The study originates from the idea of developing a predictive crash model relying on an SPF that accounts explicitly for the presence of AVs in traffic. As emerged from the literature review, there is a gap in this sense that needs to be filled, also in virtue of the fact that there is still not a large dataset of AV crashes to rely on for the development of an SPF. Under this optic, the main solution adopted for accounting for AV presence in traffic has been to rely on simulations (Guido et al., 2019; Morando et al., 2018; Papadoulis et al., 2019). For the sake of safety, microsimulations are the best tools for acquiring the trajectories of the vehicles and analyzing them using selected Surrogate Safety Measures, within SSAM (Safety Surrogate Analysis Model) software or by dedicated algorithms (Raju & Farah, 2021; C. Wang et al., 2021). This trajectory elaboration leads to the definition of conflicts, the number and typology of which strongly depend on the Surrogate Safety Measures (SSMs) used (Sinha et al., 2020; Virdi et al., 2019). The first step was to find the most suitable simulation model for research, i.e., modelling the presence of AVs in traffic. Thus, different scenarios of traffic made of regular vehicles (RVs) and autonomous vehicles, partial or full (PAVs and FAVs, respectively SAE1 level vehicles 2-3 and 4-5) were conceived, as detailed in paragraph 2.2. The traffic scenarios were applied to a sample of roads in the Province of Bari, which had been found to be critical for road safety issues in previous studies (Coropulis et al., 2023, 2024). Based on road typologies, environmental conditions, and promiscuous traffic consisting of AVs and RVs, the most suitable traffic models were identified for both car-following and lane-changing interactions, i.e., the Gipps models (Coropulis et al., 2023). The latter models are the basis of the Aimsun Next simulation software, which is why this software was used for this analysis (Coropulis et al., 2023; Vrbanić et al., 2021) rather than others more commonly used, such as Vissim (Morando et al., 2018) or SUMO, which is more specific and suitable for urban scenarios (Kusari et al., 2022).

After the selection of the traffic models to be used, the different parameters associated with vehicle typologies were defined to depict the different vehicle behaviors in traffic and their interactions. In this sense, a set of parameters was defined considering the stochastic variability of RVs and the variability of PAV and FAV behaviors in a deterministic way, see Table 1 (Coropulis et al., 2024). The main characteristics of each type of vehicle (RVs, FAVs and PAVs) were based on previous research (Ims & Pedersen, 2021; Morando et al., 2018) considering FAVs with an assertive driving behavior, characterized by reduced time headway and very small reaction time; PAVs with a cautious behavior due to the coexistence of human and sensors in detecting issues and completing driving tasks; RVs with a great variability of driving behaviors denoted by local and consolidated hypotheses on driving styles (Barceló et al., 2005; Coropulis et al., 2024).

The maximum desired speed of FAVs is the lowest one, since it reflects the posted speed limits on the investigated network. The speeds of PAVs and RVs, on the other hand, are not limited unlike the speed of FAVs, since these types of vehicles are still under human driver control. Therefore, drivers can adjust their speeds according to their own decisions. The speed values used for PAVs and RVs were obtained by means of traffic and speed surveys using traffic counters over the investigated network.

