Numerical simulations of traffic data via fluid dynamic approach

Transcription

1 Numerical simulations of traffic data via fluid dynamic approach S. Blandin a, G. Bretti b, A. Cutolo c, B. Piccoli d a Department of Civil and Environmental Engineering, University of California, Berkeley, CA , USA b Me.Mo.Mat Department of University La Sapienza, Via A. Scarpa 16, Rome, Italy c Diima Department of the University of Salerno, Via Ponte Don Melillo, Fisciano(Sa), Italy d Istituto per le Applicazioni del Calcolo M. Picone, Viale del Policlinico 137, Rome, Italy Abstract In this paper we introduce a simulation algorithm based on fluid-dynamic models to reproduce the behavior of traffic in a portion of the urban network in Rome. Numerical results, obtained comparing experimental data with numerical solutions, show the effectiveness of our approximation. Key words: scalar conservation laws, 65M06, traffic flow, 90B20, fluid-dynamic models, 90B10, finite difference schemes, 35L65, boundary conditions, 34B45. 1 Introduction In this paper we develop an algorithm to determine the evolution in time of traffic quantities, such as flux, density and cars speed. The algorithm is then applied to a portion of the urban area in Rome usually subject to congestion, namely Viale del Muro Torto, for which measured traffic data are avail- The authors would like to thank Roberto Natalini for the interesting discussions and suggestments addresses: blandin@berkeley.edu (S. Blandin), bretti@dmmm.uniroma1.it (G. Bretti), a.cutolo@diima.unisa.it (A. Cutolo), b.piccoli@iac.cnr.it (B. Piccoli). Preprint submitted to Elsevier Science 22 June 2012

2 able. Two fluid dynamic approaches are considered: the Lighthill-Whitham- Richards (briefly LWR, see [30], [32]) model with Newell-Daganzo flux and a phase transition second order model (briefly PT), slight modification of the Colombo model ([10]). Our work consists in the calibration of system parameters in such a way to give a good reconstruction of traffic behavior. The numerical scheme used is the Godunov scheme, already proposed for road networks (see [3],[4], [26], [29]). The basic idea is to compute approximate flux and density on a single road, assuming as boundary conditions traffic data measured at the endpoints of it. An estimate of the validity of this procedure is obtained by comparing solutions produced numerically and experimental data detected on the road. Measured data are provided by the municipal society for traffic monitoring and control of Rome, namely ATAC S.p.A. Traffic is observed through an Intelligent Transport System technology, where each subsystem in it, represented by a sensor placed along roads of the city, acquires every minute ( t = 1 is the sensor time unit) traffic data such as the flux f, the velocity ṽ and the occupation rate õ. Since each sensor generates a magnetic field, the flux is intended as the number of cars crossing it per minute, the velocity is the average speed of cars at every minute, while the occupation rate is given by the time passed by cars on a sensor, hence it is the time interval in which cars pass through the magnetic field. Considering the whole road as a sequence of segments, the theory is based on LWR and on phase transition model applied to networks. The LWR theory on networks was proposed and developed in various papers, see [8], [9], [16], [18], [21], [22], [24], [28], [29]. The network models of transportation systems are assumed to be static in classical approaches, but these models do not allow a correct simulation of heavily congested urban road networks. For this reason, traffic engineers have been studying dynamic traffic assignment or within-day models, thus rendering necessary the use of traffic simulation models. Such models, principally created from static network traffic assignments, can be divided in microscopic, mesoscopic and macroscopic (see [1] and the references therein). The main problems of the static approach are that it does not properly reproduce the backward propagation of shocks and the difficulty of collecting experimental data to test the validity of the models. In the 1950s James Lighthill and Gerald Whitham, two experts in fluiddynamics, and independently P. Richards, [30]-[32], thought that the equations describing the flow of water could also describe the flow of car traffic. Fluid-dynamic models for traffic flow seem the most appropriate to detect some phenomena as shocks formation and propagation on roads, since solutions can show discontinuities in finite time even starting from smooth initial data (see [2]). This nonlinear formulation, based on the conservation of cars, is given by: t ρ + x f(ρ) = 0, (1.1) 2

