Dynamics of particulate emissions in the presence of autonomous vehicles

2.1 A model for particulate emissions

Focus on the microscopic emission model described in [30]. Given a generic petrol vehicle j with instantaneous speed v j ( t ) [m/s] and acceleration a j ( t ) [ m ⁄ s 2 ], the corresponding PM emissions are estimated as:

where ψ i , i = 1 , … , 4 , are as follows for an internal combustion engine:

while ψ 5 represents an emission lower-bound, assumed zero in case of using ψ i , i = 1 , … , 4 . The overall PM emission for a fleet of N vehicles is as follows:

Note that (1) is a combination of polynomials in velocities and accelerations. This allows possible extensions and adaptations to other different emission models (see [38] for further details).

2.2 Production of pollutants

Chemical reactions for particulate deal with different gases, including NO x and SO x . The formation of nitrogen oxides involves atomic oxygen (O), oxygen ( O 2 ), and nitrogen ( N 2 ) with consequent production of ozone O 3 . We focus on the following reactions for the ground-level ozone formation due to nitrogen oxides:

where, indicating by mol the number of molecules, k 1 , k 2 , and k 3 are the following kinetic constants [39]:

From reaction (2), NO 2 is photo-dissociated into O. The creation of O 3 is due to (3). Finally, from (4), O 3 is transformed into NO 2 and O 2 . Further remarks about the principal chemical processes for the cycles of O 3 due to NO x are fully described in [32] and [33].

For the deterioration of NO x and the evolution of SO x in the atmosphere, we also consider the following reactions [40,41]:

with [39]:

Note that, while (6) and (7) describe the whole cycle for the dynamics of sulfur oxides, equation (5) indicates the decay of nitrogen dioxide, hence completing the atmospheric process involving NO x and ozone. Representing by [ ⋅ ] the concentration (in terms of weight unit/volume unit) vs time of a generic chemical species, we have the following ODEs from reactions (2), (3), and (4):

From (5), (6), and (7), we obtain

Equations (13)–(20) show that concentrations of O, O 2 , O 3 , NO x , and SO x evolve following the dynamics of [ SO ] [ O 2 ] , [ SO 2 ] [ O 3 ] , and [ NO 2 ] [ O ] . In equations (17) and (19), we also consider the damping terms, k n [ NO 2 ] and k s 2 [ SO 2 ] , respectively, to model the decay phenomena of the principal pollutant agents. These additive terms are necessary to avoid unbounded growth and reflect the natural decay of molecules. As sulfur oxides depend on O, O 2 , and O 3 , that also appear in reactions for nitrogen oxides, equations (13)–(20) provide a unique model for the contemporary evolution of NO x and SO x and, hence, for PM 10 and PM 2.5 . With this aim, assuming that all reactions occur in a volume of dimension Δ x 3 , we define the source term S P ( t ) for the pollutant P ∈ { NO x , SO x , PM } as follows:

where E P ( t ) coincides with (1) in case of particulate, while the corresponding expressions for nitrogen and sulfur oxides are discussed in [30].

The particulate has primary components due to S PM ( t ) and secondary contributions mainly depending on NO x and SO x in the atmosphere. Concentrations of nitrogen and sulfur oxides are not always precisely predictable for PM 10 , see [8], hence we impose that:

where the coefficients 0 < α i < 1 , i = 1 , 2, α 1 + α 2 = 1 , modulate the contributions of NO 2 and SO 2 , and the term k n [ PM 10 ] describes the damping phenomena which model the natural decay of the pollutant over time. For PM 2.5 , we have that [42]:

where β = 0.62 . Coupling ODEs (8)–(12) for the cycles of O 3 due to NO x to equations (13)–(20), we obtain the following final system for the concentrations of pollutants at ground level:

where p = 0.15 (see [43]) indicates the percentages of NO 2 , and q i ∈ [ 0 , 1 ] , q 1 + q 2 + q 3 modulate the contributions of SO x .

Note that system (24) can be rewritten as:

where Γ and s are the vectors that contain, respectively, the concentrations of pollutants and the source terms, the latter being of the type (21), while G is the matrix that describes the reactions.

