# Biform Game Approach to Strategy Optimization of Autonomous Vehicle Lane Changes on Highway Ramps
## 2 Lane-Change Model for Autonomous Vehicles on Highways
### Analysis of Relevant Parameters  
When an autonomous vehicle approaches the ramp section and becomes aware of the need to change lanes, the probability of executing a forced lane change is influenced by the distance to the exit ramp. The lane change process of the autonomous vehicle is illustrated in Figure 1a-c:  
We represent the probability of a forced lane change through a uniform distribution function \(\mu_{i}\), as shown in Equation (1):  
\[\mu_{i}=\left\{ \begin{array}{ccc}0,&\mathrm{d}_{i}\geq\mathrm{d}_{0}\\ \frac{\mathrm{d}_{i}}{\mathrm{d}_{0}},&\frac{\mathrm{d}_{0}}{2}<\mathrm{d}_{i }<\mathrm{d}_{0}\\ \frac{\mathrm{d}_{0}}{\mathrm{d}_{i}},&0<\mathrm{d}_{i}\leq\frac{\mathrm{d}_{0 }}{2}\\ \mathrm{Pointless},&\mathrm{d}_{i}=0\end{array} \right. \tag{1}\]  
In this equation, \(\mathrm{d}_{0}\) represents the safe lane-change distance for autonomous vehicles from the entrance of the ramp and \(\mathrm{d}_{i}\) is the instantaneous actual distance of the lane-changing vehicle (including the lane-changing vehicle and nearby target lane vehicles during the lane change) from the ramp entrance. The upcoming ramp segment is divided into three regions: when \(\mathrm{d}_{i}\geq\mathrm{d}_{0}\), the lane-changing vehicle is in the third region and has not entered the ramp segment, and does not take forced lane-change measures; when \(\frac{\mathrm{d}_{0}}{2}<\mathrm{d}_{i}<\mathrm{d}_{0}\), the lane-changing vehicle enters the second region, entering the latter half of the ramp segment and the corresponding probabilities for forced lane-change measures should be applied; when \(0<\mathrm{d}_{i}\leq\frac{\mathrm{d}_{0}}{2}\), the lane-changing vehicle enters the first region, that is, the first half of the ramp segment, and forced lane-change measures must be taken to  
Figure 1: Automatic lane-change process of a vehicle. strengthen the acceleration trend; and when \(d_{i}=0\), the vehicle has missed the opportunity to change lanes and there is no need to change lanes, making this parameter meaningless. During the lane change process, the autonomous lane-changing vehicle \(i\) interacts with vehicles in the target lane, where the interaction between the lane-changing vehicle \(i\) and the vehicles ahead and behind in the target lane is most pronounced. When the lane-changing vehicle \(i\) begins to change to the target lane, the vehicle ahead in the target lane \(i-1\) compares the actual headway distance \(\Delta d\) of the lane-changing vehicle \(i\) with the distances between the vehicles ahead and behind \(d_{safe}\) to determine the speed change (acceleration, constant speed, or deceleration), where \(\Delta d\) is given by Equation (2):  
\[\Delta d=d_{i-1}-d_{i} \tag{2}\]  
During the lane change of the autonomous lane-changing vehicle, the degree of influence from it and the vehicles ahead and behind in the target lane continuously changes over time. Based on the degree of influence, corresponding measures are taken for the vehicle, such as acceleration, deceleration, or maintaining the original speed, with the degree of change represented by a probability function \(\theta_{i}\), as shown in Equation (3):  
\[\theta_{i}=\left\{ \begin{array}{cl}0,&\Delta d_{(i-1,i)}>d_{safe}\\ \frac{\Delta d_{(j-1,i)}}{d_{safe}-\Delta d_{(i-1,i)}},&\Delta d_{(i-1,i)}<d_{ safe}\\ 1,&\Delta d_{(i-1,i)}=d_{safe}\end{array} \right.