3 where ρ = ρ(t,x) is the car density, with ρ [0,ρ max ], (t,x) R 2 and ρ max is the maximum car density. The flux f(ρ) is given by ρv, where v is the average velocity of cars. Assuming v to be a smooth decreasing function of the density ρ, also f depends only on ρ and its graph is called the fundamental diagram. We always further assume that f, as function of ρ, is concave. Many other ideas have been developed by researchers studying traffic from other perspectives, see for instance [12], [13], [14], [15], [23], [25], [27], [31], [33]. In more detail, the procedure followed in our simulations consists of the steps: 0. data capturing; 1. data cleaning; 2. calibration of flux parameters; 3. generation of approximate solution; 4. computation of errors. The main results achieved with the mentioned algorithm are: the calibration of traffic parameters, providing the maximum velocity v max very close to measured one ṽ max ; the description of traffic behavior, namely of flux and density, with a good approximation. Focusing on the latter, the percentage error for LWR model is 10% in the free phase and 17% in the congested phase of traffic, while for PT model one gets 9% in the free phase and 22% in the congested phase. Note that in both cases the percentage error for the congested phase is comparable to detection errors by sensors, which is up to 20%. Another important byproduct is the reconstruction of the traffic data on the whole road. In particular, this allows to reconstruct the queues evolutions thus permitting a good estimate of the travelling times. Other numerical algorithms, based on fluid dynamic approaches, were considered by researcher in transportation systems, see for istance [6], [13], [17], [26]. The papers is organized as follows: Sections 2 and 3 are devoted to the description of the mathematical models, while the approximation algorithm is presented in Section 4. In Section 5 the results obtained for the segments composing the road are showed and a comparison between numerical and measured solutions is established. Some animations are reported on the web page [5]. 3

4 2 LWR on sequence of roads We consider a long road as a sequence of segments modeled by intervals [a i,b i ] that meet at some junctions, corresponding to endpoints. In order to describe the evolution in time of traffic we use the LWR model on each segment and describe the dynamics at junctions. A Riemann problem for a scalar conservation law is a Cauchy problem for an initial data of Heaviside type, that is piecewise constant with only one discontinuity. Once Riemann problems are solved, a solution to Cauchy problems can be obtained by wave front tracking, see [2]. Since the flux is concave, the Riemann solutions are of two types: continuous waves called rarefactions and traveling discontinuities called shocks. For a junction, a Riemann problem is a Cauchy problem with an initial data that is constant on each road. Junctions play a fundamental role, as the system at a junction is under-determined, even after prescribing the conservation of cars. Due to finite speed of waves in solutions to (1.1), it is enough assigning the dynamics at each junction separately to obtain an evolution on the whole network. In our case we have only simple junctions with one incoming and one outgoing road. For instance, consider the case where the incoming road is occupied by cars with maximum density, while the outgoing road is empty, see Figure 1. Let us denote the initial datum on road i by ρ i,0, i = 1, 2, then: ROAD 1 ROAD 2 ρ 1,0 = ρ max ρ 2,0 = 0 Fig. 1. A particular situation in a junction with one incoming and one outgoing road. ρ 1,0 (x) = ρ max, ρ 2,0 (x) = 0. (2.2) Two possible extreme behaviors of cars are expected, namely: all cars flow towards the outgoing road (entropy solution) or all cars do not pass through the junction (non-entropy solution). In this way two different solutions ρ and ρ are determined. Namely: ρ max, if x < f (ρ max )t ρ 1 (t,x) = (f ) ( ) 1 x t, if f (ρ max )t < x < 0, (f ) ( ) 1 x t, if 0 x < f (0)t, ρ 2 (t,x) = 0, if x > f (0)t, 4

