**2. Effect of Floating DC Bus on System Stability**

This section discusses the stability degradation that can possibly arise in a floating DC bus microgrid supplying a CPL. The present paper is based on a previous work [33] published by the authors about the criticality of a DC floating bus powering a hybrid electric vehicle. Nevertheless, the same conclusions can be extended to any islanded DC microgrid with a floating bus, supplying a high-bandwidth destabilizing load. In this regard, the simplified microgrid under study is described in Section 2.1, while initial considerations about the effect of a floating DC bus on the stability margin are proposed in Section 2.2. Then, two methodologies for large-signal analysis are introduced in Section 2.3, whereas numerical simulations are proposed in Section 2.4 for validating, through the system dynamic response analysis, the methodological approach used for the stability evaluation. In Section 2.5, the dynamic numerical simulations are used to verify the basin of attraction (BA) and demonstrate the validity of the Lyapunov analytical approach.

### *2.1. DC Microgrid Topology* equivalent linear resistance *RL*. Conversely, the latter category is made up of loads requiring tight

*2.1. DC Microgrid Topology* 

used in this study is based on the following elements:

• Energy storage system, i.e., an electrochemical battery (B);

• Generic static DC load (L), fed by a DC–DC converter (C);

The simplified power system shown in Figure 1 was chosen as a case study to analyze the voltage stability in a floating-bus DC microgrid in presence of a destabilizing CPL. The islanded microgrid used in this study is based on the following elements: control by their input power converter (C or I in Figure 1), to either provide a constant voltage (for static loads) or a constant speed/torque (for rotating loads). The final effect is a constant power delivery from the DC bus to the loads. These loads can be modeled through a single equivalent nonlinear current *IL* = *P*/*V*, where *P* is the CPL power*.* It has to be noticed that both static and rotating

that do not require strict voltage regulation for proper operation (e.g., resistive heaters, loads with an integrated input conversion stage, etc.). Therefore, their global effect can be modeled through an

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• DC power-generating system, composed of an internal combustion engine (IC), an alternator

(G), and a controlled rectifier (P), to supply loads and/or recharge batteries;

• LC input filter (F), to assure proper voltage and current quality on the load bus;

• Generic rotating load (M), supplied by means of a controlled inverter (I).

The simplified power system shown in Figure 1 was chosen as a case study to analyze the voltage stability in a floating-bus DC microgrid in presence of a destabilizing CPL. The islanded microgrid

In order to save space, the system avoids using a battery charger unit, relying upon the controlled rectifier (P) to regulate battery charging. Thus, the DC bus voltage is floating when the power generating system is offline, depending on the battery SoC (the battery is the only source of power). As already shown in Reference [33], in this case voltage stability issues may affect the DC bus. Particular attention has to be paid to the loads; depending on the control bandwidth of the converters used to supply loads from the DC bus, it is possible to classify them into two classes: (i)


stage components, and *R* is the inductor physical resistance.

• Generic rotating load (M), supplied by means of a controlled inverter (I). circuit of Figure 2, where *E* is the battery voltage, *V* is the voltage on the CPL, *L* and *C* are the filtering

**Figure 1.** Proposed direct current (DC) microgrid; the power system section in battery-only operation is shown in black. **Figure 1.** Proposed direct current (DC) microgrid; the power system section in battery-only operation is shown in black.

