Evaluation of Single Lane Parallel-Type Entrance Ramp Terminal Using Vehicle Dynamics Model for Freeways

A parallel-type design was used for the acceleration lane. A simple horizontal curve was used at the beginning of the roadway, connecting to the acceleration lane and a 90-meter taper. Acceleration lane lengths were equal to the recommended minimum acceleration lengths of the AASHTO Green Book, shown in Table 1. The curve radius at the beginning of the roadway was selected as the minimum radius proposed by AASHTO for a design speed of 60 km/h. For grades, values of 0, ±3, and ±5 percent were used to analyze the adjusted grade factors shown in Table 2. The friction coefficient factor between the tire and the pavement was assumed to be 0.9.

Driver controls in the software have four parts: speed control, brake control, gear-shifting control, and steering control. The truck travels at given speeds. The driver starts to change lane before the taper without braking and enters the mainline of the freeway. Initial speeds of 40, 50, and 60 km/h at the beginning of the acceleration lane and the merge speeds of 74, 88, and 92 km/h were employed at the end of the acceleration lane before starting the taper. In addition, automatic gear-shifting control was used in this study.

The proposed acceleration length of AASHTO is based on experimental studies, and the vehicle’s dynamic characteristics are not considered in the results. Heavy vehicles have a lower acceleration rate than passenger cars for merging into the highway. Therefore, this study tried to investigate the acceleration lane length for heavy trucks by considering the dynamic behavior of different heavy trucks.

Four types of trucks with weight to power ratios of 61, 67, 86, and 108 kg/kW were selected for simulation and their specifications are shown in Table 3. The dimensions of the 108 kg/kW truck are illustrated in Figure 2. The 108 kg/kW truck’s sprung mass is 4455 kg.

Figure 2.

After identifying vehicles’ characteristics, road geometry, and driver behavior, the inputs were defined in the TruckSim software. One of the advantages of TruckSim is having the capability to animate the vehicle performance to create a clear perception of simulation for users. TruckSim is universally the preferred tool for analyzing vehicle dynamics, developing active controllers, calculating a truck’s performance characteristics, and engineering next-generation active safety systems. TruckSim provides open-loop and closed-loop driver models with advanced features to help engineers quickly discover a truck’s limitations or its optimal path through complex maneuvers. Parameters, such as geometric roadway, design, inertial properties, and vehicle characteristics, are user-defined. Although vehicle dynamics simulations have long been standard in vehicle design, highway engineers rarely use them in roadway designs [15].

After inserting data on road geometry, driver behavior, and vehicle characteristics, the simulation process’s final step is plot definition. The defined plots in this study are the longitudinal acceleration and longitudinal speed versus station. Since outputs are represented as stations in TruckSim software, longitudinal distance or x is calculated by Equation 1. Finally, linear regression analysis was considered for the simulation’s outcome after the simulation process to make them more transparent and valuable in highway design. Therefore, models are based on the vehicle’s initial and merge speed, grade, and truck weight to power ratio. Statistical Package for the Social Sciences (SPSS) software was used to analyze the results.

S_i: Station i, S_i-1: Station i-1, X_i: Horizontal distance point i, X_i-1: Horizontal distance point i-1, Y_i: Vertical distance point i, Y_i-1: Vertical distance point i-1

Longitudinal speed versus station of the 61 kg/kW truck is shown in Figure 3. According to Figure 3, the speed controller software results are shown in Diagram 1. In this test, the initial speed was chosen to be 60 km/h, and the merge speed was equal to 74 km/h at level grade. Diagram 2 shows the speed of the vehicle and their responses to acceleration. As shown in Figure 3, curve number 2 at the end of the acceleration lane reached the input merge speed in the software, indicating that the acceleration lane’s length at these initial and merge speeds is sufficient for a truck at 61 kg/kW.

Figure 3.

The output of the 108 kg/kW truck is shown in Figure 4. The initial speed was chosen to be 60 km/h, and the merging speed was equal to 74 km/h at level grade. According to Figure 4, Diagram 2 meets with a lag of curve 1, which means that the speed of 108 kg/kW truck does not reach the merge speed during the designed length. In this output, by converting the length shown in Figure 4 to the horizontal distance using Equation 1, it was determined that the heavy vehicle reached 74 km/h after 63 meters at the end of the acceleration lane. As a result, by adding this length to the design length of 205 meters, the minimum length of 268 meters was calculated.

