Multiple-phase train trajectory optimization with signalling and operational constraints

The train trajectory optimization problem aims at finding the optimal speed profiles and control regimes for a safe, punctual, comfortable, and energy-efficient train operation. This paper studies the train trajectory optimization problem with consideration of general operational constraints as well as signalling constraints. Operational constraints refer to time and speed restrictions from the actual timetable, while signalling constraints refer to the influences of signal aspects and automatic train protection on train operation. A railway timetable provides each train with a train path envelope, which consists of a set of positions on the route with a specified target time and speed point or window. The train trajectory optimization problem is formulated as a multiple-phase optimal control model and solved by a pseudospectral method. This model is able to capture varying gradients and speed limits, as well as time and speed constraints from the train path envelope. Train trajectory calculation methods under delay and no-delay situations are discussed. When the train follows the planned timetable, the train trajectory calculation aims at minimizing energy consumption, whereas in the case of delays the train trajectory is re-calculated to track the possibly adjusted timetable with the aim of minimizing delays as well as energy consumption. Moreover, the train operation could be affected by yellow or red signals, which is taken into account in the train speed regulation. For this purpose, two optimization policies are developed with either limited or full information of the train ahead. A local signal response policy ensures that the train makes correct and quick responses to different signalling aspects, while a global green wave policy aims at avoiding yellow signals and thus proceed with all green signals. The method is applied in a case study of two successive trains running on a corridor with various delays showing the benefit of accurate predictive information of the leading train on energy consumption and train delay of the following train.     

A dual decomposition method for sector capacity constrained traffic flow optimization

An aggregate air traffic flow model based on a multicommodity network is used for traffic flow management in the National Airspace System. The problem of minimizing the total travel time of flights in the National Airspace System of the United States, subject to sector capacity constraints, is formulated as an Integer Program. The resulting solution achieves optimal delay control. The Integer Program implemented for the scenarios investigated has billions of variables and constraints. It is relaxed to a Linear Program for computational efficiency. A dual decomposition method is applied to solve the large scale Linear Program in a computationally tractable manner. A rounding algorithm is developed to map the Linear Program solution to a physically acceptable result, and is implemented for the entire continental United States. A 2-h traffic flow management problem is solved with the method.     

Globally energy-optimal speed planning for road vehicles on a given route

This paper provides a globally optimal solution to an important problem: given a real-world route, what is the most energy-efficient way to drive a vehicle from the origin to the destination within a certain period of time. Along the route, there may be multiple stop signs, traffic lights, turns and curved segments, roads with different grades and speed limits, and even leading vehicles with pre-known speed profiles. Most of such route information and features are actually constraints to the optimal vehicle speed control problem, but these constraints are described in two different domains. The most important concept in solving this problem is to convert the distance-domain route constraints to some time-domain state and input constraints that can be handled by optimization methods such as dynamic programming (DP). Multiple techniques including cost-to-go function interpolation and parallel computing are used to reduce the computation of DP and make the problem solvable within a reasonable amount of time on a personal computer.     

Ferry service network design: optimal fleet size,routing, and scheduling

The study formulated a ferry network design problem by considering the optimal fleet size, routing, and scheduling for both direct and multi-stop services. The objective function combines both the operator and passengers’ performance measures. Mathematically, the model is formulated as a mixed integer multiple origin–destination network flow problem with ferry capacity constraints. To solve this problem of practical size, this study developed a heuristic algorithm that exploits the polynomial-time performance of shortest path algorithms. Two scenarios of ferry services in Hong Kong were solved to demonstrate the performance of the heuristic algorithm. The results showed that the heuristic produced solutions that were within 1.3% from the CPLEX optimal solutions. The computational time is within tens of seconds even for problem size that is beyond the capability of CPLEX.     

A note on the optimal control of an oversaturated intersection

An existence condition for the solution of an optimal control problem for a single traffic intersection is derived in this note. The optimal control problem has been initially formulated and solved in the pioneering work of Gazis (1963, 1964) and coworkers. It is shown that existence conditions are fairly restrictive if control is required to dissolve all intersection queues at the same time.     

