Estimate of traffic emissions through multiscale second order models with heterogeneous data

A sample of a GPS trajectory dataset (left) and of flux data coming from a fixed sensor (right). The data was provided by Autovie Venete S.p.A and are not publicly available

Left: flux function $ {\mathcal{F}}(x,t,\rho) $ at fixed $ t $ as $ \rho $ and $ x $ change; the circles represent $ (\sigma(x,t),{\mathcal{F}}^{\max}(x,t)) $ as $ x $ changes. Right: comparison of the flow function given by the (CV) and (ACVs) approaches, assuming to have three vehicles travelling with different speed and close enough to have $ \phi(x,t)>1 $; the solid line represents the flux obtained by the (ACVs) formula; the dotted lines represent the flux obtained with the (CV) formula with closest vehicle moving with velocity $ \dot{p}_\kappa = 90, 50, 5\,{\mathrm{km/h}} $

Comparison between model (1) with velocity function $ {\mathcal{U}} $ defined in (2) and (5), red-solid and blue-dotted lines respectively, at different times of the simulation starting from a rarefaction wave (top) and shock one (bottom)

Left: Trajectories generated by the second order microscopic model used in [12]. Right: Evolution in space and time of the macroscopic density $ \rho = \rho(x,t) $ obtained by the multiscale model (14)-(2) with the $ N = 50 $ trajectories given on the left

Evolution of the density $ \rho = \rho(x,t) $ along a road at time $ t = 50{\Delta t} $ (left) and $ t = T $ (right), obtained by solving (14) with $ {\mathcal{U}} $ defined in (2) and (5), represented by red-solid and blue-dotted lines, respectively

Left: A sample of the trajectories drawn in Figure 4-left. Right: Evolution in space and time of the density $ \rho = \rho(x,t) $ obtained by solving (14)-(2) with only the $ N = 5 $ trajectories given on the left

Comparison between the microscopic E-max-formula and the E-exp-formula (see Section 5.2)

Effects of monitored slow vehicles on the second order model (15), see Section 5.1

Vehicle trajectories (25) on a stretch of the road

Section 5.2 tests. Density profile at $ t = 10\,{\mathrm{min}} $ (top) and total amount of emissions associated with traffic dynamics using the E-max-formula (22) (center) and the E-exp-formula (23) (bottom) for different $ {\widetilde N} $

Section 5.3 test. Sketch of the highway network, where the roads are numbered from 1 to 6, the triangles represent the fixed sensors, the diverge junctions are represented by points D1, D2, D3 and the merge ones by points M1, M2, M3

Section 5.3 test. Example of real trajectory data recorded on 27/08/2021. The size of the space-time circles is proportional to vehicles velocity. The data were provided by Autovie Venete S.p.A and are not publicly available

Section 5.3 test. Variation in time of the flux per minute of heavy vehicles recorded by the three sensors on 27/08/2021. The data were provided by Autovie Venete S.p.A and are not publicly available

Section 5.3 test. Density of vehicles at different times of the simulation

Section 5.3 test. Total emissions on road 2 (left) and road 3 (right)

Section 5.3 test. Density, speed, acceleration and $ {\mathrm{NO_{x}}} $ emissions on the last 15 km of road 3 at different times

Section 5.3 test. Domain $ \Omega $ built around the junctions involving the four horizontal roads

Section 5.3 test. Source term of $ {\mathrm{NO_{x}}} $ emissions (top) and diffusion of $ {\mathrm{NO_{x}}} $ concentration into the air (bottom) at different times. The dashed red lines are used to identify the four roads in $ \Omega $