# Small Signal Stability Analysis of a Microgrid in Grid-Connected Mode

## Abstract

**:**

## 1. Introduction

- Deriving a state space model of a MG in grid-connected mode, which involves a LCL filter, line impedance, and control system.
- Investigation of the impact of the LCL filter and line impedance parameters on the stability of a MG in grid-connected mode.
- Sensitivity analysis of MG parameters was performed to investigate the most sensitive parameters that affect the SSS of a MG in grid-connected mode.

## 2. Proposed Controller Design

## 3. State Space Model

#### 3.1. Phase-Locked Loop

#### 3.2. Droop Controller

#### 3.3. Current Controller

#### 3.4. LCL Filter

#### 3.5. Bus

#### 3.6. Full System

## 4. Simulation Model

^{®}Core™ i7-8750H CPU @ 2.2 GHz processor and memory (RAM) of 16.00 GB.

## 5. Results and Discussion

_{g}and L

_{f}. Therefore, it was decided to study the impact of changing their values on the stability condition.

_{f}that destabilized the system was 6.2 mH. In order to assess the accuracy of the mathematical model, it was decided to simulate the system with different values of L

_{f}and measure the d-component of the voltage at the PCC. It was found in the simulation that the system became unstable at 6.2 mH of L

_{f}, as shown in Figure 9. The figure also shows that when reducing the L

_{f}value by 0.3 mH, the system was stable, while the system was unstable when it was increased by 0.3 mH.

## 6. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Three-phase voltage source inverter connected to a grid via a control method and a LCL filter.

**Figure 8.**Trace of the eigenvalues of the linearized mathematical model under the variation of L

_{f}in the range of 0.16 mH and 0.01 H.

**Figure 10.**Trace of the eigenvalues of the linearized mathematical model under the variation of L

_{g}in the range of 3.44 µH and 0.01 H.

**Figure 12.**Trace of the eigenvalues of the linearized mathematical model under the variation of L

_{c}in the range of 4.77 mH and 0.01 H.

**Table 1.**Parameters of the grid-tied inverter system [26].

Parameter | Symbol | Value |
---|---|---|

Inverter-side parasitic resistance | ${R}_{c}$ | $0.25\mathsf{\Omega}$ |

Grid-side parasitic resistance of the LCL filter | ${R}_{f}$ | $0.25\mathsf{\Omega}$ |

Transmission line parasitic resistance | ${R}_{g}$ | $0.25\mathsf{\Omega}$ |

Inverter-side inductance of the LCL filter | ${L}_{c}$ | $4.77\mathrm{mH}$ |

Grid-side inductance of the LCL filter | ${L}_{f}$ | $0.16\mathrm{mH}$ |

Transmission line inductance | ${L}_{g}$ | $3.44\mathsf{\mu}\mathrm{H}$ |

Filter capacitance | ${C}_{f}$ | $14.7\mathsf{\mu}\mathrm{F}$ |

Proportional gain of the current controller | ${k}_{pc}$ | $14.24$ |

Integral gain of the current controller | ${k}_{ic}$ | $2.11\times {10}^{4}$ |

Frequency droop gain | m | $1\times {10}^{-6}$ |

Voltage droop gain | n | $1\times {10}^{-6}$ |

Inverter input DC voltage | ${v}_{dc}$ | $180\mathrm{V}$ |

Switching frequency | ${f}_{s}$ | $8\mathrm{kHz}$ |

Grid voltage magnitude | ${v}_{g}$ | $120\mathrm{V}$ |

Fundamental frequency | ${f}_{n}$ | $50\mathrm{Hz}$ |

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Alnuman, H.
Small Signal Stability Analysis of a Microgrid in Grid-Connected Mode. *Sustainability* **2022**, *14*, 9372.
https://doi.org/10.3390/su14159372

**AMA Style**

Alnuman H.
Small Signal Stability Analysis of a Microgrid in Grid-Connected Mode. *Sustainability*. 2022; 14(15):9372.
https://doi.org/10.3390/su14159372

**Chicago/Turabian Style**

Alnuman, Hammad.
2022. "Small Signal Stability Analysis of a Microgrid in Grid-Connected Mode" *Sustainability* 14, no. 15: 9372.
https://doi.org/10.3390/su14159372