# A Stability Preserving Criterion for the Management of DC Microgrids Supplied by a Floating Bus

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

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

#### 2.1. DC Microgrid Topology

- 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;
- Energy storage system, i.e., an electrochemical battery (B);
- LC input filter (F), to assure proper voltage and current quality on the load bus;
- Generic static DC load (L), fed by a DC–DC converter (C);
- Generic rotating load (M), supplied by means of a controlled inverter (I).

_{L}. 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

_{L}= 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.

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

_{L}= ∞). Indeed, the equivalent linear resistance R

_{L}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

_{0}= 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

_{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 limits and the battery specifications.

#### 2.3. Basin of Attraction versus Region of Asymptotic Stability

_{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 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

_{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.

#### 2.4. Numerical Simulation

_{0}= 0.8 and i

_{0}= 1.25 p.u.) was perturbed at t = t

_{0}= 0.2 s by a voltage impulse 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 Figure 4 and Figure 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.).

#### 2.5. Validation of Methodological Approach

_{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 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).

_{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 very easy.

## 3. Stability Preserving Criterion

#### 3.1. Stability Index

_{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 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.

_{0}decreases and i

_{0}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

_{0t}, which is the battery voltage needed to supply the CPL rated power (p = 1 p.u.) at the rated load voltage (v

_{0}= 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

_{0}is defined through Equation (4).

_{0}.

_{0}results in Equation (7), while the total battery voltage e

_{0t}is defined in Equation (8), once the rated condition (p = v

_{0}= 1 p.u.) and full battery (e

_{%}= 1.0) are applied to Equation (4):

_{%}, 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).

_{%}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.

_{%}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.

#### 3.2. Smart CPL Management

_{%}is decreasing). By multiplying Equation (9) with the steady-state voltage v

_{0}, a second-order equation can be obtained:

_{0}and Δ is

_{0}expression (Equation (12)) is substituted into Equation (15), one obtains

_{0}, Δ) can be derived:

_{0}becomes lower.

_{L}and τ

_{C}. The K term allows simplifying Equation (22).

_{0}voltage definition in Equation (4) into Equation (25), one obtains

_{0}) toward a new point with the given (desired) stability index Δ (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 Δ 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 [11,12,13,14,15,16,17,18,19,20], 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).

#### 3.3. Migration of RAS and BA

_{%}and p diminish. Moreover, their area remains constant, due to the stability index invariance (Δ = 0.516) guaranteed by applying the proposed power reduction function.

## 4. Application Example

_{%}= 1, while 336 V at 20% SoC corresponds to e

_{%}= 0.83. For what concerns the stability index, the value Δ = 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,

_{m}is the force applied by the electric motor to the wheels, F

_{f}the force due to the wheel–road friction, and F

_{a}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

_{f}and the air drag force F

_{a}were determined using the following equations:

_{a}(v) the drag force due to the air, and F

_{md}(v) is the dynamic force applied by the electric propulsion motor to the wheels.

_{md}may be obtained.

