposed to reduce the fuel consumption by optimizing the trajectories of vehicles at intersections [7] and on highways [8]. ...

In this paper, an ecological driving (eco-driving) algorithm considering queue effects is proposed for connected and automated vehicles (CAVs) at unsaturated intersections in order to reduce fuel consumption and travel time. Firstly, after the traffic flow parameters are obtained using vehicle-to-infrastructure communication technology, the kinematic shockwave model is used to predict the vehicle queue length at the saturated intersections. Secondly, to decrease fuel consumption, a fuel-saving optimization problem is formulated using the estimated queue length, and for real-time control, the formulated optimization problem is decomposed into two subproblems depending on whether the CAV will stop to queue. Then, to reduce fuel consumption and travel time simultaneously, the eco-driving algorithm is designed, where the trajectory re-optimization process is implemented in order not to block the upstream vehicles. Finally, extensive simulations are carried out on VISSIM to demonstrate the control performance of the proposed eco-driving algorithm on a single CAV and on the entire traffic flow. Simulation results show that the proposed eco-driving algorithm can significantly decrease the fuel consumption and travel time of both CAVs and the traffic flow, and higher market penetration rate of CAVs can result in better control performance.

Vehicular network technology is frequently used to provide several services and applications for drivers on road networks. The proposed applications in the environment of road networks are classified into three main categories based on their functions: safety, traffic efficiency, and entertainment. The traffic efficiency services are designed to enhance the moving fluency and smoothness of traveling vehicles over the road network. The grid layout architecture of the downtown areas provides several routes toward any targeted destination. Moreover, since several conflicted traffic flows compete at the road intersections, many vehicles have to stop and wait for safe situations to pass the road intersection without coming into conflict with other vehicles. The traffic efficiency applications in this scenario are designed to select the most efficient path for vehicles traveling toward their targeted destination/destinations. Moreover, other applications aimed to decrease the queuing delay time for vehicles at road intersections. In this article, we review several recently proposed mechanisms that worked to enhance the fluency of traffic over downtown road networks and point to the expected future trends in this field.

Traffic signal control (TSC) plays an important role in intelligent transportation system. It is helpful to improve the efficiency of urban transportation by controlling the traffic signal intelligently. Recently, various deep reinforcement learning methods have been proposed to solve TSC. However, most of these methods ignore the fusion of spatial and temporal features in traffic roadnets. Besides, these methods pay no attention to the correlations of the intersections in several local areas. This paper proposes a novel multi-agent deep reinforcement learning method with spatio-temporal feature fusion to solve TSC. The proposed method firstly calculates the correlations among different time steps to capture their temporal dependencies. Secondly, the proposed method constructs connected subnetworks to capture interactive relations among intersections in the subnetwork. Experimental results demonstrate that our method achieves state-of-the-art performance on synthetic and real-world datasets.

With the development of modern intelligent traffc system technology, the travel time information can be collected and processed to provide route-choice suggestions to travellers. However, due to the complex nature of a traffc system, the feedback of traffc information might lead to undesired congestion in some concerned areas (such as the central business district). In this paper, an improved time shortest path strategy (ITSP) based on advanced travel time information feedback is proposed and applied in a Manhattan-like urban traffc system. With the strategy, the link travel time in the concerned area is adjusted by a travel-cost-related coeffcient before being sent to final users. We study the effects of ITSP on traffc performance based on cellular automaton model of urban traffc. What we found most interesting is that when providing the traffc time information with an slightly larger than 1.0 (typically 1.3 1.5), the performance of the system will be enhanced as compared to the situations of no information feedback or providing the real travel time information. Simulation results show that, ITSP can increase the average arrival rate and the traffc flow in scenario of fixed total number of vehicles. Vehicle density decreases with ITSP strategy under scenario with varying total number of vehicles, which helps to avoid traffc gridlock and improve the system reliability. Furthermore, the effects of different size of core areas and different origin-destination patterns are also explored. All the results show that ITSP can improve the traffc performance of network systems.

