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A Cell-Based Traffic Control Formulation: Strategies and Benefits of Dynamic Timing Plans
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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.
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... Similar to the MFD, the CTM is also employed to assess traffic system performance and has been widely utilized in optimizing traffic signals at freeway ramps and urban intersections. Its applications range from early-stage local freeway ramp metering (Gomes and Horowitz, 2006) to the coordinated control of multiple ramps (Pang and Yang, 2020), and from optimizing signals at a single intersection (Lo, 2001) to distributed control across multiple intersections (Timotheou et al., 2014). ...
Perimeter control is a traffic management approach aimed at regulating vehicular accumulation within urban regional networks by managing flows on all border-crossing roads. Methods based on the macroscopic fundamental diagram (MFD) fall short in providing specific metering for individual roads. Recent advancements in the cell transmission model (CTM) have attempted to address this limitation but are hindered by their reliance on centralized control, which requires the availability of full information and authority over traffic generation sites. Our study proposes an innovative decentralized, game-theoretical framework for perimeter control to address these practical challenges. It is structured around two key groups of agents: perimeter agents, tasked with managing border roads, and interior agents, focused on traffic within generation sites. The framework also incorporates mechanisms for interactions between these agents and the road network, aiming to optimize their individual utilities. Additionally, we have developed a multi-agent reinforcement learning (RL) algorithm, extending the mean-field theory concept, to address the complexity of simultaneous learning by multiple agents.
... (25)-(26) in model M2 are non-linear. We introduce a set of binary variables to transform them into equivalent linear constraints, similar to those in Lo (1999Lo ( , 2001 and Wang and Lo (2010). Eq. (26) can be expressed as below: ...
This paper develops optimization models for Pareto-improving seat allocation schemes in high-speed railway (HSR) networks, where time-dependent demand and equilibrium travel choices are incorporated. Under the Pareto-improving seat allocation scheme, at least some passengers are strictly better off, and no passenger is worse off, when compared to the situation with no seat allocation scheme. In particular, two scenarios with fixed and elastic passenger demands are considered. The passenger travel choice equilibrium conditions are taken as constraints for the seat allocation optimization model, which can be further reformulated into a mixed-integer linear programming (MILP) with the help of linearization, relaxation and outer-approximation techniques. The optimal solution then can be obtained by solving the proposed MILP. The numerical examples on two real-world regional HSR systems are presented to illustrate the proposed Pareto-improving seat allocation optimization approach.
... Obviously, there are products of decision variables, which are non-linear and non-concave and difficult to solve. We introduce a set of binary variables to transform them into an equivalent set of constraints, similar to those in, e.g., Lo (1999Lo ( , 2001 and Wang and Lo (2010). Eq. (12) can be expressed as ...
This paper examines the integrated optimization of train stopping plan and seat allocation scheme in railway systems, where equilibrium passenger travel choice under elastic demand is considered. The integrated optimization problem is formulated as a non-concave and non-linear mixed-integer mathematical model, where the objective is to maximize the system net benefit considering ticket revenue, consumer surplus, and cost associated with train stoppings. The integrated optimization model can be reformulated into a mixed-integer linear programming (MILP) model based on a series of linearization, relaxation, and outer-approximation techniques, which can then be solved by commercial MILP solvers (e.g., GUROBI). We also compare the integrated optimization approach with that when the train stopping plan and seat allocation are optimized separately and identify the potential benefits. Numerical studies have been conducted on a small-scale example, the Zhengzhou-Xi’an and Shanghai-Beijing high-speed railway corridors to illustrate the proposed model and solution approach.
... SO-DTA quantifies the best possible performance of a traffic network based on the dynamic extension of Wardrop's second principle (Wardrop 1952). For its advantage of exploring the spatial-temporal traffic dynamics to achieve the best performance of traffic networks, the research of the application of SO-DTA problem focuses on many different areas, such as emission reduction (Lu et al. 2016, Ma et al. 2017, Long et al. 2018, Tan et al. 2021, congestion mitigation (Levin 2017, Samaranayake et al. 2018, Liu et al. 2020, parking services (Levin 2019, Qian & Rajagopal 2014, traffic signal control (Lo 2001, Han et al. 2016, emergency evacuation (Liu et al. 2006, Chiu et al. 2007) and network design (Waller & Ziliaskopoulos 2001, Waller et al. 2006. Different from human-driven vehicles (HVs), AVs show great potential in improving control efficiency and may play an important role in achieving SO-DTA (Wang et al. 2018). ...