In order to save space, the system avoids using a battery charger unit, relying upon the controlled rectifier (P) to regulate battery charging. Thus, the DC bus voltage is floating when the power generating system is offline, depending on the battery SoC (the battery is the only source of power). As already shown in Reference [33], in this case voltage stability issues may affect the DC bus. Particular attention has to be paid to the loads; depending on the control bandwidth of the converters used to supply loads from the DC bus, it is possible to classify them into two classes: (i) conventional loads (CLs), and (ii) constant power loads (CPLs). The former category represents loads that do not require strict voltage regulation for proper operation (e.g., resistive heaters, loads with an integrated input conversion stage, etc.). Therefore, their global effect can be modeled through an equivalent linear resistance *RL*. Conversely, the latter category is made up of loads requiring tight control by their input power converter (C or I in Figure 1), to either provide a constant voltage (for static loads) or a constant speed/torque (for rotating loads). The final effect is a constant power delivery from the DC bus to the loads. These loads can be modeled through a single equivalent nonlinear current *I<sup>L</sup>* = *P*/*V*, where *P* is the CPL power. It has to be noticed that both static and rotating loads can be classified as CL or CPL, depending on their operating characteristics. The effect of several CLs and CPLs can be modeled using two equivalent aggregated loads. Given the interest in evaluating voltage stability in a floating-bus system, the simplified microgrid shown in Figure 1 was considered as being supplied by batteries only (power system section depicted in black in Figure 1). Thus, it is possible to model the overall power system in battery-only operation using the equivalent circuit of Figure 2, where *E* is the battery voltage, *V* is the voltage on the CPL, *L* and *C* are the filtering stage components, and *R* is the inductor physical resistance.

*Appl. Sci.* **2018**, *8*, x FOR PEER REVIEW 4 of 18

**Figure 2.** DC microgrid equivalent circuit model. **Figure 2.** DC microgrid equivalent circuit model.

### *2.2. Definition of Hard Lower Bound for DC Load Voltage 2.2. Definition of Hard Lower Bound for DC Load Voltage*

The analysis in Reference [33] introduced the stability problem of a DC floating bus. In particular, an islanded power system supplied only by batteries may be affected by voltage instability in the presence of a perturbation capable of moving the system state outside the BA. This issue becomes critical when the battery has a low SoC and, consequently, its voltage output is low. In this case, the BA shrinks as the bus voltage decreases. This limits the operating margin of the power The analysis in Reference [33] introduced the stability problem of a DC floating bus. In particular, an islanded power system supplied only by batteries may be affected by voltage instability in the presence of a perturbation capable of moving the system state outside the BA. This issue becomes critical when the battery has a low SoC and, consequently, its voltage output is low. In this case, the BA shrinks as the bus voltage decreases. This limits the operating margin of the power system.

