Explanatory Analysis of the Safety of Short Passing Zones on Two-Lane Rural Highways

Abstract

Passing zones afford sufficient passing distance for passing vehicles on two-lane rural highways. This study aimed to assess the effects of geometric and traffic variables on the safety of short passing zones using the rate of passing maneuvers ending in the no-passing zone as a surrogate safety measure. A Poisson regression model was applied to aerial data collected by drone at seven passing zones. The findings showed that increase in the length of the passing zone corresponds to decrease in the rate of passing maneuvers ending in the no-passing zone. The passing rate is also affected by the lane width, the percentage of heavy vehicles in the subject direction, and the directional split in the subject direction. The rate of passing maneuvers ending in no-passing zones reaches a peak as the two-way traffic flow rate increases to 600 vehicles per hour regardless of the directional split, or as the absolute vertical grade increases to 4.8%–6% depending on the percentage of heavy vehicles. After the peak value, the rate decreases. The presented model can serve as a tool for evaluating and improving the safety of short passing zones.

In Iran, about 70% of traffic fatalities occur on rural highways ( 1 ), and the main two-lane rural highways constitute 30% of the road network ( 2 ). The most serious causes of injuries and fatalities in two-lane highways are passing maneuvers ( 1 , 3 ). Kashani and Mohaymany ( 2 ) showed that whereas passing maneuvers accounted for only about 20% of the total accidents on two-lane rural highways in Iran, they caused 30% of injuries and 50% of fatalities. Passing maneuvers increase the likelihood of fatality by 601.6% ( 4 ).

The passing zone (PZ) affords a sight distance for conducting a passing maneuver, which allows the faster vehicle to travel at its own desired speed. Therefore, PZs improve the operational effectiveness of two-lane rural highways based on the Highway Capacity Manual ( 5 ). PZs are divided into short (<240 m) and long (>300 m) groups ( 6 ). Previous studies showed that short PZs do not contribute to the operational effectiveness of two-lane highways ( 6 , 7 ). In spite of the operational effectiveness of PZs, only a few studies have examined the safety of short PZs.

Jones ( 8 ) studied the safety of three short PZs and subjectively rated the severity of the return of the passing vehicle to the subject lane at the end of the maneuver. The three rating levels were smooth return, forced return, and violent return. By increasing the length of the PZ, the percentage of hazardous maneuvers decreased. Forced or violent returns occurred in 63%, 45%, and 10% of cases for the PZ length of 120, 200, and 270 m, respectively. Jones ( 8 ) also recorded the end position of passing maneuvers as a measurement of safety. The percentage of the passing maneuvers ending in the no-passing zone (NPZ) was higher in shorter PZs. Harwood et al. ( 6 ) determined that the passing rate was 0.77 passes per hour in short PZs, in comparison with 2.95 passes per hour in long PZs; however, the maximum traffic volume was 260 vehicles per hour (vph). They also found that the percentage of passing maneuvers ending in NPZ was 92% for short PZs and 21% for long PZs.

Moreno et al. ( 7 ) performed a reliability analysis to determine the probability that a PZ length would be lower than the required passing distance, which was named the probability of noncompliance. The authors determined the expected number of noncompliant passing maneuvers by multiplying the number of passing maneuvers obtained from a traffic simulation study by the probability of noncompliance. They used the noncompliant passing rate as a safety measure to evaluate the safety of short PZs. The contributing factors included traffic flow and PZ length. The PZ length ranged from 100 to 500 m. They found that the maximum rate of noncompliant maneuvers fell between 275 and 350 m, with respect to the traffic flow. However, many assumptions were made in the simulation study, such as constant acceleration until reaching the desired speed, constant speed after that, and executing passing maneuvers if there is a minimum safety margin at the end of maneuvers.

Previous studies did not investigate the effects of variables such as directional split, the percentage of heavy vehicles, free-flow speed, lane width, vertical grade, and upstream NPZ length, on the safety of short PZs. Also, the effects of PZ length and traffic volume were not investigated using observational data. To fill this gap in knowledge, this study examined the effects of geometry- and traffic-related variables on the rate of passing maneuvers ending in NPZ at short PZs using observational data.

The objectives of this study were to determine the effect of geometric (i.e., short PZ length, lane width, vertical grade, and length of upstream NPZ) and traffic-related variables (i.e., two-way traffic flow rate, directional split, the percentage of heavy vehicles, and free-flow speed) on the safety of short PZs. The safety of short PZs was evaluated using the rate of passing maneuvers ending in NPZ as a surrogate safety measure.