This study develops the headway control framework in a fully automated road network, as we believe headway of Automated Vehicles (AVs) is another influencing factor to traffic dynamics in addition to conventional vehicle behaviors (e.g. route and departure time choices). Specifically, we aim to search for the optimal time headway between AVs on each link that achieves the network-wide system optimal dynamic traffic assignment (SO-DTA). To this end, the headway-dependent fundamental diagram (HFD) and headway-dependent double queue model (HDQ) are developed to model the effect of dynamic headway on roads, and a dynamic network model is built. It is rigorously proved that the minimum headway could always achieve SO-DTA, yet the optimal headway is non-unique. Motivated by these two findings, this study defines a novel concept of maximin headway, which is the largest headway that still achieves SO-DTA in the network. Mathematical properties regarding maximin headway are analyzed and an efficient solution algorithm is developed. Numerical experiments on both a small and large network verify the effectiveness of the maximin headway control framework as well as the properties of maximin headway. This study sheds light on deriving the desired solution among the non-unique solutions in SO-DTA and provides implications regarding the safety margin of AVs under SO-DTA.
Model-based traffic state estimation is used to reproduce traffic flow states from available observation data to assist traffic control and management. Owing to temporal and spatial observation limitations, numerous unknown traffic states and parameters exist in a nonlinear traffic flow model. These cannot be accurately estimated using filter methods with unavoidable parametric assumptions. In addition, the difficulty of estimation increases owing to an increased number of diverse traffic flow states and more unfavorable observation conditions at signalized intersections compared with those on the freeway. To overcome these problems, in this study, we developed switching state–space models to approximate the description of dynamic traffic flows at signalized intersections. By setting the traffic flow rate at the upstream and downstream boundaries of the road as observation data, we constructed a Bayesian learning framework in which all unknown variables in the models can be learned from the observed data. Finally, the utility of our method is demonstrated with the synthetic and the real NGSIM data. The result demonstrated that the proposed method could perform reasonably well in estimating the traffic flow dynamic process and states at signalized intersections, and could constitute an efficient scheme that avoids critical dependence on parametric calibrations and assumptions.
Connected and Autonomous Vehicles (CAVs) are a promising technology that is ready to be deployed in the near future to improve the traffic efficiency and safety as well as environment. Extensive studies have been done to investigate the potential performance of CAVs on freeways, at roundabouts, and conventional intersections. Nevertheless, innovative intersections, as an important component of today’s transportation infrastructure, have been seldom investigated in relation to the performance of CAVs. Hence, this research is designed to examine how CAV technologies can influence the performance of a superstreet, one of the popular innovative intersection designs. In this research, the car-following model, platooning, trajectory planning, and adaptive signal control are specified for CAVs and signal controllers in a superstreet. An equivalent conventional intersection with the same lane configurations is also constructed in the simulation environment to make a fair comparison and gain important insights. More importantly, the findings from this research may provide references for studies on other innovative intersections which share similar design characteristics.
We introduce a stochastic discrete automaton model to freeway traffic. Monte-Carlo simulations of the model show a transition from laminar traffic flow to start-stop-waves with increasing vehicle density, as is observed in real freeway traffic. For special cases analytical results can be obtained.
The formulation and solution of a new algorithm for queue management and coordination of traffic signals along oversaturated arterials are presented. Existing traffic-control and signal-coordination algorithms deal only with undersaturated steady-state traffic flow conditions. No practical algorithms are readily available for oversaturated flow conditions. The main idea of the procedure is to manage queue formation and dissipation on system links so that traffic flow is maximized by efficiently using all green time, preventing formation of de facto red, accounting for the non-steady-state conditions, and providing time-dependent control measures. The problem is formulated as a throughput maximization problem subject to state and control variables. The solution is then obtained using genetic algorithms. The results show that the control procedure can produce dynamic and responsive control so that traffic progression is attained and all undesirable conditions such as queue spill-back and de facto red are avoided.
The dynamics of formation and dissipation of queues at isolated signalized intersections are investigated by analyzing the vehicle conservation equation along the street. Solution of the conservation equation is obtained by the method of characteristics for general initial and boundary conditions, and the effect of the control variables (cycle length, green and red intervals) and system parameters (arrival rates, capacity) on the length of the queue is examined. The theoretical results are obtained by examining the dynamics of the shock waves which are formed because of the intermitted service of traffic by the signal. These results include analytical criteria for the development of saturated or unsaturated situations, analytical expressions for the maximum queue length and limits on the control variables so that the queues remain bounded. The basic findings of this study can be employed for the solution of the traffic signal optimal control problem.
