Operating speed is a critical indicator for road alignment consistency design and safety evaluation. Although extensive studies have been conducted on operating speed prediction, few models can finish practical continuous prediction at each point along alignment on multilane highways. This study proposes a novel method to estimate the operating speed for multilane highways in China from the aspect of the threedimensional alignment combination. Operating speed data collected in field experiments on 304 different alignment combination sections are detected by means of Global Positioning System. First, the alignment comprehensive index (ACI) is designed and introduced to describe the function accounting for alignment continuity and driving safety. The variables used in ACI include horizontal curve radius, change rate of curvature, deflection angle of curve, grade, and lane width. Second, the influence range of front and rear alignment on speed is determined on the basis of drivers’ fixation range and dynamical properties of vehicles. Furthermore, a prediction model based on exponential relationships between road alignment and speeds is designed to predict the speed of passenger cars and trucks. Finally, three common criteria are utilized to evaluate the effectiveness of the prediction models. The results indicate that the prediction models outperform the other two operating speed models for their higher prediction accuracy.
Humanorient and safety supremacy are currently the new guidance during the period of highway construction. The traditional design speedbased alignment design approach usually only specifies the minimum value of one isolated alignment element. This designing method is prone to be inconsistent with successive elements of a road. Large amount of practical studies highlights the fact that inconsistent alignment might cause a sudden change in the characteristics of the roadway, which would lead to critical driving errors and crash risks [
There are extensive literatures on operating speed prediction models in which the variables and the model constructions vary considerably. Most models focus on horizontal curve by assuming constant speed on curves and therefore deceleration and acceleration that occur entirely on the approach tangent and on the departure tangent [
Meanwhile, previous works introduced the vertical alignment influences on operating speed [
Numerous studies have been completed for passenger car operating speed prediction and design consistency on rural twolane highways [
In terms of limitations in aforementioned methods, it is challenging to design different approaches to explore a comprehensive representation of the operating speed. The report EC151 of the Transportation Research Board [
The main objective of the research in this paper is to propose a continuous operating speed prediction model for passenger cars and trucks on multilane highways. This new methodology, for the first time, formulates a threedimensional alignment comprehensive index (ACI) combined with driver’s visual characteristics and vehicle dynamic properties to achieve higher accurate and reliable speed estimation at each point along the roadway. This could be useful for researchers to evaluate alignment design consistency and determine alignment features.
Operating speed is affected by multiple factors. How to find the key information from complex influence factors is critical for accurate prediction. Based on the analysis of the relation among operating speed, alignment, and other influence factors, the basic hypotheses are summarized as follows:
Operating speed varies with the change of road condition along the driving direction.
The comprehensive influence of alignment on operating speed is not only mutually independent, but also not equivalent to a simple linear overlay. As a quantitative indicator to characterize horizontal, vertical, and cross section alignment, the road alignment comprehensive index is related to the speed variation.
Operating speed on a certain section is related to alignment features on this section and also affected by the range of a certain length of alignment between rear segment and front segment.
These three assumptions which focus on the influences of front and rear alignment on operating speed are in line with the general driving rules of vehicles run on highway. Meanwhile, the continuity of operating speed in space is also taken into account.
An ACI is defined as a mathematical indicator
It is indicated that one point corresponds to a unique value of
The key idea of method lies in setting up the horizontal, vertical, and cross section alignment model, respectively, and then integrating them into the ACI description model. According to the definition of ACI and the works in [
Three variables including radius, change rate of curvature, and deflection angle of curve are considered in the horizontal alignment description model. Generally, these three variables can represent lateral force, rotation rate of the steering wheel, and deflection angle of driver’s vision. When vehicles travel on a horizontal curve with greater curvature, the worse lateral stability may be generated due to the larger centrifugal [
The relationship between each individual indices and the intermediary variable is applied to transform and unify the change laws of each index and the comprehensive index. The vertical and cross section correction model also use the same research ideas, in which speed is often taken as the intermediary variable.
In some traditional regression models, the speed at a given radius is formulated as an ordinary linear model (
In the vertical alignment description model, grade is considered as the main variable. It can be concluded that the driving safety would become worse as the grade increases no matter on downhill or uphill due to the insufficient sighting distance or speeding. From the point of the definition of the consistent relationship between
In the cross section alignment model, the five independent variables are utilized in model, including lane width, lane number, widths of right and left shoulder, and the adjustment coefficient which represents the variation of pavement width because of the transition from common road to bridge or tunnel. Generally, the interaction between adjacent vehicles along the driving direction is smaller on the wider roadway. Such driving environment also offers greater driving convenience and freedom due to a wider vision field. It indicates that wider roadway is more favorable to the traffic. In other words,
