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.     

Hierarchical perimeter control with guaranteed stability for dynamically coupled heterogeneous urban traffic

Perimeter control based on the Macroscopic Fundamental Diagram (MFD) is widely developed for alleviating or postponing congestion in a protected region. Recent studies reveal that traffic conditions might not be improved if the perimeter control strategies are applied to unstable systems where high demand generates heavy and heterogeneously distributed traffic congestion. Therefore, considering stability of the targeted traffic system is essential, for the sake of developing a feasible and then optimal control strategy. This paper sheds light on this direction. It integrates a stability characterization algorithm of MFD system equations into the Model Predictive Control (MPC) scheme, and features respectively an upper and a lower bound of the feasible control inputs, to guarantee system stability. Firstly, the dynamics of traffic heterogeneity and its effect on the MFD are analyzed, using real data from Guangzhou in China. Piecewise affine functions of average flow are proposed to capture traffic heterogeneity in both regional and subregional MFDs. Secondly, stability of a three-state two-region system is investigated via stable equilibrium and surface boundaries analysis. Finally, a three-layer hierarchical control strategy is introduced for the studied two-region heterogeneous urban networks. The first layer of the controller calculates the stable surface boundaries for the given traffic demands and then determines the bounds of control input (split rate). An MPC approach in the second layer is used to solve an optimization problem with two objectives of minimizing total network delay and maximizing network throughput. Heterogeneity among the subregions is minimized in the last layer by implementing simultaneously a subregional perimeter flow control and an internal flow control. The effectiveness and stability of the proposed control approach are verified by comparison with four existing perimeter control strategies.     

Cooperative traffic control of a mixed network with two urban regions and a freeway

Currently most optimization methods for urban transport networks (i) are suited for networks with simplified dynamics that are far from real-sized networks or (ii) apply decentralized control, which is not appropriate for heterogeneously loaded networks or (iii) investigate good-quality solutions through micro-simulation models and scenario analysis, which make the problem intractable in real time. In principle, traffic management decisions for different sub-systems of a transport network (urban, freeway) are controlled by operational rules that are network specific and independent from one traffic authority to another. In this paper, the macroscopic traffic modeling and control of a large-scale mixed transportation network consisting of a freeway and an urban network is tackled. The urban network is partitioned into two regions, each one with a well-defined Macroscopic Fundamental Diagram (MFD), i.e. a unimodal and low-scatter relationship between region density and outflow. The freeway is regarded as one alternative commuting route which has one on-ramp and one off-ramp within each urban region. The urban and freeway flow dynamics are formulated with the tool of MFD and asymmetric cell transmission model, respectively. Perimeter controllers on the border of the urban regions operating to manipulate the perimeter interflow between the two regions, and controllers at the on-ramps for ramp metering are considered to control the flow distribution in the mixed network. The optimal traffic control problem is solved by a Model Predictive Control (MPC) approach in order to minimize total delay in the entire network. Several control policies with different levels of urban-freeway control coordination are introduced and tested to scrutinize the characteristics of the proposed controllers. Numerical results demonstrate how different levels of coordination improve the performance once compared with independent control for freeway and urban network. The approach presented in this paper can be extended to implement efficient real-world control strategies for large-scale mixed traffic networks.     

Model based urban traffic control,part II: Coordinated model predictive controllers

The present paper describes how to use coordination between neighbouring intersections in order to improve the performance of urban traffic controllers. Both the local MPC (LMPC) introduced in the companion paper (Hao et al., 2018) and the coordinated MPC (CMPC) introduced in this paper use the urban cell transmission model (UCTM) (Hao et al., 2018) in order to predict the average delay of vehicles in the upstream links of each intersection, for different scenarios of switching times of the traffic lights at that intersection. The feedback controller selects the next switching times of the traffic light corresponding to the shortest predicted average delay. While the local MPC (Hao et al., 2018) only uses local measurements of traffic in the links connected to the intersection in comparing the performance of different scenarios, the CMPC approach improves the accuracy of the performance predictions by allowing a control agent to exchange information about planned switching times with control agents at all neighbouring intersections. Compared to local MPC the offline information on average flow rates from neighbouring intersections is replaced in coordinated MPC by additional online information on when the neighbouring intersections plan to send vehicles to the intersection under control. To achieve good coordination planned switching times should not change too often, hence a cost for changing planned schedules from one decision time to the next decision time is added to the cost function. In order to improve the stability properties of CMPC a prediction of the sum of squared queue sizes is used whenever some downstream queues of an intersection become too long. Only scenarios that decrease this sum of squares of local queues are considered for possible implementation. This stabilization criterion is shown experimentally to further improve the performance of our controller. In particular it leads to a significant reduction of the queues that build up at the edges of the traffic region under control. We compare via simulation the average delay of vehicles travelling on a simple 4 by 4 Manhattan grid, for traffic lights with pre-timed control, traffic lights using the local MPC controller (Hao et al., 2018), and coordinated MPC (with and without the stabilizing condition). These simulations show that the proposed CMPC achieves a significant reduction in delay for different traffic conditions in comparison to these other strategies.     