The paper is organized as follows. The second section presents the material and methods, the objectives of the study, the definition and selection of variables, the statistical procedure to model the rate of passing maneuvers ending in NPZ, and the study sites and data collection methods. The third section provides the results of the study, while the discussion of the results is presented in the fourth section. Finally, the fifth section presents the main findings and addresses implications, recommendations, and requirement of future research.

Material and Methods

Variables

Dependent Variable

Passing maneuvers ending in PZs have a sufficient sight distance based on design policy manuals; therefore, the passing and opposing vehicles can prevent a potential collision. The main assumption is that the driver starts the passing maneuver close to the beginning of the PZ and ends it before the NPZ ( 9 ). However, because of the presence of opposing vehicles or high-speed catch-up, the passing maneuver could start away from the beginning of the PZ and end in NPZ ( 10 ). This type of passing maneuver is not safe because of inadequate passing sight distance. Consequently, the counts of passing maneuvers ending in NPZ could be used as a surrogate measure to evaluate the safety of PZs, as was done by several studies ( 6 – 8 , 10 ). In this study, the counts of passing maneuvers per 15 min ending in NPZ were regarded as the dependent variable.

Explanatory Variables

Explanatory variables utilized in developing models were derived from previous works, and directly or indirectly affected the rate of passing maneuvers ending in NPZ. The explanatory variables are discussed below.

The traffic volume in the subject and opposing direction affect the passing rate ( 7 , 10 – 16 ). It is expected that by increasing the traffic volume in the subject direction, the desire to pass should increase. However, the passing rate increases to a peak and after that decreases because of the formation of long platoons and inadequate gaps in the subject direction, which makes it difficult for the passing vehicle to merge back to its operating lane. By increasing the traffic volume in the opposing direction, the opportunity to pass decreases, which leads to a lower passing rate. Another way to account for both the effects of subject and opposing traffic volume in the model is to use the total traffic volume in both directions and the split direction ( 11 , 12 , 17 ).

Length of PZ is an important variable that affects the safety and operation of PZs ( 6 , 10 , 12 ). A short PZ constrains the driver from starting the passing maneuver close to the beginning of the PZ. If the driver delays in starting the maneuver, they are more likely to end the maneuver in NPZ as the length of the PZ is shorter. A passing maneuver with high-speed catch-up between passing and passed vehicles is more likely to end in NPZ as the PZ is shorter. Thus, by increasing the length of short PZs, it is theoretically expected that the counts of passing maneuvers ending in NPZ would be reduced.

Other explanatory variables include absolute vertical grade, the 85th percentile and deviation of free-flow speed in the subject direction, the percentage of heavy vehicles ( 10 , 17 ), and the lane width ( 17 ). It is expected that the rate of passing maneuvers would increase by increasing these four variables. As the absolute vertical grade increases, the speed of heavy vehicles significantly decreases, which leads to an increase in the passing rate. Increasing the lane width encourages drivers to pass in the absence of adequate passing sight distance, because a wider lane width could provide more room for passing and opposing vehicles to avoid a collision. Topography of the road (mountainous versus level) may also have a significant effect on the rate of maneuvers ending in NPZ. In mountainous terrain, PZs have limited right of way with the risky roadside environment (e.g., cliffs and embankments), which makes driving more demanding ( 18 ). Therefore, it is expected that drivers overtake less often on PZs on mountainous roads than level roads.

Statistical Analysis

The counts of passing maneuvers ending in NPZ as the count data could be estimated by the Poisson regression model. In this model, the probability that y_i passing maneuvers would end in NPZ observed in the ith 15 min interval is given in Equation 1.

P (y_{i}) = \frac{\exp (- λ_{i}) λ_{i}^{y_{i}}}{y_{i}!}

(1)

where y_i is the counts of passing maneuvers ending in NPZ observed in a 15 min i, and λ_i is the expected counts of passing maneuvers ending in NPZ per 15 min i.

The Poisson regression model is estimated by specifying λ_i as a function of explanatory variables, as shown in Equation 2.

λ_{i} = E [y_{i}] = \exp (β X_{i})

(2)

where X_i represents the vector of explanatory variables, and β denotes the vector of model parameters. The model parameters could be estimated using the standard maximum likelihood method ( 19 ).

The Poisson regression model assumes that the mean and the variance are equal. The case in which the variance is larger than the mean is known as overdispersion. Overdispersion leads to inflation in the estimated standard errors of the estimated parameters, but it does not affect the magnitude of the estimated parameters ( 20 ). The negative binomial regression model is a way to account for overdispersion, which is defined by Equation 3.