A horizontal alignment in a roadway refers to the alignment or how “straight” the roadway section is. A vertical alignment refers to a roadway’s change in elevation or the “flatness” of the roadway. With respect to the road information perceived by drivers, it is not only related to alignment itself but also involved operating speed. In this paper, the challenge is how to quantify the road alignment information and integrate the horizontal, vertical, and cross section alignment ACI into a 3D ACI description model serving for the operating speed prediction. Because people’s perception to the distance, shape, and speed of the objects in real space depends on continuous learning and experience [
It is worth mentioning that tangent is a radial ray expanded from a vanishing point in the fields of vision of drivers [
During construct ACI model, we consider the following several reasons: first, in some traditional regression models, the speed at a given radius, change rate of curvature, and deflection angle of curve are formulated as an ordinary linear model, power model, or exponential model [
Second, we found speed decreased as grade varied from downhill to uphill. So we analyze the correlation between grade and the vertical ACI using the linear regression firstly [
Third, in the findings of Harwood et al. [
Moreover, the challenge in this study is how to integrate the horizontal, vertical, and cross section alignment ACI into a 3D ACI description model. By considering alignment design features, several research findings and the cross section alignment adjustment form mentioned in Highway Capacity Manual 2010 [
After repeated trial calculation and parameters calibration, the threedimensional alignment comprehensive index description function is set up finally. The ACI description model is put forward based on the sensitivity to each alignment index as shown in
The reasons we choose these indicators are shown as follows: First, on the basis of data analysis, we studied the correlation among the single index, operating speed, and traffic safety, including length of tangent, radius of horizontal curve (curvature), curvature rate, curve length deflection angle of horizontal curve, grade, length of vertical grade, and lane width. Then, we selected the indexes which were often used to establish operating speed model in the related achievements at home and aboard. In summary, the indexes which have great influence on operating speed and safety were selected preliminarily.
Secondly, according to the characteristics of road alignment, these indexes can be divided into two categories. One category is the section design index corresponding to the milepost, mainly including radius, curvature rate, curve length deflection angle of horizontal curve, grade, lane number, lane width, and shoulder width. Another category is the indexes along the roadway, such as tangent length, curve length, length of vertical curve length, and spiral length.
The alignment comprehensive index is based on the road section, so the model of ACI mainly considers the first category indexes, and the second category indexes will be selected in the operating speed prediction model.
This study emphasizes a continuous speed prediction which is more accurate than other researches on operating speed with a single alignment index. The section speed is selected as objects through discretizing the continuously variable operating speed. Although the alignment comprehensive index and operating speed are divided into points, the operating speed on a certain section is always related to the front and rear alignments within a certain length. The speed of one point is the cumulative result of speed variation on rear alignment that has already traveled. On the other hand, a certain range of alignment ahead decides the driver’s expectation of acceleration and deceleration based on the perception of the visual information obtained at the present moment. So the influence range of front and rear alignment on current section speed should be determined.
The visual characteristics of drivers are the most important factor affecting the change in operating speed. The key step to determine the influence range of front alignment is to quantify environmental factors of visual information to a digital index, then using this digital index to analyze the influence of front alignment on operating speed.
Road alignment forms a visual sensitive area in the drivers’ view plane, generally known as fixation range [
Operating speed on a current section is the result of cumulative speed change of the reartraveled sections. The speed differences existing between the front and rear sections are induced by the acceleration and deceleration of a vehicle. Thus, the influence range of rear alignment can be approximately characterized by the acceleration and deceleration distance. According to several previous studies [
Thanks to the continuous speed profiles observed for each individual trajectory, 15th and 85th percentile speeds are, respectively, 102 km/h and 123 km/h for car compared to 69 km/h and 81 km/h for truck. Because the probability of occurrence of speed decelerating from 85th percentile to 15th percentile is generally low, it is relatively conservative and safe to consider these speed intervals as the speed differences in deceleration process. Consequently, according to the recommended deceleration of 0.9 m/s^{2} for cars and 0.35 m/s^{2} for trucks in our project report [
From the given analysis, the operating speed
In (
Speed data were collected on Shenda Highway (eightlane in two directions, design speed of 120 km/h), Shenshan Highway (sixlane in two directions, design speed of 120 km/h), and Shendan Highway (fourlane in two directions, design speed of 100 km/h, 80 km/h, and 60 km/h on different sites) for both directions in two time periods: from June to October 2013 and from March to May 2014, to establish prediction model. The testing data set used to validate the proposed model was collected on Taijiu Highway from April to May 2014.
The test sites consists of two types alignment combinations including 158 sections of a sag curve combined with a horizontal curve and 146 sections of a crest curve combined with a horizontal curve. In all cases, there exists a spiral transition between tangent and circular curve. The geometric design data were acquired from road alignment design documents. The data include radius of horizontal curve, deflection angle of horizontal curve, length of horizontal, vertical and spiral curve, grade, length of tangent, lane width, number of lane, shoulder width, and the milepost of each feature points. Table
Site geometric design characteristics.
Variable  Minimum  Maximum  Mean  Standard deviation 