Optimal control of oversaturated signals is one of the major concerns of traffic engineering practice today. Although some control strategies have been proposed in the past, due to their limitations they have not been used in practice. One of their major deficiencies is that the effects of queue length constraints on the signal operation have been inadequately investigated. Thus, the optimal control strategy with state variable constraints is studied in this paper. The optimal policy minimizing intersection delay subject to queue length constraints is to switch the signals as soon as the queues are at their limits so that the input and output flows are balanced. Cycle length can remain constant if only one queue length constraint is imposed but it must be free to vary if constraints are imposed on more than one queue. The conditions under which the problem is impossible are stated. A numerical solution method is proposed for determining the optimal control for the case in which the intersection demand is predictable for the entire control period.
In this paper and in part II, we give the theory of a distinctive type of wave motion, which arises in any one-dimensional flow problem when there is an approximate functional relation at each point between the flow q (quantity passing a given point in unit time) and concentration k (quantity per unit distance). The wave property then follows directly from the equation of continuity satisfied by q and k. In view of this, these waves are described as 'kinematic', as distinct from the classical wave motions, which depend also on Newton's second law of motion and are therefore called 'dynamic'. Kinematic waves travel with the velocity partial q/partial k, and the flow q remains constant on each kinematic wave. Since the velocity of propagation of each wave depends upon the value of q carried by it, successive waves may coalesce to form 'kinematic shock waves'. From the point of view of kinematic wave theory, there is a discontinuous increase in q at a shock, but in reality a shock wave is a relatively narrow region in which (owing to the rapid increase of q) terms neglected by the flow-concentration relation become important. The general properties of kinematic waves and shock waves are discussed in detail in section 1. One example included in section 1 is the interpretation of the group-velocity phenomenon in a dispersive medium as a particular case of the kinematic wave phenomenon. The remainder of part I is devoted to a detailed treatment of flood movement in long rivers, a problem in which kinematic waves play the leading role although dynamic waves (in this case, the long gravity waves) also appear. First (section 2), we consider the variety of factors which can influence the approximate flow-concentration relation, and survey the various formulae which have been used in attempts to describe it. Then follows a more mathematical section (section 3) in which the role of the dynamic waves is clarified. From the full equations of motion for an idealized problem it is shown that at the 'Froude numbers' appropriate to flood waves, the dynamic waves are rapidly attenuated and the main disturbance is carried downstream by the kinematic waves; some account is then given of the behaviour of the flow at higher Froude numbers. Also in section 3, the full equations of motion are used to investigate the structure of the kinematic shock; for this problem, the shock is the 'monoclinal flood wave' which is well known in the literature of this subject. The final sections (section section 4 and 5) contain the application of the theory of kinematic waves to the determination of flood movement. In section 4 it is shown how the waves (including shock waves) travelling downstream from an observation point may be deduced from a knowledge of the variation with time of the flow at the observation point; this section then concludes with a brief account of the effect on the waves of tributaries and run-off. In section 5, the modifications (similar to diffusion effects) which arise due to the slight dependence of the flow-concentration curve on the rate of change of flow or concentration, are described and methods for their inclusion in the theory are given.
The existing network traffic signal optimization formulations usually do not include traffic flow models, except for control schemes such as SCOOT that use simulation for heuristic optimization. Other conventional models normally use isolated intersection optimization with traffic arrival prediction using detector information, or optimization schemes based on green bandwidth. A complete formulation of the problem is presented, including explicit constraints to model the movement of traffic along the streets between the intersections in a time-expanded network, as well as constraints to capture the permitted movements from modern signal controllers. The platoon dispersion model used is the well-known Robertson's model, which forms linear constraints. Thus, it is a rare example of a traffic simulation being analytically embedded in an optimization formulation. The formulation is an integer-linear program and does not assume fixed cycle lengths or phase sequences. It assumes full information on external inputs but can be incorporated in a sensor-based environment as well as in a feedback control framework. The formulation is an integer-linear program that may not be efficiently solved with standard simplex and branch and bound techniques. Network programming formulations to handle the linear platoon dispersion equations and the integer constraints at the intersections are discussed. A special-purpose network simplex algorithm for fast solutions is also mentioned.