Model based urban traffic control,part I: Local model and local model predictive controllers

This paper is the first in a series of reports presenting a framework for the hierarchical design of feedback controllers for traffic lights in urban networks. The goal of the research is to develop an easy to understand methodology for designing model based feedback controllers that use the current state estimate in order to select the next switching times of traffic lights. In this paper we introduce an extension of the cell transmission model that describes with sufficient accuracy the major causes of delay for urban traffic. We show that this model is computationally fast enough such that it can be used in a model predictive controller that decides for each intersection, taking into account the vehicle density as estimated along all links connected to the intersection, what switching time minimizes the local delay for all vehicles over a prediction horizon of a few minutes. The implementation of this local MPC only requires local online measurements and local model information (unlike the coordinated MPC, to be introduced in the next paper in this series, that takes into account interactions between neighbouring intersections). We study the performance of the proposed local MPC via simulation on a simple 4 by 4 Manhattan grid, comparing its delay with an efficiently tuned pretimed control for the traffic lights, and with traffic lights controlled according to the max pressure rule. These simulations show that the proposed local MPC controller achieves a significant reduction in delay for various traffic conditions.     

Traffic guidance–perimeter control coupled method for the congestion in a macro network

Macroscopic fundamental diagram (MFD) describes the macro relationship between a network vehicle density and a network space mean flow, without requiring the mastery of complex origin to destination data. Thus, MFD provides an opportunity for the macro control of urban road network. However, most of the existing MFD control methods ignore the active role of traffic guidance in solving congestion problems. This study presents a traffic guidance–perimeter control coupled (TGPCC) method to improve the performance of macroscopic traffic networks. The method considers the optimal cumulative volume of a network as the goal and establishes a programming function according to the network equilibrium rule of traffic flow amongst multiple MFD sub-regions, which regards the minimum delay of network, as the objective. The Logit model for the compliance rate of driver route guidance is established by the stated preference survey. Moreover, the perimeter control (PC) method is proposed for adjusting the phase split of intersections. Finally, three schemes, namely, the TGPCC, PC and the method without PC and guidance are tested on a network with four well-defined MFD sub-regions. Results show that the TGPCC addresses the issue of congestion and decreases the total delay accordingly.     

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.     

Clustering of heterogeneous networks with directional flows based on “Snake” similarities

Aggregated network level modeling and control of traffic in urban networks have recently gained a lot of interest due to unpredictability of travel behaviors and high complexity of physical modeling in microscopic level. Recent research has shown the existence of well-defined Macroscopic Fundamental Diagrams (MFDs) relating average flow and density in homogeneous networks. The concept of MFD allows to design real-time traffic control schemes specifically hierarchical perimeter control approaches to alleviate or postpone congestion. Considering the fact that congestion is spatially correlated in adjacent roads and it propagates spatiotemporaly with finite speed, describing the main pockets of congestion in a heterogeneous city with small number of clusters is conceivable. In this paper, we propose a three-step clustering algorithm to partition heterogeneous networks into connected homogeneous regions, which makes the application of perimeter control feasible. The advantages of the proposed method compared to the existing ones are the ability of finding directional congestion within a cluster, robustness with respect to parameters calibration, and its good performance for networks with low connectivity and missing data. Firstly, we start to find a connected homogeneous area around each road of the network in an iterative way (i.e. it forms a sequence of roads). Each sequence of roads, defined as ‘snake’, is built by starting from a single road and iteratively adding one adjacent road based on its similarity to join previously added roads in that sequence. Secondly, based on the obtained sequences from the first step, a similarity measure is defined between each pair of the roads in the network. The similarities are computed in a way that put more weight on neighboring roads and facilitate connectivity of the clusters. Finally, Symmetric Non-negative Matrix Factorization (SNMF) framework is utilized to assign roads to proper clusters with high intra-similarity and low inter-similarity. SNMF partitions the data by providing a lower rank approximation of the similarity matrix. The proposed clustering framework is applied in medium and large-size networks based on micro-simulation and empirical data from probe vehicles. In addition, the extension of the algorithm is proposed to deal with the networks with sparse measurements where information of some links is missing. The results show the effectiveness and robustness of the extended algorithm applied to simulated network under different penetration rates (percentage of links with data).     