This paper addresses a bi-objective urban traffic light scheduling problem (UTLSP), which requires minimizing both the total network-wise delay time of all vehicles and total delay time of all pedestrians within a given finite-time window. First, a centralized model is employed to describe the UTLSP, where the cost functions and constraints of the two objectives are presented. A non-domination strategy-based metric is used to compare and rank solutions based on the two objectives. Second, metaheuristics, such as harmony search (HS) and artificial bee colony (ABC), are implemented to solve the UTLSP. Based on the characteristics of the UTLSP, a local search operator is utilized to improve the search performance of the developed optimization algorithms. Finally, experiments are carried out based on the real traffic data in Jurong area of Singapore. The HS, ABC, and their variants with the local search operator are evaluated in 19 case studies with different scales and time windows. To the best of our knowledge, this paper is the first of its kind to solve bi-objective traffic light scheduling problems in the literature. To demonstrate the effectiveness of the proposed algorithms in dealing with bi-objective optimization in traffic light scheduling, they are compared to the classical non-dominated sorting genetic algorithm II (NSGAII) with and without the local search operation. The comparisons indicate that our algorithms outperform the NSGAII algorithm with and without the local search operator for solving the UTLSP.

Boosting the concept of smart cities for implementing an intelligent management of traffic congestion while reducing cybersecurity concerns will not only be more efficient for reducing traffic congestion but also more resilient to cyber incidents. In this paper we proposed a framework that can act as a generalized firewall and work interactively with several critical infrastructures in a smart city to protect the respective operations from a variety of cyber threats. The objective is to develop several steps for a comprehensive traffic management framework in smart cities that facilitates the cooperation among drivers and between drivers and the traffic management authority. The transformative nature of the proposed study supports its applications to a variety of networked critical infrastructures, including electricity, gas, water, rails, and telecommunications, as they intend to respond effectively to a wide range of weather- or human-related disruptions. The contributions of this paper include: Improving the traffic management performance in urban transportation systems, assessing and mitigating the cybersecurity risk in urban traffic management, and facilitating efficient and cyber-secure traffic management in metropolitan areas; Developing and testing an interactive simulation platform for evaluating the traffic management performance under various traffic conditions; Validating and demonstrating the applications in a practical urban transportation system; Disseminating the proposed study results to a wide range of concerned audiences via user-group meetings, detailed education forums, and a close collaboration with the local traffic management authority.

Transportation infrastructure is undergoing major revolutions in most metropolitan areas, which demands for improved operational strategies to meet requirements of smart cities. Such requirements include more convenience more travelers, and higher levels of security, reliability, economics, and societal sustainability in our communities. Given that the wide-area situational awareness is enabled by advanced information and communication technologies, this paper develops a hierarchical operation framework for regulating traffic signals effectively and flexibly in dynamic traffic conditions. The proposed framework which is based on the multi-agent system manages to mitigate potential traffic congestions and minimize drivers’ average travel time in metropolitan areas. Further traffic efficiency improvements can be achieved by the utilization of a closed-loop management system. Interactive simulations are conducted in this paper to examine the performance of the proposed framework in a real-world transportation system.

Software-based methods are common approaches for detecting faults in chemical processes. In this paper, new software-based methodologies were developed for locating multiple leaks in a natural gas pipeline. Two types of multiple leaks, subsequent and simultaneous multiple leaks, from a natural gas pipeline were studied, separately. For both subsequent and simultaneous leaks, case studies with two leak occurrences were demonstrated using MATLAB® simulation. For detecting and locating subsequent multiple leaks, an unknown input observer was designed and applied, which was modified from our previous study. New optimization methodology for locating simultaneous multiple leaks was demonstrated. Leak locations were estimated by solving a nonlinear global optimization problem. The global optimization problem contained constraints of partial differential equation, integer variable, and continuous variable. A new discretization approach was proposed and demonstrated, which required significant less computation time comparing to conventional global optimization algorithm.

This article is devoted to traffic flow networks including traffic lights at intersections. Mathematically, we consider a nonlinear dynamical traffic model where traffic lights are modeled as piecewise constant functions for red and green signals. The involved control problem is to find stop and go configurations depending on the current trafic volume. We propose a numerical solution strategy and present computational results.