Design speed (km/h)  60.00  120.00  108.32  15.62 
Horizontal curve radius (m)  400.00  3500.00  868.05  218.65 
Curve length (m)  284.22  1155.48  902.65  89.37 
Deflection angle of horizontal curve  6.51  71.12  37.22  12.36 
Length of spiral curve  135.00  1800.00  728.93  106.32 
Tangent length (m)  121.00  2245.00  1253.28  185.21 
Grade (%)  −5.40  5.00  0.63  2.57 
Length of vertical grade (m)  160.00  1300.00  776.89  165.79 
Lane number  4.00  8.00  6.75  0.93 
Lane width (m)  3.50  3.75  3.70  0.12 
Left shoulder width [hard/soft] (m)  [1.00/0.75]  [1.25/0.75]  [1.17/0.75]  [1.02/0] 
Right shoulder width [hard/soft] (m)  [3.00/0.75]  [3.50/0.75]  [3.35/0.75]  [0.56/0] 
There are several instruments for speed data collection, including Global Positioning System (GPS), radar gun, loop detector, video detection system, and infrared detector. By contrast with the features of each instruments, this study applied GPS devices placed on passenger cars and trucks to obtain the individual continuous operating speed profiles. Drivers were proved to be not biased by the presence of GPS device [
The passenger cars or trucks which would travel through the observed sections were recruited in toll stations to participant this project. All the participants were informed that speed data would be used only for research purpose; thus they were free to select their speed according to their driving habits.
The experiments were carried out during daytime, offpeak periods, in sunny day, under freeflow conditions which are typically defined as having time headways of at least 5 or 6 s [
In order to explore the operating speed prediction models, the following speed data were processed subsequently based on the initial analysis of continuous speed profile of each vehicle and the reference of data collection position proposed by Gibreel et al. [
Points were 0 m, 50 m, and 100 m on the approach tangent before the beginning of the spiral curve where drivers may change speed but not completely because of the effect of the 3D combination ahead.
Point was the start point of a horizontal curve where drivers could finish the speed selection from a tangent to transition of a curve.
Points were the middle point of tangent, spiral curve, and horizontal curve.
Point was the end point of a horizontal curve and the beginning of spiral curve.
Points were 0 m, 50 m, and 100 m on the departure tangent after the end of spiral curve where drivers may select speed according to the transition from curve to tangent.
If the length of tangent is short, the processed points were reduced correspondingly. Furthermore, the 3
The distribution characteristics of speed data are analyzed based on histogram features of overall frequency of speed sample. Normal, Weibull, Gamma, and Logistic distribution are applied to finish distribution fitting and frequency testing. Normal probability plot is applied to accomplish qualitative test and determine the preliminary distribution form. In Figure
Normal probability plot of cars. (a) Cumulative probability distribution. (b) Deviation from Normal.
First, according to the regulations of minimum and maximum radius of curve on highway, the influence weight in (
After repeated trial calculation and parameters calibration, the threedimensional alignment comprehensive index description function is set up as follows:
In order to analyze the sensitivity of each index to
Relation between single index and
It can be seen from Figure
Figure
Meanwhile, in order to further validate the sensitivity of ACI to alignment condition, K63 + 000–K83 + 000 and K63 + 300–K65 + 100 of Shendan Highway are selected to conduct relation analysis (Figures
Alignment comprehensive index variations. (a) ACI at each point. (b) Distribution of cumulated alignment comprehensive index.
From Figure
The alignment comprehensive indices are divided into two categories (i.e., I uphill and other indices and II downhill and other indices) due to the different influence degree of grade in vertical description model on operating speed.
Considering the point in integration theory, the cumulative value of the ACI in a certain length range using small spacing as a unit is equal to the integral of ACI in this length range. So during the calculation of the cumulative value of ACI, it can approximately take 1 m as a unit to calculate one ACI value and then to solve the accumulation value in a certain length range. It is worthy noting that the radius value is taken as 3000 on the tangent or on the curve with more than 3000 m radius since it may has little influence on driving behavior. Some operating speed data and the cumulative value of I alignment comprehensive index are shown in Table
Test data for cars.
Milepost  Grade (%) 


