Robust constrained control of uncertain macroscopic fundamental diagram networks

This paper considers modeling and control of uncertain Macroscopic Fundamental Diagram (MFD) systems for multiple-region networks. First, the nonlinear vehicle conservation equations based on MFD dynamics, presented in earlier publications, are transformed to linear equations with parameter uncertainties. The parameter uncertainties include the destination decomposition fractions, that are difficult to estimate in reality. Then, the uncertain linear model is utilized to design a robust feedback controller by an interpolation-based approach. This approach (i) guarantees robustness against all parameter uncertainties, (ii) handle control and state constraints, and (iii) present a computationally cheap solution. The main idea is to interpolate between (i) a stabilizing outer controller that respects the control and state constraints and (ii) an inner robustly stable controller designed by any method. The robust control is further challenged to deal with different relative locations of reference accumulation points on the MFD diagrams. Numerical results for a two-region system show that the uncertain linear model can replace the nonlinear model for modeling and control. Moreover, the robust control law is presented as implicit and explicit solutions, where in the implicit case one linear programming (LP) problem is solved at each time instant, while in the explicit case, the control law is shown as a piecewise affine function of state. Finally, a comparison between the interpolating controller and other controllers in the literature is carried out. The results demonstrate the performance advantages from applying the robust interpolating controller.     

Multi-agent model predictive control of signaling split in urban traffic networks

The operation of large dynamic systems such as urban traffic networks remains a challenge in control engineering to a great extent due to their sheer size, intrinsic complexity, and nonlinear behavior. Recently, control engineers have looked for unconventional means for modeling and control of complex dynamic systems, in particular the technology of multi-agent systems whose appeal stems from their composite nature, flexibility, and scalability. This paper contributes to this evolving technology by proposing a framework for multi-agent control of linear dynamic systems, which decomposes a centralized model predictive control problem into a network of coupled, but small sub-problems that are solved by the distributed agents. Theoretical results ensure convergence of the distributed iterations to a globally optimal solution. The framework is applied to the signaling split control of traffic networks. Experiments conducted with simulation software indicate that the multi-agent framework attains performance comparable to conventional control. The main advantages of the multi-agent framework are its graceful extension and localized reconfiguration, which require adjustments only in the control strategies of the agents in the vicinity.     

Controller design for gating traffic control in presence of time-delay in urban road networks

Recent studies demonstrated the efficiency of feedback-based gating control in mitigating congestion in urban networks by exploiting the notion of macroscopic or network fundamental diagram (MFD or NFD). The employed feedback regulator of proportional-integral (PI)-type targets an operating NFD point of maximum throughput to enhance the mobility in the urban road network during the peak period, under saturated traffic conditions. In previous studies, gating was applied directly at the border of the protected network (PN), i.e. the network part to be protected from over-saturation. In this work, the recently developed feedback-based gating concept is applied at junctions located further upstream of the PN. This induces a time-delay, which corresponds to the travel time needed for gated vehicles to approach the PN. The resulting extended feedback control problem can be also tackled by use of a PI-type regulator, albeit with different gain values compared to the case without time-delay. Detailed procedures regarding the appropriate design of related feedback regulators are provided. In addition, the developed feedback concept is shown to work properly with very long time-steps as well. A large part of the Chania, Greece, urban network, modelled in a microscopic simulation environment under realistic traffic conditions, is used as test-bed in this study. The reported results demonstrate a stable and efficient behaviour and improved mobility of the overall network in terms of mean speed and travel time.     

Robust identification of air traffic flow patterns in Metroplex terminal areas under demand uncertainty

Multi-Airport Systems (MAS), or Metroplexes, serve air traffic demand in cities with two or more airports. Due to the spatial proximity and operational interdependency of the airports, Metroplex airspaces are characterized by high complexity, and current system structures fail to provide satisfactory utilization of the available airspace resources. In order to support system-level design and management towards increased operational efficiency in such systems, an accurate depiction of major demand patterns is a prerequisite. This paper proposes a framework for the robust identification of significant air traffic flow patterns in Metroplex systems, which is aligned with the dynamic route service policy for the effective management of Metroplex operations. We first characterize deterministic demand through a spatio-temporal clustering algorithm that takes into account changes in the traffic flows over the planning horizon. Then, in order to handle uncertainties in the demand, a Distributionally Robust Optimization (DRO) approach is proposed, which takes into account demand variations and prediction errors in a robust way to ensure the reliability of the demand identification. The DRO-based approach is applied on pre-tactical (i.e. one-day planning) as well as operational levels (i.e. 2-h rolling horizon). The framework is applied to Time Based Flow Management (TBFM) data from the New York Metroplex. The framework and results are validated by Subject Matter Experts (SMEs).     