system. To introduce the concept of hard lower bound for voltage stability, in this section, the case study was based on the data reported in Reference [33]. The adoption of per unit (p.u.) notation [34] allows performing a numerical continuation analysis [32,33], following the hypothesis of neglecting the presence of the CL (i.e., *RL* = ∞). Indeed, the equivalent linear resistance *RL* determines an increase in the damping factor, thus enhancing the system's voltage stability. Conversely, by neglecting CL, it is possible to assess the stability degradation in the worst case [35], thereby making the negative effect of the floating bus more apparent. The method used here to assess voltage stability relies on the evaluation of the basins of attraction (BAs) in the regions close to the stable operating point (Figure 3). Each stable operating point is defined by the couple of variables (*v*0, *i*0), where *v*0 is the steady-state voltage on the capacitor (in p.u.), and *i*0 is the steady-state nonlinear current in the filter inductor (in p.u.). The progressive reduction in the battery voltage *e* (due to the battery SoC decrease) causes the shift of the equilibrium point toward the upper left-hand side of Figure 3 (i.e., lower *v*0 and higher current *i*0 = *p*/*v*0), with a consequent shrinking in the BA. In particularly, it can be noted that the progressive reduction in BA area (which can be assumed as a measure of system stability) becomes significantly faster for *v*0 below 0.8 p.u. (refer to the area with a red boundary in Figure 3). Based on this consideration, it is possible to consider *v*<sup>0</sup> *=* 0.8 p.u. as a sort of hard lower bound for the steadystate voltage on the load bus (blue basin of Figure 3). Clearly, this limit on *v*0 corresponds to a lower bound also for the battery voltage *e*, whose value depends on the voltage drop in the filter resistance component. Consequently, for each set of input data (i.e., filter components, CPL power, and rated voltage of the CPL), it is also possible to define a lower limit for the battery SoC, using the voltage To introduce the concept of hard lower bound for voltage stability, in this section, the case study was based on the data reported in Reference [33]. The adoption of per unit (p.u.) notation [34] allows performing a numerical continuation analysis [32,33], following the hypothesis of neglecting the presence of the CL (i.e., *R<sup>L</sup>* = ∞). Indeed, the equivalent linear resistance *R<sup>L</sup>* determines an increase in the damping factor, thus enhancing the system's voltage stability. Conversely, by neglecting CL, it is possible to assess the stability degradation in the worst case [35], thereby making the negative effect of the floating bus more apparent. The method used here to assess voltage stability relies on the evaluation of the basins of attraction (BAs) in the regions close to the stable operating point (Figure 3). Each stable operating point is defined by the couple of variables (*v*0, *i*0), where *v*<sup>0</sup> is the steady-state voltage on the capacitor (in p.u.), and *i*<sup>0</sup> is the steady-state nonlinear current in the filter inductor (in p.u.). The progressive reduction in the battery voltage *e* (due to the battery SoC decrease) causes the shift of the equilibrium point toward the upper left-hand side of Figure 3 (i.e., lower *v*<sup>0</sup> and higher current *i*<sup>0</sup> = *p*/*v*0), with a consequent shrinking in the BA. In particularly, it can be noted that the progressive reduction in BA area (which can be assumed as a measure of system stability) becomes significantly faster for *v*<sup>0</sup> below 0.8 p.u. (refer to the area with a red boundary in Figure 3). Based on this consideration, it is possible to consider *v*<sup>0</sup> *=* 0.8 p.u. as a sort of hard lower bound for the steady-state voltage on the load bus (blue basin of Figure 3). Clearly, this limit on *v*<sup>0</sup> corresponds to a lower bound also for the battery voltage *e*, whose value depends on the voltage drop in the filter resistance component. Consequently, for each set of input data (i.e., filter components, CPL power, and rated voltage of the CPL), it is also possible to define a lower limit for the battery SoC, using the voltage limits and the battery specifications.

### limits and the battery specifications. *2.3. Basin of Attraction versus Region of Asymptotic Stability*

The numerical continuation analysis is a method for studying the stability of nonlinear dynamic systems [32,33]. On the one hand, it allows identifying the Hopf bifurcation [36], which appears when the unstable limit cycle and the equilibrium point coalesce, thus making the equilibrium point unstable. Such a bifurcation was displayed at *v*<sup>0</sup> = 0.696 p.u. for the case studied in Reference [33]. On the other hand, the numerical continuation analysis makes it possible to evaluate the BA for several different equilibrium points. In fact, it allows determining the lower voltage stability bound once the magnitude of the possible perturbations is known. In particular, in Reference [33], the BA for *v*<sup>0</sup> = 0.8 p.u. was recognized as the smallest acceptable region for assuring stability in the presence of an impulse perturbation with a reasonable magnitude [37]. Actually, the defined lower bound allows keeping voltage stability after a perturbation constituted by an instantaneous voltage drop of

up to ~30% with respect to the actual working point *v*<sup>0</sup> = 0.8 and *i*<sup>0</sup> = 1.25 p.u. (i.e., the system state moved to *v* = 0.5768 p.u. and *i* = 1.25 p.u.). This can be demonstrated by considering Figure 3, where the perturbed state (yellow triangle) is still in the calculated BA for the starting equilibrium point (blue-bounded area). Conversely, if a lower equilibrium point is assumed (e.g., *v*<sup>0</sup> = 0.7 p.u., resulting in the red-bounded area in Figure 3), the related BA is so small that the system can be considered unstable for any realistic perturbation. *Appl. Sci.* **2018**, *8*, x FOR PEER REVIEW 5 of 18

**Figure 3.** Basins of attraction (BAs) corresponding to shifted equilibrium points. **Figure 3.** Basins of attraction (BAs) corresponding to shifted equilibrium points.