λ_{i} = \exp (β X_{i} + ε_{i})

(3)

where exp(ε_i) indicates a disturbance term, which has a gamma distribution with mean of one and variance α; this term makes it possible to have a variance different from the mean, as shown in Equation 4.

VAR [y_{i}] = E [y_{i}] + α E {[y_{i}]}^{2} = λ_{i} + α {λ_{i}}^{2}

(4)

where α represents the overdispersion parameter.

When α is equal to zero, there is no difference between the negative binomial and Poisson regression models. Therefore, a selection between these two models depends on the value of α. The parameters of the negative binomial model could be estimated using the maximum likelihood method ( 20 ).

To consider possible correlations among observations in each PZ, the Poisson random-effects model is applied, which is defined as:

Ln (λ_{ij}) = β X_{ij} + η_{j}

(5)

where λ_ij is the expected counts of maneuvers that ended in NPZ for the ith 15 min belonging to the jth PZ, X_ij is a vector of explanatory variables, β is a vector of corresponding parameters, and η_j is a random effect for the observations of the jth PZ. For the random-effect Poisson model, the mean value is not equal to variance, and the variance-to-mean ratio is 1+λ_ij/(1/ φ). If φ is zero, the random-effect and pooled (Poisson) models are not significantly different ( 21 ).

A common problem in count data models is that there is an over-abundance of zero counts in the data, which decreases the mean and inflates the variance. The zero-inflated Poisson (ZIP) regression model has been developed to consider the excess of zeros in the data ( 20 ). The ZIP model assumes that the events (i.e., $Y = (y_{1}, y_{2}, . . ., y_{n})$ ) are independent and the model is defined as follows:

P (y_{i}) = {\begin{matrix} p_{i} + (1 - p_{i}) EXP (- λ_{i}) & y_{i} = 0 \\ \frac{(1 - p_{i}) EXP (- λ_{i}) λ_{i}^{y}}{y!} & y_{i} = y \end{matrix}

(6)

where p_i is the probability of being the zero state, and y_i shows the counts of passing maneuvers that end in NPZ in the ith 15 min interval. The maximum likelihood method is adopted to estimate the parameters of a ZIP regression model ( 19 ), and a logit (logistic) or probit (normal) probability process is used to calculate the p_i.

Study Sites and Data Collection

Data were collected from seven PZs on three two-lane rural highways: Jiroft Faryab, Jiroft-Baft, and Jiroft-Kerman in Iran. Observations were collected on working days between 8:00 a.m. and 6:00 p.m. Table 1 presents the geometric characteristics and posted speed limits of PZs. The length of the PZs varied from 164 to 345 m. Moreover, the roadway width was 3–3.75 m, and the absolute vertical grade was 0.58%–9.5%, both having a wide range. PZ no.7 (Table 1) is the only one with a shoulder that is paved. Table 1 indicates length of upstream NPZ and radius of horizontal curves after PZ for both directions (i.e., Dir 1, Dir 2). The sight distances at the end of PZs were limited by horizontal curves, vertical crest curves, or a combination of them.

Table 1.

Geometric Characteristics of the Studied PZs

PZ ID (m)	Highway	Length (m)	Lane width (m)	Shoulder width (m)	Absolute vertical grade (%)	Posted speed limit (km/h)	Length of upstream NPZ (m)		Radius of horizontal curves after PZ (m)
							Dir1	Dir2	Dir1	Dir2
1	Jiroft-Faryab	225	3.45	-	0.85	85	130	930	-	1330
2	Jiroft-Faryab	309	3.45	-	0.58	85	948	1780	550	1000
3	Jiroft-Faryab	180	3.25	-	1.1	85	430	620	236	426
4	Jiroft-Faryab	204	3.25	-	2.5	85	780	430	-	600
5	Jiroft-Baft	164	3.00	-	5.5	95	225	527	650	500
6	Jiroft-Baft	231	3.00	-	3.5	95	520	280	680	300
7	Jiroft-Kerman	345	3.75	1.2	9.5	85	1716	5010	850	472

Note: PZ = passing zone; NPZ = no-passing zone.

Data were collected by a Phantom 4 Pro drone. The drone was equipped with a 1 in. 20 megapixel sensor capable of shooting a 4K/60fps video and a three-axis gimbal to stabilize the oscillation of the camera. Since the minimum altitude of the drone during video recording was 150 m, there was no impact on drivers’ behaviors ( 22 ). The passing maneuvers for each direction were collected from 38 flights with a duration of 15 min each. The sky was clear during data collection.