Fuel and time savings by flying continuous cruise climbs: Estimating the benefit pools for maximum range operations

This paper estimates the benefits, in terms of fuel and time, which continuous climb operations can save during the cruise phase of the flights, assuming maximum range operations. Based on previous works, a multiphase optimal control problem is solved by means of numerical optimization and using accurate aircraft performance data from the manufacturer. Optimal conventional trajectories (subject to current air traffic management practices and constraints) are computed and compared with ideal continuous operations only subject to aircraft performance constraints. Trip fuel and time for both concepts of operations are quantified for two aircraft types (a narrow-body and a wide-body airplane) and a representative set of different trip distances and landing masses. Results show that the continuous cruise phase can lead to fuel savings ranging from 0.5% to 2% for the Airbus A320, while for an Airbus A340 the dispersion is lower and savings lie in between 1% and 2%. Interestingly, trip time is also reduced between 1% and 5%.     

A highway corridor planning model: QROAD

This model calculates an optimal investment plan for a highway corridor or number of corridors, subject to budget constraints. The available options include upgrading the current alignment, constructing a bypass highway over a different alignment, or various combinations. The budget constraints can be specified as a total budget restriction, or as an available budget each period. The highway system is described by K different road links. Each link consists of the current alignment which may be described by any number of sections, and a bypass section over a new alignment. The model finds the construction plan for each link that maximizes discounted benefits, subject to the financial constraints on the maintenance and capital expenditures. The problem is formulated as a large combinatorial optimization problem. A Lagrangian relaxation of the budget constraints is used, and the problem decomposes by link. A dynamic programming (DP) model is used to solve for the optimal expansion path for each link, given the dual variables. The sub-gradient dual optimization problem is a linear programming problem which is solved for the optimal dual variables. An application is presented based on the World Bank's Third National Highway Project in India, which is a US$1.3 billion project for upgrading approximately 2000 km of the Indian National Highway System. The project was approved based on results from this model.     

Total unimodularity and decomposition method for large-scale air traffic cell transmission model

In an earlier work, Sun and Bayen built a Large-Capacity Cell Transmission Model for air traffic flow management. They formulated an integer programming problem of minimizing the total travel time of flights in the National Airspace System of the United States subject to sector capacity constraints. The integer program was relaxed to a linear program for computational efficiency. In this paper the authors formulate the optimization problem in a standard linear programming form. We analyze the total unimodular property of the constraint matrix, and prove that the linear programming relaxation generates an optimal integral solution for the original integer program. It is guaranteed to be optimal and integral if solved by a simplex related method. In order to speed up the computation, we apply the Dantzig–Wolfe Decomposition algorithm, which is shown to preserve the total unimodularity of the constraint matrix. Finally, we evaluate the performances of Sun and Bayen’s relaxation solved by the interior point method and our decomposition algorithm with large-scale air traffic data.     

A new algorithm for optimal signal control in congested networks

Optimization of traffic lights in a congested network is formulated as a linear programming problem. The formulation adapted here takes into account particular capacity constraints for road links and for intersections. A necessary prerequisite for the determination of optimal green times is that representative a-priori information about the origin-destination and route choice pattern inside the network is available. Because any particular control strategy temporarily alters the effective turning rates at intersections, an iterative procedure is proposed here which accomplishes convergence of optimal signal control and resulting O-D flows. The efficiency of this optimization procedure is demonstrated in a case study for a network with fifteen intersections.     

Optimal joint distance and time toll for cordon-based congestion pricing

This paper addresses the optimal toll design problem for the cordon-based congestion pricing scheme, where both a time-toll and a nonlinear distance-toll (i.e., joint distance and time toll) are levied for each network user’s trip in a pricing cordon. The users’ route choice behaviour is assumed to follow the Logit-based stochastic user equilibrium (SUE). We first propose a link-based convex programming model for the Logit-based SUE problem with a joint distance and time toll pattern. A mathematical program with equilibrium constraints (MPEC) is developed to formulate the optimal joint distance and time toll design problem. The developed MPEC model is equivalently transformed into a semi-infinite programming (SIP) model. A global optimization method named Incremental Constraint Method (ICM) is designed for solving the SIP model. Finally, two numerical examples are used to assess the proposed methodology.     