5 while ρ 1 = ρ 1,0, ρ 2,0 = ρ 2. Hence, the conservation of cars quantity through the junction, which reads as: incoming roads incoming fluxes = outgoing roads outgoing fluxes, holds for both solutions: In other words the solely conservation of cars is not sufficient to ensure uniqueness. Therefore, a map assigning solutions to initial data, called a Riemann solver at junctions, is needed. A classification of possible choices for Riemann solvers, in the above case, have been recently presented in [19]. To maximize the entropy flux, as for conservation laws, see [2], we assign the following rule: (R) The number of cars passing the junction is the maximum possible. For a single incoming road, it is easy to check that rule (R) is equivalent to maximize the average velocity. For a general junction with n incoming and m outgoing roads see [9], [15], [22], [28]. We recall briefly the procedure for constructing solutions, for further details see [9]. Assume that f is concave with a unique maximum point σ ]0,ρ max [. Define the map τ : [0,ρ max ] [0,ρ max ] such that τ(σ) = σ and for ρ σ it holds τ(ρ) ρ, f(τ(ρ)) = f(ρ). Call ρ 1,0 and ρ 2,0 the initial data on, respectively, road 1 and road 2. To correctly solve the Riemann problem, we want to determine new states ˆρ i, i = 1, 2, at the junction in such a way that the wave (ρ 1,0, ˆρ 1 ) has negative speed and the wave (ˆρ 2,ρ 2,0 ) has positive speed. This implies restrictions on the maximal possible fluxes γi max, in particular: If ρ 1,0 < σ then γ1 max = f(ρ 1,0 ), otherwise γ1 max = f(σ); if ρ 2,0 > σ then γ2 max = f(ρ 2,0 ), otherwise γ2 max = f(σ). To respect rule (R), the through flux is given by: ˆγ 1 = ˆγ 2 = min{γ max 1,γ max 2 }. The new states ˆρ i are uniquely determined from ˆγ i by inverting the relations: f(ˆρ 1 ) = ˆγ 1, f(ˆρ 2 ) = ˆγ 2, {ρ 1,0 } ]τ(ρ 1,0 ), 1], if 0 ρ 1,0 σ, with ˆρ 1 [σ, 1], if σ ρ 1,0 ρ max. [0,σ], if 0 ρ 2,0 σ, with ˆρ 2 {ρ 2,0 } [0,τ(ρ 2,0 )[, if σ ρ 2,0 ρ max. (2.3) (2.4) 5

6 From now on we set ρ max = 1 and the flux function f = vρ is assumed to be: f(ρ) = f max σ ρ if 0 ρ σ, ) if σ ρ ρ max, f max ( ρmax ρ ρ max σ (2.5) where σ is the value of density corresponding to the maximum flux f max, see Fig. 2. Such fundamental diagram is usually called Newell-Daganzo flux 1 f(ρ) f max 0 0 σ ρ max ρ Fig. 2. The Newell-Daganzo flux function. function. Modelization of the congested phase is complicated, since in this case the flux assumes a scattering behavior, therefore there are many possible choices for flux function. Our approach is to use a simple model with reasonable properties: 1) there are only two characteristic velocities; 2) it is able to reproduce empirical phenomena of backward moving clusters. 3 Colombo s Phase Transition Model A hyperbolic phase transition second order model for traffic was recently introduced by Colombo, see [10], [11]. Two phases corresponding to the free and congested flow are considered. In the free flow the LWR equation ρ t + (ρv) x = 0, v = v f (ρ), (3.6) is assumed to hold; see Section 2. When the car density ρ is high, the assumption that the speed v is a function only of the density is no longer taken. As the speed v reaches a certain value, the density-flow points are scattered in a two-dimensional region. Therefore, the traffic evolution in the congested region is described by the equations t ρ + x [ρ v] = 0, (3.7) t q + x [(q q )v] = 0, 6

