Acceleration and Deceleration Calibration of Operating Speed Prediction Models for Two-Lane Mountain Highways

Over the past two decades, design-consistency analysis methods based on operating speed profiles have become primary tools for road geometry safety evaluation because of their advantages of fast processing, high efficiency, and good user convenience. Such methods have played a critical role in limiting the frequency of accidents on mountain highways and reducing the formation of accident-prone highway sections. The function of operating speed prediction models is to predict a continuous-speed curve for vehicles driving along a road. In addition to the horizontal geometry, such models are also able to consider three-dimensional (3D) geometric effects. The rationality of the forms and coefficients of a model determines the accuracy of its speed prediction, which influences the reliability of the evaluation results used for ensuring design consistency in highway alignments. Therefore, improving existing operating-speed models and proposing new models are subjects of interest in the field of highway engineering.

A 3D road surface can be decomposed into horizontal, vertical, and cross-sectional components. The horizontal alignment can be further decomposed into straight, spiral, and circular components. Most existing models have been developed for a specific type of road section or single/multiple geometric elements. Operating speed models currently in use include horizontal curve models (Christopher and Mason 1995; Easa and Mehmood 2007; Jacob and Anjaneyulu 2013), straight road models (Luca et al. 2014), curved slope models (Gibreel et al. 2001; Xu et al. 2015), desired speed models (Crisman et al. 2005), roadside interference models (Fitzpatrick et al. 2005), traffic impact models (Shao et al. 2015), tunnel models (Fang et al. 2010), and interchange ramp models (Zhang et al. 2014).

Operating speed prediction models proposed roughly over the last 30 years by researchers from the United States (U.S.), Canada, and some European countries listed in Table 1, along with the independent variables (predictors) used in the model. As seen in the table, $R$, CCR, $L_c$, and $L_T$ are the most commonly used predictors irrespective of the location (United States, Canada, or Europe). However, factors such as the terrain conditions, road characteristics, traffic composition, and automobile performance may differ for different countries and regions; therefore, the road geometry features (i.e., the meaning of variables) that have a major impact on the operating speed can have significant differences, which is represented as follows. First, in the United States and Canada, researchers often use the predictor $DC$ to establish the model, whereas in Europe, this predictor is not used. Second, in addition to the predictors $R$ and CCR, the operating speed model for a horizontal curve in Europe usually contains $V_T$ (operating speed on the approach tangent of the curve) or $V_A$ (speed at curve entry). In contrast, these two predictors are not employed in the models from the United States and Canada probably because the terrain in North America is flatter than that in Europe; therefore, in North America, the ratio of the curve length to the total road length is smaller, and the average spacing between adjacent bends is larger. However, in mountainous European countries such as Italy and Spain, two-lane highways generally have intensive horizontal curves, and hence, the operating speed on a bend is greatly impacted by the presence of an adjacent bend. In addition, the structure of operating speed models from Italy and Spain is more complex (e.g., Montella et al. 2014; Castro et al. 2011; Camacho-Torregrosa et al. 2013). Note that the models proposed by Donnell et al. (2001) from the United States and Gibreel et al. (2001) from Canada have more predictors, but that is because the former model is intended for heavy vehicles, whereas the latter model is intended for 3D highway alignment. To understand what the notations in Table 1 indicate, refer to the Appendix at the end of this paper.

The operating speed varies on a road depending on two factors: the speed on different types of road segments (such as circular section or tangent), and speed adjustments made when transitioning between adjacent road segments. An example of the second factor is the deceleration made while entering a curve and the acceleration provided while exiting a curve; hence, this factor can be described in terms of vehicular acceleration/deceleration. As Fig. 1 shows, for a given road section, the operating speed profiles will have different shapes for different acceleration/deceleration rates. Extensive research has been conducted on this subject in China and other countries. In several of these studies, the acceleration and deceleration rates are both assumed to be equal to $0.8\mathrm{m}/{\mathrm{s}}^2$ (Lamm et al. 1988) or $0.85\mathrm{m}/{\mathrm{s}}^2$ (Echaveguren and Basualto 2003; Laurel and Tarek 2011). In other studies, they are both assumed to be within a certain range [$a_{\text{min}}$, $a_{\text{max}}$], and it is common practice to employ linear interpolation within this range based on the current operating speed, which usually yields a smaller acceleration at a higher speed (Liu et al. 2010). Another hypothesis assumes constant acceleration/deceleration between two adjacent road elements, with the magnitude of the acceleration/deceleration determined by the difference in speeds on the two adjacent elements and the distance between them (Montella et al. 2014; Hashim et al. 2016). Researchers have also used experimental data to model the acceleration rate–speed relationship (Moon and Yi 2008) and acceleration rate–acceleration distance relationship (Gao et al. 2004). However, because these models do not differentiate between acceleration and deceleration, they are unable to capture the significant differences in the magnitudes of acceleration and deceleration caused by differences in the dynamic properties of vehicles (Fitzpatrick 2000; Camacho-Torregrosa et al. 2013; Montella et al. 2014). Thus, the results are not likely to accurately reflect actual vehicle operating characteristics.