Figure 4.

By using the longitudinal speed diagrams versus the station and converting them to the longitudinal distance, the results are presented in Table 4 to Table 8.

According to Table 4, when the input speed is 60 km/h and the merge speed is 74 km/h, 61 and 67 kg/kW trucks can accelerate along the design length. In Figure 5, obtained acceleration lane lengths at level grade and the minimum lengths of AASHTO for all trucks are shown.

Figure 5.

In Table 5, related to the 3% upgrade, the 108 kg/kW and 86 kg/kw trucks have been unable to achieve the merge speed and have had a poor performance. The 108 kg/kW truck has a poor performance on upgrades, and added length for this truck was too long. For example, at an initial speed of 50 km/h and a merge speed of 74 km/h, the truck reached a distance of 3000 meters to a merge speed of 74 km/h.

In Table 6, on a 3% downgrade, all heavy trucks over 90 meters of taper reached the desired speed when the initial speed of 50 and 60 km/h and the merge speed of 74 km/h were considered. Only 61 and 67 kg/kW trucks could accelerate during taper at an initial speed of 50 km/h and merge speeds of 88 and 92 km/h. The obtained acceleration lane lengths at a 3% downgrade and minimum lengths of AASHTO for all trucks are shown in Figure 6.

Figure 6.

Trucks with weight to power ratios of 86 and 108 kg/kW, at a 5% upgrade cannot accelerate to the desired speed (Table 7). Only 61 and 67 kg/kW trucks in this grade accelerated at a merge speed of 74 km/h with a length of 1.8 to 3.9 times greater than the recommended acceleration lane length of the AASHTO Green Book.

In a 5% downgrade, all trucks could reach the merge speed along the 90 meters taper when the initial speed was assumed to be 60 km/h. Also, all trucks except 108 kg/kW at an initial speed of 40 km/h could accelerate during the taper when the merge speed of 74 km/h was considered (Table 8). The acceleration lane lengths at 5% downgrade and the minimum lengths of AASHTO for all trucks are shown in Figure 7.

Figure 7.

The speeds that four types of vehicles reached at the end of the acceleration lane in level grade are presented in Table 9. By considering the acceleration lane length according to the AASHTO Green Book, the speeds reached by trucks in the TruckSim software are shown in Figure 8.

Figure 8.

One of the design problems is to assume the constant acceleration rate of vehicles; the TruckSim software can simulate vehicles with variables of the acceleration rates. The output is related to the longitudinal acceleration for the 61 kg/kW truck, shown in Figure 9. As shown in Figure 9, the acceleration rate of the truck is not constant.

Figure 9.

Given the unsteady acceleration rate of the vehicle’s output, the maximum longitudinal acceleration rate experienced by the vehicle through the acceleration lane is given in Table 10. The 61 kg/kW truck on downgrades had a higher acceleration rate than other vehicles. The acceleration rate at downgrades is about 1.6 times higher than at upgrades. Lower initial speeds have a higher acceleration than other scenarios. Maximum longitudinal acceleration rates are shown in Figure 10.

Figure 10.

Regression models are a valuable tool for describing the relations between different variables. These models are widely used for descriptions, estimations, predictions, and control. In general, a multiple linear regression model can be considered as follows:

Where Y is the response variable and X₁… X_k is an independent variable. Also, the values a₁… a_k are regression coefficients, and ε is known as residual.

In the present study, four variables were considered as the independent variables, including I.S.: Initial speed (km/h), M.S.: Merge speed (km/h), W.P.: Weight to power ratio (kg/kW), and G: Grade (%). Also, A_cclength: Acceleration lane length (m) as the response variable. By using the Equation 2, the following multiple regression model with R² =0.742 to the data was fitted:

The validity of some assumptions in the regression model should be checked. If they are adequate, the regression model is valid, and other statistical inferences such as prediction and control can be made. The basic assumptions in regression analysis are as follows:

Residuals are normally distributed; 2. Residuals are uncorrelated; 3. Residuals have the constant variance; 4. Independent variables are uncorrelated. In the following, all of these assumptions are checked:

Normality of residuals: A histogram and PP plot can be used for the graphical checking of this assumption. Figure 11 gives the histogram and PP plot for the regression model in Equation 3. This figure can verify the normality of residuals. Another theoretical way to check the normality of the residuals is the Kolmogorov-Smirnov test. Since the “p-value” is 0.056, which is more than 0.05, the normality of the residuals is accepted.