7 where v = v c (ρ,q) is a known function depending on the density ρ and on the linearized momentum q, while q is a given parameter. From the traffic point of view, q is characterized by the phenomenon of wide jams, for further details see [11]. 3.1 Choice of the velocity In the original Colombo s model, the velocity is assumed to obey the Greenshields law ([20]): v f (ρ) =v max ( 1 ρ ) ρ max, v c (ρ) = v max ( 1 ρ ) q ρ max ρ. (3.8) In the free phase, we have v v max, while in the congested phase we have v < v max and we aim at determining the state (ρ,q) of the system. Since the choice (3.8) in the congested phase infers some complexity to Riemann solver, we are motivated to find an alternative specification of the velocity. In particular, we will set v f (ρ) =v max, (3.9) ( v c (ρ,q) = (ρ max ρ) A + B ) ( + (q q )(ρ max ρ) A + B ). (3.10) ρ ρ The second term is a perturbation term: Depending on the flow q, the quantity (q q ) may be positive or negative. Notice that with the choice (3.10) the flux is quadratic. We further set B = σ v max ρ max σ Aσ, with A ( 0, σ v max (ρ max σ) 2 ] and σ ρmax /2. Then A and B are positive and for q = q we have the following features: for ρ = σ the speed equals v max ; for ρ = ρ max the speed equals 0; the velocity is a decreasing function of the density, while the flux is concave and decreasing for ρ σ. We remove the gap between the phases of the original model, thus we have the following domains: Ω f = {(ρ,q) : ρ [0,σ + ],q = ρv max }, 7

8 { Ω c = (ρ,q) [σ,ρ max ] [Q,Q + ] : v c (ρ,q) v max, Q q R q q ρ } Q + q R respectively for the free and congested phase. The parameters satisfy Q q Q +, σ σ σ + and v c (σ,q ) = v c (σ +,Q + ) = v max. Therefore Ω f Ω c = {(ρ,q) : σ ρ σ +,v c (ρ,q) = v max } and we call such states metastable. The intersection of the phases is possible because of the special choice (3.9), (3.10). We now determine further restrictions on the parameters to achieve specific properties of the model. Positivity of speed. We want to ensure v c 0. If (q q ) 0, then this is granted. For the other case, we impose: 1 (Q q ). (3.11) Properties of characteristic fields. The eigenvalues are λ 1 = v+(q q )v q +ρv ρ, and λ 2 = v, and the corresponding eigenvectors are, respectively, r 1 = (ρ,q q ) and r 2 = (v q, v ρ ). Thus: ( λ 1 r 1 = 2Aρ + 2 (q q ) A (ρ max + σ 3ρ) σ v ) max, λ 2 r 2 = 0. ρ max σ We would like the first expression to be strictly negative for the first characteristic field to be genuinely nonlinear. Now 2Aρ 0 is decreasing, while A (ρ max + σ 3ρ) σ vmax 2 σ ρ max σ 0 and is decreasing for ρ. So the sum is 3 always negative for q q 0. To ensure negativity for q q 0 we require: Q q ρ max A A (σ 2ρ max ) σ vmax ρ max σ. (3.12) Hyperbolicity. In order to ensure the hyperbolicity, the eigenvalues must satisfy the relation λ 1 < λ 2, which can be expressed as: Aρ+ σρ max ρ ( A v ) ( max +(q q ) A(ρ max + σ 2ρ) ρ max σ σv ) max < 0, ρ max σ which is the sum of a negative increasing term and, for q q > 0, a negative term decreasing in ρ. Hence we require: Waves velocity sign. To ensure λ 1 0, we impose ) v max ρ max σ Q q > A ( σ ρ max A ρ max A (σ ρ max ) σ vmax (ρ max ρ) ( A + B ρ +(q q ) [ 2(ρ max ρ) ( A + B ρ ρ max σ ) [ ( ) ] + ρ A + B (ρmax ρ) B ρ 2 ρ ) + ρ [ ( A + B ρ. (3.13) ) (ρmax ρ) B ρ 2 ]] 0, 8