In some studies, acceleration/deceleration rate–curve radius models have been constructed based on small-scale experimental data (Shao et al. 2011; Fitzpatrick 2000). However, because of the limited number of road samples considered, these models are restricted in their range of applicability. Other studies have considered acceleration/deceleration distribution characteristics based on observed speed or acceleration data. For instance, Altamira et al. (2014) used vehicle acceleration/deceleration data on horizontal curves and curved slopes to derive a relationship between the acceleration/deceleration rate and the distance to a curve entry/exit. Tokunaga et al. (2005) used a small passenger car driven on mountain roads to collect continuous acceleration/deceleration data, which were analyzed to assess the effects of day and night, summer and winter, and the use of electronic navigation devices on driver behavior. Based on speed measurements of vehicles on different cross sections and the distances between adjacent cross sections, Perco and Robba (2012) determined the distribution characteristics of acceleration and deceleration of vehicles on curved road segments.

As described above, previous research on automobile acceleration/deceleration suffers from the following limitations. First, many of the assumptions made in the previous studies were not based on actual driving behavior. Instead, they were made primarily to simplify calculations, and thus, the predicted speed could be significantly different from real-world driving. Second, although some researchers have developed acceleration/deceleration regression models, the sizes of the sample datasets used were relatively small, which limits the applicability of these models to other roads until their reliability and appropriateness can be improved. Third, almost all previous research on this subject has been limited to observing small passenger cars; little research has been conducted on the acceleration/deceleration characteristics of heavy trucks or large coaches on mountain highways, particularly on two-lane mountain highways. In this study, acceleration data were collected from both natural driving tests (onboard tests) and roadside observation tests for characterizing the statistical distributions of the acceleration/deceleration of small passenger cars, heavy trucks, and coaches. Models relating acceleration/deceleration to road geometry elements were constructed for these different types of vehicles. The results obtained provide a means for calibrating acceleration/deceleration rates in operating speed models for mountain highways, thereby remedying the shortcomings of the existing models presented in the literature.

In many operating speed models, acceleration/deceleration is simplified as a constant, and it is common practice in Europe and North America to set this value to $0.85\mathrm{m}/{\mathrm{s}}^2$ for passenger cars on two-lane highways. China’s Highway Safety Audit Guidelines (JTG/TB05-2015) suggest ranges for passenger cars ($0.15–0.50\mathrm{m}/{\mathrm{s}}^2$) and large cargo vehicles ($0.20–0.25\mathrm{m}/{\mathrm{s}}^2$) and assume that the acceleration and deceleration characteristics are the same. The highway capacity manual (HCM 2010) recommends a rate of $1.22\mathrm{m}/{\mathrm{s}}^2$ ($4.0\mathrm{ft}/{\mathrm{s}}^2$) for both acceleration and deceleration if these rates are unknown for straight urban roads; for right turns, it recommends an acceleration rate of $1.067\mathrm{m}/{\mathrm{s}}^2$ and a deceleration rate of $1.22\mathrm{m}/{\mathrm{s}}^2$. In Sweden, acceleration and deceleration rates are prescribed as $0.8\mathrm{m}/{\mathrm{s}}^2$ in Swiss Norm 640 080b (Vereingung Schweizerischer Strassenfachleute 1992) to estimate the speed profile on the tangents between curves. In a report published by the U.S. Federal Highway Administration (1995, Report No. FHWA-RD-94-034), deceleration and acceleration are assumed to occur on the tangents approaching and leaving the curve, and both are set to $0.85\mathrm{m}/{\mathrm{s}}^2$, which is slightly greater than the rate for Sweden (Krammes 1995). In the present study, the authors examined these approaches to characterize acceleration and assessed their applicability to mountain highway conditions, vehicle performance, and driver behavior on two-lane mountain highways.