_{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.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Hansen, J.F.; Wendt, F. History and State of the Art in Commercial Electric Ship Propulsion, Integrated Power Systems, and Future Trends. Proc. IEEE
**2015**, 103, 2229–2242. [Google Scholar] [CrossRef] - Emadi, A. Transportation 2.0. IEEE Power Energy Mag.
**2011**, 9, 18–29. [Google Scholar] [CrossRef] - Zubieta, L.E. Are Microgrids the Future of Energy? DC Microgrids from Concept to Demonstration to Deployment. IEEE Electrif. Mag.
**2016**, 4, 37–44. [Google Scholar] [CrossRef] - Patterson, B.T. DC, Come Home: DC Microgrids and the Birth of the “Enernet”. IEEE Power Energy Mag.
**2012**, 10, 60–69. [Google Scholar] [CrossRef] - IEEE Std. 1709-2010. IEEE Recommended Practice for 1 to 35 kV Medium Voltage DC Power Systems on Ships; IEEE: Piscataway, NJ, USA, 2010. [Google Scholar]
- Meng, L.; Shafiee, Q.; Trecate, G.F.; Karimi, H.; Fulwani, D.; Lu, X.; Guerrero, J.M. Review on Control of DC Microgrids and Multiple Microgrid Clusters. IEEE J. Emerg. Sel. Top. Power Electron.
**2017**, 5, 928–948. [Google Scholar] - Jin, Z.; Sulligoi, G.; Cuzner, R.; Meng, L.; Vasquez, J.C.; Guerrero, J.M. Next-Generation Shipboard DC Power System: Introduction Smart Grid and dc Microgrid Technologies into Maritime Electrical Netowrks. IEEE Electrif. Mag.
**2016**, 4, 45–57. [Google Scholar] [CrossRef] [Green Version] - Emadi, A.; Williamson, S.S.; Khaligh, A. Power Electronics Intensive Solutions for Advanced Electric, Hybrid Electric, and Fuel Cell Vehicular Power Systems. IEEE Trans. Power Electron.
**2006**, 21, 567–577. [Google Scholar] [CrossRef] - Dragičević, T.; Lu, X.; Vasquez, J.C.; Guerrero, J.M. DC Microgrids—Part I: A Review of Control Strategies and Stabilization Techniques. IEEE Trans. Power Electron.
**2016**, 31, 4876–4891. [Google Scholar] - Dragičević, T.; Lu, X.; Vasquez, J.C.; Guerrero, J.M. DC Microgrids—Part II: A Review of Power Architectures, Applications, and Standardization Issues. IEEE Trans. Power Electron.
**2016**, 31, 3528–3549. [Google Scholar] [CrossRef] [Green Version] - Kwasinski, A.; Onwuchekwa, C.N. Dynamic behavior and stabilization of DC microgrids with instantaneous constant-power loads. IEEE Trans. Power Electron.
**2011**, 26, 822–834. [Google Scholar] [CrossRef] - Emadi, A.; Khaligh, A.; Rivetta, C.H.; Williamson, G.A. Constant power loads and negative impedance instability in automotive systems: Definition, modeling, stability and control of power electronic converters and motor drives. IEEE Trans. Veh. Technol.
**2006**, 55, 1112–1125. [Google Scholar] [CrossRef] - Rahimi, A.M.; Emadi, A. Active Damping in DC/DC Power Electronic Converters: A Novel Method to Overcome the Problems of Constant Power Loads. IEEE Trans. Ind. Electron.
**2009**, 56, 1428–1439. [Google Scholar] [CrossRef] - Rahimi, A.M.; Williamson, G.A.; Emadi, A. Loop-Cancellation Technique: A Novel Nonlinear Feedback to Overcome the Destabilizing Effect of Constant-Power Loads. IEEE Trans. Veh. Technol.
**2010**, 59, 650–661. [Google Scholar] [CrossRef] - Bosich, D.; Giadrossi, G.; Sulligoi, G. Voltage control solutions to face the CPL instability in MVDC shipboard power systems. In Proceedings of the AEIT Annual Conference 2014, Trieste, Italy, 18–19 September 2014. [Google Scholar]
- Sulligoi, G.; Bosich, D.; Giadrossi, G.; Zhu, L.; Cupelli, M.; Monti, A. Multiconverter Medium Voltage DC Power Systems on Ships: Constant-Power Loads Instability Solution using Linearization via State Feedback Control. IEEE Trans. Smart Grid
**2014**, 5, 2543–2552. [Google Scholar] [CrossRef] - Cupelli, M.; Ponci, F.; Sulligoi, G.; Vicenzutti, A.; Edrington, C.S.; El-Mezyani, T.; Monti, A. Power Flow Control and Network Stability in an All-Electric Ship. Proc. IEEE
**2015**, 103, 2355–2380. [Google Scholar] [CrossRef] [Green Version] - Bosich, D.; Sulligoi, G.; Mocanu, E.; Gibescu, M. Medium Voltage DC Power Systems on Ships: An Offline Parameter Estimation for Tuning the Controllers’ Linearizing Function. IEEE Trans. Energy Convers.