Contending with excessive delays at signalized intersections due to traffic flow fluctuation has been recognized as one of the most challenging issues for traffic researchers and engineers. Due to the uncertainty of vehicle arrivals, a signal timing plan optimized with a fixed demand pattern may lead to ineffective control. In response to this need, this study adopts the theory of interval analysis and defines a set of demand intervals to represent the demand fluctuations. Depending on the demand interval patterns, an optimization model is proposed to maximize the overall robustness of signal design while maintaining an acceptable level of efficiency. A recursive two-stage solution procedure is also developed to solve the optimization problem. To ensure the global optimization, a modified branch-and-bound algorithm is developed for the exploration of solutions. The extensive experimental analyses in comparison with the deterministic optimization model reveal that the proposed model is quite promising for applications, especially under highly fluctuated demand patterns.

Optimizing manufacturing systems consists in generating large-quantity outputs to fulfill cus-tomers demands. But naturally machines may fail and the production process is either slowed down or completely interrupted. In order to keep production running, we are interested in assigning repair crews to currently broken-down machines. But due to the limited repair ca-pacity and the dynamics involved in the production process, we propose a scheduling problem based on ordinary differential equations for the description of buffer levels and the actu-ally available processing capacity. We discuss properties of the model and present a solution approach leading to a mixed-integer programming model.

In recent years, autonomic and organic computing have become areas of active research in the informatics community. Both initiatives aim at handling the growing complexity in technical systems by focusing on adaptation and self-optimisation capabilities. A promising application for organic concepts is the control of road traffic signals in urban areas. This article presents an organic approach to traffic light control in urban areas that exhibits adaptation and learning capabilities, allowing traffic lights to autonomously react on changing traffic conditions. A coordination mechanism for neighbouring traffic lights is presented that relies solely on locally available traffic data and communication among neighbouring intersections, resulting in a distributed and self-organising traffic system for urban areas. The organic system's efficiency is demonstrated in a simulation-based evaluation.

A simplified isolated controlled vehicular traffic intersection with two movements is considered. A discrete-event max-plus model is proposed to formulate an optimization problem for the green-red switching sequence. In the case when the criterion is a strictly increasing, linear function of the queue lengths, the problem becomes a linear programming problem. Also, in this case, the steady-state control problem can be solved analytically. A sufficient and necessary condition for steady-state control is derived, and the structure of optimal steady-state traffic control is revealed. Our condition is the same as the necessary condition in for both queue lengths to be non-increasing at an isolated intersection.

A production system which produces a large number of items in many steps can be modelled as a continuous flow problem. The resulting hyperbolic partial differential equation (PDE) typically is nonlinear and nonlocal, modeling a factory whose cycle time depends nonlinearly on the work in progress. One of the few ways to influence the output of such a factory is by adjusting the start rate in a time dependent manner. We study two prototypical control problems for this case: (i) demand tracking where we determine the start rate that generates an output rate which optimally tracks a given time dependent demand rate and (ii) backlog tracking which optimally tracks the cumulative demand. The method is based on the formal adjoint method for constrained optimization, incorporating the hyperbolic PDE as a constraint of a nonlinear optimization problem. We show numerical results on optimal start rate profiles for steps in the demand rate and for periodically varying demand rates and discuss the influence of the nonlinearity of the cycle time on the limits of the reactivity of the production system. Differences between perishable and non-perishable demand (demand versus backlog tracking) are highlighted.

In this paper we address the following traffic regulation problem: given a junction with some incoming roads and some outgoing ones, is it preferable to regulate the flux via a traffic light or via a traffic circle on which the incoming traffic enters continuously? More precisely, assuming that drivers distribute on outgoing roads according to some known coefficients, our aim is to understand which solution performs better from the point of view of total amount of cars going through the junction. To deal with this problem we consider a fluid dynamic model for traffic flow on a road network. The model is that proposed in [9] and is applied to the case of crossings with lights and with circles. For the first we consider timing of lights as control and determine the asymptotic fluxes. For the second we extend and complete the model of [9] introducing some right of way parameters. Also in this case we determine the asymptotic behavior. We then compare the performances of the two solutions. Finally, we can indicate which choice is preferable, depending on traffic level and control necessity, and give indications on how to tune traffic light timing and traffic circle right of way parameters.

We introduce a model for gas flows in pipeline networks based on isothermal Euler equations. We model the intersection of multiple pipes by posing an additional assumption on the pressure at the interface. We give a method to obtain solutions to the gas network problem, and present numerical results for sample networks.

