# A Vehicle Conflict Risk Identification Method Based on an Improved Intelligent Driver Model
## 2 Methodology
### Improved Intelligent Driver Model  
The vehicle-following model is a valuable tool for studying traffic flow characteristics in depth and has been extensively applied in traffic risk research. In 2002, [PERSON] et al. [26] introduced the intelligent driver model (IDM), which utilizes a minimal set of meaningful parameters to ensure consistency with empirical observations. The model's parameters are easily calibrated and capable of capturing real-time dynamic changes in vehicle position and speed. Subsequently, [PERSON] et al. [36] enhanced the traditional IDM by incorporating additional factors, such as non-negative expected distance and driver reaction time. The specific formula is as follows:  
\[\dot{v_{\alpha}}(s_{\alpha},v_{\alpha},\Delta v_{\alpha})=a\left[1-\left(\frac {v_{\alpha}}{v_{0}^{\alpha}}\right)^{\delta}-\left(\frac{s_{\alpha}^{*}}{s_{ \alpha}}\right)^{2}\right] \tag{3}\]  
\[s_{\alpha}^{*}=s_{0}^{\alpha}+s_{1}^{\alpha}\sqrt{\frac{v_{\alpha}}{v_{0}^{ \alpha}}}+T_{\alpha}v_{\alpha}+\frac{v_{a}\Delta v_{\alpha}}{2\sqrt{a_{\alpha} b_{\alpha}}} \tag{4}\]  
Here, \(\alpha\) represents the vehicle index, \(\alpha\) denotes the vehicle speed, \(\Delta v\) is the speed difference between the current vehicle and the preceding vehicle, \(v_{0}\) is the desired speed, \(s\) is the distance between the current vehicle and the preceding vehicle, \(s^{*}\) is the desired distance under the current driving state, T is the reaction time, a is the maximum acceleration, b is the comfortable deceleration, \(\delta\) is the acceleration exponent, \(s_{0}\) is the static safety distance parameter, and \(s_{1}\) is the speed-dependent safety distance parameter. When in free traffic flow, the distance between the vehicle and the preceding vehicle is large, i.e., \(\left(\frac{s_{\alpha}^{*}(v_{\alpha},\Delta v_{\alpha})}{s_{\alpha}}\right)^{ 2}\to 0\), and the acceleration is determined by the free-flow state, causing the vehicle speed to gradually converge toward the desired speed \(v_{0}\). However, when the IDM model is applied to mixed traffic flow conditions, it tends to generate a safe distance to prevent conflicts or collisions. This often results in an unrealistically large deceleration, which undermines the model's overall effectiveness. To address this limitation, this study proposes an improved model, as illustrated in Equation (5). \[\frac{\partial^{2}}{\partial t^{2}}x_{\alpha}=a_{\alpha}\left[1-\left(\frac{ \frac{\partial}{\partial t}x_{\alpha}}{v_{\alpha}^{0}}\right)^{\delta}-\frac{ (s_{\alpha}^{*})^{2}}{\epsilon^{2}+s_{\alpha}^{2}}\right]+(h_{f})_{i}a_{\alpha -1}\left[1-\left(\frac{\frac{\partial}{\partial t}x_{\alpha-1}}{v_{\alpha-1} ^{0}}\right)^{\delta}-\frac{(s_{\alpha-1}^{*})^{2}}{\epsilon^{2}+s_{\alpha-1 }^{2}}\right] \tag{5}\]  
The formula incorporates direct safety factors related to both the vehicle itself and the rear vehicle. Here, \(h_{f}\) represents the driver factor, which is determined based on the driver's state and is set to 1 in this study [37]; then, the influence of the front and rear vehicles can be considered to avoid vibration in speed and acceleration. Meanwhile, \(\epsilon\) denotes the braking variable. When the vehicle begins to decelerate or stop, the speed of the rear vehicle also gradually decreases. When it comes to a sufficiently low speed, a critical condition arises, as demonstrated in Equations (6) and (7). It is evident that \(\epsilon\) is directly influenced by the static safety distance and the state of the vehicle itself.