The speeds of vehicles and time headways were calculated by using road marking at the beginning of the PZs (their lengths were measured in the field) and the timestamps of vehicles determined by the open-source video analysis software Kinovea ( 23 ). The types of vehicles were visually recorded. The free-flow speeds of passenger cars were determined in the free-flow condition based on at least 6 s headways, as suggested by Al-Kaisy and Karjala ( 24 ). The geometric variables, including the lane width, vertical grade, and PZ length, were measured in the field. The PZs were located in straight sections with almost constant vertical grades. To calculate the flow rate, the volume of vehicles in 15 min was multiplied by four.

Results

Data Description

The total number of passing maneuvers conducted in the PZs was 622, of which 243 maneuvers (39%) ended in NPZ. Table 2 presents a statistical summary of the variables used in the estimation of the model. Based on Table 2, the average counts of passing maneuvers ending in NPZ in the subject direction (N_NPZ) was equal to 3.197 passes per 15 min, and it had a range of 0–20 passes per 15 min. The two-way flow rate had a range of 212–840, with a directional split up to 77%. The percentage of heavy vehicles in the subject direction (P_HVS) was between 0 and 37.93%, with a mean of 8.56%. There were, on average, 3% of motorcycles in the traffic. The mean free-flow speed for passenger cars in a 15 min interval was 85.30 km/h on average, with a standard deviation of 16.02 km/h. The percentage of passenger cars exceeding the posted speed limit at the first of PZ was 33.1%, on average.

Table 2.

Summary Statistics of Explanatory Variables

Variable (Unit)	Mean	Min.	Max.	Confidence interval (95%)
Counts of the passing maneuvers ending in NPZ in the subject direction, N_NPZ (passes per 15 min)	3.2	0	20	(2.3, 4.1)
Length of PZ, L_PZ (m)	254.6	164	345	(240.2, 269.0)
Length of upstream NPZ, L_UNPZ (m)	1265.5	130	5010	(946.4, 1584.6)
Lane width, L_W (m)	3.4	3.0	3.8	(3.3, 3.5)
Absolute vertical grade, V_G (%)	3.4	0.6	9.5	(2.6, 4.1)
Two-way traffic flow rate, V (vph)	470.4	212	840	(434.6, 506.3)
One-way traffic flow rate in the subject direction, V_S (vph)	235.2	80	648	(212.7, 257.8)
One-way traffic flow rate in the opposite direction, V_O (vph)	235.2	80	648	(212.7, 257.8)
Directional split of traffic volume, D_S (%)	50	22.9	77.1	(47.6, 52.4)
Percentage of heavy vehicles in both direction, P_HV (%)	8.2	0	30.2	(6.4, 10.1)
Percentage of heavy vehicles in the subject direction, P_HVS (%)	8.6	0	37.9	(6.4, 10.8)
Percentage of motorcycle in the subject direction, P_MCS (%)	3.0	0	13.9	(2.2, 3.7)
Average free-flow speed, M_FFS (km/h)	85.3	63.7	102.0	(83.6, 87.0)
Standard deviation of free-flow speed, σ_s (km/h)	16.0	9.5	27.6	(15.1, 17.0)
85th Percentile free-flow speed, S₈₅ (km/h)	101.3	71.0	129.4	(99.1, 103.4)
Percentage of passenger cars exceeding the speed limit, P_SP (%)	33.1	0	81.8	(29.3, 36.9)

Note: Min. = minimum; Max. = maximum; PZ = passing zone; NPZ = no-passing zone; vph = vehicles per hour.

Model Estimation

To find a suitable model for estimating the N_NPZ, several count data models were fitted to the data using the STATA statistical software ( 25 ). As shown in Table 3, the results of the likelihood ratio test of the overdispersion parameter (α) indicate that α was statistically insignificant at the 95% confidence level ( ${\bar{χ}}^{2}$ = 3.4 × 10⁻⁷, p-value = 0.500), which implies that α is statistically equal to zero, and there is no difference between negative binomial and Poisson models.

Table 3.