The continuous curves of longitudinal acceleration obtained from the LPMS units mounted in the test vehicles were processed. The data were filtered and smoothed to remove signal noise and glitches, as shown in Fig. 4(a). A peak extraction algorithm was then designed to determine the maximum deceleration and acceleration as each vehicle entered and exited each horizontal curve, as illustrated in Fig. 4(b). Next, the acceleration and deceleration peaks were collected separately and sequenced to obtain $\{(j,a_{xj})|j=1,2,\dots ,N\}$ and $\{(i,a_{bi})|i=1,2,\dots ,M\}$, where $a_{xj}$ and $a_{bi}$ = peak acceleration and peak deceleration, respectively, and $N$ and $M$ = number of peak acceleration and deceleration values, respectively. Finally, the cumulative frequency curves of acceleration and deceleration were developed for each type of vehicle.

The measured results of the longitudinal acceleration for three drivers on the test road—National Highway G212—are presented in Figs. 5(a–i), where $N_s$ is the sample size of the measured data. The results in Figs. 5(a–c) represent the distribution of longitudinal acceleration versus the driving speed, cumulative frequency curves of longitudinal acceleration, and cumulative frequency curves of acceleration and deceleration, respectively, for the first driver; the speed considered here is the initial velocity when drivers are decelerating/accelerating their vehicles. From the scatter plot in Fig. 5(a), one can see that deceleration and acceleration have two different sets of variation characteristics with an increase in the driving speed. The amplitude of acceleration decreases as the initial speed increases; in contrast, a larger deceleration rate often corresponds to a higher initial speed. Data analysis indicates that during acceleration, the vehicle dynamic performance plays a major role; that is, the reserve accelerating ability is smaller at a higher speed, which results in weak acceleration. With regard to deceleration, the distribution characteristics are determined by the actual demand while driving on curvy roads because a higher speed at curve entry will require a larger reduction in the driving speed to ensure safe bend negotiation; consequently, the driver will have to adopt a large deceleration. The distribution of longitudinal acceleration versus the initial speed for the second and third drivers is presented in Figs. 5(d and e). The results for all the three drivers are shown in Fig. 5(f); they all exhibit similar variations with the initial speed, as illustrated in Fig. 5(j).

Figs. 5(g and h) show the cumulative frequency curves of $a_x$ and $a_b$ versus the initial speed of the second and third drivers. As in the case of the first driver [Fig. 5(c)], one finds that the deceleration is larger than the acceleration for the second and third drivers; and the difference between deceleration and acceleration is significant. Moreover, the personal characteristics of drivers can cause variations in the shape of the cumulative frequency curves. The results for all the three drivers on the G212 are shown in Fig. 5(i). The results for the other test roads are not described here owing to the limitation on the paper length.

Figs. 5(k and l) show the cumulative frequency curves of acceleration and deceleration for each type of vehicle. The 85th, 90th, and 95th percentile characteristic values are also indicated. Several obvious features are seen in the figures. First, the deceleration is greater than the acceleration, and the maximum deceleration rate is approximately twice the maximum acceleration rate. The reason for this difference is that the deceleration rate produced by braking depends on the friction coefficient of the pavement, $f$. Thus, the deceleration rate can approach or even reach $f·g$. On the other hand, as long as the pavement is in reasonably good condition, a vehicle’s acceleration depends on the torque of the engine and transmission/drive mechanisms. Considering current passenger car technology, the acceleration rate will obviously be less than the deceleration rate. Therefore, existing research findings and design specifications that do not address acceleration and deceleration separately are apparently not reasonable.

Second, the breakpoints of the cumulative frequency curves do not appear at the 85th percentile, as previously assumed. Instead, they are actually very close to the 95th percentile. This discrepancy is crucial, considering the fact that the core philosophy of the current road design methods concerning the operating speed is to base the design of road geometry features on driving behavior at the 85th percentile (breakpoint). The results of this research suggest that the 85th percentile philosophy needs to be corrected.