We consider a mathematical model for fluid-dynamic flows on net-works which is based on conservation laws. Road networks are considered as graphs composed by arcs that meet at some junctions. The crucial point is represented by junctions, where interactions occur and the problem is under-determined. The approximation of scalar conservation laws along arcs is carried out by using conservative methods, such as the classical Godunov scheme and the more recent discrete velocities kinetic schemes with the use of suitable boundary conditions at junctions. Riemann problems are solved by means of a simulation algorithm which proceeds processing each junction. We present the algorithm and its application to some simple test cases and to portions of urban network.

We investigate coupling conditions for gas transport in networks where the governing equations are the isothermal Euler equations. We discuss intersections of pipes by considering solutions to Riemann problem. We introduce additional assumptions to obtain a solution near the intersection, and we present numerical results for sample networks.

This study developed a dynamic traffic-control formulation that considers the entire Fundamental Diagram. This incorporation of the Fundamental Diagram is especially important for modeling oversaturated traffic. For this purpose, traffic is modeled after the cell-transmission model (CTM), which is a convergent numerical approximation to the hydrodynamic model. We transformed CTM to a set of mixed-integer constraints and subsequently cast the dynamic signal-control problem to a mixed-integer linear program. As a dynamic platform, the formulation is flexible in dealing with dynamic timing plans and traffic patterns. It can derive dynamic as well as fixed timing plans and address preexisting traffic conditions and time-dependent demand patterns. This study produced results to show the benefit of dynamic timing plans and demonstrated that some of the existing practice on signal coordination could be further improved.

This paper deals with the optimal control of systems governed by nonlinear systems of conservation laws at junctions. The applications considered range from gas compressors in pipelines to open channels management. The existence of an optimal control is proved. From the analytical point of view, these results are based on the well posedness of a suitable initial boundary value problem and on techniques for quasidifferential equations in a metric space.

This article proposes a new method for data assimilation and data reconciliation problems applicable to systems modeled by conservation laws. The state of the system is governed by a scalar Hamilton-Jacobi partial dierential equation , for which the solution is fully characterized by a Lax-Hopf formula. Using the properties of the solution, we prove that when the data of the problem is prescribed in piecewise ane form, the resulting constraints which consist of the partial dierential equation in the assimilation and reconciliation problems are convex, and can be instantiated explicitly. This property enables us to identify a class of data assimilation and data reconciliation problems that can be formulated using convex programs.

This paper proposes a macroscopic ∞uid dynamic model dealing with the ∞ows of information on a telecommunication network with sources and destinations. The model consists of a conservation law for the packets density and a semilinear equation for tra-c distributions functions, i.e. functions describing packets paths. We describe methods to solve Riemann Problems at junctions assigning difierent tra-c distrib- utions functions and two "routing algorithms". Moreover we prove existence of solutions to Cauchy problems for small perturbations of network equilibria.

We consider a supply chain consisting of a sequence of buffer queues and processors with certain throughput times and capacities. Based on a simple rule for releasing parts, i.e. batches of product or individual product items, from the buffers into the processors we derive a hyperbolic conservation law for the part density and flux in the supply chain. The conservation law will be asymptotically valid in regimes with a large number of parts in the supply chain. Solutions of this conservation law will in general develop concentrations corresponding to bottlenecks in the supply chain.

This paper deals with a model for traffic flow based on a system of conservation laws [2]. We construct a solution of the Riemann Problem at an arbitrary junction of a road network. Our construction provides a solution of the full system. In particular, all moments are conserved.

A mathematical model describing supply chains on a network is introduced. In particular, conditions on each vertex of the network are specified. Finally, this leads to a system of nonlinear conservation laws coupled to ordinary differential equations. To prove the existence of a solution we make use of the front tracking method. A comparison to another approach is given and numerical results are presented.

Traffic congestion occurs frequently at downtown intersections during rush hours, at road construction zones as well as at accident sites. Under such circumstances, traffic flow exceeds intersection capacity causing queuing of automobiles that cannot be eliminated in one signal cycle. In this paper, we present a timing decision methodology which considers the whole oversaturation period. Discrete dynamic optimization models are developed and an algorithm to solve them is presented. The optimal cycle length and the optimal assigned green time for each approach are determined for the case of two-phase control. The application of the performance index model to certain multi-phase signals in common use is also introduced. Evaluation results indicate that the proposed discrete type performance index model is a more appropriate design for congested traffic signal timing control.