Operating speed (km/h) 

K65 + 985  1.47  1106.47  1252.86  111.82 
K66 + 049  1.47  1103.77  1354.53  108.15 
K66 + 095  1.97  1125.53  1393.17  114.22 
K66 + 139  2.61  1228.09  1419.28  109.14 
K66 + 281  4.64  1705.29  1203.53  110.35 
K66 + 423  4.90  1890.35  1488.18  98.94 
K66 + 465  4.90  1878.74  1617.33  98.76 
K66 + 513  4.90  1740.86  1624.97  92.63 
K69 + 105  1.09  1099.20  1162.30  113.32 
K69 + 263  2.42  1191.36  1330.58  113.03 
K69 + 565  5.00  1778.82  1160.20  106.55 
K69 + 664  5.00  1790.49  1554.80  101.46 
K69 + 752  5.00  1646.05  1626.90  97.16 
K69 + 829  4.68  1642.82  1585.98  99.02 
K69 + 900  4.20  1817.33  1556.34  96.14 
K69 + 974  3.71  2129.76  1276.69  92.91 
K70 + 119  3.40  2114.19  1183.25  94.90 
K76 + 370  1.50  1866.46  1056.80  103.44 
K76 + 469  1.50  1908.52  981.25  105.93 
K76 + 575  1.17  1718.65  1112.75  108.10 
K76 + 645  0.01  1456.53  1062.38  104.18 
K77 + 555  1.62  1090.20  1346.95  110.63 
K77 + 592  1.62  1096.66  1336.88  112.39 
K79 + 366  1.39  1112.55  1061.81  114.87 
K79 + 657  0.60  1450.03  1240.57  110.63 
K79 + 725  0.60  1570.39  1240.58  105.58 
K79 + 808  0.60  1644.95  1196.11  106.98 
K79 + 885  0.60  1654.10  1087.42  109.76 
K79 + 959  0.60  1654.24  952.85  107.44 
Through the analysis of operating speed variation with alignment index and coefficient calibration, the following operating speed prediction models which best fit the criteria of the regression analysis are established. The predicting speed for cars can be calculated by using (
Cars
Trucks
Four statistical validation indicators are applied to evaluate the effectiveness of prediction model including Goodness of Fit test,
Statistical test of speed prediction model.
Goodness of fit test  

Multiple 
0.8077  

0.6523  
Adjusted 
0.6330  
Standard error  0.0362  
Observed value  39  






df  SS  MS 

Significance 

Regression analysis  2  0.0887  0.0444  33.7746 


Residual  36  0.0473  0.0013  
Total  38  0.1360  






Coefficients  Standard error 


Low limit  Upper limit  
Intercept  4.9490  0.0381  129.9758 

4.8718  5.0262 

−0.0001 

−7.2385 

−0.0002 




−2.75481  0.0092  −0.0001 

Residual analysis.
In [
In the following section, the results of the proposed prediction models are compared with the models proposed by Morris and Donnell [
To consider different position of operating speed according to the GSAH model and proposed models in this paper, we finally compare the prediction results of three models at 480 points on 3D alignment with length of 12 km. From Table
Comparison results of three models.
Vehicle type  Criteria  Estimation methods  

