# Small Signal Stability of a Balanced Three-Phase AC Microgrid Using Harmonic Linearization: Parametric-Based Analysis

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## Abstract

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## 1. Introduction

## 2. System Modeling Using HL

## 3. Mathematical Modeling of the Impedances at the PCC

## 4. Simulation Results

- First, stability was assessed by changing the parallel clustering (penetration) of grid-connected active loads.
- Then, the stability was evaluated by changing the distance of active loads from the PCC.
- Afterwards, the stability at the PCC was assessed by changing the serial clustering (size) of active loads.

#### 4.1. Effect of Penetration

#### 4.2. Effect of Distance from PCC

#### 4.3. Effect of the Sizes of Active Loads

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Comparative stability analysis of grid-connected active loads at the point of common coupling (PCC) with changing size, distance, and penetration.

Model | Disadvantages | Reference |
---|---|---|

Lyapunov Methods (Time Domain) | A detailed system modeling is required for this method, so it does not work well for complex large systems. Its converter model is unable to capture the harmonic effect. | [28,29,31,33] |

Probabilistic Methods (Time Domain) | It requires huge computational effort, so this method is very time consuming. Inaccurate first approximation may lead to faulty conclusions. Not all applied schemes work for complex large systems. | [32] |

Phasor Model | It is often not differentiable due to significantly higher dimensions. | [20] |

Bifurcation Theory | It is slow in the time domain and more complicated in the frequency domain for a higher order system. | [30] |

SRF Method | Limited to only balanced three-phase systems. | [20,33] |

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

Power Source | P | 2500 MVA | V | 120 kV |

Line Section | $R\left[{r}_{1}\right]$ | [0.1153] | $L\left[{r}_{1}\right]$ | $[1.05\times {10}^{-3}]$ |

Line Section | $R\left[{r}_{0}\right]$ | [0.413] | $L\left[{r}_{0}\right]$ | $[3.32\times {10}^{-3}]$ |

Industrial Load | P | 30 MW | Q | 2 MVar |

Residential Load 1 | P | 2 MW | Q | 0 |

Residential Load 2 | P | 100 kW | Q | 0 |

Penetration | Distance at PCC | Size | Effect on Stability |
---|---|---|---|

Increasing | Constant | Constant | Improve |

Constant | Increasing | Constant | Improve |

Constant | Constant | Increasing | Deteriorate |

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**MDPI and ACS Style**

Rahman, A.U.; Syed, I.; Ullah, M.
Small Signal Stability of a Balanced Three-Phase AC Microgrid Using Harmonic Linearization: Parametric-Based Analysis. *Electronics* **2019**, *8*, 12.
https://doi.org/10.3390/electronics8010012

**AMA Style**

Rahman AU, Syed I, Ullah M.
Small Signal Stability of a Balanced Three-Phase AC Microgrid Using Harmonic Linearization: Parametric-Based Analysis. *Electronics*. 2019; 8(1):12.
https://doi.org/10.3390/electronics8010012

**Chicago/Turabian Style**

Rahman, Atta Ur, Irtaza Syed, and Mukhtar Ullah.
2019. "Small Signal Stability of a Balanced Three-Phase AC Microgrid Using Harmonic Linearization: Parametric-Based Analysis" *Electronics* 8, no. 1: 12.
https://doi.org/10.3390/electronics8010012