9 where the first term is negative decreasing and the second one is decreasing from a positive to a negative value (for q q 0). So it is sufficient to require: Q + q ρ max (σ ρ ( max) A + B σ [ ( ) ]) A + B ρ ρ (ρmax σ) B ρ 2 2(ρ max σ)(aσ + B) + σ [ ].(3.14) (Aσ + B) (ρ max σ) B ρ 3.2 Phase transition model on sequence of roads. To determine the dynamics at junctions for the phase transition model we proceed as for the LWR case. We have maximal fluxes again determined by the sign of waves. To describe this we need some notation. Call F the maximal density flux in free phase. For every (ρ,q) in congested phase (including metastable states), we indicate by f1 max (ρ, q) the maximal density flux along the first family curve through (ρ,q) and by f2 max (ρ, q) the maximal density flux along the second family curve through (ρ,q). Then the maximal fluxes are given by: γ max 1 = ρv f (ρ), (ρ,q) free not metastable, i.e. ρ σ, f max 1 (ρ,q), (ρ,q) metastable or congested. γ max 2 = F, (ρ,q) free, f max 2 (ρ,q), (ρ,q) congested. The through flux is determined as before, while the states are determined inverting relations similar to (2.4) and (2.3). 4 Description of the approximation algorithm Our procedure is composed by the following steps: 0. data capturing; 1. data cleaning; 2. calibration of flux parameters; 3. generation of approximate solution; 4. computation of errors. Let us give the details. 9

10 4.0.1 Step 0: data capturing Traffic is observed by sensors located along roads. Sensors acquire traffic data, i.e. the flux, the velocity and the occupation rate, along each road with a frequency of one minute within an entire day. In the portion of road we are considering, namely Viale del Muro Torto, described in Section 5, there are 7 sensors per direction Step 1: data cleaning For each segment, traffic measured data represented by the flux, i.e. the number of cars crossing it per minute, the velocity, considered as the average speed of cars at every minute, and the occupation rate, namely the time interval in which cars exit the magnetic field generated by sensors, are stored in a file labelled as the road itself. Due to their structure, such files cannot be used directly by simulation algorithm, hence we need to apply a standardization procedure to consent data loading to the program. Since density is not detected by sensors, we may recover its value on each road as the ratio between measured flux f i and velocity ṽ i : ρ i = f i ṽ i, if ṽi 0, for i = 1,...,T, with T the final observation time expressed in minutes. Otherwise, if ṽ i = 0, a selection and cleaning of measured data is done. More precisely, traffic data are required to verify the following admissibility conditions. When the velocity is null a flux different from zero cannot be assumed, hence in this case we exclude such values. If the flux is null and the occupation rate is also null then the density is taken equal to zero, otherwise we can assume the density to be maximal, as displayed in (4.15): if ṽ i = 0 and f if õ i = 0 ρ i = 0, i = 0 if õ i 0 ρ i = ρ max. (4.15) Since the most complete data sets are from sensors on road segments 544, 549 and 548, we mainly focused on them and, in this case, the percentage of excluded data is around 5.6% Step 2: calibration of flux parameters A calibration procedure is applied to each segment of the road. It consists of a minimization with two parameters, namely f max and σ, to determine the analytical expression of flux function (2.5), where the maximum density 10

