*3.1. Stability Index*

The Lyapunov analysis demonstrated the importance of the *v*min term (Equation (3)) in a DC microgrid supplying a CPL. As shown in Figure 6, this parameter represents the lowest voltage margin for a specific DC power system (with given *r*, *l*, and *c* parameters) supplying a CPL with power *p* and working in steady state at the voltage *v*0. In this regard, it is possible to define the distance ∆ between the equilibrium point *v*<sup>0</sup> and the lower bound *v*min as a conservative stability index. In fact, any perturbation capable of moving the voltage state inside the area defined by ∆ does not jeopardize the system stability, as the Lyapunov conservative condition is still verified.

As previously mentioned, the battery voltage reduction (due to SoC decrease) is responsible for an operating point shift toward the upper left-hand side of the *v*–*i* state plane (*v*<sup>0</sup> decreases and *i*<sup>0</sup> increases), due to the relationship between supply voltage and absorbed current in a CPL. Moreover, the *v*<sup>0</sup> drop determines an increase in the *v*min limit, as highlighted by Equation (3), which, in turn, leads to a further reduction of ∆. As this behavior is particularly important for the stability issue, the following mathematical study aims to demonstrate the relationship between the bus voltage decrease and the stability index ∆ shrinking. To study this issue, it is necessary to define the parameter *e*0*t* , which is the battery voltage needed to supply the CPL rated power (*p* = 1 p.u.) at the rated load voltage (*v*<sup>0</sup> = 1 p.u.). This parameter is representative of an optimistic scenario, with a fully charged battery. Conversely, in normal operating conditions, the battery SoC is lower, thus leading to a lower battery voltage *e*0, which can be represented as a percentage (*e*%) of the full charge voltage *e*0t. By observing Figure 2 and assuming the steady-state condition, the battery voltage *e*<sup>0</sup> is defined through Equation (4).

$$e\_0 = v\_0 + r\frac{p}{v\_0} = e\_{0t}e\_{\%}.\tag{4}$$

By rearranging the second equality of Equation (4), it is possible to derive Equation (5), and then obtain the second-order Equation (6) by multiplying Equation (5) with the unknown quantity *v*0.

$$v\_0 - e\_{0t}e\_\% + r\frac{p}{v\_0} = 0.\tag{5}$$

$$v\_0^2 - e\_{0t}e\_{\%}v\_0 + rp = 0.\tag{6}$$

By neglecting negative roots, the steady-state voltage *v*<sup>0</sup> results in Equation (7), while the total battery voltage *e*0*<sup>t</sup>* is defined in Equation (8), once the rated condition (*p* = *v*<sup>0</sup> = 1 p.u.) and full battery (*e*% = 1.0) are applied to Equation (4):

$$v\_0 = \frac{e\_{0t}e\_{\%} + \sqrt{\left(e\_{0t}e\_{\%}\right)^2 - 4rp}}{2}.\tag{7}$$

$$e\_{0t} = 1 + r.\tag{8}$$

Assuming a variable *e*%, Equations (7) and (8) can be used to delineate the *v*<sup>0</sup> voltage shift in the presence of different load powers. Consequently, the lower bound *v*min corresponding to each *v*<sup>0</sup> value can be determined using Equation (3), whereas the stability index ∆ can be found with Equation (9).

$$
\Delta = v\_0 - v\_{\rm min} = v\_0 - \frac{pl}{rcv\_0}.\tag{9}
$$

The equations explained so far allow the drawing of Figure 7, where the gradual contraction of ∆ is made evident in response to *e*% decrease. In particular, for the system under study, the *v*<sup>0</sup> curve (*e*% = 1.0) are applied to Equation (4):

intersects the *v*min trace (i.e., ∆ = 0) for *e*% ≈ 0.83 in the case of rated power (blue curve). At this specific point, both the large- and the small-signal stability are impaired. Δ is made evident in response to *e*% decrease. In particular, for the system under study, the *v*0 curve intersects the *v*min trace (i.e., Δ = 0) for *e*% ≈ 0.83 in the case of rated power (blue curve). At this specific point, both the large- and the small-signal stability are impaired.

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*Appl. Sci.* **2018**, *8*, x FOR PEER REVIEW 9 of 18

0 <sup>0</sup> <sup>0</sup> *e e v p*

By rearranging the second equality of Equation (4), it is possible to derive Equation (5), and then

0 <sup>0</sup> % <sup>0</sup>

By neglecting negative roots, the steady-state voltage *v*0 results in Equation (7), while the total battery voltage *e*0*t* is defined in Equation (8), once the rated condition (*p* = *v*0 = 1 p.u.) and full battery

> ( ) 2

*e e e e rp*

Assuming a variable *e*%, Equations (7) and (8) can be used to delineate the *v*0 voltage shift in the presence of different load powers. Consequently, the lower bound *v*min corresponding to each *v*0 value can be determined using Equation (3), whereas the stability index Δ can be found with Equation (9).

The equations explained so far allow the drawing of Figure 7, where the gradual contraction of

0 % 0 %

obtain the second-order Equation (6) by multiplying Equation (5) with the unknown quantity *v*0.

<sup>0</sup> <sup>0</sup> % <sup>−</sup> <sup>+</sup> <sup>=</sup> *<sup>v</sup>*

2

0

0 %

0

4 <sup>2</sup>

0

*p*

*e v r <sup>t</sup>* = + = . (4)

*v e e r <sup>t</sup>* . (5)

<sup>0</sup> *v* − *e e v* + *rp* = *<sup>t</sup>* . (6)

*<sup>v</sup> <sup>t</sup> <sup>t</sup>* <sup>+</sup> <sup>−</sup> <sup>=</sup> . (7)

*pl* <sup>Δ</sup> <sup>=</sup> *<sup>v</sup>* <sup>−</sup> *<sup>v</sup>* <sup>=</sup> *<sup>v</sup>* <sup>−</sup> . (9)

*e r <sup>t</sup>* = 1 + <sup>0</sup> . (8)

$$
\Delta = 0 \to v\_0 = v\_{\rm min} \to v\_0 = \frac{pl}{rcv\_0} \to v\_0 = \sqrt{\frac{pl}{rc}}.\tag{10}
$$

0

*rcv*

On the contrary, for smaller values of CPL power *p* (black/red curves), larger values of ∆ are assured along the *e*% drop, thus revealing a possible strategy for ensuring a sufficient stability margin. Actually, it is possible to conceive a smart management of the CPL, able to conveniently decrease its power *p* as the voltage *e*<sup>0</sup> decreases, in order to guarantee a proper stability margin. On the contrary, for smaller values of CPL power *p* (black/red curves), larger values of Δ are assured along the *e*% drop, thus revealing a possible strategy for ensuring a sufficient stability margin. Actually, it is possible to conceive a smart management of the CPL, able to conveniently decrease its power *p* as the voltage *e*0 decreases, in order to guarantee a proper stability margin.

**Figure 7.** Stability index ∆, with varying constant power load (CPL) power and DC bus voltage.
