A Multi-directional Aggregated Model for Large Urban Networks
Liudmila Tumasha, , Carlos Canudas-de-Wita,∗ , Maria Laura Delle Monacheba CNRS , Univ. Grenoble Alpes, GIPSA-Lab, Franceb University of California, Berkeley, USA∗ Corresponding author: carlos.canudas-de-wit@cnrs.fr

Extended abstract submitted for presentation at the 11th Triennial Symposium onTransportation Analysis conference (TRISTAN XI)
June 19-25, 2022, Mauritius Island

February 2, 2022
Macroscopic model for urban traffic, partial differential equations, simulation and validation,
continuation of ODE to PDE, conservation laws.

1Introduction

The development of traffic models has been in its majority influenced by the model structure
proposed in the the Lighthill-Whitham-Richards model (LWR) Lighthill & Whitham (1955) andRichards (1956). This model is the most simple and therefore the most popular macroscopic
model for traffic. Many variants of this model consider traffic flow that has only one direction ofmotion. The first attempt to include multiple directions in 2D continuum models was made only
several years ago, when Lin et al. (2017) took inspiration from pedestrian modeling and deployeddynamic user-optimal principle for the path choice. The drawback of this model is that the traffic
density may become unbounded. There exist also other works Mollier et al. (2019), Mollier et al.
(Jan. 2019), Aghamohammadi & Laval (2019) proposing 2D multi-layer models with bounded
densities. However, they do not include traffic mixing between different direction layers. Thesemodels are also not necessarily hyperbolic, i.e., their PDE type varies with parameters, which
exaggerates its analysis and numerical simulation. Hyperbolicity for all parameters implies thatit can be analysed like many other conservation law based models for traffic.
In this work we propose a new multi-direction traffic flow model called the NEWS model.
This macroscopic model is composed by a set of four partial differential equations (PDEs),each modelling the density propagation in one of the four cardinal directions: North, East,
West and South. The model is formally derived from the classical cell transmission model
at intersections. Knowledge about the network topology (location of the roads) and network
infrastructure parameters (roads maximal speeds, number of lanes and capacities) are used totune the model. The information about the flow direction is retrieved from the turning ratios
at the intersections, which is then aggregated in four directions using projection matrices. Thepaper also discusses some of the mathematical properties of the model, namely, hyperbolicity
and mass conservation law. We will show some trials for the model validation, using synthetic
data from a microsimulator mimicking the Grenoble downtown network. In addition, the modelis also validated using real data collected from real sensors installed in Grenoble.

2The NEWS model

The NEWS model describes the multi-directional evolution of vehicle density on a bounded 2Dcontinuum plane Ω ⊂ R2 that is a rectangular domain bounded by xmin , xmax , ymin and ymax .
TRISTAN XI Symposium

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L. Tumash, C. Canudas-de-Wit, M.-L. Delle Monache

ϕ̄in
S

2

ϕ̄out
Nϕ̄SW
ϕ̄in
W

ϕ̄out
W

ϕ̄outEϕ̄in
Eϕ̄N Eϕ̄outSϕ̄inNFigure 1 – Schematic explanation of flow directions in NEWS formulation.

It is given by the following set of partial differential equations ∀(x, y, t) ∈ Ω × R2 : ∂(cos θN ϕ̄N ) ∂(sin θN ϕ̄N )1 in∂ ρ̄Nout=ϕ̄−ϕ̄−−,NN∂tL∂x∂y ∂(cos θE ϕ̄E ) ∂(sin θE ϕ̄E )

1 in
∂ ρ̄E


=ϕ̄E − ϕ̄out−−,E

∂tL
∂x∂y
∂ ρ̄

1∂(cos θ ϕ̄ )∂(sin θ ϕ̄ )WW WW Wout=ϕ̄in−,−
W − ϕ̄W∂tL∂x∂y ∂(cos θS ϕ̄S ) ∂(sin θS ϕ̄S )∂ ρ̄S1 in.
=ϕ̄S − ϕ̄out−−S∂x∂y
 ∂t{z} ||L{z}mixing termtransportation term

