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Publication# The influence of trip length distribution on urban traffic in network-level models

Résumé

Modeling urban traffic on the level of network is a wide research area oriented to the development of ITS. In this thesis properties of models based on MFD (Macroscopic Fundamental Diagram) are studied. The idea behind MFD is to say that the state of the traffic inside an urban zone is fully determined by its accumulation (the number of traveling vehicles) and that the dynamics of accumulation is caused by the inflow of vehicles (flow of vehicles that enter the zone or start their trips from inside). Nowadays, two different philosophies of modeling dynamics of accumulation exist in the literature. The first one (outflow-MFD) postulates that the outflow of vehicles depends on accumulation. The second one (speed-MFD) postulates that the space-mean speed of vehicles depends on accumulation. The second philosophy already has strong empirical support based on observations of traffic inside many big (kilometer-scale) urban areas around the world. Thus, different speed-MFD models are of great scientific interest.

The thesis is mainly devoted to the comparison of so-called PL model (which assumes the existence of both speed-MFD and outflow-MFD) and TB model (which assumes the equality of speeds of vehicles and explicitly assumes the existence of trip length distribution). It was shown that PL model cannot accurately describe the dynamics of accumulation after the jump of inflow. In this case TB model is more preferable. Moreover, it was proven that PL model is a specific case of TB model for the exponential trip length distribution. This makes TB model more attractive than PL model for the practical usage.

TB model can be formulated mathematically either as integral equation or nonlocal PDE. Thus, the main drawback of TB model that can be an obstacle in practice is its computational complexity. In this thesis it was proposed to approximate TB model with a simpler model which does not require precise information about the trip length distribution. This so-called M model operates with only the mean and the standard deviation of distribution and has a form of ODE. The analytical comparison between PL, TB and M models proved that M model is much closer to TB model than PL model in the case of constant speed-MFD. The more realistic case of decreasing speed-MFD was studied through the numerical tests and also showed the same effect. Thus, the main conclusion of the study is that M model has practical potential as an elegant and computationally cheap approximation of TB model. Also, given that in the case of constant speed TB model is a type of LTI system, it can be expected that M model might be useful for a wide range of problems that are not related to transportation. However, in this thesis the conclusion about the small difference between M and TB models was made for the inflows that are typical for the transportation field. More precisely, only peak hour shaped (smooth and with small jumps) functions were studied.

Despite the simple form of M model it was found that there exists even more simple ODE approximation of TB model. This so-called $\alpha$ model works quite well for both smooth and jumping inflows except the case of a short period of time following the jump of inflow. Thus, it might be another good alternative to TB model. The main advantage of $\alpha$ model in the case of realistic speed-MFD function is its convex formulation which allows $\alpha$ model to be efficiently used inside optimization frameworks.

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Recent analysis of empirical data from cities showed that a macroscopic fundamental diagram (MFD) of urban traffic provides for homogenous network regions a unimodal low-scatter relationship between network vehicle density and network space-mean flow. In this paper, the optimal perimeter control for two-region urban cities is formulated with the use of MFDs. The controllers operate on the border between the two regions and manipulate the percentages of flows that transfer between the two regions such that the number of trips that reach their destinations is maximized. The optimal perimeter control problem is solved by model predictive control, where the prediction model and the plant (reality) are formulated by MFDs. Examples are presented for different levels of congestion in the regions of the city and the robustness of the controller is tested for different sizes of error in the MFDs and different levels of noise in the traffic demand. Moreover, two methods for smoothing the control sequences are presented. Comparison results show that the performances of the model predictive control are significantly better than a "greedy" feedback control. The results in this paper can be extended to develop efficient hierarchical control strategies for heterogeneously congested cities.

Nikolaos Geroliminis, Jack Haddad, Mohsen Ramezani Ghalenoei

Recent analysis of empirical data from cities showed that a macroscopic fundamental diagram (MFD) of urban traffic provides for different network regions a unimodal, low-scatter relationship between network vehicle density and network space-mean flow. In this paper, the optimal perimeter control for two-region urban cities is formulated with the tool of MFDs. The controllers operate on the border between the two regions, and manipulate the percentages of flows that transfer between the two regions such that the number of trips reach their destinations is maximized. The perimeter control problem is solved by model predictive control, where the prediction model and the plant (reality) are formulated by macroscopic fundamental diagrams. Examples are presented for different levels of congestion in the regions of the city and the robustness of the controller is tested for different size of error in the MFDs. The direct sequential method is utilized to optimize the nonlinear problem of the open-loop control. Comparison results shows that the performances of the model predictive control are significantly better than a "greedy" feedback control. The results of this paper can be extended to develop efficient hierarchical control strategies for heterogeneously congested cities.

Nikolaos Geroliminis, Jack Haddad, Mohammadreza Hajiahmadi

Since centralized control of urban networks with detailed modeling approaches is computationally complex, developing efficient hierarchical control strategies based on aggregate modeling is of great importance. The dynamics of a heterogeneous large-scale urban network is modeled as R homogeneous regions with the macroscopic fundamental diagrams (MFDs) representation. The MFD provides for homogeneous network regions a unimodal, low-scatter relationship between network vehicle density and network space-mean flow. In this paper, the optimal hybrid control problem for an R-region MFD network is formulated as a mixed-integer nonlinear optimization problem, where two types of controllers are introduced: 1) perimeter controllers and 2) switching signal timing plans controllers. The perimeter controllers are located on the border between the regions, as they manipulate the transfer flows between them, while the switching controllers influence the dynamics of the urban regions, as they define the shape of the MFDs and as a result affect the internal flows within each region. Moreover, to decrease the computational complexity due to the nonlinear and nonconvex nature of the optimization problem, we reformulate the problem as a mixed-integer linear programming (MILP) problem utilizing piecewise affine approximation techniques. Two different approaches for transformation of the original model and building up MILP problems are presented, and the performances of the approximated methods along with the original problem formulation are evaluated and compared for different traffic scenarios of a two-region urban case study.