An online speed profile generation approach for efficient airport ground movement

The precise guidance and control of taxiing aircraft based on four-dimensional trajectories (4DTs) has been recognised as a promising means to ensure safe and efficient airport ground movement in the context of ever growing air traffic demand. In this paper, a systematic approach for online speed profile generation is proposed. The aim is to generate fuel-efficient speed profiles respecting the timing constraints imposed by routing and scheduling, which ensures conflict-free movement of aircraft in the planning stage. The problem is first formulated as a nonlinear optimisation model, which uses a more flexible edge-based speed profile definition. A decomposed solution approach (following the framework of matheuristic) is then proposed to generate feasible speed profiles in real time. The decomposed solution approach reduces the nonlinear optimisation model into three tractable constituent problems. The control point arrival time allocation problem is solved using linear programming. The control point speed allocation problem is solved using particle swarm optimisation. And the complete speed profile between control points is determined using enumeration. Finally, improved speed profiles are generated through further optimisation upon the feasible speed profiles. The effectiveness and advantages of the proposed approach are validated using datasets of real-world airports.     

Constraint reformulation and a Lagrangian relaxation-based solution algorithm for a least expected time path problem

Using a sample-based representation scheme to capture spatial and temporal travel time correlations, this article constructs an integer programming model for finding the a priori least expected time paths. We explicitly consider the non-anticipativity constraint associated with the a priori path in a time-dependent and stochastic network, and propose a number of reformulations to establish linear inequalities that can be easily dualized by a Lagrangian relaxation solution approach. The relaxed model is further decomposed into two sub-problems, which can be solved directly by using a modified label-correcting algorithm and a simple single-value linear programming method. Several solution algorithms, including a sub-gradient method, a branch and bound method, and heuristics with additional constraints on Lagrangian multipliers, are proposed to improve solution quality and find approximate optimal solutions. The numerical experiments investigate the quality and computational efficiency of the proposed solution approach.     

Weather impact on containership routing in closed seas: A chance-constraint optimization approach

Weather conditions have a strong effect on the operation of vessels and unavoidably influence total time at sea and associated transportation costs. The velocity and direction of the wind in particular may considerably affect travel speed of vessels and therefore the reliability of scheduled maritime services. This paper considers weather effects in containership routing; a stochastic model is developed for determining optimal routes for a homogeneous fleet performing pick-ups and deliveries of containers between a hub and several spoke ports, while incorporating travel time uncertainties attributed to the weather. The problem is originally formulated as a chance-constrained variant of the vehicle routing problem with simultaneous pick-ups and deliveries and time constraints and solved using a genetic algorithm. The model is implemented to a network of island ports of the Aegean Sea. Results on the application of algorithm reveal that a small fleet is sufficient enough to serve network’s islands, under the influence of minor delays. A sensitivity analysis based on alternative scenarios in the problem’s parameters, leads to encouraging conclusions with respect to the efficiency and robustness of the algorithm.     

Dynamic user equilibrium with side constraints for a traffic network: Theoretical development and numerical solution algorithm

This paper investigates a traffic volume control scheme for a dynamic traffic network model which aims to ensure that traffic volumes on specified links do not exceed preferred levels. The problem is formulated as a dynamic user equilibrium problem with side constraints (DUE-SC) in which the side constraints represent the restrictions on the traffic volumes. Travelers choose their departure times and routes to minimize their generalized travel costs, which include early/late arrival penalties. An infinite-dimensional variational inequality (VI) is formulated to model the DUE-SC. Based on this VI formulation, we establish an existence result for the DUE-SC by showing that the VI admits at least one solution. To analyze the necessary condition for the DUE-SC, we restate the VI as an equivalent optimal control problem. The Lagrange multipliers associated with the side constraints as derived from the optimality condition of the DUE-SC provide the traffic volume control scheme. The control scheme can be interpreted as additional travel delays (either tolls or access delays) imposed upon drivers for using the controlled links. This additional delay term derived from the Lagrange multiplier is compared with its counterpart in a static user equilibrium assignment model. If the side constraint is chosen as the storage capacity of a link, the additional delay can be viewed as the effort needed to prevent the link from spillback. Under this circumstance, it is found that the flow is incompressible when the link traffic volume is equal to its storage capacity. An algorithm based on Euler’s discretization scheme and nonlinear programming is proposed to solve the DUE-SC. Numerical examples are presented to illustrate the mechanism of the proposed traffic volume control scheme.     