Estimated Model Parameters

Variables	β-Estimate	Z-value	p-value
L_PZ	−0.010729	−4.28	0.000
L_W	4.885655	3.57	0.000
V_G	0.857126	2.07	0.039
V ² _G	−0.088856	−1.97	0.049
V	0.020265	5.43	0.000
V ²	−16.9×10⁻⁶	−4.60	0.000
D_S	0.055669	6.81	0.000
V_G× P_HVS	0.005667	2.99	0.003
constant	−22.384570	−4.37	0.000
Test	na	χ ²	p-value
Overall model evaluation
Likelihood ratio test	na	185.91	0.0000
Goodness-of-fit
Deviance test	na	87.9476	0.0440
Pearson test	na	75.0290	0.2343
Test	na	${\bar{χ}}^{2}$	p-value
Overdispersion
LR test of α	α = 0.0000876	3.4 × 10⁻⁷	0.500
Random effect
LR test of φ	φ = 1.46 × 10^-8	0.00	1.00
Zero inflation
Voung test	na	V = 1.61	0.0541
Cragg-Uhler R²	0.681	na	na
McFadden’s R²	0.4123	na	na
AIC	282.9956	na	na
BIC	303.97	na	na
AIC (ZIP)	280.16	na	na
BIC (ZIP)	312.79	na	na
Sample size	76	na	na

Note: AIC = Akaike information criterion; BIC = Bayesian information criterion; LR = ikelihood ratio; ZIP = zero-inflated Poisson; na = not applicable.

The passing rate data were collected from seven PZs. Data from the same PZs may share unobserved effects, which could lead to correlation among observations. To account for such a correlation, the random-effect model was considered. The results of the likelihood ratio test indicated in Table 3 show that the random-effect parameter (φ) was statistically equal to zero ( ${\bar{χ}}^{2}$ = 0.00, p-value = 1.00). Therefore, there is no difference between the random-effect and pooled (Poisson) models.

Out of 76 observations, 18 observations had an N_NPZ equal to zero. The excess of zeros could lead to inflation in the variance. To address this phenomenon, the ZIP regression model was applied with a Probit zero state model, including the variables of V_G, D_S, and the topography of PZs (mountainous or level). The Vuong test ( 26 ) is a common test to examine the appropriateness of using ZIP rather than the Poisson regression model. In Table 3, the results of the Vuong test illustrate that the traditional Poisson regression model is superior to ZIP at the 95% confidence level (V = 1.61, p-value = 0.0541). However, Wilson ( 27 ) concluded that the Vuong test is inappropriate for testing zero inflation. Other evaluation criteria for maximum likelihood models are the Akaike information criterion (AIC) ( 28 ) and Bayesian information criterion (BIC) ( 29 ). AIC and BIC are defined as:

AIC = - 2 \ln (L) + 2 k

(7)

BIC = - 2 \ln (L) + \ln (N) \times k

(8)

where L is the model likelihood parameter, k is the number of parameters estimated, and N is the number of observations.

A lower AIC or BIC value indicates a better fit of the data. In Table 3, AIC and BIC are reported by the STATA software. The AIC for ZIP and Poisson models is almost equal. In the case of BIC, the Poisson had a lower value than ZIP. The results imply that the Poisson model is better than ZIP. Therefore, based on all comparisons among different models, the Poisson model was chosen for estimating N_NPZ. Different model forms were estimated using explanatory variables (presented in Table 2). Table 3 shows the estimated Poisson regression model for N_NPZ.

The overall significance of the estimated Poisson model was evaluated using the likelihood ratio test. As shown in Table 3, the model was significant at the 95% confidence level (χ² = 185.91, p-value <0.0001). The result of the Pearson test suggested that the model statistically fits the data at the significance level of 0.05. The results of the deviance test showed that the model statistically fits the data at a significance level of 0.04. Cragg-Uhler R² ( 30 ) and McFadden’s R² ( 31 ), as two descriptive measures of goodness-of-fit, were equal to 0.681 and 0.4123, respectively. As the values are closer to one, the model fits the data better.

The Wald test conducted the test of the estimated parameters. The length of upstream NPZ, 85th percentile free-flow speed, and standard deviation of free-flow speed had no significant effects on N_NPZ at the 95% confidence level. Previous studies found that the length of upstream NPZ had no significant effect on the rate of passing maneuvers ending in PZ ( 11 , 17 ); possible reasons discussed by Mwesige et al. ( 11 ). The percentage of motorcycles in the subject direction was also not significant; this could be because most passing maneuvers from motorcycles happened before or at the beginning of PZs, and also a significantly lower passing time was required to pass a motorcycle. As a result, there were few passing maneuvers from motorcycles that ended in NPZ. The results also showed that the percentage of passenger cars exceeding the posted speed limit was not significant at the 95% confidence level (p-value = 0.457).

As shown in Table 3, the PZ length was a highly significant variable (p-value <0.001) that affected the N_NPZ. The sign of the estimated coefficient revealed that the N_NPZ decreased by increasing the length of PZ. The lane width was significant at the 95% confidence level (p-value <0.001). The results presented in Table 3 indicate that as the lane width increases, N_NPZ increases.