*2.3. Basin of Attraction versus Region of Asymptotic Stability*  The numerical continuation analysis is a method for studying the stability of nonlinear dynamic systems [32,33]. On the one hand, it allows identifying the Hopf bifurcation [36], which appears when the unstable limit cycle and the equilibrium point coalesce, thus making the equilibrium point unstable. Such a bifurcation was displayed at *v*0 = 0.696 p.u. for the case studied in Reference [33]. On the other hand, the numerical continuation analysis makes it possible to evaluate the BA for several different equilibrium points. In fact, it allows determining the lower voltage stability bound once the magnitude of the possible perturbations is known. In particular, in Reference [33], the BA for *v*0 = 0.8 p.u. was recognized as the smallest acceptable region for assuring stability in the presence of an impulse perturbation with a reasonable magnitude [37]. Actually, the defined lower bound allows keeping voltage stability after a perturbation constituted by an instantaneous voltage drop of up to ~30% with respect to the actual working point *v*0 = 0.8 and *i*0 = 1.25 p.u. (i.e., the system state moved to *v* = 0.5768 p.u. and *i* = 1.25 p.u.). This can be demonstrated by considering Figure 3, where the A different approach to define the stability limits of the system is to establish the region of asymptotic stability (RAS). By applying the Lyapunov theory [38], it is possible to find a conservative region, i.e., the RAS, where a sufficient but not necessary condition for large-signal stability is verified [30]. This, in turn, means that any system state inside the RAS originates a transient that evolves toward a stable equilibrium point. Conversely, system states located outside the RAS are not guaranteed to originate an unstable evolution. Although RAS provides only a subset of the entire BA, its determination is significant. Unlike the BA, which is found numerically, the RAS is defined analytically. Therefore, it is possible to use it in algorithms and control laws able to perform a stability-oriented real-time load management in floating-bus islanded DC microgrids. In particular, the mathematical approach at the basis of the definition of the RAS is used in this paper to define a stability preserving criterion, able to suitably reduce the CPL power when the bus voltage drops. In such a way, the voltage stability is guaranteed even in the presence of low battery voltage output *e*, i.e., low SoC, assuring the correct supply of the remaining loads.

### perturbed state (yellow triangle) is still in the calculated BA for the starting equilibrium point (blue-*2.4. Numerical Simulation*

*2.4. Numerical Simulation* 

define the system's states as in Equation (1).

bounded area). Conversely, if a lower equilibrium point is assumed (e.g., *v*0 = 0.7 p.u., resulting in the red-bounded area in Figure 3), the related BA is so small that the system can be considered unstable for any realistic perturbation. By removing the conventional load CL from the equivalent circuit in Figure 2, it is possible to define the system's states as in Equation (1).

$$\begin{cases} \stackrel{\bullet}{i} = \frac{di}{dt} = \frac{1}{l}(e - ri - v) \\\stackrel{\bullet}{v} = \frac{dv}{dt} = \frac{1}{c}(i - i\_L) = \frac{1}{c}(i - \frac{p}{v}) \end{cases} \tag{1}$$

[30]. This, in turn, means that any system state inside the RAS originates a transient that evolves toward a stable equilibrium point. Conversely, system states located outside the RAS are not where *v* is the CPL voltage, and *i* is the current flowing in the inductor *l*. The system parameters were those used in Reference [33], while the CPL power was set at its rated value (i.e., *p* = 1 p.u.).