Robust calibration of macroscopic traffic simulation models using stochastic collocation

The predictions of a well-calibrated traffic simulation model are much more valid if made for various conditions. Variation in traffic can arise due to many factors such as time of day, work zones and weather. Calibration of traffic simulation models for traffic conditions requires larger datasets to capture the stochasticity in traffic conditions. In this study we use datasets spanning large time periods to incorporate variability in traffic flow, speed for various time periods. However, large data poses a challenge in terms of computational effort. With the increase in number of stochastic factors, the numerical methods suffer from the curse of dimensionality. In this study, we propose a novel methodology to address the computational complexity due to the need for the calibration of simulation models under highly stochastic traffic conditions. This methodology is based on sparse grid stochastic collocation, which, treats each stochastic factor as a different dimension and uses a limited number of points where simulation and calibration are performed. A computationally efficient interpolant is constructed to generate the full distribution of the simulated flow output. We use real-world examples to calibrate for different times of day and conditions and show that this methodology is much more efficient that the traditional Monte Carlo-type sampling. We validate the model using a hold out dataset and also show the drawback of using limited data for the calibration of a macroscopic simulation model. We also discuss the drawbacks of the predictive ability of a single calibrated model for all the conditions.     

A robust optimization approach for dynamic traffic signal control with emission considerations

We consider an analytical signal control problem on a signalized network whose traffic flow dynamic is described by the Lighthill–Whitham–Richards (LWR) model (Lighthill and Whitham, 1955; Richards, 1956). This problem explicitly addresses traffic-derived emissions as constraints or objectives. We seek to tackle this problem using a mixed integer mathematical programming approach. Such class of problems, which we call LWR-Emission (LWR-E), has been analyzed before to certain extent. Since mixed integer programs are practically efficient to solve in many cases (Bertsimas et al., 2011b), the mere fact of having integer variables is not the most significant challenge to solving LWR-E problems; rather, it is the presence of the potentially nonlinear and nonconvex emission-related constraints/objectives that render the program computationally expensive.To address this computational challenge, we proposed a novel reformulation of the LWR-E problem as a mixed integer linear program (MILP). This approach relies on the existence of a statistically valid macroscopic relationship between the aggregate emission rate and the vehicle occupancy on the same link. This relationship is approximated with certain functional forms and the associated uncertainties are handled explicitly using robust optimization (RO) techniques. The RO allows emissions-related constraints and/or objectives to be reformulated as linear forms under mild conditions. To further reduce the computational cost, we employ a link-based LWR model to describe traffic dynamics with the benefit of fewer (integer) variables and less potential traffic holding. The proposed MILP explicitly captures vehicle spillback, avoids traffic holding, and simultaneously minimizes travel delay and addresses emission-related concerns.     

Dynamics of heterogeneity in urban networks: aggregated traffic modeling and hierarchical control

Real traffic data and simulation analysis reveal that for some urban networks a well-defined Macroscopic Fundamental Diagram (MFD) exists, which provides a unimodal and low-scatter relationship between the network vehicle density and outflow. Recent studies demonstrate that link density heterogeneity plays a significant role in the shape and scatter level of MFD and can cause hysteresis loops that influence the network performance. Evidently, a more homogeneous network in terms of link density can result in higher network outflow, which implies a network performance improvement. In this article, we introduce two aggregated models, region- and subregion-based MFDs, to study the dynamics of heterogeneity and how they can affect the accuracy scatter and hysteresis of a multi-subregion MFD model. We also introduce a hierarchical perimeter flow control problem by integrating the MFD heterogeneous modeling. The perimeter flow controllers operate on the border between urban regions, and manipulate the percentages of flows that transfer between the regions such that the network delay is minimized and the distribution of congestion is more homogeneous. The first level of the hierarchical control problem can be solved by a model predictive control approach, where the prediction model is the aggregated parsimonious region-based MFD and the plant (reality) is formulated by the subregion-based MFDs, which is a more detailed model. At the lower level, a feedback controller of the hierarchical structure, tries to maximize the outflow of critical regions, by increasing their homogeneity. With inputs that can be observed with existing monitoring techniques and without the need for detailed traffic state information, the proposed framework succeeds to increase network flows and decrease the hysteresis loop of the MFD. Comparison with existing perimeter controllers without considering the more advanced heterogeneity modeling of MFD highlights the importance of such approach for traffic modeling and control.     