The parameters of the simulation presented in Table 1 were confirmed after having run a sensitivity analysis to determine the importance of each parameter on the overall result of the simulation. Each parameter was evaluated considering five different values, accounting for a large span of its variability (minimum, mean, and maximum value, and 95th and 5th percentiles). Every simulation was run setting just one parameter differently and all the others unvaried. This approach was used for all five values of each parameter and all the parameters highlighted in Table 1. More than 50 simulations were run. Each simulation contained 10 repetitions to improve the stability of results. These simulations were propaedeutic to define the influence of each parameter on the safety outcome, evaluated in terms of conflicts. After this first sensitivity analysis, three other sensitivity analysis steps were investigated. The first one related to the same parameters but tested with the values defined for FAVs and PAVs to have an insight into the influence on the safety outcome of the single parameter calibrated for the AV behaviors. The safety outcome in this case was also quantified by conflicts. The second sensitivity analysis further step was run by using different simple and hypothetical market penetration scenarios of AVs (50% RVs - 50% FAVs, 50% RVs - 50% PAVs; 50% FAVs - 50% PAVs; 100% FAVs; 100% PAVs). Vehicles were recreated using the values in Table 1. These scenarios are simplified and they were used as a basis to test the impact of the combination of different parameters to outline the safety of a certain type of vehicle and the interaction mechanism between vehicles of different types. It emerged that for all the conflict types investigated (lane-changing, rear-end, crossing), the coexistence of PAVs and FAVs led to benefits compared to promiscuous scenarios since the vehicles reduced risky interactions. The simultaneous presence of FAVs and RVs, on the other hand, led to a greater value of rear-end and crossing conflicts since FAVs tended to drive more closely to other vehicles and to have reduced waiting times, based on the sensors’ performance. On the other hand, RVs can be more prone to misunderstanding driverless vehicle behaviors. These results were in line with previous literature about the interactions of different types of vehicles, suggesting that promiscuity can be associated with increased uncertainty and reduced safety (Morando et al., 2018; Wen et al., 2022). The last sensitivity analysis investigated a fictitious vehicle modelled using the most relevant values, in terms of conflict increase or decrease, derived from the previous analyses, for each parameter (Coropulis et al., 2024). The results highlight the adequacy of the selected parameters as well as the different behaviors of the different types of vehicles and their interactions. These aspects were crucial to define the consequent studies about safety, as detailed in the next sections.

Before starting the simulations with the different vehicle typologies, the simulation framework was validated for both traffic and safety. The former was investigated by comparing the available traffic data on each section with the traffic obtained from the simulations in an RV only regime (considering the hourly and seasonal traffic fluctuations and the presence of heavy vehicles) using the GEH value (GEH is an acronym standing for Geoffrey E. Havers, the name of the inventor of the metric). The Origin-Destination matrices were applied to each site, and the traffic at each intersection and segment has been compared to the one available from the SUMP (Sustainable Urban Mobility Plan) and onsite monitoring campaign using the GEH values. This condition was useful to define whether the input data could realistically depict the real situation occurring on the selected roads (two-way two-lane rural roads in the Province of Bari). Therefore, each site was precisely characterized in terms of geometric features and traffic components to realistically represent the situation occurring at the sites. The same approach, but with different market penetration of vehicle types (RVs, FAVs, and PAVs), was adopted for all the simulations. After the validation for traffic, the safety validation was performed. This was done by comparing the obtained simulated crashes with the recorded ones in the period 2015-2019 (ACI-ISTAT dataset) on the same roads used in the simulations. Bearing in mind the final purpose of the study, i.e., developing an SPF for AVs, and the available tool to pursue it, one important choice was made. Namely, only multiple-vehicle crashes were investigated by the simulations, since it is possible to acquire them from the trajectories analyses, thanks to the selected SSM. Single-vehicle crashes were neglected for different reasons. Firstly, the high level of uncertainty in defining a single-vehicle crash by the trajectory elaboration derived from a microsimulation tool. Secondly, there are several types of unpredictable single-vehicle crashes that can occur in the presence of AVs, as the ones due to technological failures, sensor damage, or cybersecurity attacks. Lastly, the quantification of the benefits introduced by the AVs in the interaction with RVs is more reliable in the definition of multiple vehicle crashes. Therefore, considering multiple vehicle crashes only, the mean annual crash frequency over the 2015-2019 period was compared with the crashes obtained from the elaboration of the trajectories coming from 1-year simulations of RV traffic for each of the selected roads. The trajectories were transformed into conflicts using the Time To Collision, TTC, as SSM (Arun et al., 2021; Zheng et al., 2019) with a threshold of 1.5 s for RVs (Morando et al., 2018; Papadoulis et al., 2019; Shahdah et al., 2015). The conflicts were correlated by means of linear correlations without any significant results. This discrepancy was because conflicts are more prone to happen than crashes since evasive maneuvers can avoid most of them. It is difficult to understand how many conflicts can become crashes since it depends on human capabilities. Human perception and capabilities are not fixed in the real world. Even if they are depicted in a probabilistic way in the simulator, it is always possible to account for slightly different behaviors in crash avoidance schemes. Even if the simulator has already been validated for traffic output, the crash statistics are more random than the traffic ones. Therefore, conflicts were converted into crashes to achieve variable comparability. The method selected for this conversion was the Univariate Extreme Value approach based on the TTC threshold of 1.5 s, widely used for its simple implementation (Jonasson & Rootzén, 2014; Tarko, 2018; Zheng et al., 2019). The detailed procedure used for this conversion is provided in section 2.4. Once this conversion had been performed, the simulated crashes were compared to the observed ones, covering a period of 1 year. The results allowed us to validate the simulation procedures also in terms of safety assessment. Once this validation had been obtained, the two-step procedure involving the simulations (simulation and elaboration of trajectories for safety outcomes) was applied to the different scenarios, including AVs. The scenarios were investigated, always simulating one year of traffic with seasonal and hourly variability of traffic. Each scenario was defined according to some criteria, as explained in the following subsection.

