*3.2. 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 ∆ 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*<sup>0</sup> and ∆ 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

$$
\left(\frac{\Delta}{2} - \sqrt{\frac{\Delta^2}{4} + \frac{pl}{rc}}\right) \tag{14}
$$

a new equation is found as a special product (i.e., difference of two squares):

$$e\_0 = v\_0 + \frac{r^2 cp}{-pl} \left(\frac{\Delta}{2} - \sqrt{\frac{\Delta^2}{4} + \frac{pl}{rc}}\right). \tag{15}$$

Once the *v*<sup>0</sup> expression (Equation (12)) is substituted into Equation (15), one obtains

$$e\_0 = \frac{\Delta}{2} + \sqrt{\frac{\Delta^2}{4} + \frac{pl}{rc}} - \frac{r^2 c \Delta}{2l} + \frac{r^2 c}{l} \sqrt{\frac{\Delta^2}{4} + \frac{pl}{rc}},\tag{16}$$

$$
\epsilon\_0 + \left(\frac{r^2c}{l} - 1\right) \cdot \frac{\Delta}{2} = \left(\frac{r^2c}{l} + 1\right) \sqrt{\frac{\Delta^2}{4} + \frac{pl}{rc}}.\tag{17}
$$

Then, by squaring Equation (17), Equation (18) is obtained and, consequently, after straightforward manipulation, Equations (19) and (20).

$$
\epsilon\_0^2 + \left(\frac{r^2c}{l} - 1\right) \cdot \Delta e\_0 + \left(\frac{r^2c}{l} - 1\right)^2 \cdot \frac{\Delta^2}{4} = \left(\frac{r^2c}{l} + 1\right)^2 \left(\frac{\Delta^2}{4} + \frac{pl}{rc}\right),
\tag{18}
$$

$$
\epsilon\_0^2 + \left(\frac{r^2c}{l} - 1\right) \cdot \Delta \epsilon\_0 + \left[\left(\frac{r^2c}{l} - 1\right)^2 - \left(\frac{r^2c}{l} + 1\right)^2\right] \frac{\Delta^2}{4} = \left(\frac{r^2c}{l} + 1\right)^2 \frac{pl}{rc} \tag{19}
$$

$$e\_0^2 + \left(\frac{r^2c}{l} - 1\right)\Delta e\_0 - \frac{r^2c}{l}\Delta^2 = \left(\frac{r^2c}{l} + 1\right)^2 \frac{pl}{rc}.\tag{20}$$

Finally, starting from

$$
\Delta \epsilon\_0^2 + \left(r^2 c - l\right) \Delta \epsilon\_0 - r^2 c \Delta^2 = \left(r^2 c + l\right)^2 \frac{p}{rc},\tag{21}
$$

the function *p* = *f*(*e*0, ∆) can be derived:

$$p = \frac{rc}{\left(r^2c + l\right)^2} \left[le\_0^2 + \left(r^2c - l\right)\Delta e\_0 - r^2c\Delta^2\right].\tag{22}$$

Equation (22) explains how to modify the CPL power *p* to maintain a desired ∆ when *e*<sup>0</sup> becomes lower.

To better analyze Equation (22), the filter parameters *r*, *l*, and *c* may be gathered in the *K* term,

$$K = \frac{\tau\_L}{\tau\_\mathbb{C}} = \frac{l}{r} \cdot \frac{1}{rc} = \frac{l}{r^2 c'} \tag{23}$$

through the definition of the time constants *τ<sup>L</sup>* and *τC*. The *K* term allows simplifying Equation (22).

$$p = \frac{rc}{\left(r^2c\right)^2 \left(1 + K\right)^2} \left[\mathcal{K}r^2c\mathcal{e}\_0^2 + r^2c(1 - \mathcal{K})\Delta\epsilon\_0 - r^2c\Delta^2\right];\tag{24}$$

$$p = \frac{K\epsilon\_0^2 + (1 - K)\Delta\epsilon\_0 - \Delta^2}{r(1 + K)^2}.\tag{25}$$

Finally, by substituting the *e*<sup>0</sup> voltage definition in Equation (4) into Equation (25), one obtains

$$p = \frac{e\_{\%}^2 \left[e\_{0t}^2 K\right] + e\_{\%} \left[\Delta e\_{0t} (1 - K)\right] - \Delta^2}{r(1 + K)^2}.\tag{26}$$

Such 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 ∆ (Figure 8). In particular, the green curve in Figure 8 is related to the power reduction function capable of guaranteeing the rated stability index, whereas greater/smaller values of ∆ are ensured when the power reduction follows the red/blue curves. Clearly, if the requested stability index exceeds the rated value (0.6 vs. 0.516, red curve), then the rated power is not reachable even with a fully charged battery. Conversely, the DC microgrid can feed the rated CPL power with a partially charged battery (e.g., *e*% = 0.95) if the stability performance is downgraded (0.4 vs. 0.516, blue curve). As stated in References [11–20], the stability of a DC power system supplying a CPL is closely related to the filter parameters. For this reason, Figure 9 depicts the influence of the parameter *K* on the power function. In particular, by halving *K* (i.e., the capacitance *c* is doubled with respect to the inductance *l*, keeping the resistance *r* constant), the range of voltages in which the rated power can be supplied is extended (blue curve). On the contrary, more critical scenarios are given by a double *K* (red curve), where the DC microgrid can never provide the rated power to the CPL while keeping voltage stability (i.e., *p* < 0.5 p.u. when *e*% = 0.9). *Appl. Sci.* **2018**, *8*, x FOR PEER REVIEW 12 of 18 *Appl. Sci.* **2018**, *8*, x FOR PEER REVIEW 12 of 18

**Figure 8.** Power reduction function for preserving different values of stability index Δ. **Figure 8.** Power reduction function for preserving different values of stability index ∆. **Figure 8.** Power reduction function for preserving different values of stability index Δ.

**Figure 9.** Power reduction function in the presence of different filtering solutions. **Figure 9.** Power reduction function in the presence of different filtering solutions. **Figure 9.** Power reduction function in the presence of different filtering solutions.

= 0.516) guaranteed by applying the proposed power reduction function.

= 0.516) guaranteed by applying the proposed power reduction function.

The effects of the proposed power reduction function on RASs and BAs of the system are

The effects of the proposed power reduction function on RASs and BAs of the system are

*3.3. Migration of RAS and BA* 

*3.3. Migration of RAS and BA* 