11 ρ max can be theorical, and in this case it is fixed to 333 on the whole road, or measured and eventually different on each road. The optimized parameters are computed in the original scale and the maximal speed is consequently derived as v max = f max /σ. Flux parameters obtained by the calibration procedure determine different flux functions on each road, thus adapting its shape to their different features. To measure calibration errors we consider the functional J, given by the sum of squares of differences between analytical and measured fluxes. Separating into free and congested phase, we get: J free = ρ ( fmax J = i σ σ ρi f i) 2, J congested = ρ ( ( ) ) ρmax ρ i 2 (4.16) i >σ fmax fi. ρ max σ Step 3: generation of approximate solution We produce approximate solutions on the whole road solving problem (1.1) on each segment. To this aim we make a space-time discretization introducing a numerical grid in R N (0,T), where: x is the space grid size; t is the time grid size; (t l,x m ) = (l t,m x), for l,m varying, respectively on a subset of N and Z, are the grid points. For a function v defined on the grid we write v l m = v(t l,x m ) for l = 0,...,N, and m = 0,...,M, with N the number of iterations in time and M the number of space steps. We apply the algorithm based on Godunov scheme and presented in [3]. The resulting formulas leads to consider the local traffic supply and demand concepts, introduced by J.P. Lebacque, see [28]. Roughly speaking, the initial datum is firstly approximated by a piecewise constant function; then the corresponding Riemann problems are solved exactly and a global solution is simply obtained by piecing them together; finally, one takes the mean and proceeds by induction. A piecewise constant approximation of the initial datum is derived by averaging over the intervals [x m,x m+1 ], while v l m is defined recursively. Under the CFL condition { } t sup sup f (u) m,l u I(u l m 1/2,ul m+1/2 ) x, (4.17) the waves, generated by the Riemann solution to initial data (v l m 1,v l m) from x m 1/2, do not reach the points x m 3/2 and x m+1/2 for t (t l,t l+1 ). Then the 11

12 scheme can be written as: v l+1 m = vm l t ( g G (v l x m,vm+1) l g G (vm 1,v l m) ) l. (4.18) where g G (u,v) = f(w R (0;u,v)), with W R ( x t ;v,v + ) the self-similar solution with data v and v +. The precise expression of the numerical LWR flux is min g G z [u,w] f(z) if u w, (u,w) = max z [w,u] f(z) if w u. Boundary conditions are imposed as explained in [3]. The Godunov scheme can be used also for the Colombo s model. However, since the admissible region in the state space, union of the free and congested regions, is not convex, some corrections should be used. The latter involve a projection on the admissible region, introducing a further numerical error. For the details we refer the reader to [7]. Input data of the algorithm provided by previous steps of the procedure are: 1. the optimized parameters f max, σ and, eventually, ρ max (if the maximal density is not fixed), to be inserted in the analytical expression (2.5); 2. initial and boundary conditions. Since the simulation starts during night, we assume an empty configuration of density and, as boundary conditions, we use measured values of density detected by sensors located in the initial and final endpoints of the segment, see Fig. 3. The simulation algorithm produces numerical values of flux and density along the entire road, therefore we can derive the flux, denoted by f num,l, computed at a certain time t l in the point of the road corresponding to the sensor position. OUTGOING BOUNDARY INCOMING BOUNDARY b f num,l a RECONSTRUCTED DENSITY Fig. 3. Schematization of the approximation procedure on a segment at time t l. 12

13 4.0.5 Step 4: computation of errors Let us introduce the approximation error. To this aim we define f max as the minimal flux value such that 90% of measured fluxes are below it and, at each time step, we compute errors as the differences between numerical flux f num and measured flux f: E free = k E = 1 K free f num,k 1 f k 1 / f max, E cong = k 2 K cong f num,k 2 f k 2 / f max, (4.19) where K free and K cong are, respectively, the sets of effective indexes of admissible values of densities in free and in congested part, obtained excluding densities corresponding to non-admissible fluxes. In particular, we are interested in the average errors: E f = E free /#K free, E c = E cong /#K cong. 5 Simulation results Here we report some results obtained by the application of the algorithm described in Section 4, first focusing on the LWR case. Input data of simulation algorithm, provided by ATAC S.p.A., refer to an area of the city of Rome, Viale del Muro Torto, which links the historical center with the northern area of the city. We consider the path covered moving from Corso d Italia towards Piazza del Popolo and we fix within the network a single segment of length 776 meters, see Fig. 4. Notice that black circles on the map represent sensors. In the following Fig. 5 we represent a diagram of the measured flux during an Fig. 4. Viale del Muro Torto. 13