Proposed model  GSAH  Morris  
Cars  MAE  2.69  4.45  10.29 
RMSE  3.54  6.03  13.67  
MARE (%)  4.38  6.98  10.28  
SD  10.7  15.32  18.21  


Trucks  MAE  3.57  6.89  16.32 
RMSE  7.10  10.22  21.04  
MARE (%)  4.93  6.34  15.39  
SD  8.85  14.17  17.33 
Comparison between predicted speed and actual speed. (a) Comparison for cars. (b) Comparison for trucks.
With respect to the consideration of vehicle dynamic properties into the model, operating speed on a current section is the result of cumulative speed change of the reartraveled sections. The speed differences existing between the front and rear sections are induced by the acceleration and deceleration of a vehicle. Thus, the influence range of rear alignment can be approximately characterized by the acceleration and deceleration distance which is related to vehicle dynamic properties. Based on each vehicle trajectory and recommended deceleration rate, the influence range of the rear alignment can be determined as 200 m.
The related researches on how to quantify driver’s visual information to a digital index are lacking. The deceleration rate of vehicles in China has significant differences from that in worldwide. At the same time, the influence range of rear alignment is also determined based on data analysis. Thus, we did not discuss more about these two considerations. In the case study, the other two models did not consider the influence of visual characteristics and vehicle dynamics; they just establish the relation between operating speed and alignment indexes. We think that the comparison results may reflect the accuracy of our model with these considerations.
One significant limitation in previous research work on highway alignment design consistency is that the existing operating speed prediction models are established mainly on 2D alignment or single index. Particularly when the road is characterized by different alignment combinations, the models may be inaccurate. Only some feature points, such as the middle point of a horizontal curve or the end point of a grade can be predicted correspondingly. Therefore, the speed variation of passenger cars and trucks along each point of the road is studied using the actual test data and alignment indices.
The achievements of this study are twofold. The first one is the threedimensional alignment comprehensive index description functions. These functions select curvature, change rate of curvature, curve angle, grade, and lane width as variables rather than a single index. Based on the principle of spatial geometry and the design characteristics of road alignment, the alignment description model is established by taking the horizontal and vertical indices as the primary models and the crosssectional index as the correction model. The second process is to set up the relationship between alignment comprehensive index and operating speed for continuous prediction. During the establishment of this model, the visual requirement of driver and the different features of acceleration and deceleration of vehicles are also considered. This modeling procedure makes it possible to predict a reliable and continuous operating speed profile at each point along the alignment and to significantly improve the performance of consistency alignment design and safety evaluations. The prediction performance of the proposed model demonstrates its higher accuracy when compared with other models using the actual observed data.
Owing to the test data mainly collected on highways (bidirectional four to eight lanes) in plain area in China, the models reported in the paper can be used to predict continuous operating speed for passenger cars and trucks on the condition of threedimensional alignment indexes along roadway which can be obtained. However, the models cannot predict the operating speeds in mountainous area or other type roads accurately. Application of the model outside China would require a new calibration based on local speed surveys because of the differences in driver behavior, roadway systems, and vehicle performances. Although design speed and speed limit have effect on operating speed to some extent, these factors are not considered in models. Substantially, highway alignment is a threedimensional curve in Euclidean space. The interaction mechanism of multiple alignment indices on speed is very complicated. Hence, the methods of alignment comprehensive modeling are required further study. With the development of automotive technology, fusion of data collected from mobile and fixed sensors [
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The project is supported by National Natural Science Foundation of China (51308059 and 71701215), China Postdoctoral Science Foundation (2015M581585), the Special Fund for Basic Scientific Research of Central Colleges, Chang’an University (310822152007), and Western Transportation Science and Technology Project of Ministry of Communications (2012318361110).