guaranteed to originate an unstable evolution. Although RAS provides only a subset of the entire BA, its determination is significant. Unlike the BA, which is found numerically, the RAS is defined analytically. Therefore, it is possible to use it in algorithms and control laws able to perform a stability-oriented real-time load management in floating-bus islanded DC microgrids. In particular, the mathematical approach at the basis of the definition of the RAS is used in this paper to define a Using Matlab Simulink, Equation (1) was implemented to perform simulations to assess the *v*–*i* transients following a perturbation. The results were used to provide a dynamic validation of the stability limits previously calculated. The studied DC microgrid in steady-state condition (working point *v*<sup>0</sup> = 0.8 and *i*<sup>0</sup> = 1.25 p.u.) was perturbed at *t* = *t*<sup>0</sup> = 0.2 s by a voltage impulse

By removing the conventional load CL from the equivalent circuit in Figure 2, it is possible to

i.e., low SoC, assuring the correct supply of the remaining loads.

capable of instantaneously moving the voltage *v* applied to the CPL to the new *v*(*t*0) voltage. After the perturbation, the state variables were free to evolve. The study developed in Reference [37] previously demonstrated that this perturbation can be employed for effectively testing the large-signal stability. Therefore, in the following sections, a voltage impulse was considered proper for evaluating the capability offered by the two approaches. The variation in the perturbed initial state *v*(*t*0) allows comparing the consequent *v*–*i* transients shown in Figures 4 and 5. In particular, it is possible to notice an unstable behavior when *v*(*t*0) = 0.56 p.u. (black curves), whilst red/green transients (*v*(*t*0) = 0.59 p.u.) are stable and converge toward the pre-disturbance working point. Thus, the performed simulations verified the lower voltage stability limit for the system calculated using BA analyses (*v* ≈ 0.57 p.u.). perturbation, the state variables were free to evolve. The study developed in Reference [37] previously demonstrated that this perturbation can be employed for effectively testing the largesignal stability. Therefore, in the following sections, a voltage impulse was considered proper for evaluating the capability offered by the two approaches. The variation in the perturbed initial state *v*(*t*0) allows comparing the consequent *v*–*i* transients shown in Figures 4 and 5. In particular, it is possible to notice an unstable behavior when *v*(*t*0) = 0.56 p.u. (black curves), whilst red/green transients (*v*(*t*0) = 0.59 p.u.) are stable and converge toward the pre-disturbance working point. Thus, the performed simulations verified the lower voltage stability limit for the system calculated using BA analyses (*v ≈* 0.57 p.u.).

instantaneously moving the voltage *v* applied to the CPL to the new *v*(*t*0) voltage. After the

*Appl. Sci.* **2018**, *8*, x FOR PEER REVIEW 6 of 18

( )

<sup>=</sup> <sup>=</sup> <sup>−</sup> <sup>=</sup> <sup>−</sup>

where *v* is the CPL voltage, and *i* is the current flowing in the inductor *l*. The system parameters were

*L* 1 1

Using Matlab Simulink, Equation (1) was implemented to perform simulations to assess the *v*–*i* transients following a perturbation. The results were used to provide a dynamic validation of the stability limits previously calculated. The studied DC microgrid in steady-state condition (working

*<sup>p</sup> <sup>i</sup> <sup>c</sup> <sup>i</sup> <sup>i</sup>*

 

, (1)

*v*

( )

those used in Reference [33], while the CPL power was set at its rated value (i.e., *p* = 1 p.u.).

*dt c*

= = − −

1

*<sup>e</sup> ri <sup>v</sup> dt <sup>l</sup>*

  •

*dv <sup>v</sup>*

*di i*

 

•

**Figure 4. Figure 4.** Voltage transients due to a vo Voltage transients due to a voltage impulse perturbation at ltage impulse perturbation at *t t* = 0.2 s. = 0.2 s. *Appl. Sci.* **2018**, *8*, x FOR PEER REVIEW 7 of 18

**Figure 5.** Current transients due to a voltage impulse perturbation at *t* = 0.2 s. **Figure 5.** Current transients due to a voltage impulse perturbation at *t* = 0.2 s.