Third, differences exist between the acceleration and deceleration rates for different types of vehicles. Comparing Figs. 5(k and l), it is apparent that vehicles that exhibit high acceleration rates also exhibit high deceleration rates. However, an examination of the acceleration data only indicates that the 85th-percentile acceleration rates are greater than $1\mathrm{m}/{\mathrm{s}}^2$ for all vehicles, except for the high-seating-capacity Toyota HIACE, which has a relatively low specific power. The 85th-percentile decelerations are even higher. Therefore, the adoption of a value of $0.85\mathrm{m}/{\mathrm{s}}^2$ for both acceleration and deceleration is clearly inconsistent with actual driver behavior on two-lane mountain highways.

Table 3 lists the maximum and average acceleration and deceleration rates along with the 50th, 85th, 90th, and 95th percentiles extracted from the cumulative frequency curves. It is noticed that the 85th-percentile acceleration and deceleration rates, in particular, are 1.051 and $1.357\mathrm{m}/{\mathrm{s}}^2$, respectively, for all test vehicles. The characteristic percentile values can be used to calibrate the acceleration and deceleration rates in operating-speed prediction models for passenger cars on mountain highways. In the calibration, suitable adjustments can be made depending on the road conditions and vehicle types. The maximum values from the table can be used to establish boundary conditions for driving simulations.

The operating speed models that have been proposed by researchers in Europe, North America, and Australia have been developed primarily for passenger cars (Misaghi and Hassan 2005; Abbas et al. 2011; Castro et al. 2011). The reasons for this are that passenger cars have higher speeds and have a greater effect on the design of horizontal alignments. However, in recent years, the transportation safety situation in China has been quite grim, with large-vehicle accidents, especially ones causing multiple deaths/casualties, becoming the most serious threat to road users. Hence, there is a pressing need for developing large-vehicle operating speed models to assess the suitability of roadway geometric designs for large vehicles.

Fig. 6 summarizes the longitudinal acceleration measurements obtained from off-road observations for large cargo vehicles (trucks weighing 20 t or more) on two-lane highways. In the figure, $N_s$ is the sample size of measured data. Positive values represent acceleration, and negative values represent deceleration. The scatter plot in Fig. 6(a) reveals a negative correlation between longitudinal acceleration and the driving speed. Fig. 6(b) shows the cumulative frequency curve for longitudinal acceleration. Taking the absolute values of the deceleration points and superimposing them on the acceleration points yields Fig. 6(c), and this figure shows that for heavy trucks, the deceleration values are clearly higher than the acceleration values.

Fig. 7 summarizes the longitudinal acceleration results for large coaches (37 seats or more) obtained using the same method as that used for heavy trucks. Note that the acceleration data collection tasks for both trucks and coaches were carried out simultaneously on the same sections of the test roads; in terms of total traffic volume, the percentage of coaches is smaller than that of trucks, and therefore, the number of coaches that passed though the observation stations was less than the number of trucks. Two special features should be noted. First, the acceleration and deceleration rates for large coaches are at least 50% higher than those for heavy trucks. Large coaches carry lighter loads than heavy trucks and have higher specific powers per unit mass, so their performance is better in terms of acceleration and braking. Second, the difference between the acceleration and deceleration values is negligible. Therefore, for this type of vehicle, the same value can be used to calibrate both acceleration and deceleration in operating-speed models. Table 4 presents the maximum, average, and characteristic values for these two types of vehicles. These values can be used to calibrate the acceleration/deceleration rates in operating speed models for large vehicles on mountain highways and to establish boundary conditions for large-vehicle driving simulations.

The main motivation for drivers to adjust their speeds on mountain highways is to adapt to the constantly changing horizontal curvature of the roadway. They tend to remain comfortably within the critical safe speed limit for horizontal curves, especially sharp curves with small radii. Fig. 8 shows the speed and acceleration of a passenger car as it traverses a short, complex section of Highway S102 between Wuxi and Yunyang. It is observed that every curve of the road causes a change in speed and hence a change in the longitudinal acceleration. This suggests that there may be a close correlation between the longitudinal acceleration and the geometric parameters of horizontal curves.