The two-way traffic flow rate and its quadratic term were significant at the 95% confidence level (p-value <0.001). The positive sign of the two-way flow rate and the negative sign of its quadratic term demonstrates that N_NPZ reaches a peak (at the flow rate of 600 vph) and then decreases. The directional split of the traffic volume in the subject direction had a highly significant effect (p-value <0.001). As the directional split of the traffic volume in the subject direction increased, the N_NPZ increased. The interaction term between the two-way flow rate and the directional split of traffic volume was insignificant at the 95% confidence level, suggesting that the peak passing rate was regardless of directional split and occurred at a flow rate of 600 vph.

The absolute vertical grade and its quadratic term were significant at the 95% confidence level (p-values <0.05). The N_NPZ rose to a peak as the absolute vertical grade increased; after the peak, N_NPZ decreased for higher values of absolute vertical grade. The interaction term between absolute vertical grade and the percentage of heavy vehicles in the subject direction was significant at the 95% confidence level (p-value = 0.003), which indicated that the effect of absolute vertical grade on the N_NPZ also depends on the percentage of heavy vehicles. To evaluate the effect of upgrade and downgrade on the passing rate, a dummy variable (upgrade = 1/downgrade = 0) was used; however, the results showed that this dummy variable was not significant at the 95% confidence level (p-value = 0.45). Therefore, the effect of vertical grade on the rate of passing maneuvers ending in NPZ is similar for both upgrade and downgrade.

Discussion

A sensitivity analysis of the estimated model was conducted to understand how explanatory variables affected the rate of maneuvers ending in NPZ, and also to compare the results with those of the previous studies. To present the 15 min passing frequency as the hourly passing rate, the N_NPZ is multiplied by four; the summarized model is presented in Equation 9.

P_{NPZ} = 4 \times N_{NPZ} = 4 \times \exp (\begin{matrix} - 22.38457 - 0.0107293 L_{PZ} + 4.885655 L_{W} \\ + 0.8571255 V_{G} - 0.0888559 {V_{G}}^{2} + 0.0202651 V \\ - 16.9 \times 10^{- 6} V^{2} + 0.0556691 D_{S} + 0.0056665 V_{G} . P_{HVS} \end{matrix})

(9)

where P_NPZ is the hourly rate of passing maneuvers ending in NPZ, and the other variables have already been defined.

Figures 1 and 2 are developed for L_PZ = 300 m, V_G = 2%, P_HVS = 5%, D_S = 50, L_W = 3.25 in Equation 9, except for the variables that change in each figure. Figure 1 is produced at four two-way traffic flow rates (i.e., 200, 400, 600, and 840 vph) and Figure 2 for the two-way traffic flow rate equal to 600 vph.

Figure 1.

The effect of explanatory variables on the rate of passing maneuvers ending in no-passing zone (NPZ): (a) passing zone length, and (b) lane width.

Figure 2.

The effect of absolute vertical grade on the rate of passing maneuvers ending in no-passing zone (NPZ) at different percentages of heavy vehicles.

Figure 1a illustrates the effect of PZ length on the P_NPZ. The figure shows that by increasing the PZ length, the P_NPZ declined at a decreasing rate, indicating that a certain increase in PZ length had a more significant effect on safety for shorter PZs. When the PZ is shorter, the traffic flow rate has a more substantial effect on its safety. In a study conducted by Mwesige et al. ( 10 ), the PZ length was not significant even at the 90% confidence level. In their study, the range of PZ length was from 0.3 to 2 km, and all PZs were long. These results implied that PZ length is a significant variable on safety in short PZs, which affects the P_NPZ. The maximum P_NPZ occurred at a traffic flow rate of 600 vph; after this value, the passing rate declined.

The results of the simulation study conducted by Moreno et al. ( 7 ) are in contrast to the results of the present study. They concluded that as the length of PZ increased, the rate of passing maneuvers ending in NPZ increased to a peak at the range of 275–350 m long. They calculated the rate of passing maneuvers ending in NPZ by multiplying the rate of passing maneuvers ending in PZ by the probability of the passing maneuver ending in NPZ, calculated by reliability analysis. The authors did not consider the effects of delayed passing maneuvers, which are the main reason for a maneuver ending in NPZ. Their studies were also based on a simulation study, which is unable to consider driver behavior, which is the most important factor in dangerous maneuvers.

