Driver Behavior on Combination of Vertical and Horizontal Curves of Mountainous Freeways

The study of driver behavior is of great importance to the traffic safety ofmountainous freeways. In order to study the characteristics of driver behavior on combination of vertical and horizontal curves (CVHCs) of mountainous freeways in free flow conditions, designated speed measurement tests of two typical segments of the upgrade direction of Xi’an-Hanzhong freeway were carried out. After data processing, vehicles in free flowwere screened out and classified into two groups byK-means clusteringmethod, and then the driver behavior with different lanes, different size vehicles, and different CVHCswas analyzed, respectively. Finally, a vehicle distribution predictionmodel and a speed predictionmodel were built which were applied to CVHCs, and a verification test wasmade to test the accuracy of themodels. Research results show that the driver behavior ismainly different among vehicle size, longitudinal slopes, and horizontal curves, and the characteristics of speed control and lane distribution on CVHCs vary according to lanes and combination of road alignment. Also, the prediction results of the models are highly consistent with the measured test results.


Background
In recent years, China's highway construction has made great progress.The total mileage of highways in China has reached 4,106,400 km in 2011, including 84,946 km of freeways, and the total number of motor vehicle drivers has reached 173,814,000 [1].At the same time, China is facing a great challenge in reducing traffic fatalities and injuries, especially on freeways.From 2009 to 2011, the number of accidents and casualties continuously increased [1][2][3].In 2011 alone, 9,583 accidents happened on freeways, and 6,448 persons were killed.
According to the geometrical conditions, freeways can be divided into three types: plain, hilly, and mountainous section.In 2011, mountainous sections accounted for only 11.85 percent of the total mileage of freeways.However, the percentage of deaths and injuries was 16.08 and 13.34 percent, respectively [1].Related to financing cost, technical level, and so on, the horizontal and vertical indicators of mountainous freeways are limited to some extent, especially for mountainous freeways with complex geological conditions.Compared to the traffic environment in the plain area, drivers can be easily misled into making erroneous judgments leading to accidents.Thus, mountainous sections of freeways can be considered more dangerous than hilly and plain sections.
Driver behavior includes many aspects, particularly related to speed control behavior and lane-changing behavior.In 2011, there were 1,419 accidents caused by misoperation in these two aspects on freeways, accounting for 14.81 percent of the total.In these accidents, 980 were killed and 2,244 were injured [1].Clearly, a study of driver behavior on mountainous freeway is important for reducing traffic accidents and improving the freeway's operating level.

Literature Review
A review of the literature revealed that numerous studies have focused on driver behavior on CVHCs, primarily studies on traffic flow simulation systems (TFSS) and the mechanics of traffic accidents.TFSS has rapidly developed in recent years, and it has brought convenience to test and optimize traffic planning and design schemes [4].For example, the N-S model made a great contribution in TFSS based on cellular automation [5], and the PIEV (Perception Intellection Evaluation Volition) model made the car-following model much closer to reality [6].At present, traffic flow simulation is mainly focused on arterial roads and level crossing of an urban traffic network, and the parameters of TFSS such as vehicle speed and trajectory are obtained by the method of macrostatistics [7].In TFSS, the speed of free flow vehicles is usually set as a constant velocity, and the vehicles seldom have lane-changing characteristic.Road condition is a primary reason of traffic accidents; the lack of coordination between drivers and the road can lead to an increase in driver's reaction time and miscarriage of justice and thus increases the risk of traffic accidents [8].Driver behavior models have incorporated compensation follow theory, preview follower theory, and direction and speed integrated control theory since the mid-20th century [8,9].In explaining the reason of traffic accidents, driver behavior models can be divided into descriptive models, information processing models, motivational models, and so on [10].Among these, some are representative, such as driver behavior in a cognitive architecture by Salvucci [11], RHT (Risk Homeostasis Theory) by Wilde [12], and TPB (Theory of Planned Behavior) by Ajzen [13].Related to driver behavior on CVHCs, Lamm et al. [14] advanced a V 85 prediction model, which was related to the radius of the horizontal curve.Warner and Åberg [15] advanced driver behavior of speed control, which was based on TPB.Spacek [16], by observing vehicle moving trajectories, divided driver behavior on curves into six categories: ideal, normal, correcting, cutting, swing, and drifting.However, at present, the study on driver behavior on CVHCs mainly focuses on two-lane highways and rarely involves mountainous freeways with four lanes and above.