The state of this model is given by the four-dimensional vector ρ̄ = ρ̄(x, y, t) = (ρ̄N , ρ̄E , ρ̄W , ρ̄S ),representing densities of the traffic flows in four direction layers. Functions ϕ̄(x, y, ρ̄), ϕ̄in (x, y, ρ̄)and ϕ̄out (x, y, ρ̄) are defined via fundamental diagrams and are space- and state-dependent. Finally, L(x, y), cos θ(x, y) and sin θ(x, y) are space-dependent parameters which aggregate thetopology of a particular traffic network.
The right-hand side of the model consists of two parts: transportation term and mixing term.
The transportation term consists of spatial derivatives and thus describes the transportation oftraffic density in the same way as it is done in the original LWR model. Therefore, the NEWSmodel can be seen as a continuous generalisation of LWR to a 2-dimensional multidirectional urban traffic networks. The mixing term accounts for coupling between traffic
flows in different direction layers. In particular, ϕ̄in (x, y, ρ̄) − ϕ̄out (x, y, ρ̄) is the net flow at roads:
 in ϕ̄N − ϕ̄out
ϕ̄EN + ϕ̄W N + ϕ̄SN − ϕ̄N E − ϕ̄N W − ϕ̄N SN ϕ̄in − ϕ̄out   ϕ̄N E + ϕ̄W E + ϕ̄SE − ϕ̄EN − ϕ̄EW − ϕ̄ES E  Eϕ̄in − ϕ̄out  = ϕ̄N W + ϕ̄EW + ϕ̄SW − ϕ̄W N − ϕ̄W E − ϕ̄W S  .
WWoutϕ̄inϕ̄N S + ϕ̄ES + ϕ̄W S − ϕ̄SN − ϕ̄SE − ϕ̄SWS − ϕ̄SThereby ϕ̄EN , ϕ̄W N , ϕ̄SN and so on are partial flows, e.g., ϕ̄SN (x, y, ρ̄S , ρ̄N ) is a flow of cars thatwere going to the South and then turned to the North. This information should be retrieved
from turning ratio measurements. Figure below illustrates the concept of partial flows on anexample of one intersection.
This simple analytic representation of urban traffic can facilitate the analysis for explicit
control design, as it was shown in Tumash et al. (2021). Details on the model derivation and theparticular flow functions can be found in Tumash et al. (2021).
TRISTAN XI Symposium

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L. Tumash, C. Canudas-de-Wit, M.-L. Delle Monache(a) Fixed sensors

3(b) Bluetooth devices

Figure 2 – Sensor location in Grenoble downtown: (a) fixed flow sensors: R denote radars
and L denote induction loops; (b) automatic vehicle identifiers using Bluetooth installed at 12intersections of Grenoble during a measurement campaign lasting for one week. These figures
are taken from Rodriguez-Vega (2021).

3Model validation with real data

The model was validated with real data, using the Grenoble Traffic Lab experimental platform,
see http://gtlville.inrialpes.fr/. The platform collects real-time traffic data from Grenoble downtown as shown Fig. 2. The sensor measure boundary flows (from radars and loops) andturning ratios at some intersections. Boundary flows are used as a flow boundary condition in
the model. Turning ratios are used in the model as a parameters. The boundary flows are also
used separately to estimated the internal density of the network, to be compared with the one
predicted by the model.
The maximal densities ρmax,j , capacities ϕmax,j , road lengths lj and orientations θj arecomputed from the network topology. In addition, and to complete the model we also use
average velocities obtained from a data provider. Finally, the method for estimating: the freeflow speed, the full set of turning ratios, the vehicle density and the boundary flows are describedin more details in Rodriguez-Vega et al. (June 2021).
In Fig. 2a) the sensors marked in blue are those giving boundary inflows and red sensors
give boundary outflows. Sensors marked in green were used for the validation of state estimation
procedure.
The results are depicted in Fig. 3, where the comparison of two density distributions is shown.
We see that in both cases the distributions look quite similar. Part of the differences between themodel predicted density evolution, may come from: the NEWS model does not include traffic
lights, and does not take into account parking lots. Another source of mismatch could be inducedby data on inflows and outflows, namely the data represents measured flows in the city that we cannot enforce in our model, since there is always a demand-supply problem that not always matchwith the actual measure. The source code used for model validation is an open source projectthat can be found here: https://github.com/Lyurlik/multidirectional-traffic-model.

Acknowledgment
The Scale-FreeBack project has received funding from the European Research Council (ERC)
under the European Unions Horizon 2020 research and innovation program (grant agreement N694209).
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L. Tumash, C. Canudas-de-Wit, M.-L. Delle Monache

4Figure 3 – Evolution of traffic density in Grenoble downtown on 8th of January 2021 from t = 6
am to t = 9 pm: numerical simulation of NEWS model (left plots) and density estimated fromsensor data (right plots). Weighting parameter η = 20. These figures are taken from Tumash(2021)

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