*3.3. Migration of RAS and BA*

The effects of the proposed power reduction function on RASs and BAs of the system are depicted in Figures 10 and 11, respectively. As expected, both RAS and BA shift along the *v–i* plane, as *e*% and *p* diminish. Moreover, their area remains constant, due to the stability index invariance (∆ = 0.516) guaranteed by applying the proposed power reduction function. *Appl. Sci.* **2018**, *8*, x FOR PEER REVIEW 13 of 18 *Appl. Sci.* **2018**, *8*, x FOR PEER REVIEW 13 of 18

**Figure 10.** RAS migration imposed by power reduction function. **Figure 10.** RAS migration imposed by power reduction function. **Figure 10.** RAS migration imposed by power reduction function.

**Figure 11.** BA migration imposed by power reduction function. **Figure 11.** BA migration imposed by power reduction function. **Figure 11.** BA migration imposed by power reduction function.

### **4. Application Example 4. Application Example**

**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., 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). 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).

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 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 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.

Sections. The data of a commercial electric car (2011 Nissan Leaf) were inferred from several online

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

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 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 ∆ = 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,

$$F\_m = F\_f + F\_{a\nu} \tag{27}$$

where *F<sup>m</sup>* is the force applied by the electric motor to the wheels, *F<sup>f</sup>* the force due to the wheel–road friction, and *F<sup>a</sup>* 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<sup>f</sup>* and the air drag force *F<sup>a</sup>* were determined using the following equations:

$$F\_f = \mu\_d RMg\_\prime \tag{28}$$

$$F\_a = \frac{1}{2} \left( r\_0 A \mathcal{C}\_x v^2 \right) . \tag{29}$$

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).

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, *Fa*(*v*) the drag force due to the air, and *Fmd*(*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 transferrable from wheels to the asphalt in bad weather conditions and a traction control safety coefficient (refer to Table 3), the term *Fmd* may be obtained.

$$F\_{md}(v) = \begin{cases} F\_m(v) & \text{if } F\_m \le F\_{\text{lim}} \\ & F\_{\text{lim}} \quad \text{if } F\_m > F\_{\text{lim}} \end{cases} \tag{31}$$

$$F\_{\rm lim} = \frac{2}{4} k\_{\rm tc} \mu\_{\rm s} \mathcal{g} \mathcal{M}\_{\prime} \tag{32}$$

where *F*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 ∆).


**Table 1.** Application example: electric vehicle parameters. SoC—state of charge.

**Table 2.** Application example: power reduction function effect and resulting power at the wheels.


**Table 3.** Application example: system parameters.


*Appl. Sci.* **2018**, *8*, x FOR PEER REVIEW 16 of 18

**Figure 12.** Maximum achievable speed and 0–100 km/h acceleration time as a function of power at the wheels. **Figure 12.** Maximum achievable speed and 0–100 km/h acceleration time as a function of power at the wheels.

### **5. Conclusions 5. Conclusions**

The paper proposed a smart power limitation in order to preserve system stability in a DC microgrid feeding a CPL. Such a management is crucial when the battery is the only power source for the microgrid. The battery operation determines a floating DC bus, whose unregulated voltage varies over time depending on the SoC of the battery. When the battery SoC is decreasing, i.e., the DC bus voltage is diminishing, the stability region shrinking can be evaluated by means of the Lyapunov theory and numerical continuation analysis. The analytical method demonstrated its effectiveness in defining the stability index, which has the main role in the definition of the stability preserving criterion. For a given power-quality filter, the power function expresses the mathematical relationship among bus voltage, stability index, and CPL power. Therefore, a proper management can be designed to limit the power of the CPL to assure stability when the steady-state DC bus voltage decreases. This power reduction was analyzed on the basis of several results, which confirm the invariance of both the area and shape of the stability regions (i.e., RASs and BAs), while the operating point moves in the state plane. The paper finally demonstrated the effectiveness of the proposed criterion by means of a suitable application example. The paper proposed a smart power limitation in order to preserve system stability in a DC microgrid feeding a CPL. Such a management is crucial when the battery is the only power source for the microgrid. The battery operation determines a floating DC bus, whose unregulated voltage varies over time depending on the SoC of the battery. When the battery SoC is decreasing, i.e., the DC bus voltage is diminishing, the stability region shrinking can be evaluated by means of the Lyapunov theory and numerical continuation analysis. The analytical method demonstrated its effectiveness in defining the stability index, which has the main role in the definition of the stability preserving criterion. For a given power-quality filter, the power function expresses the mathematical relationship among bus voltage, stability index, and CPL power. Therefore, a proper management can be designed to limit the power of the CPL to assure stability when the steady-state DC bus voltage decreases. This power reduction was analyzed on the basis of several results, which confirm the invariance of both the area and shape of the stability regions (i.e., RASs and BAs), while the operating point moves in the state plane. The paper finally demonstrated the effectiveness of the proposed criterion by means of a suitable application example.

**Author Contributions:** D.B. conceived the presented idea and developed the theory. S.G. verified the analytical method. A.V. developed the case study and G.S. supervised the research activity. All the authors discussed the **Author Contributions:** D.B. conceived the presented idea and developed the theory. S.G. verified the analytical method. A.V. developed the case study and G.S. supervised the research activity. All the authors discussed the results and contributed to the final manuscript.

results and contributed to the final manuscript. **Funding:** This research received no external funding.

IEEE: Piscataway, NJ, USA, 2010.

**Funding:** This research received no external funding. **Acknowledgments:** Authors wish to acknowledge Giovanni Giadrossi for the valuable contributions given in **Acknowledgments:** Authors wish to acknowledge Giovanni Giadrossi for the valuable contributions given in revising this work.

revising this work. **Conflicts of Interest:** The authors declare no conflicts of interest.

### **Conflicts of Interest:** The authors declare no conflicts of interest. **References**


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