# Stability Analysis of DC Distribution Systems with Droop-Based Charge Sharing on Energy Storage Devices

^{*}

## Abstract

**:**

## 1. Introduction

_{2}emissions, a task that renewable energy sources such as wind and photovoltaic (PV) power systems can effectively satisfy by providing a large part of “clean” power production. At the consumer side, other innovations such as electric vehicles (EVs) can also play a key role [1]. In light of this evolution, and due to the fact that many of the distributed energy resources (DERs) are or may be DC devices, and since DC-supplied loads are continuously increased, adoption of DC distribution systems and DC microgrids is now feasible. Furthermore, since most of the DERs and DC loads are locally connected through a DC/DC power electronic converter, this provides a significant opportunity for locally controlling the power absorbed or injected at desired rates. In such a scheme, the superiority of DC distribution over AC distribution is efficiency, reliability and economy of DC distribution, since intermediate DC/AC/DC conversion stages are eliminated [2]. Emerging new sources and loads such as batteries, PVs, data centers, office and home appliances (such as computers and printers), as well as different industrial applications (e.g., electrochemical processing) are natively DC-supplied while others such as industrial drives and traction are implemented by using controlled DC sources via DC/AC power conversion (e.g., inverted fed AC motors) [3,4]. Finally, as reactance has little effect on DC transmission, and reactive power does not exist, cables can carry more power with reduced losses.

## 2. System Modeling

## 3. The Proposed Control Scheme

## 4. Stability Analysis of the Nonlinear Closed-Loop System

## 5. The Examined Case Study

## 6. Conclusions

## Author Contributions

## Conflicts of Interest

## Appendix A

## Appendix B

**Theorem**

**A.1.**

**k**$\infty $ functions, $\mathit{\rho}$ is a class

**k**function, and ${\mathit{W}}_{\mathit{3}}$ is a continuous positive definite function on ${\mathit{R}}^{\mathit{n}}$. Then the system is input-to-state stable (ISS).

**Assumption**

**A.1.**

- For any trajectory $\mathit{x}\left(\mathit{t}\right)\in \mathit{\Omega}\subset {\mathit{R}}^{\mathit{n}}$, for all $\mathit{t}\ge \mathit{0}$, the matrix $\mathit{A}\left(\mathit{x}\right)$ is locally Lipchitz and Hurwitz.
- Matrix $\mathit{B}\left(\mathit{x}\right)$ is constant, i.e., $\mathit{B}\left(\mathit{x}\right)=\mathit{B}$.
- Input ${\mathit{u}}_{\mathit{d}}$ is assumed to be constant, i.e., ${\mathit{u}}_{\mathit{d}}\left(\mathit{t}\right)=\mathit{c}$, $\mathit{t}\ge \mathit{0}$.

**Assumption**

**A.2.**

- System (A1), (A2) is passive with respect to the input $\mathit{u}$ and output $\mathit{y}$, for some storage function $\mathit{V}\left(\mathit{x}\left(\mathit{t}\right)\right)\ge \mathit{0}$.
- There exist non-zero equilibrium points for (A1): ${\mathit{x}}_{\mathit{e}}\in \mathit{M}\subset \mathit{\Omega}$ that are distinct, each satisfying the equation $\dot{\mathit{V}}\left({\mathit{x}}_{\mathit{e}}\right)=\mathit{0}$, for some ${\mathit{u}}_{\mathit{d}}\left(\mathit{t}\right)=\mathit{c}\ne \mathit{0}$.
- No limit cycles exist in $\mathit{\Omega}$.

**Theorem**

**A.2.**

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**Figure 5.**(

**a**) The net input supply at each distributed energy resource (DER) bus and the current at the constant power load bus; (

**b**) Details of the current changes of the constant power load at the load bus (zoomed).

Parameters | Value |
---|---|

${L}_{bat}$ | 100 mH |

${R}_{ser}$ | 0.0745 Ω |

${C}_{o}$ | 4475 F |

${R}_{o,1}={R}_{o,2}$ | 0.0489 Ω |

${V}_{o,1}={V}_{o,2}$ | 200 V |

${R}_{1}={R}_{2}$ | 120 Ω |

${C}_{1}={C}_{2}$ | 1.1 mF |

$C$ | 1.1 mF |

${L}_{line}$ | 0.4 mH |

${R}_{line}$ | 0.15 Ω |

${C}_{cap}$ | 3060 F |

Gain | Value |
---|---|

${k}_{p,c}$ | 10 |

${k}_{i,c}$ | 0.1 |

${k}_{p,v}$ | 0.5 |

${k}_{i,v}$ | 25 |

${k}_{1,droop}$ | 0.005 |

${k}_{2,droop}$ | 0.001 |

${k}_{f1}={k}_{f2}$ | 0.01 |

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**MDPI and ACS Style**

Makrygiorgou, D.I.; Alexandridis, A.T. Stability Analysis of DC Distribution Systems with Droop-Based Charge Sharing on Energy Storage Devices. *Energies* **2017**, *10*, 433.
https://doi.org/10.3390/en10040433

**AMA Style**

Makrygiorgou DI, Alexandridis AT. Stability Analysis of DC Distribution Systems with Droop-Based Charge Sharing on Energy Storage Devices. *Energies*. 2017; 10(4):433.
https://doi.org/10.3390/en10040433

**Chicago/Turabian Style**

Makrygiorgou, Despoina I., and Antonio T. Alexandridis. 2017. "Stability Analysis of DC Distribution Systems with Droop-Based Charge Sharing on Energy Storage Devices" *Energies* 10, no. 4: 433.
https://doi.org/10.3390/en10040433