The time slot allocation problem under uncertain capacity

This paper presents two stochastic programming models for the allocation of time slots over a network of airports. The proposed models address three key issues. First, they provide an optimization tool to allocate time slots, which takes several operational aspects and airline preferences into account; second, they execute the process on a network of airports; and third they explicitly include uncertainty. To the best of our knowledge, these are the first models for time slot allocation to consider both the stochastic nature of capacity reductions and the problem’s network structure. From a practical viewpoint, the proposed models provide important insights for the allocation of time slots. Specifically, they highlight the tradeoff between the schedule/request discrepancies, i.e., the time difference between allocated time slots and airline requests, and operational delays. Increasing schedule/request discrepancies enables a reduction in operational delays. Moreover, the models are computationally viable. A set of realistic test instances that consider the scheduling of four calendar days on different European airport networks has been solved within reasonable – for the application’s context – computation times. In one of our test instances, we were able to reduce the sum of schedule/request discrepancies and operational delays by up to 58%. This work provides slot coordinators with a valuable decision making tool, and it indicates that the proposed approach is very promising and may lead to relevant monetary savings for airlines and aircraft operators.     

Adaptive traffic signal control with equilibrium constraints under stochastic demand

This study develops a methodology to model transportation network design with signal settings in the presence of demand uncertainty. It is assumed that the total travel demand consists of commuters and infrequent travellers. The commuter travel demand is deterministic, whereas the demand of infrequent travellers is stochastic. Variations in demand contribute to travel time uncertainty and affect commuters’ route choice behaviour. In this paper, we first introduce an equilibrium flow model that takes account of uncertain demand. A two-stage stochastic program is then proposed to formulate the network signal design under demand uncertainty. The optimal control policy derived under the two-stage stochastic program is able to (1) optimize the steady-state network performance in the long run, and (2) respond to short-term demand variations. In the first stage, a base signal control plan with a buffer against variability is introduced to control the equilibrium flow pattern and the resulting steady-state performance. In the second stage, after realizations of the random demand, recourse decisions of adaptive signal settings are determined to address the occasional demand overflows, so as to avoid transient congestion. The overall objective is to minimize the expected total travel time. To solve the two-stage stochastic program, a concept of service reliability associated with the control buffer is introduced. A reliability-based gradient projection algorithm is then developed. Numerical examples are performed to illustrate the properties of the proposed control method as well as its capability of optimizing steady-state performance while adaptively responding to changing traffic flows. Comparison results show that the proposed method exhibits advantages over the traditional mean-value approach in improving network expected total travel times.     

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

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

Optimal mainstream traffic flow control of large-scale motorway networks

The continuously increasing daily traffic congestions on motorway networks around the world call for innovative control measures that would drastically improve the current traffic conditions. Mainstream traffic flow control (MTFC) is proposed as a novel and efficient motorway traffic management tool, and its possible implementation and principal impact on traffic flow efficiency is analysed. Variable speed limits, suitably operated and enforced, is considered as one (out of several possible) way(s) for MTFC realisation, either as a stand-alone measure or in combination with ramp metering. A previously developed, computationally efficient software tool for optimal integrated motorway network traffic control including MTFC is applied to a large-scale motorway ring-road. It is demonstrated via several investigated control scenarios that traffic flow can be substantially improved via MTFC with or without integration with coordinated ramp metering actions.     

