# A Microscopic Traffic Model Considering Driver Reaction and Sensitivity
## 3 Stability Analysis
### Theoretical Stability
In unstable traffic, \(k\to 0\) so there is a negligible delay between changes in flow (traffic waves) [45]. Using Taylor series, \(Y(k)\) and \(X(k)\) can be approximated for a small delay, i.e., \(k\to 0\), as  
\[X(k)=-f_{v}-if_{\Delta v}k, \tag{27}\]  
\[Y(k)=if_{s}k+\frac{f_{s}}{2}k^{2}. \tag{28}\]  
At equilibrium [45]  
\[f_{s}=-v_{e}^{\prime}(s_{e})f_{v}. \tag{29}\]  
where \(v_{e}^{\prime}(s_{e})\) is the gradient of the equilibrium velocity (speed) with respect to the equilibrium distance headway. Substituting (29) in (28) gives  
\[Y(k)=-iv_{e}^{\prime}(s_{e})f_{v}-\frac{v_{e}^{\prime}(s_{e})}{2}f_{v}. \tag{30}\]  
Let  
\[ \begin{array}{l}X(k)=a_{1}+a_{2}k,\\ Y(k)=b_{1}k+b_{2}k^{2},\end{array}  \tag{31}\]where  
\[ \begin{array}{c}a_{1}=-f_{\
u},\\ a_{2}=-if_{\Delta\
u},\\ b_{1}=-iv_{e}^{\prime}(s_{e})f_{\
u}=iv_{e}^{\prime}(s_{e})a_{0},\\ b_{2}=-\frac{v_{e}^{\prime}(s_{e})}{2}f_{\
u}=\frac{v_{e}^{\prime}(s_{e})}{2}a_ {0}.\end{array}  \tag{32}\]  
Employing a Taylor series approximation, the square root in (26) becomes  
\[\sqrt{1-\frac{4Y(k)}{X^{2}(k)}}=1-\frac{2Y(k)}{X^{2}(k)}-\frac{2Y^{2}(k)}{X^{4 }(k)}. \tag{33}\]  
which results in  
\[\lambda_{2}=\frac{-Y(k)X^{2}(k)-Y^{2}(k)}{X^{3}(k)}. \tag{34}\]  
and from (31)  
\[\lambda_{2}=-\frac{b_{1}}{a_{1}}k+\left(\frac{b_{1}a_{2}}{a_{1}^{2}}-\frac{b_ {2}}{a_{1}}-\frac{b_{1}^{2}}{a_{1}^{3}}\right)k^{2}. \tag{35}\]  
Now using (32) we obtain  
\[\lambda_{2}=-iv_{e}^{\prime}(s_{e})k+\frac{v_{e}^{\prime}(s_{e})}{f_{\
u}} \left[\frac{-2f_{\Delta\
u}-f_{\
u}}{2}-v_{e}^{\prime}(s_{e})\right]k^{2}. \tag{36}\]  
The real part of (36) denotes the change in amplitude of the traffic oscillations (growth rate). The traffic flow is string stable if this is negative. Since  
\[v_{e}^{\prime}(s_{e})\ \geq 0\ \text{and}\ f_{\
u}<0, \tag{37}\]  
\(\left[\frac{-2f_{\Delta\
u}-f_{\
u}}{2}-v_{e}^{\prime}(s_{e})\right]\) is the string stability criterion [26], i.e. \[v_{e}^{\prime}(s_{e})\leq-\frac{f_{\
u}}{2}-f_{\Delta\
u}. \tag{38}\]  
From (37) and (38), the product of \(\left[-\frac{2f_{\Delta\
u}-f_{\
u}}{2}-v_{e}^{\prime}(s_{e})\right]\) and \(\frac{v_{e}^{\prime}(s_{e})}{f_{\
u}}\) indicates that \(\lambda_{2}\) has a negative real part. At equilibrium, \(f_{\
u}\) and \(f_{\Delta\
u}\) for the ID model (1) are  
\[f_{\
u}=a\left(-\frac{\delta v_{e}(s_{e})^{\delta-1}}{v_{max}^{\delta}}-\frac{ 2\tau\big{(}s_{j}+v_{e}(s_{e})\tau\big{)}}{s_{e}^{2}}\right), \tag{39}\]  
\[f_{\Delta\
u}=-\frac{v_{e}(s_{e})}{s_{e}}\sqrt{\frac{a}{b}}\left(\frac{s_{j}+ v_{e}(s_{e})\tau}{s_{e}}\right). \tag{40}\]  
Using (39) and (40), the string stability criterion from (38) is  
\[v_{e}^{\prime}(s_{e})\leq\frac{a\Big{(}\delta(s_{e})^{2}v_{e}(s_{e})^{\delta -1}+2\tau s_{j}v_{max}^{\delta}+2v_{e}(s_{e})\tau^{2}v_{max}^{\delta}\Big{)}} {2(s_{e})^{2}v_{max}^{\delta}}+\frac{v_{e}(s_{e})\sqrt{ab}(s_{e}+\tau v_{e}(s_ {e}))}{(s_{e})^{2}b} \tag{41}\]  
This shows that changes in velocity with the ID model are characterized by the constant \(\delta\). A traffic string is more stable with a larger value of \(\delta\), but in this case, congestion ends quickly. Thus, making \(\delta\) larger to ensure stability ignores traffic physics and results in unrealistic traffic behavior [11]. The velocity changes during traffic alignment are affected by driver sensitivity and reaction. Thus, to ensure appropriate behavior, \(\delta\) in (41) is replaced with (6), which gives the proposed model stability criteria as  
\[v_{e}^{\prime}(s_{e})\leq\frac{a\left(a\tau\left(\frac{s}{b}\right)\left(\frac{ \tau}{\tau_{b}}\right)^{2}v_{e}(s_{e})^{\mathrm{err}\left(\frac{s}{b}\right)( \frac{\tau}{\tau_{b}})-1}+2\tau s_{p}^{\mathrm{err}\left(\frac{s}{b}\right)( \frac{\tau}{\tau_{b}})}+2\tau s_{e}(s_{e})^{\mathrm{err}\left(\frac{s}{b} \right)(\frac{\tau}{\tau_{b}})}}{2(s_{e})^{2}\sigma_{\mathrm{max}}^{\mathrm{err} \left(\frac{s}{b}\right)(\frac{\tau}{\tau_{b}})}}+\right. \tag{42}\]  
Thus, with the proposed model, changes in velocity are based on driver sensitivity and reaction.