Model-based predictive control for bicycling in urban intersections

In this paper, a model predictive control approach for improving the efficiency of bicycling as part of intermodal transportation systems is proposed. Considering a dedicated bicycle lanes infrastructure, the focus in this paper is to optimize the dynamic interaction between bicycles and vehicles at the multimodal urban traffic intersections. In the proposed approach, a dynamic model for the flows, queues, and number of both vehicles and bicycles is explicitly incorporated in the controller. For obtaining a good trade-off between the total time spent by the cyclists and by the drivers, a Pareto analysis is proposed to adjust the objective function of the MPC controller. Simulation results for a two-intersections urban traffic network are presented and the controller is analyzed considering different methods of including in the MPC controller the inflow demands of both vehicles and bicycles.     

Agent-based network transmission model using the properties of macroscopic fundamental diagram

The field of research that has recently come to the fore is the perimeter control, which aims to control traffic demand for a large urban area prior to controlling internal flow inside the area. Such control concept needs to be tested by simulations, hence, it is necessary to develop a model that can appropriately estimate the network-wide flow dynamics. In this paper, agent-based network transmission model (ANTM) is proposed for describing the aggregated flow dynamics over an urban area of multiple large-scale networks. The proposed model is the combination of the cell transmission model (CTM), macroscopic fundamental diagram (MFD), and agent concept. The CTM-based simulation is adopted for the simplicity considering the computation requirements for real-time feasibility. The MFD concept is applied for representing the network properties, and a new approach is taken particularly for estimating network outflow affected by both demand patterns and boundary capacity. The agent concept is applied for representing drivers’ travel behaviors. The model is compared with microscopic simulations and shows reasonable accuracy for large areas. In addition, various travel direction choice behaviors are applicable to this model. Various perimeter control policies are applicable as well, thus, the proposed model can be a useful tool for testing various control methods, in terms of reducing the congestion in urban areas.     

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.     

Nonlinear gating control for urban road traffic network using the network fundamental diagram

This work proposes a nonlinear model predictive controller for the urban gating problem. The system model is formalized based on a research on existing models of the network fundamental diagram and the perimeter control systems. For the existing models, modifications are suggested: additional state variables are allocated to describe the queue dynamics at the network gates. Using the extended model, a nonlinear model predictive controller is designed offering a ‘non‐greedy’ policy compared with previous, ‘greedy’ gating control designs. The greedy and non‐greedy nonlinear model predictive control (NMPC) controllers are compared with a greedy linear feedback proportional‐integral‐derivative (PID) controller in different traffic situations. The proposed non‐greedy NMPC controller outperforms the other two approaches in terms of travel distance performance and queue lengths. The performance results justify the consideration of queue lengths in dynamic modeling, and the use of NMPC approach for controller design. Copyright © 2014 John Wiley & Sons, Ltd.     

Recent developments in traffic flow modeling using macroscopic fundamental diagram

ABSTRACT

This paper presents an overview of the recent developments in traffic flow modelling and analysis using macroscopic fundamental diagram (MFD) as well as their applications. In recent literature, various aggregated traffic models have been proposed and studied to analyse traffic flow while enhancing network efficiency. Many of these studies have focused on models based on MFD that describes the relationship between aggregated flow and aggregated density of transport networks. The analysis of MFD has been carried out based on experimental data collected from sensors and GPS, as well as simulation models. Several factors are found to influence the existence and shape of MFD, including traffic demand, network and signal settings, and route choices. As MFD can well express the traffic dynamics of large urban transport networks, it has been extensively applied to traffic studies, including the development of network-wide control strategies, network partitioning, performance evaluation, and road pricing. This work also presents future extensions and research directions for MFD-based traffic modelling and applications.     

A three-dimensional macroscopic fundamental diagram for mixed bi-modal urban networks

Recent research has studied the existence and the properties of a macroscopic fundamental diagram (MFD) for large urban networks. The MFD should not be universally expected as high scatter or hysteresis might appear for some type of networks, like heterogeneous networks or freeways. In this paper, we investigate if aggregated relationships can describe the performance of urban bi-modal networks with buses and cars sharing the same road infrastructure and identify how this performance is influenced by the interactions between modes and the effect of bus stops. Based on simulation data, we develop a three-dimensional vehicle MFD (3D-vMFD) relating the accumulation of cars and buses, and the total circulating vehicle flow in the network. This relation experiences low scatter and can be approximated by an exponential-family function. We also propose a parsimonious model to estimate a three-dimensional passenger MFD (3D-pMFD), which provides a different perspective of the flow characteristics in bi-modal networks, by considering that buses carry more passengers. We also show that a constant Bus–Car Unit (BCU) equivalent value cannot describe the influence of buses in the system as congestion develops. We then integrate a partitioning algorithm to cluster the network into a small number of regions with similar mode composition and level of congestion. Our results show that partitioning unveils important traffic properties of flow heterogeneity in the studied network. Interactions between buses and cars are different in the partitioned regions due to higher density of buses. Building on these results, various traffic management strategies in bi-modal multi-region urban networks can then be integrated, such as redistribution of urban space among different modes, perimeter signal control with preferential treatment of buses and bus priority.     