Accounting for dynamic speed limit control in a stochastic traffic environment: A reinforcement learning approach

This paper proposes a novel dynamic speed limit control model accounting for uncertain traffic demand and supply in a stochastic traffic network. First, a link based dynamic network loading model is developed to simulate the traffic flow propagation allowing the change of speed limits. Shockwave propagation is well defined and captured by checking the difference between the queue forming end and the dissipation end. Second, the dynamic speed limit problem is formulated as a Markov Decision Process (MDP) problem and solved by a real time control mechanism. The speed limit controller is modeled as an intelligent agent interacting with the stochastic network environment stochastic network environment to assign time dependent link based speed limits. Based on different metrics, e.g. total network throughput, delay time, vehicular emissions are optimized in the modeling framework, the optimal speed limit scheme is obtained by applying the R-Markov Average Reward Technique (R-MART) based reinforcement learning algorithm. A case study of the Sioux Falls network is constructed to test the performance of the model. Results show that the total travel time and emissions (in terms of CO) are reduced by around 18% and 20% compared with the base case of non-speed limit control.     

Perimeter and boundary flow control in multi-reservoir heterogeneous networks

In this paper, we macroscopically describe the traffic dynamics in heterogeneous transportation urban networks by utilizing the Macroscopic Fundamental Diagram (MFD), a widely observed relation between network-wide space-mean flow and density of vehicles. A generic mathematical model for multi-reservoir networks with well-defined MFDs for each reservoir is presented first. Then, two modeling variations lead to two alternative optimal control methodologies for the design of perimeter and boundary flow control strategies that aim at distributing the accumulation in each reservoir as homogeneously as possible, and maintaining the rate of vehicles that are allowed to enter each reservoir around a desired point, while the system’s throughput is maximized. Based on the two control methodologies, perimeter and boundary control actions may be computed in real-time through a linear multivariable feedback regulator or a linear multivariable integral feedback regulator. Perimeter control occurs at the periphery of the network while boundary control occurs at the inter-transfers between neighborhood reservoirs. To this end, the heterogeneous network of San Francisco is partitioned into three homogeneous reservoirs and the proposed feedback regulators are compared with a pre-timed signal plan and a single-reservoir perimeter control strategy. Finally, the impact of the perimeter and boundary control actions is demonstrated via simulation by the use of the corresponding MFDs and other performance measures. A key advantage of the proposed approach is that it does not require high computational effort and future demand data if the current state of each reservoir can be observed with loop detector data.     

Paradoxes of reservation-based intersection controls in traffic networks

Reservation-based intersection control is a revolutionary idea for using connected autonomous vehicle technologies to improve intersection controls. Vehicles individually request permission to follow precise paths through the intersection at specific times from an intersection manager agent. Previous studies have shown that reservations can reduce delays beyond optimized signals in many demand scenarios. The purpose of this paper is to demonstrate that signals can outperform reservations through theoretical and realistic examples. We present two examples that exploit the reservation protocol to prioritize vehicles on local roads over vehicles on arterials, increasing the total vehicle delay. A third theoretical example demonstrates that reservations can encourage selfish route choice leading to arbitrarily large queues. Next, we present two realistic networks taken from metropolitan planning organization data in which reservations perform worse than signals. We conclude with significantly positive results from comparing reservations and signals on the downtown Austin grid network using dynamic traffic assignment. Overall, these results indicate that network-based analyses are needed to detect adverse route choices before traffic signals can be replaced with reservation controls. In asymmetric intersections (e.g. local road-arterial intersections), reservation controls can cause several potential issues. However, in networks with more symmetric intersections such as a downtown grid, reservations have great potential to improve traffic.     

Modeling chain collisions in vehicular networks with variable penetration rates

The vehicular ad hoc network has great potential in improving traffic safety. One of the most important and interesting issues in the research community is the safety evaluation with limited penetration rates of vehicles equipped with inter-vehicular communications. In this paper, a stochastic model is proposed for analyzing the vehicle chain collisions. It takes into account the influences of different penetration rates, the stochastic nature of inter-vehicular distance distribution, and the different kinematic parameters related to driver and vehicle. The usability and accuracy of this model is tested and proved by comparative experiments with Monte Carlo simulations. The collision outcomes of a platoon in different penetration rates and traffic scenarios are also analyzed based on this model. These results are useful to provide theoretical insights into the safety control of a heterogeneous platoon.     

Traffic noise and traffic light control

An optimal control problem of traffic light duration is considered. The traffic noise level is introduced as a state variable in a dynamical optimization problem. A closed loop control system is designed which influences the green duration of the lights according to the equivalent noise level. Real time considerations lead to sub-optimal control implementation. This control policy decreases the noise levels at intensive traffic intersections. The traffic lights adapt their duration according to the noise pollution. Simulation and experimental results are discussed.     