14 entire week. The first part of the graph, i.e. up to density ρ 55, represents the free phase of traffic, while the second part reproduces the congested phase ρ FREE 2000 Flux CONGESTED Density Fig. 5. Measured flux-density diagram. 5.1 Calibration of Fundamental diagrams for road segments Here we consider all segments composing the road of Viale del Muro Torto. The fitting procedure, characterized by the steps 0)-1)-2) of the algorithm described in Section 4, is performed assuming different initial conditions of flux parameters. A two-parameters constrained optimization is obtained by the following procedure organized in four steps: i) in the congested phase we apply a least square method (regression line) and we call f k 2 the values approximating f k 2 ; ii) setting E regr = ( f k2 f k2 ) 2 k 2 K cong as the mean square error, we compute the average error: iii) indicating by δ the variance: δ = µ = E regr #K cong ; k 2 K cong (( f k 2 f k 2 ) 2 µ) #K cong we discard the values of congested flux not satisfying the condition: ( f k 2 f k 2 ) 2 < µ + δ. 14

15 iv) taking ρ max as the maximal measured density on each road a constrained optimization on the screening data is operated to determine σ and f max. Simulations run starting from different initial values within the intervals: 10 < σ < 70, 1500 < f max < (5.20) Since the maximum velocity is defined as the ratio f max /σ, we derive v max from the optimized values. Then we compare it to the maximum velocity ṽ max computed as the average of maximum speed measured by sensors in the free phase of traffic. In Table 1 we report the results of the calibration procedure. From calibra- Initial values Optimized values Road code ρ max σ f max σ f max v max ṽ max Table 1 Two-parameters constrained optimization subject to (5.20) for different initial values with ρ max the maximum density observed on each segment. tion of data produced by sensor located inside the road, we get optimized 15

16 parameters in the original scale: σ = 38.87, f max = , (5.21) hence v max is about 58 km/h. Setting x = 77.6 the examined road is divided into 10 sub-intervals and the CFL condition t v max < x (reducing all quantities to the same scale, e.g. meters per second) reads: t < 77.6 t < t < Recalling that t corresponds to 1 min (60 sec ): t t < 4.81sec 60sec Assuming t = 1 = , with time expressed in minutes, the CFL required 16 by Godunov scheme is respected. In order to impose boundary conditions we use traffic data provided by sensors on the right and on the left endpoint of considered segment as, respectively, incoming ρ inc b (road code 548) and outgoing boundary data ρ out b (road code 544), see Fig. 6. Since the number of Fig. 6. A road segment in Viale del Muro Torto. iterations in time of the simulation algorithm is N = 16 T, we need to fill the gaps in the density values. This can be done using, for example, a linear interpolating procedure: ρ num,l = ( 1 r ) ρ l1 + r ρ l 1 +1, l = 1,...,N, with l 1 = [l/16] and r = l l 1. The average errors for free and congested phase are respectively: E f = , E c = Therefore the percentage error is around 10% for free phase and around 17% for congested phase, thus it is comparable to the sensor error which is, in the latter case, around 20%. The following Figures 7, 8, and 9, show a comparison between computed and given fluxes. In the mentioned figures we also report 16

17 the density values ρ, depicted according to the scale on the right side of y- axis. This was done to have an immediate distinction between the two traffic phases, namely free phase and the congested phase, for ρ > σ = From the analysis of Figures 7 and 8, it can be noticed that the approximate f num f meas ρ meas Flux Minutes Fig. 7. Comparison between f num and f in the first day, from 0:00 to 6:00 a.m. - free phase only f num f meas ρ meas Flux σ Minutes Fig. 8. Comparison between f num and f in the first day, from 16:00 to 20:00. solution f num seems to substantially follow the profile of the curve of experimental data, except in some zones where the distance between f num and f is quite high. We could consider the curve of approximate solution as a sort of average of the curve given by measured data. In order follow the behavior of our approximation more accurately, in Fig. 8 we restrict to the period 18:00-19:00 and we represent it in Fig