### *2.5. Validation of Methodological Approach 2.5. Validation of Methodological Approach*

*v*(*t*0) = 0.59 p.u. (red point).

very easy.

which the Lyapunov first derivative (Equation (2)) is negative.

•

 

1 1

•

For the microgrid under study, such a limit is expressed by Equation (3).

Dynamic transients are useful for validating both approaches with regards to the large-signal stability, i.e., BA and RAS. To this aim, BA and RAS for the hard lower bound case (i.e., working point *v*0 = 0.8 p.u. and *i*0 = 1.25 p.u.), together with the dynamic transients after the perturbation, are depicted in the *v*–*i* state plane of Figure 6. As can easily be seen, the BA (blue line) can correctly assess system stability. As expected, the transient starting from outside the basin at *v*(*t*0) = 0.56 p.u. (black point) diverges, whereas the red trace of the transient starting inside the basin at *v*(*t*0) = 0.59 p.u. (red point) converges toward the stable working point. Conversely, the RAS (green line) covers only part Dynamic transients are useful for validating both approaches with regards to the large-signal stability, i.e., BA and RAS. To this aim, BA and RAS for the hard lower bound case (i.e., working point *v*<sup>0</sup> = 0.8 p.u. and *i*<sup>0</sup> = 1.25 p.u.), together with the dynamic transients after the perturbation, are depicted in the *v*–*i* state plane of Figure 6. As can easily be seen, the BA (blue line) can correctly assess system stability. As expected, the transient starting from outside the basin at *v*(*t*0) = 0.56 p.u. (black point) diverges, whereas the red trace of the transient starting inside the basin at *v*(*t*0) = 0.59 p.u. (red point) converges toward the stable working point. Conversely, the RAS (green line) covers only part

 

( ) <sup>0</sup> : *<sup>v</sup> rcv pl* <sup>Ψ</sup> *<sup>v</sup>* <sup>≤</sup> <sup>⇔</sup> *<sup>v</sup>* <sup>≥</sup> <sup>=</sup>

( ) *cv*

0

min

*<sup>p</sup> <sup>v</sup> <sup>v</sup> <sup>l</sup>*

( )

0

By assuming the system parameters used in Reference [33]—CPL power *p* = 1 p.u. and working point *v*0 = 0.8 p.u.—the voltage limit *v*min = 0.6055 p.u. can be determined (represented by the "×" in Figure 6). Actually, *v*min and the basin's voltage limit (yellow triangle located at *v* = 0.5768 p.u.) are very close, having a difference smaller than 0.03 p.u. Considering this gap negligible, the voltage limit *v*min can act as an effective, yet still conservative, margin for the large-signal stability in the presence of the class of perturbations envisaged in Section 2.4. Moreover, it has to be noted that Equation (3) is a simple equation that can be evaluated immediately, thus making the voltage limit assessment

 

<sup>Ψ</sup> <sup>=</sup> <sup>+</sup> <sup>−</sup> <sup>⋅</sup> <sup>−</sup> <sup>−</sup> <sup>−</sup> <sup>+</sup>

*<sup>r</sup> <sup>e</sup> <sup>v</sup> <sup>v</sup> rp*

However, although the RAS can only partially analyze the large-signal behavior, it can constitute a valuable method for designing a stability preserving criterion for floating-bus DC microgrids. Indeed, the determination of the RAS is possible after the identification of the voltage limit above

> 

. (3)

0

*p*

1

*c v*

*lc <sup>v</sup>* . (2)

of the actual BA. This is expected, as the RAS is based only on a sufficient condition. Therefore, it is impossible to predict the system stability through Lyapunov analysis for the transient starting from *v*(*t*0) = 0.59 p.u. (red point). *Appl. Sci.* **2018**, *8*, x FOR PEER REVIEW 8 of 18

**Figure 6.** BA and region of asymptotic stability (RAS); validation when *v*0 = 0.8 p.u. **Figure 6.** BA and region of asymptotic stability (RAS); validation when *v*<sup>0</sup> = 0.8 p.u.

