•Macroscopic Traffic flow modeling in heterogeneous networks partitioned in reservoirs.•Integrating issues of efficiency, heterogeneity and equity in perimeter flow control.•Developing perimeter and ...boundary control through multivariable feedback regulators.•Improving traffic performance and reducing scatter and hysteresis in the network MFD.
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.
► We partition transportation networks based on spatial features of congestion. ► A three-step partitioning mechanism for transportation networks is designed. ► The spatial compactness of the ...clusters is obtained after partitioning. ► Several homogeneity metrics to evaluate the partitioning results are specified. ► Scatter of partitioned MFDs decreases for heterogeneous density of networks.
It has been recently shown that a macroscopic fundamental diagram (MFD) linking space-mean network flow, density and speed exists in the urban transportation networks under some conditions. An MFD is further well defined if the network is homogeneous with links of similar properties. This collective behavior concept can also be utilized to introduce simple control strategies to improve mobility in homogeneous city centers without the need for details in individual links. However many real urban transportation networks are heterogeneous with different levels of congestion. In order to study the existence of MFD and the feasibility of simple control strategies to improve network performance in heterogeneously congested networks, this paper focuses on the clustering of transportation networks based on the spatial features of congestion during a specific time period. Insights are provided on how to extend this framework in the dynamic case. The objectives of partitioning are to obtain (i) small variance of link densities within a cluster which increases the network flow for the same average density and (ii) spatial compactness of each cluster which makes feasible the application of perimeter control strategies. Therefore, a partitioning mechanism which consists of three consecutive algorithms, is designed to minimize the variance of link densities while maintaining the spatial compactness of the clusters. Firstly, an over segmenting of the network is provided by a sophisticated algorithm (Normalized Cut). Secondly, a merging algorithm is developed based on initial segmenting and a rough partitioning of the network is obtained. Finally, a boundary adjustment algorithm is designed to further improve the quality of partitioning by decreasing the variance of link densities while keeping the spatial compactness of the clusters. In addition, both density variance and shape smoothness metrics are introduced to identify the desired number of clusters and evaluate the partitioning results. These results show that both the objectives of small variance and spatial compactness can be achieved with this partitioning mechanism. A simulation in a real urban transportation network further demonstrates the superiority of the proposed method in effectiveness and robustness compared with other clustering algorithms.
•Developing a static clustering that minimises heterogeneity and guarantees connectivity.•Formulating and solving an MILP that truncates a feasible set of snakes with the aforementioned ...objectives.•Extending the method to dynamic clustering that considers spatial interactions.•Merging or Splitting allows for a better fine-tuning in the dynamic framework.•Fast computation and proper integration of physical properties of congestion propagation.
The problem of clustering in urban traffic networks has been mainly studied in static framework by considering traffic conditions at a given time. Nevertheless, it is important to underline that traffic is a strongly time-variant process and it needs to be studied in the spatiotemporal dimension. Investigating the clustering problem over time in the dynamic domain is critical to better understand and reveal the hidden information during the process of congestion formation and dissolution. The primary motivation of the paper is to study the spatiotemporal relation of congested links, observing congestion propagation from a macroscopic perspective, and finally identifying critical pockets of congestion that can aid the design of peripheral control strategies. To achieve this, we first introduce a static clustering method to partition the heterogeneous network into homogeneous connected sub-regions. The proposed framework guarantees connectivity of the cluster in different steps, which eases the development of a dynamic framework. The proposed clustering approach has 3 steps; firstly, it obtains a set of homogeneous connected components in the network. Each component has a form of sequence which is built by sequentially adding neighboring links with similar level of congestion. Secondly, the major skeleton of clusters is obtained out of this feasible set by minimizing a heterogeneity index. Thirdly, a fine-tuning step is designed to complete the clustering task by assigning the unclustered links of the network to proper clusters while keeping the connectivity. The optimization problem in both second and third step is formulated as a mixed integer linear programming. The approach is also extended to capture spatiotemporal growth and formation of congestion. The dynamic clustering is based on an iterative and fast procedure that considers the spatiotemporal characteristics of congestion propagation and identifies the links with the highest degree of heterogeneity due to time dependent conditions and finally re-cluster them to guarantee connectivity and minimize heterogeneity. An implementation of the developed methodologies in a megacity based on more than 20,000 taxis with GPS highlights the quality of the method due to its fast computation and proper integration of physical properties of congestion.
•Modeling the dynamics of heterogeneity with a parsimonious model.•Developing a hierarchical control for heterogeneously congested networks.•Considering regional and a sub-regional interactions in an ...MFD framework.•Optimizing system delays by combining an MPC and a feedback homogeneity controller.
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.
► A strict homogeneity of traffic states on the network is not necessary to observe a well-defined MFD. ► The spatial distribution of car density in the network is one of the key components that ...affect the scatter of an MFD and its shape. ► Investigate the scatter of an arterial MFD in terms of errors in probability density function of spatial occupancy and errors of individual links’ fundamental diagram. ► Provide an analytical derivation of the spatial distribution of link density in networks that considers correlation between adjacent links.