18 For sake of completeness we investigated how the frequency with which sensors f num f meas ρ meas Flux σ Minutes 1140 Fig. 9. Comparison between f num and f in the first day, from 18:00 to 19:00. collect data affect our approximation. To this aim, we applied the optimization procedure supposing that data are provided with a lower frequency, i.e. for a time interval of 2 and 4 minutes. We obtained the results displayed in Table 5.1. It can be noticed that a lower frequency in collecting data means an t = 1 t = 2 t = 4 E f E c E f E c E f E c Table 2 Two-parameters optimization obtained setting differently the frequency of detecting data by sensors. inferior quantity of informations: t = 2 corresponds to half of data, t = 4 corresponds to quarter of data. However, from results in Table 5.1, we just observe a slight worsening in the approximation: the error in the congested case increases less than one percentage point if t = 2 and less than two percentage points if t = Data reconstruction Our algorithm permits to reconstruct the data on the whole road for every time. Some graphs describing the evolution of the density from 6:00 to 10:00 18

19 are reported in Fig. 10. An important consequence is the possibility of computing and visualizing the queues forming at the end of the road and moving backwards. Using only the data from sensors, we can just determine if the queue reached one sensor or did not. Since sensors are placed every 400 meters (more or less), the average exptected error for the queue length can be of 200 meters. This in turn may give rise to big errors in the estimation of the travelling time (usually communicated to drivers through panels at the entering of the road). On the contrary our algorithm permits to reconstruct the queue end position at every time, with a low error, thus giving good travelling time estimates. Related animations are reported on the web page [5]. 100 Density at h. 6: Density at h. 8: Road Density at h. 9: Road Density at h. 10: Road Road Fig. 10. Reconstruction of the density on the segment between 6:00 and 10: Hyperbolic phase transition model We give here the results for data reconstruction for the hyperbolic phase transition model described in Section 3. The fundamental diagram for this model is defined by the optimal value for σ and f max given by (5.21). They represent here the equilibrium critical density and maximal flux respectively. We take also the same value of the maximal density, which is a physical constraint. The parameter A is chosen to be 1e 02. The parameters Q and Q + are maximized in absolute value under the constraints described in Section 3. Using the same metric as for the LWR model, the average errors for free and 19

20 congested phase are respectively: E f = , E c = Strictly speaking, the LWR model performs equally in the free phase and better in the congested phase than the hyperbolic PT model. This is explained by two factors: in the middle of the section where the values are compared the congestion occurs only 17 percent of the time, so it is not a system where the dynamics in congestion phase plays a central role; the Colombo diagram needs the boundary values to be in its two dimensional domain, thus the boundary data needs to be projected on the fundamental diagram. For the LWR diagram, the observed densities can be directly used as boundary conditions. Indeed, if we compute the error E proj between the original boundary data and the values used by the scheme, for the LWR model we have clearly E proj = 0 because the exact values for density can be used in the scheme. For the PT model, with the projection used, the error is E proj (ρ) = 0.06 and E proj (v) = 0.22, thus the advantage of having a more complex dynamic in congestion is reduced by the restriction imposed on the size of the fundamental diagram and the necessity to project the exact data. In order to give a richer picture of the compared performances of these two schemes, we propose to compute the relative L 1 -error for the two models, for flux, density, velocity, as reported in Table 3. LWR PT q ˆq ˆq ρ ˆρ ˆρ v ˆv ˆv Table 3 Comparison between L 1 -errors obtained by LWR and PT models. Notice that the difference is more significative for densities and velocities and in favor of the PT model. 20

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