Trial-and-error method for congestion pricing scheme under side-constrained probit-based stochastic user equilibrium conditions

A toll pattern that can restrict link flows on the tolled links to some predetermined thresholds is named as effective toll solution, which can be theoretically obtained by solving a side-constraint traffic assignment problem. Considering the practical implementation, this paper investigates availability of an engineering-oriented trial-and-error method for the effective toll pattern of cordon-based congestion pricing scheme, under side-constrained probit-based stochastic user equilibrium (SUE) conditions. The trial-and-error method merely requires the observed traffic counts on each entry of the cordon. A minimization model for the side-constrained probit-based SUE problem with elastic demand is first proposed and it is shown that the effective toll solution equals to the product of value of time and optimal Lagrangian multipliers with respect to the side constraints. Then, employing the Lagrangian dual formulation of the minimization method, this paper has built a convergent trial-and-error method. The trial-and-error method is finally tested by a numerical example developed from the cordon-based congestion pricing scheme in Singapore.     

A link-node complementarity model and solution algorithm for dynamic user equilibria with exact flow propagations

In this paper, we propose a link-node complementarity model for the basic deterministic dynamic user equilibrium (DUE) problem with single-user-class and fixed demands. The model complements link-path formulations that have been widely studied for dynamic user equilibria. Under various dynamic network constraints, especially the exact flow propagation constraints, we show that the continuous-time dynamic user equilibrium problem can be formulated as an infinite dimensional mixed complementarity model. The continuous-time model can be further discretized as a finite dimensional non-linear complementarity problem (NCP). The proposed discrete-time model captures the exact flow propagation constraints that were usually approximated in previous studies. By associating link inflow at the beginning of a time interval to travel times at the end of the interval, the resulting discrete-time model is predictive rather than reactive. The solution existence and compactness condition for the proposed model is established under mild assumptions. The model is solved by an iterative algorithm with a relaxed NCP solved at each iteration. Numerical examples are provided to illustrate the proposed model and solution approach. We particularly show why predictive DUE is preferable to reactive DUE from an algorithmic perspective.     

Impact analysis of cordon-based congestion pricing on mode-split for a bimodal transportation network

This paper investigates the impact of cordon-based congestion pricing scheme on the mode-split of a bimodal transportation network with auto and rail travel modes. For any given toll-charge pattern, its impact on the mode-split can be estimated by solving a combined mode-split and traffic-assignment problem. Using a binary logit model for the mode-split, the combined problem is converted into a traffic-assignment problem with elastic demand. Probit-based stochastic user equilibrium (SUE) principle is adopted for this traffic-assignment problem, and a continuously distributed value of time (VOT) is assumed to convert the toll charges and transit fares into time-units. This combined mode-split and traffic-assignment problem is then formulated as a fixed-point model, which can be solved by a convergent Cost Averaging method. The combined mode-split and traffic-assignment problem is then used to analyze a multimodal toll design problem for cordon-based congestion pricing scheme, with the aim of increasing the mode-share of public transport system to a targeted level. Taking the fixed-point model as a constraint, the multimodal toll design problem is thus formulated as a mathematical programming with equilibrium constraints (MPEC) model. A genetic algorithm (GA) is employed to solve this MPEC model, which is then numerical validated by a network example.     

Optimal vehicle speed trajectory on a signalized arterial with consideration of queue

Vehicle speed trajectory significantly impacts fuel consumption and greenhouse gas emissions, especially for trips on signalized arterials. Although a large amount of research has been conducted aiming at providing optimal speed advisory to drivers, impacts from queues at intersections are not considered. Ignoring the constraints induced by queues could result in suboptimal or infeasible solutions. In this study, a multi-stage optimal control formulation is proposed to obtain the optimal vehicle trajectory on signalized arterials, where both vehicle queue and traffic light status are considered. To facilitate the real-time update of the optimal speed trajectory, a constrained optimization model is proposed as an approximation approach, which can be solved much quicker. Numerical examples demonstrate the effectiveness of the proposed optimal control model and the solution efficiency of the proposed approach.