Figure 1b shows how P_NPZ depends on the lane width. The passing rate significantly increased at an increasing rate as the lane width increased. The effect of lane width on P_NPZ was not investigated in the previous studies. A wider lane persuades drivers to take more risk and start the passing maneuver at an insufficient sight distance. Although a wider lane width increases the P_NPZ (i.e., decreasing the safety), it provides more room for passing and opposing vehicles to ward off a collision (i.e., increasing the safety). Therefore, the lane width is a double-edged sword. Three passing vehicles were observed in the PZ with a lane width of 3.75 m that met the opposing vehicle before completely leaving the opposing lane. Some of the previous works showed that as the lane width increased, the total accident frequency decreased in two-lane rural highways ( 32 – 34 ). However, Wang et al. ( 35 ) showed that a wider lane width is associated with more opposite-direction crashes, but fewer single-vehicle crashes. Shariat-Mohaymany et al. ( 36 ) also showed that surface width increased the risk of head-on traffic conflicts.

Figure 2 illustrates the effect of the absolute vertical grade on P_NPZ. As the interaction term between absolute vertical grade and the percentage of heavy vehicles in the subject direction was statistically significant, the effect of absolute vertical grade on P_NPZ also depended on the percentage of heavy vehicles in the subject direction. The figure indicates that P_NPZ reached a peak with increases in the absolute vertical grade. The peak of P_NPZ depended on the percentage of heavy vehicles in the subject direction, which occurred in an absolute vertical grade between 4.8% and 6%. By increasing the absolute vertical grade, the speed of heavy vehicles substantially decreases at both upgrade and downgrade because of gravity and braking harder to maintain stability, respectively; therefore, the desire to pass and, as a result, the passing rate will increase. However, after a vertical grade value (the peak value), some drivers found it more dangerous to conduct a passing maneuver and ended it in NPZ. Some of the previous works found that the absolute vertical grade increases the frequency and severity of accidents along two-lane rural highways ( 37 , 38 ). As shown in Figure 2, increases in the percentage of heavy vehicles in the subject direction correspond to an increase in P_NPZ. Kardar and Davoodi ( 39 ) showed that the percentage of heavy vehicles corresponded to more severe injuries. It should be noted that trucks in Iran were not equipped with speed limiters.

Figure 3 indicates the graph of P_NPZ and the two-way traffic flow rate. The graph was plotted considering L_PZ = 300 m, V_G = 2%, P_HVS = 15%, D_S = 50, L_W = 3.5. The observed and predicted values for P_NPZ are also illustrated in the figure.

Figure 3.

Sensitivity analysis of two-way traffic flow rate and directional split on the rate of passing maneuvers ending in no-passing zones (NPZ), and comparison of the model with previous studies: Moreno et al. ( 7 ), Mwesige et al. ( 10 ), Karimi et al. ( 17 ).

As shown in Figure 3, the peak of P_NPZ occurred at the two-way traffic flow rate of 600 vph, regardless of the directional split. After that value, the passing rate decreased with increases in the two-way traffic flow rate. Moreno et al. ( 7 ) found the peak rate to fall in the range of 300–600 vph. In the study conducted by Mwesige et al. ( 10 ) on long PZs, the rate of passing maneuvers ending in NPZ strictly increased as the traffic volume in both directions changed between 112 and 426 vph. The figure also shows that increases in the directional split in the subject direction correspond to increases in P_NPZ.

Figure 3 also has the models of Moreno et al. ( 7 ) and Mwesige et al. ( 10 ) at directional split of 50/50 for comparison. Moreno et al. ( 7 ) performed a reliability analysis based on a simulation study. As shown in Figure 3, the rate predicted by Moreno et al. ( 7 ) is significantly lower than that of other models developed by the field data. The authors estimated the difference between the lengths of PZ and required passing distance. The negative differences were used as noncompliance. A possible reason for the significantly lower passing rate for the present study could be different driver behavior.

In the model proposed by Mwesige et al. ( 10 ), only the traffic volume was significant at the level of 95%. However, the authors kept the variables of PZ length, absolute vertical grade, directional split, and the percentage of heavy vehicles to estimate the rate of passing maneuvers ending in NPZ for long PZs.

Figure 3 displays that the estimation of the present study is closer to the model of Mwesige et al. ( 10 ) in lower traffic volumes, and these differences are more significant in higher traffic volumes. Mwesige et al. ( 10 ) used hourly periods for analysis, which makes peak traffic volumes and traffic proportions more smoothed. Another reason could be differences in driver behavior between the study locations of Iran and Uganda. Also, the model of the present study was estimated based on PZs that were relatively short, while the model estimation of Mwesige et al. ( 10 ) was conducted on long PZs; driver behavior may differ in short and long PZs.