**2017**, 32, 748–758. [Google Scholar] [CrossRef] - Hossain, E.; Perez, R.; Nasiri, A.; Padmanaban, S. A Comprehensive Review on Constant Power Loads Compensation Techniques. IEEE Access
**2018**, 6, 33285–33305. [Google Scholar] [CrossRef] - Su, M.; Liu, Z.; Sun, Y.; Han, H.; Hou, X. Stability Analysis and Stabilization Methods of DC Microgrid with Multiple Parallel-Connected DC–DC Converters Loaded by CPLs. IEEE Trans. Smart Grid
**2018**, 9, 132–142. [Google Scholar] [CrossRef] - Riccobono, A.; Santi, E. Comprehensive Review of Stability Criteria for DC Power Distribution Systems. IEEE Trans. Ind. Appl.
**2014**, 50, 3525–3535. [Google Scholar] [CrossRef] - Riccobono, A.; Cupelli, M.; Monti, A.; Santi, E.; Roinila, T.; Abdollahi, H.; Arrua, S.; Dougal, R.A. Stability of Shipboard DC Power Distribution: Online Impedance-Based Systems Methods. IEEE Electrif. Mag.
**2017**, 5, 55–67. [Google Scholar] [CrossRef] - Javaid, U.; Freijedo, F.D.; Dujic, D.; van der Merwe, W. Dynamic Assessment of Source–Load Interactions in Marine MVDC Distribution. IEEE Trans. Ind. Electron.
**2017**, 64, 4372–4381. [Google Scholar] [CrossRef] [Green Version] - Belkhayat, M.; Cooley, R.; Witulski, A. Large Signal Stability Criteria for Distributed Systems with Constant Power Loads. In Proceedings of the 26th IEEE Annual Power Electronics Specialists Conference, Atlanta, GA, USA, 18–22 June 1995; pp. 1333–1338. [Google Scholar]
- Griffo, A.; Wang, J.; Howe, D. Large Signal Stability Analysis of DC Power Systems with Constant Power Loads. In Proceedings of the IEEE Vehicle Power and Propulsion Conference (VPPC), Harbin, China, 3–5 September 2008; pp. 1–6. [Google Scholar]
- Herrera, L.; Zhang, W.; Wang, J. Stability Analysis and Controller Design of DC Microgrids with Constant Power Loads. IEEE Trans. Smart Grid
**2017**, 8, 881–888. [Google Scholar] - Aboushady, A.A.; Ahmed, K.H.; Finney, S.J.; Williams, B.W. Lyapunov-based high-performance controller for modular resonant DC/DC converters for medium-voltage DC grids. IET Power Electron.
**2017**, 10, 2055–2064. [Google Scholar] [CrossRef] - Mukherjee, N.; Strickland, D. Control of Cascaded DC–DC Converter-Based Hybrid Battery Energy Storage Systems—Part II: Lyapunov Approach. IEEE Trans. Ind. Electron.
**2016**, 63, 3050–3059. [Google Scholar] [CrossRef] [Green Version] - Kabalan, M.; Singh, P.; Niebur, D. Large Signal Lyapunov-Based Stability Studies in Microgrids: A Review. IEEE Trans. Smart Grid
**2017**, 8, 2287–2295. [Google Scholar] [CrossRef] - Bosich, D.; Gibescu, M.; Sulligoi, G. Large-signal stability analysis of two power converters solutions for DC shipboard microgrid. In Proceedings of the 2017 IEEE Second International Conference on DC Microgrids (ICDCM), Nuremburg, Germany, 27–29 June 2017; pp. 125–132. [Google Scholar]
- Sulligoi, G.; Bosich, D.; Giadrossi, G. Linearizing voltage control of MVDC power systems feeding constant power loads: Stability analysis under saturation. In Proceedings of the 2013 IEEE Power & Energy Society General Meeting, Vancouver, BC, Canada, 21–25 July 2013. [Google Scholar]
- Grillo, S.; Musolino, V.; Sulligoi, G.; Tironi, E. Stability enhancement in DC distribution systems with constant power controlled converters. In Proceedings of the IEEE 15th International Conference on Harmonics and Quality of Power (ICHQP), Hong Kong, China, 17–20 June 2012; pp. 848–854. [Google Scholar]
- Bosich, D.; Giadrossi, G.; Sulligoi, G.; Grillo, S.; Tironi, E. More Electric Vehicles DC Power Systems: A Large Signal Stability Analysis in presence of CPLs fed by Floating Supply Voltage. In Proceedings of the IEEE International Electric Vehicle Conference (IEVC), Florence, Italy, 17–19 December 2014; pp. 1–6. [Google Scholar]
- Chen, M.; Rincón-Mora, G.A. Accurate Electrical Battery model capable of predicting runtime and I-V performance. IEEE Trans. Energy Convers.
**2006**, 21, 504–511. [Google Scholar] [CrossRef] - Rahimi, A.M.; Emadi, A. An Analytical Investigation of DC/DC Power Electronic Converters with Constant Power Loads in Vehicular Power Systems. IEEE Trans. Veh. Technol.
**2009**, 58, 2689–2702. [Google Scholar] [CrossRef] - Kuznestov, Y.A. Elements of Applied Bifurcation Theory, 3rd ed.; Springer: New York, NY, USA, 2004. [Google Scholar]
- Arcidiacono, V.; Monti, A.; Sulligoi, G. Generation control system for improving design and stability of medium-voltage DC power systems on ships. IET Electr. Syst. Transp.
**2012**, 2, 158–167. [Google Scholar] [CrossRef] - Khalil, H.K. Nonlinear Systems; Prentice Hall: Upper Saddle River, NJ, USA, 1992. [Google Scholar]