A field experiment in Yokohama (Japan) revealed that a macroscopic fundamental diagram (MFD) linking space-mean flow, density and speed exists on a large urban area. It was observed that when the highly scattered plots of flow vs. density from individual fixed detectors were aggregated the scatter nearly disappeared and points grouped along a well defined curve. Despite these and other recent findings for the existence of well-defined MFDs for urban areas, these MFDs should not be universally expected. In this paper we investigate what are the properties that a network should satisfy, so that an MFD with low scatter exists. We show that the spatial distribution of vehicle density in the network is one of the key components that affect the scatter of an MFD and its shape. We also propose an analytical derivation of the spatial distribution of congestion that considers correlation between adjacent links. We investigate the scatter of an MFD in terms of errors in the probability density function of spatial link occupancy and errors of individual links’ fundamental diagram (FD). Later, using real data from detectors for an urban arterial and a freeway network we validate the proposed derivations and we show that an MFD is not well defined in freeway networks as hysteresis effects are present. The datasets in this paper consist of flow and occupancy measures from 500 fixed sensors in the Yokohama downtown area in Japan and 600 loop detectors in the Twin Cities Metropolitan Area Freeway network in Minnesota, USA.
► Stability analysis of perimeter control is conducted for two urban regions with macroscopic fundamental diagrams. ► Necessary and sufficient conditions for equilibrium points are derived. ► A new ...algorithm is proposed to characterize the stable and unstable regions. ► State-feedback control strategy is derived to stabilize and maximize the output of the system. ► Numerical examples are presented.
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.
We study the dynamical process of congestion formation for large-scale urban networks by exploring a unique dataset of taxi movements in a megacity. We develop a dynamic model based on a reaction and ...a diffusion term that properly reproduces the cascade phenomena of traffic. The interaction of these two terms brings the values of the speeds on road network in self-organized patterns and it reveals an elegant physical law that reproduces the dynamics of congestion with very few parameters. The results presented show a promising match with an available real data set of link speeds estimated from more than 40 millions of GPS coordinates per day of about 20,000 taxis in Shenzhen, China.
A field experiment in Yokohama (Japan) reveals that a macroscopic fundamental diagram (MFD) linking space-mean flow, density and speed exists on a large urban area. The experiment used a combination ...of fixed detectors and floating vehicle probes as sensors. It was observed that when the somewhat chaotic scatter-plots of speed vs. density from individual fixed detectors were aggregated the scatter nearly disappeared and points grouped neatly along a smoothly declining curve. This evidence suggests, but does not prove, that an MFD exists for the complete network because the fixed detectors only measure conditions in their proximity, which may not represent the whole network. Therefore, the analysis was enriched with data from GPS-equipped taxis, which covered the entire network. The new data were filtered to ensure that only full-taxi trips (i.e., representative of automobile trips) were retained in the sample. The space-mean speeds and densities at different times-of-day were then estimated for the whole study area using relevant parts of the detector and taxi data sets. These estimates were still found to lie close to a smoothly declining curve with deviations smaller than those of individual links – and entirely explained by experimental error. The analysis also revealed a fixed relation between the space-mean flows on the whole network, which are easy to estimate given the existence of an MFD, and the trip completion rates, which dynamically measure accessibility.
► Macroscopic fundamental diagrams should not be expected for all networks. ► Higher flows are observed in the onset than the offset of congestion for the same network density of freeway systems. ► ...Spatial distribution of congestion and synchronized occurrence of transitions from individual detectors cause hysteresis.
Observations of traffic pairs of flow vs. density or occupancy for individual locations in freeways or arterials are usually scattered about an underlying curve. Recent observations from empirical data in arterial networks showed that in some cases by aggregating the highly scattered plots of flow vs. density from individual loop detectors, the scatter almost disappears and well-defined macroscopic relations exist between space-mean network flow and network density. Despite these findings for the existence of well-defined relations with low scatter, these curves should not be universal. In this paper we investigate if well-defined macroscopic relations exist for freeway network systems, by analyzing real data from Minnesota’s freeways. We show that freeway network systems not only have curves with high scatter, but they also exhibit hysteresis phenomena, where higher network flows are observed for the same average network density in the onset and lower in the offset of congestion. The mechanisms of traffic hysteresis phenomena at the network level are analyzed in this paper and they have dissimilarities to the causes of the hysteresis phenomena at the micro/meso level. The explanation of the phenomenon is dual. The first reason is that there are different spatial and temporal distributions of congestion for the same level of average density. Another reason is the synchronized occurrence of transitions from individual detectors during the offset of the peak period, with points remain beneath the equilibrium curve. Both the hysteresis phenomenon and its causes are consistently observed for different spatial aggregations of the network.
Spatial systems that experience congestion can be modeled as weighted networks whose weights dynamically change over time with the redistribution of flows. This is particularly true for urban ...transportation networks. The aim of this work is to find appropriate network measures that are able to detect critical zones for traffic congestion and bottlenecks in a transportation system. We propose for both single and multi-layered networks a path-based measure, called dynamical efficiency, which computes the travel time differences under congested and free-flow conditions. The dynamical efficiency quantifies the reachability of a location embedded in the whole urban traffic condition, in lieu of a myopic description based on the average speed of single road segments. In this way, we are able to detect the formation of congestion seeds and visualize their evolution in time as well-defined clusters. Moreover, the extension to multilayer networks allows us to introduce a novel measure of centrality, which estimates the expected usage of inter-modal junctions between two different transportation means. Finally, we define the so-called dilemma factor in terms of number of alternatives that an interconnected transportation system offers to the travelers in exchange for a small increase in travel time. We find macroscopic relations between the percentage of extra-time, number of alternatives and level of congestion, useful to quantify the richness of trip choices that a city offers. As an illustrative example, we show how our methods work to study the real network of a megacity with probe traffic data.
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