Fig. 9 shows the scatter plots generated from the data collected from the onboard passenger car tests. The sample size of the data is 534, and the sample size of the bends is 132 because two or three drivers make a round trip on the test roads (i.e., one bend can produce multiple test data points). Driving speed of the passenger car is not sensitive to the slope of the roadway (Xu et al. 2012), so the relationship between the acceleration/deceleration and horizontal alignment rather than the vertical alignment is analyzed.

The plots in Figs. 9(a and b) show a rather strong negative correlation between acceleration/deceleration and the curve radius. This indicates that a driver entering a curve with a smaller radius will brake harder and will tend to accelerate faster as the vehicle leaves the curve. The reason for this is that curves with smaller radii have lower critical safe speed limits, so drivers must reduce their speed to a greater degree, which requires greater deceleration. When exiting one of these sharper curves, the difference between the driver’s desired speed and the vehicle’s actual speed is greater, so the driver will often use greater acceleration to increase his speed more quickly. The trend that is evident in the figure provides a reasonable basis for developing a model that relates acceleration/deceleration to the curve radius $R$. After comparing the fitting accuracies of functions with different forms, the following two equations were found to yield fairly good results:

Figs. 9(c and d) show the scatter plots of the acceleration/deceleration rate and curve deflection angle. In contrast to the correlation observed for the curve radius, a positive correlation is evident between acceleration/deceleration and the curve angle. The larger the deflection angle, the harder the driver tends to brake and accelerate while entering and exiting the curve, respectively. The reason for this can be explained as follows. Curves with larger deflection angles are likely to give the driver the impression of being sharp and to adversely affect driving observations from the vehicle. Thus, drivers tend to choose a lower speed while negotiating such curves, resulting in greater deceleration. Eqs. (4) and (5) represent the regression models obtained from the scatter plots using the least-squares method.

In Eqs. (2)–(5), $a_x$ and $a_b$ = deceleration while entering a curve and the acceleration while exiting a curve, respectively; $R$ = horizontal curve radius; and $\mathrm{\Delta }$ = deflection angle of the horizontal curve, expressed in degrees.

The four formulas mentioned above can be used in operating speed model calculations and calibrations for the acceleration/deceleration of passenger cars on two-lane mountain highways, instead of setting the acceleration/decoration rate to a fixed value, as mentioned previously in “Acceleration/Deceleration Distribution Characteristics for Passenger Cars.” Together with horizontal curve speed models and desired speed models, these formulas make up a relatively complete system of operating speed models.

Fig. 10 illustrates the differences in operating speed profiles produced by three acceleration/deceleration calibration methods for a 5.5-km segment of the Anlong–Ceheng Section of Highway S313. The first method uses $a_x=0.85\mathrm{m}/{\mathrm{s}}^2$ and $a_b=0.85\mathrm{m}/{\mathrm{s}}^2$ (Echaveguren and Basualto 2003; Perco and Robba 2012), the second method uses $a_x=0.2\mathrm{m}/{\mathrm{s}}^2$ and $a_b=0.4\mathrm{m}/{\mathrm{s}}^2$ (China’s Highway Safety Audit Guidelines, JTG/TB05-2015), and the third method uses the values of $a_x$ and $a_b$ determined from Eqs. (2) and (3) developed in this study.

The calibration and modeling of vehicle acceleration/deceleration rates are important aspects of constructing a complete operating speed model system. To determine the actual acceleration characteristics of vehicles, onboard driving tests and external observations were carried out on six two-lane mountain highways with complex shapes. The acceleration/deceleration characteristics of different types of vehicles were determined, and based on these characteristics, models relating acceleration/deceleration to road features were developed. The main conclusions are as follows:

Future work: In the present study, only passenger cars were considered while developing the models relating acceleration/deceleration to curve features. The authors are currently carrying out continuous onboard driving tests using heavy trucks. Models relating the acceleration/deceleration of heavy trucks to the geometric features of roads will be established when sufficient data have been obtained.

This research was supported by the National Natural Science Foundation of China (Grant No. 51278514), Applied Basic Research Program of the Ministry of Transport of China (Grant No. 2015319814050), and Science and Technology Planning Project of ChongQing Municipality, China (Grant No. Cstc2014jcyjA30024).