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

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

Electric motor power | P_{n} | 80 kW |

Battery capacity | 24 kWh | |

Nominal battery voltage | 360 V | |

Battery voltage at full charge | 403 V | |

Battery voltage at 20% SoC | 336 V | |

Mass | M | 1500 kg |

Frontal area | A | 2.28 m^{2} |

Wheel diameter (205/55 R16) | R | 63.16 cm |

Maximum speed (unlimited) | 170 km/h | |

Maximum speed (software limited) | 144 km/h | |

0–100 km/h (unofficial tests) | ~9 s | |

Aerodynamic penetration coefficient | C_{x} | 0.32 |

Battery Voltage (%) | Electric Motor Power (per Unit) | Electric Motor Power (kW) | Wheels Power (kW) |
---|---|---|---|

100 | 1 | 80.00 | 68.00 |

95 | 0.875 | 70.00 | 59.50 |

90 | 0.757 | 60.56 | 51.48 |

85 | 0.646 | 51.68 | 43.93 |

80 | 0.543 | 43.44 | 36.92 |

Static friction coefficient (rubber–asphalt) wet conditions | μ_{s} | 0.5 |

Rolling friction coefficient (rubber–asphalt) | μ_{d} | 0.035 |

Traction control safety coefficient | k_{tc} | 0.75 |

Power loss (from motor to wheels) | 15% | |

Minimum battery SoC | 20% | |

Air density (kg/m^{3}) | r_{0} | 1.29 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bosich, D.; Vicenzutti, A.; Grillo, S.; Sulligoi, G.
A Stability Preserving Criterion for the Management of DC Microgrids Supplied by a Floating Bus. *Appl. Sci.* **2018**, *8*, 2102.
https://doi.org/10.3390/app8112102

**AMA Style**

Bosich D, Vicenzutti A, Grillo S, Sulligoi G.
A Stability Preserving Criterion for the Management of DC Microgrids Supplied by a Floating Bus. *Applied Sciences*. 2018; 8(11):2102.
https://doi.org/10.3390/app8112102

**Chicago/Turabian Style**

Bosich, Daniele, Andrea Vicenzutti, Samuele Grillo, and Giorgio Sulligoi.
2018. "A Stability Preserving Criterion for the Management of DC Microgrids Supplied by a Floating Bus" *Applied Sciences* 8, no. 11: 2102.
https://doi.org/10.3390/app8112102