Figure 11.

Uncorrelated residuals: The Durbin-Watson statistic can be used to check this assumption. This assumption is verified when the Durbin-Watson statistics are placed within the range 1 to 2. For Equation 3, this assumption is acceptable because the Durbin-Watson statistic is 1.283

Homoscedasticity (same variance) of residuals: The Pagan-Godfrey test is used to check of homoscedasticity for the variance of residuals. For this aim, another regression model should be fitted between the square of residuals as a response variable and the independent variables of the original model. When nR′2<νχν2, this assumption is accepted, where (n) is the number of observations, ν is the degree of freedom for the model, χν2 is the value of the chi-square distribution, and R^′2 is R² of this model. In Equation 3, as 132×0.277<4×9.49, we accept this assumption.

Uncorrelated independent variables: the non-correlation of the independent variables should be checked by T=1-R², tolerance measure, or Variance Inflation Factor (VIF=T1) where T>0.1 (VIF<10). According to Table 11, we accept this assumption. Table 11.

Results in assumption 4

Variable	Tolerance	VIF
Grade	0.902	1.109
Weight to Power ratio	0.941	1.063
Merge Speed to Initial Speed	0.759	1.318

The acceleration lane length of the 108 kg/kW truck at level grade is compared with other studies. For a more accurate comparison with previous studies, given the AASHTO’s Green Book and NCHRP report 505, an initial speed of 35 km/h was considered for a 40 km/h entrance curve design. The results of the TruckSim software output are shown at an initial speed of 35 km/h and merge speeds of 74 and 88 km/h (Table 12).

According to Table 12, acceleration lane lengths for 108 kg/kW of these two tests differed by 1% compared to those found in NCHRP Report 505. In the NCHRP 505 report, the Truck Speed Profile Model (TSPM) was used, in which the acceleration lane length was 1.8 times greater than the recommended length in AASHTO Green Book.

Table 13 summarizes the compared acceleration rate in this study and in the AASHTO green book.

It should be noted that the acceleration rates reported in this study are the highest accelerations that vehicles experience during the acceleration lane. Also, the values obtained from the AASHTO Green Book results are based on the initial speed, merge speed, and acceleration lane length, which assumes that acceleration rate values are constant and based on kinematic equations. The initial design speed for 40, 50, and 60 km/h were considered 35, 42, and 51 km/h, respectively. A more accurate comparison is shown in Table 13. The results of the research are also presented with the mentioned initial speeds. The acceleration rates reported in this study are affected by vehicle performance, which has an internal resistance, such as internal friction and gradient resistance. In an observational study for an initial speed of 51 to a merge speed of 74 km/h (NCHRP report 730), the average acceleration rate was calculated at about 1.75 km/h/s. In this study, the acceleration rate for a 109 kg/kW truck reaches about 1.58 km/h/s.

The comparison of NCHRP report 505 and the results obtained in this study are shown in Table 14. In this study, at the initial speed of 35 km/h and the merge speed of 74 km/h, the heaviest vehicle that could accelerate was the 67 kg/kW truck.

Table 15 compares the highway research program results at the 2% upgrade and this study at the 3% upgrade. As shown in Table 15, in the 3% upgrade, the lightest truck in this research (61 kg/kW) could not accelerate at the recommended length of AASHTO’s Green Book except in the 51 to 74 km/h test by adding a 90-meter taper. In NCHRP report 505, a truck with a maximum weight to power ratio of 52 kg/kW at a 2% upgrade can reach the merge speed at the recommended length of AASHTO’s Green Book. The difference between this study and the NCHRP505 studies is due to considering the vehicle’s internal deterrent forces in the model.