## 4 Calculation and Results
### Execution of Lane-Change Distance \(S_{3}\)  
Upon the emergence of an available gap within the curb lane of the mainline, vehicles initiate lane-changing maneuvers. Building upon the analysis, it becomes evident that the adapted hyperbolic tangent function lane-changing model more accurately captures the lane-changing patterns exhibited by vehicles navigating the interchange diversion area, particularly concerning the RTRL. At this stage, the mathematical representation of the vehicle's lateral vehicle speed, lateral acceleration, and rate of change of lateral acceleration is encapsulated by Equation (16). \[ \begin{array}{l}x\prime(t)=\frac{W\tau V_{d}}{2L\tanh(\tau/2)}\left[\tanh^{2 }\left(\frac{\tau V_{d}}{L}t-\frac{\tau}{2}\right)-1\right]\\ x^{\prime\prime}(t)=\frac{W}{\tanh(\tau/2)}\left(\frac{\tau V_{d}}{L}\right)^{ 2}\left[\tanh^{3}\left(\frac{\tau V_{d}}{L}t-\frac{\tau}{2}\right)-\tanh\left( \frac{\tau V_{d}}{L}t-\frac{\tau}{2}\right)\right]\\ x^{\prime\prime\prime}(t)=\frac{W}{\tanh(\tau/2)}\left(\frac{\tau V_{d}}{L} \right)^{3}\left[\tanh^{4}\left(\frac{\tau V_{d}}{L}t-\frac{\tau}{2}\right)-1 \right]\end{array}  \tag{16}\]  
Considering passengers' comfort throughout the vehicle's lane-change maneuver, the lateral acceleration and its rate of change are required to conform to the constraints delineated in Equations (17) and (18). \[x^{\prime\prime}(t)\leq\frac{2\sqrt{3}W}{9\tanh(\tau/2)}\left(\frac{\tau V_{d }}{L}\right)^{2}\leq a_{\max} \tag{17}\]  
\[x^{\prime\prime\prime}(t)\leq\frac{W}{\tanh(\tau/2)}\left(\frac{\tau V_{d}}{L} \right)^{3}\leq\sigma_{\max} \tag{18}\]  
Integrating the aforementioned equation, the expression for the length of the lane change is detailed in Equation (19) as follows:  
\[S_{3}=\max\left\{\sqrt{\frac{2\sqrt{3}W}{9a_{\max}\tanh(\tau_{\max}/2)}}\tau_{ \max}V_{d},\sqrt[3]{\frac{W}{\sigma_{\max}\tanh(\tau_{\max}/2)}}\tau_{\max}V_{ d}\right\} \tag{19}\]  
where represents the maximum value of lateral acceleration (m/s\({}^{2}\)); \(\tau_{\max}\) signifies the maximum value of the emergency coefficient of lane changing; \(W\) is the lateral displacement (m) during the lane-changing process, taking 3.75 m; \(V_{d}\) symbolizes the vehicle speed (m/s) in the process of lane changing, takes the mainline diversion vehicle speed. \(\sigma_{\max}\) denotes the maximum value of the rate of change of lateral acceleration (m/s\({}^{3}\)). Various countrieshave adopted distinct values within their road specifications. For instance, the United States prescribes 0.61 m/s\({}^{3}\), West Germany designates 0.5 m/s\({}^{3}\), while the United Kingdom typically utilizes 0.305 m/s\({}^{3}\). In Japan, the range varies from 0.5 m/s\({}^{3}\) to 0.75 m/s\({}^{3}\), and in China, the usual range is 0.5 m/s\({}^{3}\) to 0.6 m/s\({}^{3}\), with the discretion exercised in actual application. By consolidating the findings from the aforementioned research, this paper computes the intermediate value of the provided range, adopting 0.6 m/s\({}^{3}\) as the value \(\sigma_{\text{max}}\). _The Green Book_ outlines the maximum permissible superelevation as either 8% or 10%, while the _Intersections Guidelines_ specify that for design speeds exceeding 80 km/h, the superelevation should not exceed 3% and 4%, respectively. These specifications form the foundation for establishing both the general minimum radius value of the mainline curve and limit of the minimum radius value of the interchange. Thus, accounting for the most unfavorable scenario, the calculation of lateral acceleration utilizes 4% of the reverse superelevation by Equation (20), and the resultant value is considered as the maximum lateral acceleration:  
\[a=(\mu-i_{h})g \tag{20}\]  
where \(\mu\) denotes the side friction factors, it is valued according to _Technical Standard of Highway Engineering_; \(i_{h}\) represents cross slope; \(g\) signifies gravitational constant, 9.8 m/s\({}^{2}\).