## 4 Application Example  
To demonstrate the applicability of the proposed smart CPL management (see Section 3.2) and to test the possible outcomes of the power derating on system performance, an application example was chosen. Specifically, a full electric vehicle was selected. The electric propulsion motor and its inverter constitute the CPL. Such a choice allows clearly showing the impact on performance of the proposed smart CPL management by evaluating indices that are evident and easy to understand (i.e., maximum achievable speed and 0-100 km/h acceleration time). In order to study the worst case for DC voltage stability, the power system of the application example was simplified by removing the CLs and considering battery-only operation. Thus, a battery, an LC filter, and a CPL (the propulsion system) constituted the power system to be analyzed. The result was an islanded DC microgrid with a floating bus, similar to the one analyzed in the previous Sections. Figure 11: BA migration imposed by power reduction function. Figure 10: RAS migration imposed by power reduction function. The data of a commercial electric car (2011 Nissan Leaf) were inferred from several online datasheets, and are shown in Table 1. Using standard physics equations and applying a set of simplifying hypotheses, a Matlab script was developed in order to evaluate the performance of the car with respect to the CPL power. As previously mentioned, the maximum achievable speed and the 0-100 km/h acceleration time were chosen as performance indices. Proper modeling was applied to allow the calculation of these indices starting from available propulsion power. To lower the modeling burden, the vehicle was considered as a point of mass, free from the influence of external forces (e.g., no height variations and no wind), whereas the propulsion motor and inverter system had a control bandwidth so high with respect to the overall vehicle that time constants could be considered negligible. As previously discussed, CPL power has to be decreased through the application of the power reduction function (Equation (26)), to preserve the DC voltage stability when the battery discharge causes a decrease in the bus voltage. In particular, the maximum battery voltage range was defined by means of Table 1 parameters: 403 V voltage at full charge means \(e_{\%}=1\), while 336 V at 20% SoC corresponds to \(e_{\%}=0.83\). For what concerns the stability index, the value \(\Delta=0.516\) was chosen as a feasible tradeoff between a wide stability margin and the applicable CPL power (as shown in Figure 8, green trace). Using these data, the reduced CPL power can be determined through Equation (26), leading to the results depicted in Table 2. Focusing on the performance indices, the maximum achievable speed can be determined by exploiting the steady-state force equilibrium equation,  
\[F_{m}=F_{f}+F_{a}, \tag{27}\]  
where \(F_{m}\) is the force applied by the electric motor to the wheels, \(F_{f}\) the force due to the wheel-road friction, and \(F_{a}\) is the drag force due to the air. These three forces were assessed by considering the additional system parameters reported in Table 3. In particular, to model the overall transmission losses, the wheel traction was calculated by reducing the electric motor power through a 15% loss coefficient, thus resulting in a wheel power ranging from 37 to 68 kW (Table 2). Conversely, the wheel-road friction \(F_{f}\) and the air drag force \(F_{a}\) were determined using the following equations:  
\[F_{f}=\mu_{d}RMg, \tag{28}\]  
\[F_{a}=\frac{1}{2}\left(r_{0}AC_{x}v^{2}\right), \tag{29}\]  
where the parameters are defined in Tables 1-3, whilst \(v\) is the car speed in m/s, and \(g\) the gravity acceleration constant. Finally, the maximum achievable speed was calculated as a function of the power at the wheels using Equations (27)-(29), the results of which are shown in Figure 12 (red curve, left ordinate).