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.     

On the stability of traffic perimeter control in two-region urban cities

In this paper, stability analysis of traffic control for two-region urban cities is treated. It is known in control theory that optimality does not imply stability. If the optimal control is applied in a heavily congested system with high demand, traffic conditions might not change or the network might still lead to gridlock. A city partitioned in two regions with a Macroscopic Fundamental Diagram (MFD) for each of the regions is considered. Under the assumption of triangular MFDs, the two-region MFDs system is modeled as a piecewise second-order system. Necessary and sufficient conditions are derived for stable equilibrium accumulations in the undersaturated regimes for both MFDs. Moreover, the traffic perimeter control problem for the two-region MFDs system is formulated. Phase portraits and stability analysis are conducted, and a new algorithm is proposed to derive the boundaries of the stable and unstable regions. Based on these regions, a state-feedback control strategy is derived. Trapezoidal shape of MFDs are also addressed with numerical solutions.     

Equilibrium analysis and route guidance in large-scale networks with MFD dynamics

Recent studies have demonstrated that Macroscopic Fundamental Diagram (MFD), which provides an aggregated model of urban traffic dynamics linking network production and density, offers a new generation of real-time traffic management strategies to improve the network performance. However, the effect of route choice behavior on MFD modeling in case of heterogeneous urban networks is still unexplored. The paper advances in this direction by firstly extending two MFD-based traffic models with different granularity of vehicle accumulation state and route choice behavior aggregation. This configuration enables us to address limited traffic state observability and to scrutinize implications of drivers’ route choice in MFD modeling. We consider a city that is partitioned in a small number of large-size regions (aggregated model) where each region consists of medium-size sub-regions (more detailed model) exhibiting a well-defined MFD. This paper proposes a route guidance advisory control system based on the aggregated model as a large-scale traffic management strategy that utilizes aggregated traffic states while sub-regional information is partially known. In addition, we investigate the effect of equilibrium conditions (i.e. user equilibrium and system optimum) on the overall network performance, in particular MFD functions.     

A comparative study of Macroscopic Fundamental Diagrams of arterial road networks governed by adaptive traffic signal systems

Using a stochastic cellular automaton model for urban traffic flow, we study and compare Macroscopic Fundamental Diagrams (MFDs) of arterial road networks governed by different types of adaptive traffic signal systems, under various boundary conditions. In particular, we simulate realistic signal systems that include signal linking and adaptive cycle times, and compare their performance against a highly adaptive system of self-organizing traffic signals which is designed to uniformly distribute the network density. We find that for networks with time-independent boundary conditions, well-defined stationary MFDs are observed, whose shape depends on the particular signal system used, and also on the level of heterogeneity in the system. We find that the spatial heterogeneity of both density and flow provide important indicators of network performance. We also study networks with time-dependent boundary conditions, containing morning and afternoon peaks. In this case, intricate hysteresis loops are observed in the MFDs which are strongly correlated with the density heterogeneity. Our results show that the MFD of the self-organizing traffic signals lies above the MFD for the realistic systems, suggesting that by adaptively homogenizing the network density, overall better performance and higher capacity can be achieved.     

A model‐based demand‐balancing control for dynamically divided multiple urban subnetworks

Traffic control is an effective and efficient method for the problem of traffic congestion. It is necessary to design a high‐level controller to regulate the network traffic demands, because traffic congestion is not only caused by the improper management of the traffic network but also to a great extent caused by excessive network traffic demands. Therefore, we design a demand‐balance model predictive controller based on the macroscopic fundamental diagram‐based multi‐subnetwork model, which can optimize the network traffic mobility and the network traffic throughput by regulating the input traffic flows of the subnetworks. Because the transferring traffic flows among subnetworks are indirectly controlled and coordinated by the demand‐balance model predictive controller, the subnetwork division can variate dynamically according to real traffic states, and a global optimality can be achieved for the entire traffic network. The simulation results show the effectiveness of the proposed controller in improving the network traffic throughput. Copyright © 2016 John Wiley & Sons, Ltd.     

Network traffic flow optimization under performance constraints

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