2.3 Horizontal diffusion

To model the horizontal diffusion of pollutants, we focus on a horizontal domain Ω = [ 0 , L x ] × [ 0 , L y ] , where L x and L y are, respectively, the length of the road and of the area transversal to the road where, in [ 0 , T ] , the pollutant agents spread. Assume that Γ , G , and s represent the same quantities described in the previous paragraph, and that Γ is constant along the third direction so that the model becomes two-dimensional. We obtain the following reaction-diffusion problem for the horizontal diffusion of pollutants:

where μ is the diffusion coefficient, assumed the same for all chemical species and equal to 1 0 − 8 km 2 ⁄ h for aerosols, see [44]; w = ( w x , w y ) is a vector that models the wind direction. We consider assigned initial data, while the boundary of Ω considers homogeneous boundary Neumann conditions for all chemical species in consideration. For a suitable treatment of the source term s , that involves contributions for NO x , SO x and particulate, see details in [33].

For our analysis, we integrate equation (25) over a uniformly discretized, rectangular domain with holes, that correspond to “green barriers,” to estimate the entity of diffusion of the various pollutant agents. Note that we may consider other types of vegetations besides green barriers. For instance, high-level green infrastructures (such as trees) represent an interesting case study to mitigate the spread of pollutants. However, in [45], it is shown that trees may harm air quality, especially in street canyon environments. Hence, further investigations are needed.

3.1 Control types

The presence of a unique AV inside the fleet proves experimentally that stop-and-go waves vanish or are dampened. The entity of such a phenomenon depends on the implemented control for the AV. As the primary aim is to stabilize traffic conditions to reach safety with a defined velocity [46], for each possible control strategy we have the following basic idea: from the evolution of traffic, a desired speed U is estimated and a commanded speed v cmd is provided as input to a low-level controller that guides the AV. The different controllers of the AV are as follows: Follower Stopper (Experiment A), human-implemented control (Experiment B), Proportional Integral (PI) with saturation (Experiment C).

In Experiment A, following some instructions that are established by an external infrastructure, the AV tracks U in safety conditions and v cmd < U when safety is needed. In Experiment B, the human pilot maintains an assigned speed and slows down only for possible collisions. In this case, the desired speed U , estimated through the length of the ring road and the time the AV needs for a complete pass, is communicated to the driver via radio. In Experiment C, the controller of the AV uses a PI logic to track v cmd from information about the average speed of the vehicles in front. A saturation effect is also useful to ensure the safety distance.

3.2 Features of the experiments

In what follows, we give details about some features of the various experiments. In Experiment A, stop-and-go waves appear at 79 s and the control by the AV starts at 126 s. The commanded velocity, communicated to the driver by an external infrastructure, varies step by step from v cmd = 6.50 m/s up to v cmd = 8.00 m/s. The autonomy period and the experiment end are at 463 and 567 s, respectively.

For Experiment B, traffic waves occur at 55 s and the control, established by the human pilot of the AV, starts at 112 s with v cmd = 6.25 m/s. The velocity has just a variation at 202 s. After the control is disabled at 300 s, the experiment ends at 409 s.

Experiment C has the first traffic wave at 161 s. The autonomy phase of the AV is activated at 218 s and v cmd is adjusted by the automatic controller at each time instant until the end of the experiment (413 s).

Tables 1 and 2 report the main characteristics of the experiments, where we distinguish: an initial phase, IPH; an interval, TW, for traffic waves; intervals, indicated by C i , where the control of the AV is activated; an interval DC, where the control is disabled and the experiment ends.

We provide information due to data analysis in Experiment A. For Experiments B and C, the obtained results are similar.

Table 3 presents an overview of the mean velocities for some vehicles along the ring track in Experiment A. Vehicle 21 is the AV, while 5 and 15 refer to vehicles that are within the fleet, approximately the same distance from the AV. From the results, obtained by not considering the outliers in the first and last intervals of the experiment, it follows that the mean velocities are not exactly the desired ones. The differences are due to the time required for all cars to adjust to the desired speed established by the AV. Note that, when stop-and-go waves arise, the mean velocities are, as expected, quite variable. During the control intervals, the regularization effect is more obvious. For instance, in C 2 , C 4 , and C 5 all vehicles have almost the same mean velocity. When the control is disabled in DC, a regularization phenomenon is still present and does not disappear immediately. This is confirmed by the almost similar values of the mean velocities.

Figure 2 shows the dynamics of velocities for the AV (black line) and vehicles 5 and 15 (blue and orange curves, respectively) in Experiment A. In the interval TW, velocities have high variations due to stop-and-go waves and the AV moves following an almost regular sinusoidal tune. For the other vehicles, the situation is similar but the amplitude of the oscillation is less marked for the car that is further away from the AV. The traffic is drastically dampened when the control is activated for the AV. In particular, in C 1 stop-and-go waves are reduced but this effect is not completely evident as the commanded velocity for the AV is 6.50 m/s, different from that of the overall traffic. In C 2 , v cmd becomes 7.00 m/s, a value comparable to the average speed of the entire fleet. This in turn creates more regularity in traffic dynamics, with a consequent reduction of possible variations in velocities.