The definition of the further scenarios with AVs might follow the hypothetical trends studied for penetration in the market of such technology. The market penetration was selected according to the assumptions presented in Garcia et al. (2021), for Austroads 2021. Even if the contexts of applications and the continents are different from those of the proposed study, Garcia et al. (2021) provided a wide and deep approach for market penetration estimation that can be applied to other case studies.

This study hypothesizes that there are three different curves for AV penetration, one more realistic and two other curves that consider the limit conditions, i.e., the worst-case scenario of slow implementation and the best-case scenario of rapid implementation of them in traffic. The chosen curve, among the three, is the realistic one, with a gradual market penetration rate in traffic, projected from the current date to 2050. Three macro-categories of vehicle typologies were analyzed, namely Regular Vehicles -RVs-, Partially Automated Vehicles -PAVs- (SAE level 2-3), and Fully Automated Vehicles -FAVs- (SAE level 4-5). The definition of market penetration is propaedeutic to estimate specific risks of promiscuous traffic situations, in which not only RVs travel but also different types of AVs with different penetration rates (see Table 2).

Apart from these penetration rates, a 100% FAVs scenario is still a utopistic in the immediate future; therefore, it was neglected in this analysis that considers temporal horizons until 2050. Despite this consideration, 100% FAVs is supposed to be a more homogeneous scenario because of the absence of humans (Desta & Toth, 2022; ElSahly & Abdelfatah, 2020). Therefore, in future analysis, this scenario will be tackled.

A sample of road sites was used as a testbed for the prediction of future AV scenarios, in which different traffic penetration rates are considered. In this work, 16 sites (that are small road networks, composed of individual segments and intersections) in the Province of Bari were selected. These roads are two-way, two-lane rural roads with no significant pedestrian or cyclist flows. The sites were chosen among the ones already investigated in the context of the Sustainable Urban Mobility Plans (SUMP) for the Province of Bari, based on their highlighted safety- or congestion-related issues. Hence, data about fatal and injury crashes and traffic volumes were already available. The sites are highlighted in the figure below (Figure 1).

As indicated in the introduction, exposure variables are needed in order to estimate an SPF. Moreover, other information about road site geometric configurations is preferrable. In this case, traffic volumes, total length of road sites, as well as the type and density of intersections for each road site were taken into account. Furthermore, given the specific research questions of this study, the different types and penetration of vehicles (RVs, Partially and Fully AVs) in traffic were also considered. Given that, as described in the next subsection, the definition of the crash frequency dependent variable is also based on traffic simulations, it was not possible to use a pure micro-level approach for the SPF development, in which geometric characteristics of individual segments and intersections are considered. However, the approach was also different from macro-level SPFs (see e.g., Huang et al., 2016) in which geographic, socio-economic and transport-related variables are used to predict crash frequencies in different geographic areas, possibly also considering spatial methods (Lord et al., 2021). These methodological aspects are clarified below.

Regularly, SPFs present in their formulation (1) a coefficient dealing with the exposure of vehicles to crashes, represented by the annual average daily traffic, AADT, at the site. In this specific case, the proposal is to include in this variable not only the traditional indication about the volume (AADT) travelling on the road, but also about the market penetration of each vehicle type and its intrinsic contribution to road safety performances, given the specific aims of this study. Considering these points, this exposition variable should be defined with the twofold purpose in mind of accounting for both the traffic volume of each vehicle typology and its impact on safety. This variable is called Tr1 in the model, and it is defined as detailed below.