Field Data Collection
If the radius of the horizontal curve of a freeway is less than 1000 m and the longitudinal grade is more than 1.5 percent, this can be considered as CVHC.In this study, a section of Xi'an-Hanzhong freeway in Qinling Mountain was chosen as the test intervals.In this section, nearly 70 percent are CVHCs.By means of designated speed measurement tests to collect data, driver behavior included speed control characteristic and track distribution characteristic on CVHCs in this section was analyzed.
Two typical intervals of the "up" direction of Xi'an-Hanzhong freeway were taken for the tests.Their stake marks were as follows: k1139+530∼k1139+920 and k1143+520∼ k1143+940.In these two intervals, 10 observation crosssections were chosen and marked as S 1 -S 10 .The instruments of the designated speed tests were NC200, which were placed in the middle position of the inner lane and curb lane of each observation cross-section.The NC200 in the inner lane was marked as "" and in the curb lane was marked as "." The test instrument arrangement is shown in Figure 1, and the parameters of each observation cross-section of the two intervals are shown in Table 1.The designated speed measurement test for each test interval lasts for 3 hours.
As shown in Figure 1, each test interval included a complete CVHC and was marked as C1 and C2.Road alignment before C1 was straight line and after C1 was a reverse curve with a large radius.Road alignment before C2 was a reverse curve with a large radius and after C2 was a straight line.

Data Processing
4.1.Screening Vehicles in Free Flow.In this paper, all analysis of driver behavior on CVHCs was in free flow.In this condition, drivers have a desired driving environment and can control the vehicle speed and change lanes on their own.Headway is always taken as the measurement index in the screening of vehicles in free flow [17].Referring to relevant content about the free flow condition of the Chinese Road Research Institute of the Ministry of Transport and combined with the average speed of different size vehicles in the test, in the study, taking 6 s headway as the judging criteria of the free flow vehicles, namely, if a vehicle's headway is larger than 6 s, the vehicle is then judged as a free flow vehicle [18].Data were collected by NC200 and HD videos, and statistical results of the free flow vehicles of all the observation cross-sections in the two test intervals are shown in Table 2.

Vehicle Classification.
To study driver behavior on CVHCs in detail, classification of the vehicles in the designated speed measurement tests is necessary."Method of Vehicle Classification" was published by the Ministry of Transport China in 2010, and vehicles were classified into four groups according to their rated load, dimensions, number of axles, and so on.In reference to this method, a quick classification of the tested vehicles was taken on the basis of the vehicle length, and the vehicles were classified into two groups: small (marked as S) and large (marked as L).Speed is an important manifestation of driver behavior, and on mountainous freeways, the speed of different sizes vehicles is significantly different.Considering that a certain size vehicle would have different speed characteristic in different test intervals, the speed data that was collected in the same interval was chosen as the basis of the vehicle classification.In this paper, the data collected in the five observation crosssections of the k1139+530∼k1139+920 interval was taken for reference.A K-means clustering method was used to classify the speed data.After nine iterations, the data came into convergence and the clustering centers were 59.14 and 95.38.The iteration process is shown in Table 3.In accordance with the principle of the nearest to the clustering center, the vehicles were classified into two groups, and the average value and the standard deviation of each category were calculated; the results are shown in Table 4.According to Table 4 and the length distribution in the test, 8.5 m was chosen as the demarcation point of the small and large size vehicles.The statistics of the free flow vehicles of each observation cross-section are shown in Table 5.With the video materials that were recorded in the test intervals, it can be seen that passenger cars accounted for most of the small size vehicles, and trucks and buses were the majority of the large size vehicles.