This paper presents the joint optimization of signal setting parameters and dynamic user equilibrium (DUE) traffic assignment for the congested urban road network. The simulation-based approach is employed to obtain the DUE condition for the case of multiple-origin multiple-destination traffic flows. The dynamic traffic assignment simulation program (DTASP), developed in C language is used to assign the traffic dynamically on the road network, whereas method of successive averages (MSA) is modified and used to arrive at the DUE condition. The artificial intelligence technique of genetic algorithms (GAs) is applied to obtain the optimal signal setting parameters and path flow distribution factor for DUE condition. The methodology developed in such a way that joint optimization of signal setting parameters with DUE is obtained. The proposed method is applied to the real network data of Fort Area of Mumbai city comprising of 17 nodes and 56 unidirectional links with 72 Origin–Destination pairs, where all the 17 nodes are signalized intersections. The traffic flow condition for the optimized signal setting parameters is considerably improved compared to the existing signal settings. The results prove that the GA is an effective technique to solve the joint optimization problem for the real network data.

In this article, we discuss the optimization of a linearized traffic flow network model based on conservation laws. We present two solution approaches. One relies on the classical Lagrangian formalism (or adjoint calculus), whereas another one uses a discrete mixed-integer framework. We show how both approaches are related to each other. Numerical experiments are accompanied to show the quality of solutions.

For area traffic control road network under realization of uncertain travel demand, a robust signal setting is investigated in this paper. Due to certain hierarchy in a decision-making order, a min-max bilevel program is proposed. A new solution method is presented to determine a Nash-Stackelberg solution where a proposed signal setting is found for area traffic control under demand uncertainty. In order to investigate the robustness of the proposed signal settings, numerical computations are performed for various initial data sets in a medium-sized example road network. Good computational results indicated that the proposed signal settings can successfully reduce a worst-case travel cost substantially while incurring a relatively slight loss of optimality with respect to the optimal deterministic solutions for nominal travel demands. Particularly, our computation results showed that the proposed signal settings become even attractive as demand growth increases under a worst-case realization taken by uncertain travel demands.

As traffic congestion rises within urban centers around the world, the intelligent control of traffic signals within cities is becoming increasingly important. Previous research within the area of intelligent traffic signal control has several shortcomings, including a reliance on historical data, the use of centralized systems which cannot handle city-sized problem instances and solutions which are not capable of addressing real-world traffic scenarios (e.g., constantly varying volumes and complex network structures). The research reported here proposes algorithms capable of controlling traffic signals that rely on traffic observations made by available sensor devices and local communication between traffic lights. This solution allows signals to be updated frequently to match current traffic demand, while also allowing for significantly large problem sizes to be addressed. To evaluate the developed system, a realistic traffic model was developed using information supplied by the City of Ottawa, Canada. It was found, through simulation within the SUMO traffic simulation environment, that the proposed adaptive system resulted in higher overall network performance when compared to the current fixed signal plan controllers, which were recreated using information from the City of Ottawa. This work also includes examples of why fixed signal controllers are inferior to an adaptive control system.

Optimal “on–off” laws for the traffic signals are developed based on the bilinear control problem with the binary constraints. A Lyapunov function based feedback law for regulating traffic congestions is developed. Also, a real-time optimal signal law is developed using a novel binary optimization method. Both methods are tested and compared, and our tests demonstrate that the both methods provide very effective and efficient traffic control laws.

In this article, we propose a computational method for solving the Lighthill–Whitham–Richards (LWR) partial differential equation (PDE) semi-analytically for arbitrary piecewise-constant initial and boundary conditions, and for arbitrary concave fundamental diagrams. With these assumptions, we show that the solution to the LWR PDE at any location and time can be computed exactly and semi-analytically for a very low computational cost using the cumulative number of vehicles formulation of the problem. We implement the proposed computational method on a representative traffic flow scenario to illustrate the exactness of the analytical solution. We also show that the proposed scheme can handle more complex scenarios including traffic lights or moving bottlenecks. The computational cost of the method is very favorable, and is compared with existing algorithms. A toolbox implementation available for public download is briefly described, and posted at http://traffic.berkeley.edu/project/downloads/lwrsolver.