Figure 3 presents the model of Karimi et al. ( 17 ), which estimated the rate of passing maneuvers ending in the PZ. The authors estimated the model using data collected at the same PZs as the present study. The results showed that at low traffic volumes, most passing maneuvers end in the PZ. By increasing the two-way traffic flow rate, the rate of passing maneuvers ending in both PZ and NPZ increased. However, the rate of passing maneuvers ending in NPZ rose dramatically so that, in the range of 500–600 vph, it was almost equal to the rate of passing maneuvers ending in PZ. Increasing the two-way traffic flow rate increased the initiation delay of the passing maneuvers and the probability of completing them outside of the PZ. After the traffic flow rate of 600 vph, the rate of passing maneuvers ending in NPZ decreased sharply; however, the rate of passing maneuvers ending in the PZ started to decline gradually from a traffic flow rate of 680 vph. At the highest levels of the two-way traffic flow rate, smaller gaps in the opposite direction led to a reduction in the rate of passing maneuvers. The delayed vehicles found it more dangerous to accept a small opposite gap in the condition of limited sight distance, which led to a greater reduction in the rate of passing maneuvers ending in NPZ than those ending in PZ. Therefore, Figure 3 indicates that although the passing maneuvers ending in NPZ decreased safety substantially, they also had a significant positive effect on the traffic operation of PZs, especially at a traffic flow rate of about 550 vph.

Conclusion

This study aimed to evaluate the effects of geometric- and traffic-related variables on the safety of short PZs. The rate of passing maneuvers ending in NPZ was used as a surrogate safety measure to evaluate the safety of short PZs. The data were collected from seven PZs using a drone. Poisson, negative binomial, Poisson random-effects, and ZIP regression models were applied to estimate the rate of passing maneuvers ending in NPZ. Finally, the Poisson regression model was selected to calculate the effectiveness of the explanatory variables.

The results confirmed the hypothesis that increasing the length of short PZs improves safety. By decreasing the length of short PZs, safety dramatically decreased. The lane width also had significant effects on the rate of passing maneuvers ending in NPZ. However, a wider lane width provides more room for passing and opposing vehicles to prevent a potential collision. The sensitivity analysis showed that increases in the absolute vertical grade corresponded to increases in the rate of passing maneuvers ending in NPZ to a peak value. The peak value occurred between 4.8% and 6%, depending on the percentage of heavy vehicles. After the peak value, the rate of passing maneuvers ending in NPZ decreased. The percentage of heavy vehicles also corresponded to an increase in the rate of passing maneuvers ending in NPZ. The rate of passing maneuvers ending in NPZ increased to its maximum value with increases in the two-way traffic flow rate up to 600 vph, regardless of the directional split. After the traffic flow rate of 600 vph, the passing rate decreased. Increases in the directional split in the subject direction corresponded to increases in the rate of passing maneuvers ending in NPZ. The proposed model can potentially be applied in the planning, design, and safety evaluation of two-lane rural highways.

In future research, the rate of passing maneuvers ending in NPZ as a surrogate safety measure should be validated using traffic collisions resulting from passing maneuvers occurring in NPZ and close to short PZs. Centerline rumble strips decrease head-on collisions ( 40 ); therefore, using them at the end part of PZs could warn and decrease the rate of passing maneuvers ending in NPZ, which should be investigated in future studies. In this study, since the posted speed limit had an insufficient variation, it was not considered in the analysis. However, the posted speed limit could also affect the rate of passing maneuvers ending in NPZ. The gross weight of heavy vehicles, especially in combination with vertical grades, could have a significant effect, which is not considered in this study. The effects of weather conditions, night and day conditions, quality of pavement, shoulder width, and lateral clearance of road can also be evaluated in future works. The rate of passing maneuvers aborted at the end of PZs could also be considered as a surrogate safety measure.

Footnotes

Acknowledgements

The authors thank Tarbiat Modares University for funding this research.

Author Contributions

The authors confirm contribution to the paper as follows: study conception and design: Arastoo Karimi, Amin Mirza Boroujerdian; data collection: Arastoo Karimi, Amin Mirza Boroujerdian; analysis and interpretation of results: Arastoo Karimi; draft manuscript preparation: Arastoo Karimi. All authors reviewed the results and approved the final version of the manuscript.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was funded by Tarbiat Modares University.

Data Accessibility Statement

The data that support the findings of this study are available from the corresponding author, on reasonable request.

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