The market penetration stands for the quantity of vehicles of each proposed type (RVs, PAVs, FAVs) travelling on roads. The sum of the AADT of each vehicle type (RVs, PAVs, FAVs) on the road should give the total AADT travelling at the site. The AADT was then converted into an equivalent AADT. This latter variable combines a baseline traffic volume with a specific safety potential related to each vehicle type, by means of a Hazard Index (HI), a factor multiplying each vehicle type’s AADT. The HI represents the tendency of each vehicle type to be involved in a crash, determined through traffic simulations. Traffic simulations were run once each with just one vehicle category on the road network (100% FAVs; 100% PAVs; 100% RVs). In this way, each simulation provided the number of conflicts (lane-changing, crossing, rear-end) for each of the three types of vehicles (FAV, PAV, and RV). Vehicle trajectories were extracted from the simulation output. Once a surrogate safety measure had been selected, i.e., the Time To Collision (Coropulis et al., 2024), the number of conflicts was estimated. This step was run using the SSAM (Safety Surrogate Analysis Model) algorithm. Of course, the detected conflicts, as previously mentioned, are only related to multiple vehicle dynamics, excluding single-vehicle ones, even if they could represent a big portion of the possible AV conflicts. This is due to the capabilities of the SSM selected in defining conflicts and the purpose of the study, which is to investigate the impact of AVs on traffic safety. Finally, for each simulation, the number of conflicts was converted into the number of expected crashes using the Lomax Distribution (Tarko, 2018):

(2) N c r a s h e s = k ⋅ P ( C | N , s ) ⋅ N s

Where:

This Equation (2) contains the P(C|N, s) term that can be explained as follows: when the generic separation value for each detected conflict, s_N (generic separation value obtained by the simulation for each detected conflict), is greater than a fixed threshold, s, the probability P that a conflict, N, becomes a Crash, C, is null. For all the values of s_N less than the threshold, s, the probability increases from 0 to 1 depending on the recorded separation value. Thus, the outcome of Equation (2) is obtained by multiplying the calculated probability by the number of conflicts and the conversion coefficient.

The separation threshold was diversified according to the different scenarios investigated. In the scenarios including more than half of RVs and PAVs, the TTC was set equal to 1.5 s; otherwise, equal to 0.5 s (Morando et al., 2018; Papadoulis et al., 2019). This difference is justified not only by simulation outcomes (Papadoulis et al., 2019) but also by analyses of the human adaptability to AVs in a driving context. As highlighted by de Zwart et al. (2023), RVs in mixed traffic with several AVs may tend to adapt their style to that of the AVs, i.e., reducing the time headway and head distance (with AVs more than 50% of the total traffic). Wen et al. (2022) demonstrated the opposite, i.e., that in a promiscuous scenario characterized by a great variability of RVs and AVs, RVs may tend to increase their TTC, compared to their TTC against other RVs. Moreover, from investigating the California DMV (Department of Motor Vehicles) Autonomous Vehicle Collision Reports, it emerged that AVs are involved in minor but frequent collisions with RVs, due to misbehavior of RVs that do not adequately perceive the maneuvers of the Autonomous Vehicle, when stopping/slowing down, while crossing, or in parking areas. As also highlighted by previously mentioned studies, RVs seem to be more prone to reduce headway in the presence of AVs without high promiscuity. The specific analysis of interactions between Level 4 AV-RV (Morando et al., 2018) highlighted that as AV market penetration increases, road safety increases accordingly. These results from literature and crash reports are the basis for the choices made for the selection of the TTC threshold for the Lomax distribution approach. This choice implied accounting for the interactions of AVs and RVs in the definition of conflicts and then crashes.