A mixed-integer linear programming formulation is proposed to solve the combined system optimal dynamic traffic assignment and signal optimization problem. Traffic conditions are modeled with the cell transmission model, a convergent numerical approximation to the hydrodynamic model of traffic flow. The formulation is suited to respond to oversaturated traffic conditions. It also can be adapted to account for turning movements, protected and permissive phases (gap acceptance), and multiple signal controller types: dynamic (traffic adaptive) and pretimed. Trials with a test network validated the formulation and achieved promising results. Specifically, dynamic signal control proved to be substantially more effective than pretimed control for incident conditions. In addition, potential benefits of rerouting vehicles in both directions of a roadway were revealed even when only one direction is closed.

We formulate the L(2)-gain control problem for a general nonlinear, state-space system with projection dynamics in the state evolution and hard constraints on the set of admissible inputs. We develop specific results for an example motivated by a traffic signal control problem. A state-feedback control with the desired properties is found in terms of the solution of an associated Hamilton-Jacobi-Isaacs equation (the storage function or value function of the associated game) and the critical point of the associated Hamiltonian function. Discontinuities in the resulting control as a function of the state and due to the boundary projection in the system dynamics lead to hybrid features of the closed-loop system, specifically jumps of the system description between two or more continuous-time models. Trajectories for the closed-loop dynamics must be interpreted as a differential set inclusion in the sense of Filippov. Construction of the storage function is via a generalized stable invariant manifold for the flow of a discontinuous Hamiltonian vector-field, which again must be interpreted in the sense of Filippov. For the traffic control model example, the storage function is constructed explicitly. The control resulting from this analysis for the traffic control example is a mathematically idealized averaged control which is not immediately implementable; implementation issues for traffic problems will be discussed elsewhere.

A new model for highway traffic networks based on a detailed description of the junctions is presented. To obtain suitable conditions at the junctions, multilane equations are introduced and investigated. The new model is compared with currently known models for traffic flow networks for several situations. Finally, the model is used for network simulation and optimization.

A mixed-integer model based on a coupled system of differential equations is presented in order to optimize design and material
distribution of production networks. Due to many binary variables arising in this model and in order to guarantee feasible
solutions several starting heuristics, which provide incumbents for the branch and cut algorithm, are developed and compared.

This paper details models and algorithms which can be applied to evacuation problems. While it concentrates on building evacuation many of the results are applicable also to regional evacuation. All models consider the time as main parameter, where the travel time between components of the building is part of the input and the overall evacuation time is the output. The paper distinguishes between macroscopic and microscopic evacuation models both of which are able to capture the evacuees’ movement over time. Macroscopic models are mainly used to produce good lower bounds for the evacuation time and do not consider any individual behavior during the emergency situation. These bounds can be used to analyze existing buildings or help in the design phase of planning a building. Macroscopic approaches which are based on dynamic network flow models (minimum cost dynamic flow, maximum dynamic flow, universal maximum flow, quickest path and quickest flow) are described. A special feature of the presented approach is the fact, that travel times of evacuees are not restricted to be constant, but may be density dependent. Using multicriteria optimization priority regions and blockage due to fire or smoke may be considered. It is shown how the modelling can be done using time parameter either as discrete or continuous parameter. Microscopic models are able to model the individual evacuee’s characteristics and the interaction among evacuees which influence their movement. Due to the corresponding huge amount of data one uses simulation approaches. Some probabilistic laws for individual evacuee’s movement are presented. Moreover ideas to model the evacuee’s movement using cellular automata (CA) and resulting software are presented. In this paper we will focuss on macroscopic models and only summarize some of the results of the microscopic approach. While most of the results are applicable to general evacuation situations, we concentrate on building evacuation.

The problem of optimizing the control of two oversaturated traffic intersections is solved by using the semi-graphical methods employed in a previous paper for an isolated intersection. As in the case of a single intersection the optimum control involves values of the control variables that lie along edges of the control region, which in this case is defined by the permissible ranges of the green phase splits. An analytical formulation of the method using Pontryagin’s control theory is also given.