**3. Stability Preserving Criterion**  As previously observed, the system stability is negatively affected by the DC bus voltage decreasing in islanded microgrids feeding a CPL. A possibility for avoiding the consequent instability is to reduce the CPL. In this perspective, a suitably designed CPL management system may be useful However, although the RAS can only partially analyze the large-signal behavior, it can constitute a valuable method for designing a stability preserving criterion for floating-bus DC microgrids. Indeed, the determination of the RAS is possible after the identification of the voltage limit above which the Lyapunov first derivative (Equation (2)) is negative.

$$\overset{\bullet}{\Psi}(\upsilon) = \frac{1}{lc} \left\{ \upsilon + rp\frac{1}{\upsilon} - e \right\} \cdot \left\{ -\frac{r}{l}(\upsilon - \upsilon\_0) - \frac{p}{c}\frac{1}{\upsilon} + \frac{p}{c\upsilon\_0} \right\}.\tag{2}$$

important to understand how the chosen stability index depends on both the CPL power and the floating-bus voltage (Section 3.1). Then, a strategy for decreasing the CPL power can be introduced For the microgrid under study, such a limit is expressed by Equation (3).

$$\stackrel{\bullet}{\Psi}(v) \le 0 \Leftrightarrow v \ge \frac{pl}{rcv\_0} := v\_{\min}.\tag{3}$$

decreases, and the operating point moves consequently. *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 Δ between the equilibrium point *v*0 and the lower bound *v*min as a conservative stability By assuming the system parameters used in Reference [33]—CPL power *p* = 1 p.u. and working point *v*<sup>0</sup> = 0.8 p.u.—the voltage limit *v*min = 0.6055 p.u. can be determined (represented by the "×" in Figure 6). Actually, *v*min and the basin's voltage limit (yellow triangle located at *v* = 0.5768 p.u.) are very close, having a difference smaller than 0.03 p.u. Considering this gap negligible, the voltage limit *v*min can act as an effective, yet still conservative, margin for the large-signal stability in the presence of the class of perturbations envisaged in Section 2.4. Moreover, it has to be noted that Equation (3) is a simple equation that can be evaluated immediately, thus making the voltage limit assessment very easy.

the presence of a sufficiently large stability region (either BA or RAS) in the *v–i* state plane as the SoC

### index. In fact, any perturbation capable of moving the voltage state inside the area defined by Δ does not jeopardize the system stability, as the Lyapunov conservative condition is still verified. **3. Stability Preserving Criterion**

Equation (4).

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*0 decreases and *i*<sup>0</sup> increases), due to the relationship between supply voltage and absorbed current in a CPL. Moreover, the *v*0 drop determines an increase in the *v*min limit, as highlighted by Equation (3), which, in turn, leads to a further reduction of Δ. 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 Δ 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 As previously observed, the system stability is negatively affected by the DC bus voltage decreasing in islanded microgrids feeding a CPL. A possibility for avoiding the consequent instability is to reduce the CPL. In this perspective, a suitably designed CPL management system may be useful for re-establishing a proper stability margin when the bus voltage (which depends on battery SoC in battery-only operation) is low. In order to define such a stability preserving criterion, it is firstly important to understand how the chosen stability index depends on both the CPL power and the floating-bus voltage (Section 3.1). Then, a strategy for decreasing the CPL power can be introduced (see Section 3.2), to guarantee a stable operation even when battery SoC is low. Finally, it is possible

observing Figure 2 and assuming the steady-state condition, the battery voltage *e*0 is defined through

to assess the stability performance ensured by the proposed criterion (see Section 3.3) by verifying the presence of a sufficiently large stability region (either BA or RAS) in the *v*–*i* state plane as the SoC decreases, and the operating point moves consequently.