First of all, the crashes were collected for the simplest scenarios (100% FAVs; 100% PAVs; 100% RVs), simulating 1 year. Hence, the crash frequency for each site in each of the simulated 100% scenarios was calculated according to Equation (2). The same approach was pursued for all the other scenarios proposed in Table 2, not only for the 100% ones that were propaedeutic for the HI calculation. As for the latter calculation (HI value), the crash frequency for each 100% scenario was averaged over the investigated sites. In this way, it was possible to get the riskiness associated with each type of vehicle (FAVs, PAVs, RVs). Therefore, the calibration of the HI index is derived from safety calculations. The HI index accounting for the AV penetration through the equivalent AADT (Tr1 variable) required the use of a benchmark to weigh the safety potential of the vehicle typology. Thus, the proposed HI index was calculated as follows:

(3) H I , j = N c r a s h e s j 100 % N c r a s h e s P A V 100 %

Where:

According to Equation (3), it is immediately clear that three HIs needed to be calculated: one for 100% FAVs, one for 100% RVs, one for 100% PAVs (HI = 1, since the PAVs was used as the benchmark). It is evident that HI determination derived from strict calculations. Then, based on the HIs, the j-th AADT of each site was converted into the equivalent AADT for each j-th vehicle type, assigning to each vehicle category its impact on safety, multiplying by the related HI. The sum of each j-th vehicle type provided the overall equivalent AADT, i.e., Tr1.

(4) T r 1 = A A D T e q u i v a l e n t = H I F A V ⋅ F A V s A A D T + 1 ⋅ P A V s A A D T + H I R V ⋅ R V s A A D T

In this way, the market penetration and the potential safety impact of vehicles were considered together by means of only one variable.

The results showed how the absence of RVs improved the safety of each site, especially if 100% of FAVs are deployed on roads. In fact, the FAVs have an HI (Hazard Index) equal to 0.76 if compared to the PAVs. The RVs have an HI of 3.59 compared to the PAVs. In this optic, the scenarios 2030, 2040, and 2050 were represented using the equivalent AADT, Tr1 (see Table 3).

As is blatantly obvious from Table 3, Tr1 values are averagely low, always lower than 20,000 vehicle/day except for one case.

The simulations to investigate the 2030, 2040, and 2050 scenarios, as the ones for the HI formulations, were run not focusing on single geometric elements, like intersections or segments, but on the entire site (that is a small road network), which includes several intersections and segments, in order to provide a realistic and holistic driving scenario in which vehicles have time and space to act and react to traffic variations over the sections. In this way, the recorded conflicts could have better explained the entire interaction mechanism among vehicle types.

The other aspect to consider for prediction purposes is the road site geometry. The length of the sites and the presence of intersections have been considered. In particular, a variable counting the intersection typology (4-legged, 3-legged, and roundabout), the intersection density I_k (intersection/km), and the combination of different intersection types at each site was defined.

Similarly to the HI, a combined geometric variable was defined by assigning weights to each intersection typology based on previous safety studies. Weights are determined considering CMFs and SPFs for specific intersection types, taken from the CMF Clearinghouse2 and the Pract-Repository3. As for the HI index, the calculated weights accounted for fatal and injury crashes, according to the crash typology available in the dataset used for validation and calibration of the simulation model (Coropulis et al., 2024).

The weight factor was basically a CMF (crash modification factor) assessing the different risk, in terms of crashes, between two alternatives. The CMF requires a benchmark solution to be compared to an alternative solution/countermeasure. To obtain a conversion factor, the CMF was calculated as follows:

(5) C M F k = N c r a s h e s k N c r a s h e s 3 − l e g g e d i n t e r s e c t i o n

Where:

In the case of 4-legged intersections, to obtain theCMFk weight, the value Ncrashesk in the previous equation (5) was set as the result of the application of the fatal and injury SPF taken from the HSM (2010) for two-way two-lane rural road 4-legged intersections. Whereas in the case of roundabouts, the CMFk was directly estimated based on the study made by Isebrands and Hallmark (2012) for roundabouts on rural roads, given the particular case under study. Note that other additional SPFs were developed over time for 3-/4-legged intersections (NASEM, 2021) and roundabouts (NASEM, 2019). However, the proposed references were deemed functional for the sake of simplicity and reproducibility, given their essential use for making comparisons rather than for providing precise estimates.