In this work we present a mixed-integer model for the optimal design of production/transportation systems. In contrast to
standard design problems, our model is originally based on a coupled system of differential equations capturing the dynamics
of manufacturing processes and stocks. The problem is to select an optimal parameter configuration from a predefined set such
that respective constraints are fulfilled. We focus on single commodity flows over large time scales as well as highly interconnected
networks and propose a suitable start heuristic to ensure feasibility and to speed up the solution procedure.
KeywordsProduction systems–Mixed integer models–Heuristics

The purpose of the paper is to adapt the classical LWR (Lighthill-Whitham-Richards) model, in its continuous version, to networks,
in the context of dynamic assignment. This implies severalspecific adaptations of the basic model: introduction of partial
flows, possibly inhomogeneous flows on links, and intersection modeling. The latter proves particularly difficult, and we
discuss three different modeling approaches: extended versus pointwise intersection models, and flow maximization. We show
that all three types of models are actually closely related, and compatible with the link flow models. The concepts of local
traffic supply and demand prove to be essential, both for link and for intersection modeling. A brief comparison with experimental
merge data gives some support to the phenomenological models introduced in the paper.

In this paper, three heuristic solution algorithms, (the Dive-and-Fix method, the Ratio-cluster method, and the Cumulative-departure method) are specially designed to solve the traffic signal control problem formulated as a 0-1 mixed-integer linear programming problem with cell transmission model. These three solution algorithms are based on two fundamental approaches. First, the 0-1 mixed-integer linear program is solved via linear relaxation (LR). Second, the non-integer solutions obtained from the LR are converted into the integer solutions by taking advantage of the underlying physical mechanism embedded in the LR solutions that lead to the optimal signal control. In particular, proportional capacities for different approaches and the cumulative exit flow at each intersection obtained from the LR solutions are utilized to determine green time allocation for each approach. It is demonstrated that the near-optimal solutions obtained with these algorithms are very close to the optimal solutions under both uncongested and congested traffic conditions.

A simple theory of traffic flow is developed by replacing individual vehicles with a continuous “fluid” density and applying an empirical relation between speed and density. Characteristic features of the resulting theory are a simple “graph-shearing” process for following the development of traffic waves in time and the frequent appearance of shock waves. The effect of a traffic signal on traffic streams is studied and found to exhibit a threshold effect wherein the disturbances are minor for light traffic but suddenly build to large values when a critical density is exceeded.

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.

We introduce a model that describes heavy traffic on a network of unidirectional roads. The model consists of a system of initial-boundary value problems for nonlinear conservation laws. We uniquely formulate and solve the Riemann problem for such a system and, based on this, then show existence of a solution to the Cauchy problem.

This paper is concerned with a fluidodynamic model for traffic flow. More precisely, we consider a single conservation law, deduced from the conservation of the number of cars, defined on a road network that is a collection of roads with junctions. The evolution problem is underdetermined at junctions; hence we choose to have some fixed rules for the distribution of traffic plus optimization criteria for the flux. We prove existence of solutions to the Cauchy problem and we show that the Lipschitz continuous dependence by initial data does not hold in general, but it does hold under special assumptions. Our method is based on a wave front tracking approach [A. Bressan, Hyperbolic Systems of Conservation Laws. The One-dimensional Cauchy Problem, Oxford University Press, Oxford, UK, 2000] and works also for boundary data and time-dependent coefficients of traffic distribution at junctions, including traffic lights.

The paper deals with a fluid dynamic model for supply chains. A mixed continuum-discrete model is proposed and possible choices of solutions at nodes guaranteeing the conservation of fluxes are discussed. Fixing a rule a Riemann solver is defined and existence of solutions to Cauchy problems is proved.

In this paper we first show how the Extended Linear Complementarity Problem, which is a mathematical programming problem, can be used to design optimal switching schemes for a class of switched systems with linear dynamics subject to saturation. More specifically, we consider the determination of the optimal switching time instants (the switching sequences are acyclic, but the phase sequence is pre-fixed). Although this method yields globally optimal switching time sequences, it is not feasible in practice due to its computational complexity. Therefore, we also discuss some approximations that lead to suboptimal switching time sequences that can be computed very efficiently and for which the value of the objective function is close to the global optimum. Finally we use these results to design optimal switching time sequences for a traffic signal controlled intersection so as to minimize criteria such as average queue length, worst case queue length, average waiting time, and so on.