Figure 2

Velocities of the AV (black) and vehicles 5 (in blue) and 15 (in orange) in Experiment A: comparisons among velocities in TIs TW, C 1 , and C 2 .

For Experiments B and C, we obtain similar features that are not considered here.

4.1 Estimation of PM emissions

Figures 3, 4, and 5 show the evolution of PM emissions (indicated by E TOT ) vs time. The average emissions (AEs) in the critical intervals of type TW are as follows: 2.76 mg/s for Experiment A, 2.23 mg/s for Experiment B, and 1.81 mg/s for Experiment C. Indeed, Experiment A represents the worst case and all case studies show different features before the activation of possible control strategies.

Figure 3

Red line: PM emissions for Experiment A. Dashed black lines: average values in the various TIs.

Figure 4

PM emissions for Experiment B (red line), and average values in TIs (dashed black lines).

Figure 5

Red line: evolution of PM emissions for Experiment C. Dashed black lines: average values in different TIs.

In Experiment A, traffic waves are already dampened in C 1 but the highest reduction occurs in C 2 (best control interval, green check mark in Figure 3) when the AV and the other vehicles move all at a speed that is about the commanded one, v cmd = 7.00 m/s. When the control of the AV is disabled in the interval DC, traffic waves reappear and the AE becomes higher (albeit slightly) than that of the interval IPH. Table 4 shows the emission average values (expressed in milligrams/second [mg/s] and grams/minute [g/min]) for all TIs of Experiment A.

In Experiment B, when the pilot of the AV drives in C 1 at v cmd = 6.25 m/s, traffic waves become lower and, consequently, the AE decreases. When v cmd becomes 7.15 m/s in C 2 (best control interval, green check mark in Figure 4), all vehicles have almost the same speed, i.e., the phenomenon is similar to that of Experiment A. Note that PM emissions are, indeed, almost similar in C 1 and C 2 . When the control is disabled in DC, as expected the AE increases. The corresponding values are in Table 5 for Experiment B.

For Experiment C, when the control is activated in C 1 , traffic waves are mitigated and the AE decreases. Note that the commanded velocity is highly variable due to the automatic logic implemented on the AV. This is also reflected in emissions that, although lower in the control interval, appear irregular in Figure 5. In Table 6, values of AEs are presented for Experiment C in all TIs.

For all experiments, PM emissions are dampened when control is activated on the unique AV of the fleet in consideration. The percentage reduction is different in the various experiments as a result of the used control strategy. Assume that R i indicates the percentage reduction of particulate emissions for Experiment i , i ∈ { A , B , C } , i.e., R i shows the decrease from the interval TW to the best control interval for Experiment i . We have

We note that Experiments A and C are almost similar in terms of decrease of PM emissions, unlike Experiment B for which reductions are about half those of Experiment A. This is predictable for Experiment B, for which only the human pilot determines the real control action.

Now we provide information about the variations in PM emissions. In Figure 6, we consider the normalized derivative of E TOT ( t ) in TW and in all control intervals for Experiment A. It is shown that the PM emissions have positive variations in TW and this confirms the increase of E TOT ( t ) . When the control strategy starts in C 1 , the derivative becomes abruptly negative, which indicates the dampening of stop-and-go waves and the consequent decrease of E TOT ( t ) . In C 2 , the overall traffic is more regular, hence creating a sinusoidal trend in oscillations. Although the mean values for PM emissions are slightly higher in C 3 , C 4 , and C 5 w.r.t. the phenomenon in C 2 , the control of the AV is still active and this creates almost regular variations. Indeed, the peak in C 4 derives from the deviation of v cmd from the average traffic speed. Hence, while there is still control, the damping of traffic waves is strongly due to the way the AV exerts control, and stop-and-go waves tend to reappear when the control logic obeys dynamics different from the average traffic. Finally, in C 5 , v cmd assumes the value observed in C 3 , thus stabilizing the phenomenon.

Figure 6

Normalized derivative of E TOT ( t ) , expressed in g/s 2 , in TIs TW, C 1 , C 2 , C 3 , C 4 , and C 5 .