Thanks to equation (5), the CMF (i.e., the influence of an intersection typology compared to the 3-legged one) was obtained for each intersection of each road site. In the case of 3-legged intersections, it is always equal to 1, while for the 4-legged intersection type it depends on the specific AADT values (i.e., the application of the base intersection SPFs is dependent on traffic volumes). After these calculations, the obtained CMFs were averaged over all the road sites, resulting in an average CMFk for 4-legged intersections equal to 1.872. In the case of roundabouts, CMF = 0.13, taken from Isebrands and Hallmark (2012), is related to the conversion of 4-legged intersections (there are only two 3-legged intersection cases in the study dataset) on two-way two-lane rural roads into one-lane roundabouts. Hence, given that the reference case used in equation (5) is the 3-legged intersection, this CMF was recalibrated considering the already defined weight factor CMFk for 4-legged intersections compared to 3-legged ones. Thus, finally, the calculated CMFk used for each roundabout is equal to 0.243. Given that it is a fixed value, the average over all the corresponding sites is still equal to 0.243.

The CMFs calculated in this way were used as coefficients for considering the crash risk derived from the presence of different intersection types at the same site. Hence, these coefficients were multiplied, respectively, by the intersection density for each type of intersection (expressed in the following equation as I_Roundabout; I_3-Legged; I_4-Legged). Then the products were summed up to obtain an indication of their combined effect on safety. The obtained variable was called Com2 since it is the combination of 2 factors regarding the intersections (density and typologies).

(6) C o m 2 = 0.243 ⋅ I R o u n d a b o u t + 1 ⋅ I 3 − L e g g e d + 1.872 ⋅ I 4 − L e g g e d

In this way, it was possible to consider the influence of intersections on crash occurrence (see Table 4).

The SPF for AVs was estimated using a general linear model in which the error distribution follows a negative binomial structure (see e.g., Lord & Mannering, 2010). The two independent variables are Tr1 and Com 2, and the outcome, i.e., the dependent variable, is the crash frequency. The crash frequency of F+I crashes, N, is calculated as the number of crashes (obtained from simulations with mixed traffic as expressed in Table 1), per year.

The assumed SPF model equation is the following:

(7) N S P F _ A V s = L × e β 0 + β 1 T r 1 + β 2 C o m 2 = L × e β 0 + β 1 T r 1 + β 2 ( 0.243 × I R o u n d a b o u t + 1 × I 3 − L e g + 1.872 × I 4 − L e g )

Where L is the total length of the i-esim road network site, and β₀ (intercept)_, β₁ and β₂ are the estimated coefficients.

The goodness of fit of the model was assessed by calculating the Nagelkerke R² as well as the residuals for obtaining the CuRe Plot, considering the independent exposure variable Tr1, instead of the usual AADT, used in previous research (Claros et al., 2018; Hauer & Bamfo, 1997; Intini et al., 2019).

The SPF was calculated for the period ranging from 2030 to 2050, to account for all the possible levels of market penetration.

To test the importance of the HI on the outcome of the developed SPF, a sensitivity analysis was run considering, for the same investigated years (2030, 2040, 2050), different penetration rates of FAVs, PAVs, and RVs, considering optimistic, realistic, and pessimistic projections. In this way, the Tr1 variable varied according to the different hypothesized scenarios thanks to the HI values. Intervening directly on the HI value, testing different alternatives would not have been coherent with the procedure used for the HI determination. For this reason, the sensitivity analysis was pursued, keeping the HI set to the determined values but varying the penetration rates of vehicles to allow a greater variability. The proposed penetration rates are the following (Table 5):

The goodness of fit of the model was assessed by using CuRe (Cumulative Residuals) Plots (Claros et al., 2018; Hauer & Bamfo, 1997; Intini et al., 2019) as well. The model residuals were calculated as the difference between the simulated (observed) and the predicted (by the SPF) crashes. They were arranged according to the independent variable Tr1. This variable was the most significant in the process of predicting crashes with automated vehicles, and the surrogate traffic variable is usually the one used for this kind of analysis. Then the cumulated residuals were calculated, as well as the standard deviations. All the cumulative residuals fall between the two standard deviations, showing that the variable explains the investigated phenomenon (see Figure 2).