Driver Behavior Model
In this formula,   is the vehicle distribution in the inner lane of a certain size vehicle in a certain observation cross-section (%);   is the number of a certain size vehicle passing through the cross-section in the inner lane; and   is the number of a certain size vehicle passing through the cross-section in the curb lane.Vehicle distribution in the inner lane of different size vehicles of the two test intervals is shown in Figure 2.
To study the proportion of a certain size vehicle's lane distribution of all the traffic flow passing by the observation cross-section, the concept of the distribution of a certain size vehicle in a certain lane is introduced as follows: In this formula,   is the distribution of a certain size vehicle in a certain lane (%), and   is the number of a certain size vehicle in a certain observation cross-section.Among these,  ∈ {, }, and  is small size vehicle and  is large size vehicle;  ∈ {, }, and  is the inner lane and  is the curb lane;  is the total number of vehicles passing through the observation cross-section.The distribution of different size vehicles in different lanes of the two test intervals is shown in Figure 3.
As can be seen from Figure 2, the lane distribution of a small size vehicle in the curb lane and the inner lane was generally close.However, large size vehicles primarily were driven in the curb lane, and the lane distribution in the inner lane was rarely more than 20 percent.At the same time, as seen in Figures 2 and 3, the running track of different size vehicles on CVHCs was significantly different; namely, the tendency of lane-changing of different vehicles was different.In particular, the tendency of changing lane of small size vehicle drivers on CVHCs was much more obvious but not so for large size vehicle drivers.Also from Figure 3, small size vehicles accounted for nearly 70 percent of all traffic flow, and large size vehicles accounted for only 30 percent.This distribution proportion was similar to that in plain areas.Also, combined with videos recorded in the two test intervals, it was found that the speed of trucks of large size vehicles was very low with full loads and longer wheel base, and they were driven in the curb lane for a long time and seldom their lanes were changed.In addition, it can be seen that, combined with road alignment in Figure 1, the lane distribution of small size vehicles in different cross-sections of CVHCs was closely related to the road alignment before and next to the current section.There was a regularity when driving a small size vehicle on CVHC of a mountainous freeway: the tendency of lane-changing behavior is that the vehicle switches from the lane in which the center position is far away from the center of the CVHC to the lane close to the center of the CVHC, and this behavior is called "curve-cutting" behavior.In this paper, the two test intervals were a combination of two groups of continuous reverse "S" shaped curves, so the behavior was more obvious.Typical curve-cutting behavior of continuous CVHCs on mountainous freeways is shown in Figure 4.
With the video materials, the track correction behavior of drivers was seen: when the vehicle entered into the curve, the vehicle significantly deviated from the centerline for a short time and then adjusted back to the centerline.The explanation of this behavior is when the vehicle enters into the curves at a relatively high speed, there is a certain deviation of required appropriate steering wheel angle in the large curvature of the curve in the mountainous area.This behavior occurred frequently in small size passenger vehicles  with a relatively high speed in our study.If the curvature of CVHCs was much bigger, the tendency of this behavior was much more obvious.
Combined with above analysis as well as related statistics and analysis method, a track prediction model of the vehicle distribution in the inner lane   of different size vehicles on CVHCs of mountainous freeways is shown in Table 6.The prediction model of the vehicle distribution in curb lane is   = 100% −   .In the model,   is the curve radius of the road before the current CVHC (m),   is the curve radius of the current CVHC (m),   is the curve radius of the road next to the current CVHC (m), S is the straight line, L is the left curve, and R is the right curve.

Speed Prediction Model.
When driving on CVHCs, the tendency of lane-changing behavior of different size vehicles was different, and also the speed characteristic of different size vehicles in different lanes was not the same.The average speed of different size vehicles in the inside lane and curb lane in the two test intervals is shown in Table 7.Among these, V is the average speed and  is standard variance of V.
In Table 7, it can be seen that, on the same CVHC of the mountainous freeway, the average speed in the inner lane of a certain size vehicle was significantly higher than that in the curb lane, especially for large vehicles.The speed difference between the inner lane and the curb lane was above 20 km⋅h −1 .On the same CVHC, the average speed of small size vehicles was significantly higher than that of large size vehicles, especially in the curb lane.In addition, even for the same size vehicle in the same lane, there was a great difference on different CVHCs.The reasons were that, on one hand, it was affected by the road alignment of the current CVHC, and, on the other hand, it was closely related to the road alignment before the current CVHC.Generally speaking, the curvature of the road alignment in C1 was less than that in C2.Before C1 at the range of about 3 km, the radius of the curves were large, the slopes were relatively gentle, and the longitudinal slope grade was less than 2 percent.But before C2 at the range of 3 km, small radius circular curve sections occupied a large proportion, and the longitudinal slopes were nearly all above 3 percent, so the average speed in C1 was higher than that in C2 of the same size vehicle in the same lane.The statistical results of average speed in each observation cross-section are shown in Table 8, and  is standard variance of V  .
As can be seen from Table 8, no matter the size of vehicles, the speed on CVHCs was a continuous gradient process.The speed characteristic of different size vehicles was related to not only the road alignment on the current CVHC, but also the alignment before and next to the current CVHC.Combined with the designated speed measurement tests, the speed control strategy of drivers on CVHCs was as follows: when the vehicles entered into the CVHC with a small curve radius from the road section before the current CVHC in which the curve radius was large, the speed was usually reduced; on the current CVHC, the speed control strategy of drivers mainly depended on the current CVHC's curvature.If the curvature was large, speed was reduced significantly until it reached and maintained a relatively reasonable rate.On the contrary, if the curvature was small, the speed reduction was not so obvious.And then, before the vehicles left the CVHC, the driver could see the next ease curve section, so the driver accelerated to drive the vehicle out of the CVHC.In addition, it can be seen from Table 8 that the grade of road longitudinal slope had a significant impact on the vehicle speed.Especially in an uphill section, if the slope grade was large, the influence was much more obvious.
By the above analysis and with related mathematical method, the speed prediction models which are applied to different linear combination of CVHCs for different size vehicles are shown in Table 9.To be sure, each model in Table 9 is applicable to uphill sections of mountainous freeway.
In the model, S is straight line; C is curve; V  , V  , V  are the prediction speed at the entrance, midpoint, and exit of the inner lane on the CVHC (km⋅h −1 ); V  , V  , V  are the prediction speed at the entrance, midpoint, and exit of the curb lane on the CVHC (km⋅h −1 );   ,   ,   are the curve radius of the CVHC in the front, current, and next (m);   ,   ,   are the road longitudinal slopes of the CVHC in the front, current, and after (%), and as the model is applicable for the uphill sections, the value of   ,   ,   is all negative.

