## 3 Stability Preserving Criterion
### Smart CPL Management  
As discussed above, when the islanded microgrid is in battery-only operation, a smart management of the CPL power can be useful to preserve voltage stability. To this aim, this subsection focuses on the definition of a power reduction function able to ensure a proper stability index \(\Delta\) while the battery is discharging (i.e., \(e_{\%}\) is decreasing). By multiplying Equation (9) with the steady-state voltage \(v_{0}\), a second-order equation can be obtained:  
\[v_{0}^{2}-\Delta v_{0}-\frac{pl}{rc}=0. \tag{11}\]  
Discarding the negative root, the solution for (11) is  
\[v_{0}=\frac{\Delta}{2}+\sqrt{\frac{\Delta^{2}}{4}+\frac{pl}{rc}}. \tag{12}\]  
The relationship between \(e_{0}\) and \(\Delta\) is  
\[e_{0}=v_{0}+rp\cdot\left(\frac{\Delta}{2}+\sqrt{\frac{\Delta^{2}}{4}+\frac{pl }{rc}}\right)^{-1}. \tag{13}\]  
By dividing/multiplying the second term of Equation (13) by the term  
\[\left(\frac{\Delta}{2}-\sqrt{\frac{\Delta^{2}}{4}+\frac{pl}{rc}}\right), \tag{14}\]  
Figure 7: Stability index \(\Delta\), with varying constant power load (CPL) power and DC bus voltage. a new equation is found as a special product (i.e., difference of two squares):  
\[e_{0}=v_{0}+\frac{r^{2}cp}{-pl}\Bigg{(}\frac{\Delta}{2}-\sqrt{\frac{\Delta^{2}}{ 4}+\frac{pl}{rc}}\Bigg{)}. \tag{15}\]  
Once the \(v_{0}\) expression (Equation (12)) is substituted into Equation (15), one obtains  
\[e_{0}=\frac{\Delta}{2}+\sqrt{\frac{\Delta^{2}}{4}+\frac{pl}{rc}}-\frac{r^{2}c \Delta}{2l}+\frac{r^{2}c}{l}\sqrt{\frac{\Delta^{2}}{4}+\frac{pl}{rc}}, \tag{16}\]  
\[e_{0}+\bigg{(}\frac{r^{2}c}{l}-1\bigg{)}\cdot\frac{\Delta}{2}=\bigg{(}\frac{r ^{2}c}{l}+1\bigg{)}\sqrt{\frac{\Delta^{2}}{4}+\frac{pl}{rc}}. \tag{17}\]  
Then, by squaring Equation (17), Equation (18) is obtained and, consequently, after straightforward manipulation, Equations (19) and (20). \[e_{0}^{2}+\bigg{(}\frac{r^{2}c}{l}-1\bigg{)}\cdot\Delta e_{0}+\bigg{[}\bigg{(} \frac{r^{2}c}{l}-1\bigg{)}^{2}-\bigg{(}\frac{r^{2}c}{l}+1\bigg{)}^{2}\bigg{]} \frac{\Delta^{2}}{4}=\bigg{(}\frac{r^{2}c}{l}+1\bigg{)}^{2}\frac{pl}{rc}, \tag{18}\]  
\[e_{0}^{2}+\bigg{(}\frac{r^{2}c}{l}-1\bigg{)}\cdot\Delta e_{0}+\bigg{[}\bigg{(} \frac{r^{2}c}{l}-1\bigg{)}^{2}-\bigg{(}\frac{r^{2}c}{l}+1\bigg{)}^{2}\bigg{]} \frac{\Delta^{2}}{4}=\bigg{(}\frac{r^{2}c}{l}+1\bigg{)}^{2}\frac{pl}{rc}, \tag{19}\]  
\[v_{0}=\frac{\Delta}{2}+\sqrt{\frac{\Delta^{2}}{4}+\frac{pl}{rc}}. \tag{12}\]0  
Finally, starting from  
\[v_{0}=\frac{\Delta}{2}+\sqrt{\frac{\Delta^{2}}{4}+\frac{pl}{rc}}. \tag{12}\]1  
the function \(p=f(e_{0},\,\Delta)\) can be derived:  
\[v_{0}=\frac{\Delta}{2}+\sqrt{\frac{\Delta^{2}}{4}+\frac{pl}{rc}}. \tag{12}\]2  
Equation (22) explains how to modify the CPL power \(p\) to maintain a desired \(\Delta\) when \(e_{0}\) becomes lower. To better analyze Equation (22), the filter parameters \(r,l,\) and \(c\) may be gathered in the \(K\) term,  
\[v_{0}=\frac{\Delta}{2}+\sqrt{\frac{\Delta^{2}}{4}+\frac{pl}{rc}}. \tag{12}\]3  
through the definition of the time constants \(\tau_{L}\) and \(\tau_{C}\). The \(K\) term allows simplifying Equation (22). \[v_{0}=\frac{\Delta}{2}+\sqrt{\frac{\Delta^{2}}{4}+\frac{pl}{rc}}. \tag{12}\]4  
\[v_{0}=\frac{\Delta}{2}+\sqrt{\frac{\Delta^{2}}{4}+\frac{pl}{rc}}. \tag{12}\]5  
Finally, by substituting the \(e_{0}\) voltage definition in Equation (4) into Equation (25), one obtains  
\[v_{0}=\frac{\Delta}{2}+\sqrt{\frac{\Delta^{2}}{4}+\frac{pl}{rc}}. \tag{12}\]6Such a function, equivalent to Equation (22), better highlights the relationship between voltage and power. For this reason, it can be called the \"power function\". Through Equation (26), it is possible to appreciate the reduction in CPL power needed to migrate the working point (i.e., a decrease in \(i_{0}\)) toward a new point with the given (desired) stability index \(\Delta\) (Figure 8).