Finally, in Figure 7 we present for Experiment A a comparison between NO x and PM emissions, the former described in [48]. The percentage reduction for emissions of nitrogen oxides in the best control interval C 2 is 23.30%, about one half the corresponding decrease for PM, 50%. Indeed, the used model for the emissions estimation presents similar features for either NO x or PM. In Figure 7, we show the emissions normalized variations for the two types of gases in TIs TW and C i , i = 1 , … , 5 . In the presence of stop-and-go waves, the two variations have a different evolution but the shapes are quite similar and almost coincident when the traffic becomes stable (at about 100 s). This means that particulate and nitrogen oxides have similar features, and possible phenomena due to other polluting agents are less evident. When the control of the AV is activated in C 1 , the two variations are still similar until about 160 s but then become different. This is due to the effects of the AV whose control strategy modifies the traffic and, hence, the concentration of pollutants that compose PM. Indeed, at about 250 s (in C 2 ) the presence of the AV dampens more the traffic waves, and the variations of particulate and nitrogen oxides still become similar with low differences in their peak values. In C 3 , PM and NO x have almost coincident variations. In the remaining control intervals C 4 and C 5 , we have similar situations. From the obtained results, we obtain that the control of the AV stabilizes different types of emissions in the same way.

Figure 7

PM (blue) and NO x (red) emissions normalized variations, expressed in g/s 2 , in TIs TW and C i , i = 1 , … , 5 .

For Experiments B and C, the normalized derivatives for emissions present similar characteristics.

4.2 Pollutants at ground level

The concentrations of pollutants along the ring road at street level are estimated by finding solutions to system (24). In order to show the effects of concentration reduction due to the presence of AVs, we analyze separate cases: experiment phases when high traffic waves appear (AV with disabled control); TIs when autonomous driving dampens stop-and-go waves (AV with exerted control).

By using approaches that deal with stiff features of system (24), see [33], we find numerical solutions on a TI I = [ 0 , T ] by a suitable choice of source terms (21). As we need simulations over a larger time horizon, the source term signal is prolonged by repeating the emission profile in I , namely: we obtain a periodic emission profile from the experimental data in a given phase (traffic waves or autonomy). We consider the following notations to distinguish among the traffic sources in different experiments and time intervals: S Γ , Φ P is the repetition, over I , of the source term in the TI Φ ∈ { T W , C i } for pollutant P ∈ { NO x , SO x , PM } in case of Experiment Γ ∈ { A , B , C } . For all simulations, we assume that: chemical species are estimated for each volume of size Δ x 3 , with Δ x = L π − 1 , where L = 260 m is the length of the ring track; T = 30 min; the concentrations at t = 0 , indicated by [ ⋅ ] 0 , are as follows:

where p = 0.15 . For the pollutants that compose PM 10 , see equation (22), we consider α 1 = α 2 = 0.5 . Note that, following the approach of [30], we obtain S Γ , Φ SO x ( 0 ) = 0 ∀ t ∈ [ 0 , T ] and the evolution of SO x is dependent on only the initial conditions.

In (24), using S A , TW NO x and S A , TW PM that refer to stop-and-go waves in Experiment A, we obtain Figures 8, 9 and 10 for some pollutants estimated until 90 min. In Figure 8, we have O 2 (left, dashed blue line) and O 3 (right, continuous black line). Note that the graph of oxygen remains almost constant, unlike ozone which has first a fast growth, reaches a maximum, and then decreases. Precisely, when O 3 is maximum, O 2 has its minimum. The slow evolution of oxygen confirms its property as a vital gas while, as expected, ozone tends to increase due to the high stop-and-go waves on the ring track. However, such pollutant agent decays naturally due to the deterioration reactions in system (24). In Figure 9, we see that O has a low concentration and quickly reaches a steady state, while SO 2 decreases due to the decay of O 3 . Finally, Figure 10 presents NO 2 (left, continuous red line) and the normalized variations for PM 10 , NO 2 , and O 3 (right, see legend). Nitrogen oxides increase due to the highest traffic waves, but then it reaches a steady state. On the other hand, all variations of pollutants remain bounded but the highest values occur only for particulate.

Figure 8

Concentrations (g/ km 3 ) for O 2 (left, dashed blue line) and O 3 (right, continuous black line) vs time by solving (24) via S A , TW NO x and S A , TW PM .

Figure 9

Concentrations (g/ km 3 ) for O (left, dashed black line) and SO 2 (right, magenta line) vs time in case of solution of (24) by S A , TW NO x and S A , TW PM .

Figure 10

Left: Concentrations (g/ km 3 ) for NO 2 (left, continuous red line) and normalized variation rates for PM 10 , NO 2 and O 3 (right) by solving (24) via S A , TW NO x and S A , TW PM .