Verification Test.
To evaluate the accuracy of the driver behavior model in the paper, a verification test was done in other 6 intervals of Xi'an-Hanzhong freeway.For the vehicle distribution model, the maximum prediction error was 18.9%, the minimum error was 4.3%, and the average error was 8.1%.For the speed prediction model, the maximum prediction error was 23.6%, the minimum error was 3.3%, and the average error was 9.6%.Generally speaking, the models which were built in the paper can predict the driver behavior on CVHCs of mountainous freeway accurately.

Conclusions and Discussions
The designated speed measurement test was done on two typical CVHCs of the Xi'an-Hanzhong freeway.The traffic flow in the test period was approximately 300 vehicles per hour, and the free flow vehicles were about 2/3 of all the traffic flow.Based on the analysis of the driver behavior on the CVHCs, conclusions are made as follows.
(1) According to the vehicle classification described in this paper, the distribution of small size vehicles is equal in the inner lane and curb lane on CVHCs, whereas about 80 percent of large size vehicles are driven on the curb lane.
(2) About 10 percent of small size vehicles display curvecutting behavior in free flow, and some small size vehicles display curve track correction behavior.However, for large size vehicles, these two driver behaviors are in a very low proportion.
(3) The average speed of vehicles in the inner lane and the curb lane is quite different.The average speed varies by more than 20 km⋅h −1 for large size vehicles.
In addition, the average speed of small size vehicles on the same CVHC is significantly higher than that of large size vehicles.
(4) The speed characteristic of different size vehicles is related to not only the road alignment on the current (5) The typical speed control strategy when driving from an ease curve section to a sharp curve section is as follows: first, the driver reduces the speed appropriately when the vehicle enters into a CVHC on which the curvature is large.When the vehicle is in the CVHC section, the driver maintains a relatively low speed.Finally, the speed increases in advance when driving out of the CVHC section.
(6) By verification test, the average error of vehicle track distribution prediction model is 8.1%, the average error of the speed prediction model is 9.6%, and the driver behavior model can predict the driver behavior on mountainous freeway accurately.
In the past three decades, China has made a remarkable progress in both infrastructure construction and motorization.However, road safety levels have not kept pace with these changes, especially on CVHCs of mountainous freeways.Although traffic administration has taken some precautionary measures, such as speed limit signs and speed bumps, the accident rate is still relatively high.Based on the analysis of driver behavior in this paper, more measures should be taken to improve the fault-tolerant capability of roads with CVHCs, such as increasing road widths for sharp curve sections, improving the road linear combination of CVHCs to improve the driver's vision, building auxiliary lane for trucks to improve the traffic capacity on CVHCs, and improving the drainage capacity of the CVHCs to ensure the friction force between the wheels and the road in rainy days.Additional research is also needed to improve road safety level as well as control costs.

Figure 2 :Figure 3 :
Figure 2: Vehicle lane distribution in inner lane of different size vehicles of the two test intervals.

Table 3 :
Iteration process of vehicle speed of the designated speed measurement tests in k1139+530∼k1139+920 interval.

Table 4 :
Statistical data of low-speed and high-speed group of the designated speed measurement tests in k1139+530∼k1139+920 interval.
S: small size vehicle; L: large size vehicle.

Table 6 :
Track prediction model of   on CVHCs of mountainous freeway.

Table 9 :
Speed prediction model of different size vehicles in different lanes of mountainous freeway.but also the alignment before and next to the current CVHC. CVHC,