# A Mixed-Integer Quadratic Formulation of the Phase-Balancing Problem in Residential Microgrids

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## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

## 3. Model of the Microgrid

## 4. Power Flow Linearization

## 5. Birkhoff Polytope

## 6. Numerical Validation

#### 6.1. Test System

#### 6.2. Scenario Generation

#### 6.3. Numerical Results

## 7. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Schematic representation of the Birkhoff polytope ${\mathcal{B}}_{3}$. This is only a representation since the real politope belongs to the space ${\mathbb{R}}^{3\times 3}$.

**Figure 6.**Voltage in generation and demand nodes: (

**a**) Before phase-balancing and (

**b**) after phase-balancing.

Matrix | Value | Permutation | Determinant |
---|---|---|---|

${M}_{1}$ | $\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$ | ABC | +1 |

${M}_{2}$ | $\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right)$ | BCA | +1 |

${M}_{3}$ | $\left(\begin{array}{ccc}0& 0& 1\\ 1& 0& 0\\ 0& 1& 0\end{array}\right)$ | CAB | +1 |

${M}_{4}$ | $\left(\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right)$ | ACB | −1 |

${M}_{5}$ | $\left(\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right)$ | BAC | −1 |

${M}_{6}$ | $\left(\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right)$ | CBA | −1 |

Scenario | Load/Generation | Probability |
---|---|---|

1 | low/low | 0.2210 |

2 | low/medium | 0.0443 |

3 | low/high | 0.0676 |

4 | medium/low | 0.2767 |

5 | medium/medium | 0.0554 |

6 | medium/high | 0.0845 |

7 | high/low | 0.0845 |

8 | high/medium | 0.0332 |

9 | high/high | 0.0507 |

Node | 8 | 10 | 11 | 12 | 13 | 14 | 16 | 18 | 19 |
---|---|---|---|---|---|---|---|---|---|

Permutation | ${M}_{4}$ | ${M}_{2}$ | ${M}_{4}$ | ${M}_{6}$ | ${M}_{5}$ | ${M}_{5}$ | ${M}_{4}$ | ${M}_{3}$ | ${M}_{6}$ |

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**MDPI and ACS Style**

Garces, A.; Gil-González, W.; Montoya, O.D.; Chamorro, H.R.; Alvarado-Barrios, L.
A Mixed-Integer Quadratic Formulation of the Phase-Balancing Problem in Residential Microgrids. *Appl. Sci.* **2021**, *11*, 1972.
https://doi.org/10.3390/app11051972

**AMA Style**

Garces A, Gil-González W, Montoya OD, Chamorro HR, Alvarado-Barrios L.
A Mixed-Integer Quadratic Formulation of the Phase-Balancing Problem in Residential Microgrids. *Applied Sciences*. 2021; 11(5):1972.
https://doi.org/10.3390/app11051972

**Chicago/Turabian Style**

Garces, Alejandro, Walter Gil-González, Oscar Danilo Montoya, Harold R. Chamorro, and Lazaro Alvarado-Barrios.
2021. "A Mixed-Integer Quadratic Formulation of the Phase-Balancing Problem in Residential Microgrids" *Applied Sciences* 11, no. 5: 1972.
https://doi.org/10.3390/app11051972