## 4 Application Example
On the other hand, the 0-100 km/h acceleration time can be found by means of the dynamic equation:  
\[a(v)=\frac{1}{2M}\left[F_{md}(v)-F_{f}-F_{a}(v)\right], \tag{30}\]  
where \(a(v)\) is the vehicle acceleration, \(F_{a}(v)\) the drag force due to the air, and \(F_{md}(v)\) is the dynamic force applied by the electric propulsion motor to the wheels. In particular, by taking into account not only the propulsion motor's available power (Table 2), but also the maximum force transferable from wheels to the asphalt in bad weather conditions and a traction control safety coefficient (refer to Table 3), the term \(F_{md}\) may be obtained. \[F_{md}(v)=\left\{ \begin{array}{cc}F_{m}(v)&if\,F_{m}\leq F_{\text{lim}}\\ F_{\text{lim}}&if\,F_{m}>F_{\text{lim}}\end{array} \right., \tag{31}\]  
\[F_{\text{lim}}=\frac{2}{4}k_{tc}\mu_{s}gM, \tag{32}\]where \(F_{\rm lim}\) represents the maximum force transferrable from wheels to the asphalt by a four-by-two-wheel drive car. It is worth noting that, in Equation (30), all the terms depend on speed. Therefore, corresponding to a series of different propulsion power values, Equation (30) was used to calculate the vehicle speed variation from zero to its maximum. The resulting dataset, which relates vehicle speed and time for each power value, allows evaluating the 0-100 km/h time with a simple search algorithm. The results of this procedure are shown in Figure 12 (blue curve, right ordinate), where the acceleration time is shown with respect to the power available at the wheels. By analyzing Figure 12, it is possible to evaluate the effect of the proposed smart CPL management on the performance indices for the application example. Firstly, the maximum achievable speed at the minimum available power is still in the range of the maximum speed limit set by the car software (135 km/h achievable vs. 144 km/h limit). Secondly, the acceleration performance at the lowest battery level is still satisfactory (~10.4 s) with respect to common car performance levels. In this regard, it has to be noted that these results were calculated in a worst-case condition, thus possibly leading to lower performance loss in a real system. Although these results were obtained by oversimplifying both the power system and the system's physics (i.e., not constituting a complete performance assessment), they may provide important insight into the stability criterion effect. Actually, considering the application example, the proposed criterion can represent a valuable solution for preserving the DC stability by suitably managing the CPL power without excessively impairing the system performance. Obviously, the applicability of the proposed smart CPL management depends not only on the single system under study, but also on some parameters that can be defined accordingly by designers (such as coefficient \(K\) and stability margin \(\Delta\)). \begin{table}
\begin{tabular}{c c c} \hline Electric motor power & \(P_{n}\) & 80 kW \\ Battery capacity & & 24 kWh \\ Nominal battery voltage & & 360 V \\ Battery voltage at full charge & & 403 V \\ Battery voltage at 20\% SoC & & 336 V \\ Mass & \(M\) & 1500 kg \\ Frontal area & \(A\) & 2.28 m\({}^{2}\) \\ Wheel diameter (205/55 R16) & \(R\) & 63.16 cm \\ Maximum speed (unlimited) & & 170 km/h \\ Maximum speed (software limited) & & 144 km/h \\
0–100 km/h (unofficial tests) & & \(\sim\)9 s \\ Aerodynamic penetration coefficient & C\({}_{x}\) & 0.32 \\ \hline \end{tabular}
\end{table} 
Table 1: Application example: electric vehicle parameters. SoC—state of charge. \begin{table}
\begin{tabular}{c c c c} \hline \hline Static friction coefficient (rubber–asphalt) wet conditions & \(\mu_{s}\) & 0.5 \\ Rolling friction coefficient (rubber–asphalt) & \(\mu_{d}\) & 0.035 \\ Traction control safety coefficient & \(k_{tc}\) & 0.75 \\ Power loss (from motor to wheels) & & 15\% \\ Minimum battery SoC & & 20\% \\ Air density (kg/m\({}^{3}\)) & \(r_{0}\) & 1.29 \\ \hline \end{tabular}
\end{table} 
Table 3: Application example: system parameters.