Robust stop-skipping patterns in urban railway operations under traffic alteration situation

In this paper an operation mode which is based on the stop-skipping approach is studied in urban railway lines under uncertainty. In this mode, each train follows a specific stop schedule. Trains are allowed to skip any intermediate stations to increase the commercial speed and to save energy consumption. As the commercial speed increases, the number of required trains in operation reduces and results eliminating unnecessary costs. To that end, a new mathematical model is proposed to reach the optimum stop schedule patterns. In the planning step, based on the traffic studies, the headway distributions are computed for different weekdays, and holidays. However, in practice, because of many unexpected events, the traffic may alter from what is planned. Therefore, in this condition, a robust plan is required that is optimized and immunized from uncertainty. In this paper, a new robust mathematical model, as well as two heuristic algorithms including (1) a decomposition-based algorithm and (2) a Simulated Annealing (SA) based algorithm is proposed. Finally, an Iranian metro line is studied and the optimum patterns are presented and analyzed.     

Stochastic prediction of train delays in real-time using Bayesian networks

In this paper we present a stochastic model for predicting the propagation of train delays based on Bayesian networks. This method can efficiently represent and compute the complex stochastic inference between random variables. Moreover, it allows updating the probability distributions and reducing the uncertainty of future train delays in real time under the assumption that more information continuously becomes available from the monitoring system. The dynamics of a train delay over time and space is presented as a stochastic process that describes the evolution of the time-dependent random variable. This approach is further extended by modelling the interdependence between trains that share the same infrastructure or have a scheduled passenger transfer. The model is applied on a set of historical traffic realisation data from the part of a busy corridor in Sweden. We present the results and analyse the accuracy of predictions as well as the evolution of probability distributions of event delays over time. The presented method is important for making better predictions for train traffic, that are not only based on static, offline collected data, but are able to positively include the dynamic characteristics of the continuously changing delays.     

Understanding the variability of speed distributions under mixed traffic conditions caused by holiday traffic

Understanding the variability of speed patterns and congestion characteristics of interstate freeway systems caused by holiday traffic is beneficial because appropriate countermeasures for safety improvement and congestion mitigation can be prepared and drivers can avoid traffic congestion and change their holiday travel schedules. This study evaluated the traffic congestion patterns during the Thanksgiving holiday period in 2006 using a Gaussian mixture speed distribution estimated by the Expectation–Maximization (EM) algorithm. This mathematical approach showed the potential of improving freeway operational performance evaluation schemes for holiday periods (even non-holiday periods). This study suggested that a Gaussian mixture model using the EM algorithm could be used to properly characterize the severity and the variability of congestion on certain interstate roadway systems. However, this study also pointed out that the fundamental limitations of the mixture model and the statistical significance test about the mixture components should be well understood and need to be further investigated. In addition, because this study investigated the changing patterns of speed distributions with only one interstate freeway system, I-95 northbound, other freeway systems with both directions need to be evaluated so that a more broad and confident analysis on holiday traffic can be achieved.     

Integrated model for traffic flow forecasting under rainy conditions

Inclement weather, such as heavy rain, significantly affects road traffic flow operation, which may cause severe congestion in road networks in cities. This study investigates the effect of inclement weather, such as rain events, on traffic flow and proposes an integrated model for traffic flow parameter forecasting during such events. First, an analysis of historical observation data indicates that the forecasting error of traffic flow volume has a significant linear correlation with mean precipitation, and thus, forecasting accuracy can be considerably improved by applying this linear correlation to correct forecasting values. An integrated online precipitation‐correction model was proposed for traffic flow volume forecasting based on these findings. We preprocessed precipitation data transformation and used outlier detection techniques to improve the efficiency of the model. Finally, an integrated forecasting model was designed through data fusion methods based on the four basic forecasting models and the proposed online precipitation‐correction model. Results of the model validation with the field data set show that the designed model is better than the other models in terms of overall accuracy throughout the day and under precipitation. However, the designed model is not always ideal under heavy rain conditions. Copyright © 2016 John Wiley & Sons, Ltd.