The concentrations of pollutants O 3 , NO 2 , SO 2 , and PM 10 at T are as follows:

Figure 11 shows the percentage variations at T of [ O 3 ] , [ NO 2 ] , [ SO 2 ] , and [ PM 10 ] , obtained by source terms S A , C i NO x and S A , C i PM , i = 1 , … , 5 , and the variation percentages w.r.t. S A , TW NO x and S A , TW PM . In the control intervals for the AV, there is almost a whole decrease in the concentrations of pollutants. In particular, the best control period is C 2 for O 3 , NO 2 , and SO 2 . Indeed, for the concentration of PM 10 , C 3 is the interval with highest decrease. However, as C 2 and C 3 almost show the same variation for PM 10 , we conclude that C 2 presents the best decrease for all the considered pollutants. Finally, from [28] the FC is 24.1l/100 km for the TW interval in Experiment A. It is shown that the highest variation occurs in C 3 , where FC decreases of about 39.32%.

Figure 11

Percentage variations for the concentrations, at T = 30 , of principal pollutants in cases of source terms for S A , C i NO x and S A , C i PM , i = 1 , … , 5 , compared to S A , TW NO x and S A , TW PM .

For Experiment B, computing the solutions of system (24) by the source terms S B , TW NO x and S B , TW PM , and S B , C i NO x and S B , C i PM , i = 1 , 2 , the behavior of concentrations is similar to Experiment A. For S B , TW NO x and S B , TW PM , the final values are as follows:

and, in the TW phase for Experiment B, the FC is 21.8 l/100km. Decreases, variations of pollutants, and FC in control intervals are given in Table 7. There is a unique best control period ( C 2 ) for which we obtain the highest decrease for pollutants O 3 , NO 2 , SO 2 , and PM 10 . FC has the most meaningful variation in C 2 as well.

Finally, we consider the solution to system (24) by S C , TW NO x and S C , TW PM , and S C , C 1 NO x and S C , C 1 PM (Experiment C). For S C , TW NO x and S C , TW PM , the final values of concentrations are as follows:

When the controller is activated in C 1 , we obtain at the final time T :

Hence, during the autonomy phase, the concentrations of O 3 , NO 2 , SO 2 , and PM 10 decrease by about 15%, 29%, 16%, and 10%, respectively, while FC is reduced by about 21%.

Table 8 shows comparisons among the final values of concentration for the principal pollutants during intervals of type TW and in the best autonomy phases. The variations of pollutants are higher for Experiment A and decrease of about a half in Experiment B, with the unique exception of PM 10 . For Experiment C, the lowest variations occur although, compared to Experiment B, they are almost comparable for pollutants O 3 , NO 2 , and SO 2 . We conclude that the three different control strategies, used in the experiments, allow meaningful variations in the concentration of pollutants. Experiment A is the best in terms of decrease, hence AVs, if controlled by an external input decided by an infrastructure (control of type Follower Stopper), perform the better results in terms of traffic stabilization. When simple communications guide the human pilot in Experiment B (control of type trained human driver), decreases occur but the human component alone does not guarantee high reductions. For Experiment C which deals with control strategies based on local information (control design of type PI controller with saturation), we have similar features to Experiment B. Indeed, Experiment C shows quite different variations compared to other cases, hence indicating that the results are dependent on the experimental conditions.

Therefore, the control with infrastructure communication allows the best performances, while those with human actuator or local controller have lesser, but still significant, effects.

4.3 Horizontal diffusion of pollutants

Now we study the horizontal diffusion of pollutants. We simulate a scenario (Figure 12), that illustrates the two-dimensional vertical cross-section of the roadway, with a hole in the domain corresponding to a green barrier that acts as a sink for pollutants.

Figure 12

Scenario with a point-source introduction of pollutants and a green barrier implemented as a hole (greyed region).

We consider the domain Ω described in Section 2.3, with L x = 500 m and L y = 0.5 m. A green wall of dimensions [ 250 , 480 ] m × [ 0.1 , 0.15 ] m and height 0.5 m is located in Ω . The domain is described by a numerical grid with d x = 5 m and d y = 0.02 m. The concentration of pollutants per unit of volume is defined by setting: Γ constant along the third component d z , equal to d y ; zero initial conditions, except oxygen that is assumed constant, see the previous subsection; μ = 1 0 − 8 km 2 /h for all pollutants; total simulation TI [ 0 , T ] with T = 30 min. Finally, we consider: a wind vector w that is