## 3 Methodology
### Stability Analysis  
Stability analysis is vital for car-following models. [PERSON] et al. [44] utilised frequency-domain scanning methods to analyse the stability of driver characteristics and controller gains. [PERSON] et al. [45] investigated the stability of car-following models through frequency-domain analysis tools and stochastic process theory, while [PERSON] et al. [46] derived a characteristic equation to describe the evolution of traffic density waves via nonlinear analysis. In this paper, we analyse the stability of the MIDM-SPF model using linear stability theory and simplify (8) as follows:  
\[\frac{dv_{n}(t+t_{d})}{dt}=f_{n}\big{(}s(t),v_{n}(t),\Delta v(t),a_{n-i}(t),a_{ n+j}(t),v_{n-i}(t),v_{n+j}(t),\Delta x_{i},\Delta x_{j}\big{)}, \tag{10}\]  
We assume that \(\overline{s}\) represents the average headway between neighbouring vehicles in uniform flow and \(\overline{v}\) denotes the vehicles' velocity in this flow. Initially, all vehicles travel with identical headway and velocity; hence, the position of vehicles in uniform flow can be expressed as follows:  
\[\overline{x}_{n}(t)=(N-n)\overline{s}+\overline{v}t,\ n=1,2,3\ldots,N, \tag{11}\]  
Assuming \(y_{n}(t)\) is the disturbance generated by vehicle \(n\) at time \(t\), we add this disturbance to (11). \[y_{n}(t)=ce^{\omega a_{k}n+zt}=x_{n}(t)-\overline{x}_{n}(t),\ y_{n}(t)\to 0, \tag{12}\]  
where \(a_{k}=\frac{2\pi k}{N}(k=0,1,\ldots,N-1)\) is a constant. Taking the second derivative of both sides of (12), we obtain  
\[y_{n}^{\prime\prime}(t+t_{d})=x_{n}^{\prime\prime}(t+t_{d})-\overline{x}_{n}^ {\prime\prime}(t+t_{d})=\frac{dv_{n}(t+t_{d})}{dt}, \tag{13}\]  
Substituting (10) into (13), we obtain  
\[y_{n}^{\prime\prime}(t+t_{d})=f_{n}\big{(}s(t),v_{n}(t),\Delta v(t),a_{n-i}(t),a_{n+j}(t),v_{n-i}(t),v_{n+j}(t),\Delta x_{i},\Delta x_{j}\big{)}, \tag{14}\]  
Linearize Equation (14) and substitute \(y_{n}(t)=ce^{\omega a_{k}n+zt}\) and \(y_{n}^{\prime}(t)=zce^{\omega a_{k}n+zt}\) into the difference equation. Rearrange terms, simplify, and expand in a power series to obtain the stability condition as shown in Equation (15). Here, traffic flows become unstable when \(z_{2}<0\) in response to small perturbations and stabilise when \(z_{2}>0\). For a detailed proof of the stability analysis, please refer to Appendix A, Appendix A.2. \[ \begin{split}\frac{t_{d}}{2}<-\frac{1-f_{n}^{\prime\prime}n-i}{f _{n}^{\prime\prime}n+j}&+\frac{f_{n}^{\prime\prime}n-i}{f_{n}^{ \prime\prime}n+j}{f_{n}^{\prime\prime}n+j}+\frac{f_{n}^{\prime\prime}n-i}{f_{n }^{\prime\prime}n+j}{f_{n}^{\prime\prime}n+i}+\frac{f_{n}^{\prime\prime}n-i}{f _{n}^{\prime\prime}n+i}f_{n}^{\prime\prime}}{f_{n}^{\prime\prime}n+i}+\\ \frac{\big{(}f_{n}^{\prime\prime}n+f_{n}^{\prime\prime}n-i}{f_{n} ^{\prime\prime}n+j}\big{)}&\frac{\big{(}f_{n}^{\prime\prime}n+f_{n }^{\prime\prime}n+j}{2\big{(}f_{n}^{\prime\prime}n+f_{n}^{\prime\prime}n-i} \big{)}\big{(}f_{n}^{\prime\prime}n+f_{n}^{\prime\prime}n+j}{f_{n}^{\prime \prime}n+i}\big{)},\\ \end{split}  \tag{15}\]  
Combining Equation (8) yields results for \(f_{n}^{\prime\prime},f_{n}^{\prime\prime},f_{n}^{\prime\prime},f_{n}^{\prime \prime}n,f_{n}^{\prime\prime},f_{n}^{\prime\prime},f_{n}^{\prime\prime},f_{n }^{\prime\prime},f_{n}^{\prime\prime},f_{n}^{\prime\prime},f_{n}^{\prime \prime}\) and \(f_{n}^{\Delta x_{j}}\). Substituting these results into Equation (15) gives the stability condition as shown in Equation (16). Following initialisation, the AOA employs the math optimiser accelerated (_MOA_) for numerical calculations, determining whether to enter the exploration (exp) or development (dev) phase:  
\[MOA(Iter)=Iter\times\frac{M_{max}-M_{min}}{Max_{Iter}}+M_{min}, \tag{18}\]  
\[ \begin{cases}\gamma_{1}>MOA,\ \exp\\ \gamma_{1}\leq MOA,\ \ \text{dev},\end{cases}  \tag{19}\]  
where \(Iter\) signifies the iteration number, \(a\) represents the maximum iterations, with \(M_{max}\) and \(M_{min}\) being the _MOA_'s maximum and minimum values, respectively. \(\gamma_{1}\) is a random number in the range [0, 1].