The results show that the observed crashes are greater than the predicted ones until a threshold of equivalent AADT (Tr1), equal to about 25000 vehicles/day. After this threshold, the SPF overestimates the crashes. This phenomenon is justified by the fact that the equivalent AADT increases the traffic according to the vehicle type, and due to an excess of RVs in promiscuous traffic, the hazard of the sites increases. On the contrary, when the equivalent AADT falls in the range between 10000 and 15000 vehicles/day, the simulations provide more crashes than those predicted by the model. This is because the simulations include greater traffic volumes and significantly more trajectories. More trajectories mean more potential risky situations and thus more potential conflicts. The conflicts-crashes conversion in this way overestimates the possible crashes. Therefore, the function shows some high vertical displacements in the middle ranges of Tr1.

Concluding the discussion about the developed SPF, this was also verified by running a sensitivity analysis to obtain the HI impact on the overall results. Different penetration rates were hypothesized as in Table 5 to make the Tr1 vary.

In the most pessimistic scenario, the one with 60% of RVs (2030 pessimistic), the Tr1 was 2.55 times greater than the detected AADT. The results for the ratio of Tr1 over AADT for all the scenarios are summarized in Table 7.

The impact of the different penetration rates is mitigated thanks to the selected values for the HI. Of course, these results are related to mixed traffic conditions. In fact, in situations in which RVs are the only circulating vehicle type, Tr1 increases dramatically. But this side effect is contemplated by the purpose itself of the SPF, i.e., being developed and applicable for mixed traffic conditions with AVs.

In the context of this sensitivity analysis, the selected scenarios were used for calculations of the SPF outcomes. The resulting distributions of crashes are presented in Figure 3.

These results highlight that in the case of a higher percentage of RVs, the variance of crashes is greater than in other cases. This aspect is strictly correlated with what emerged from the CuRe Plot, i.e., great values of Tr1 (in this case affected by the preponderance of RVs) result in instability and tend to overestimate the safety outcomes. To confirm the variability induced by different percentages of vehicles and the convergence of results obtained by large percentages of AVs, the following graphs have been provided. The definition of HI has provided valuable results for the study considering the presence of AVs. Of course, for scenarios with greater values of RVs, the HI definition must be recalibrated according to the scenarios and the sites. This definition of the coefficients is strictly related to the simulated scenarios since it depends on the results of the simulations. However, the structure of the Tr1 variable as well as the entire SPF can be considered reliable for all the contexts, even if the specific coefficients might be recalibrated according to the different conditions.

Thus, the validity range of the proposed SPF is, in this case, strictly related to the proposed scenarios (see Figure 4). For different scenarios, the SPF could necessitate parameter recalibrations. Moreover, it is obvious that RV percentages tending to 100% would make this SPF unsuitable for AV investigations. Another aspect to account for in this formulation is its validity for multivehicle crashes alone. The rationale behind this selection is that the study is based on traffic simulations, which can detect and record trajectories and elaborate on traffic conflicts happening between two vehicles. All the single vehicle crashes, including the ones due to technological failures while driving, have been neglected, because they were beyond the scope of the analysis, even if they can represent a big portion of the possible CAV (Connected and Autonomous Vehicle) accidents (Coropulis et al., 2025).

S. Coropulis: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Resources, Software, Writing—original draft, Writing—review & editing; N. Berloco: Investigation, Resources, Visualization, Validation, Writing—review & editing; P. Intini: Data Curation, Formal analysis, Supervision, Validation, Writing—original draft, Writing—review & editing; V. Ranieri: Conceptualization, Funding acquisition, Project administration, Resources, Supervision, Validation, Writing—review & editing.

The authors report that they have no competing interests.

The authors declare that no generative AI was used in this work.

This research was not subjected to ethical commission approval since it does not involve volunteers and people. However, all the research was conducted following ethical methodologies.

Handling editor: Nicolas Saunier, Polytechnique Montreal, Canada.

Reviewers: Carmelo D’Agostino, Lund University, Sweden; Peijie Wu, Chongqing Jiaotong University, China; Mariano Pernetti, University of Campania, Italy.