# A Step-by-Step Methodology for Obtaining the Reliability of Building Microgrids Using Fault TreeAnalysis

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## Abstract

**:**

## 1. Introduction

## 2. Proposed Methodology

#### 2.1. First Step: Definitions of MG Circuit Models

#### 2.2. Second Step: Selection of a Probability Density Function

#### 2.3. Third Step: Exponential Distribution

#### 2.4. Fourth Step: Failure Rate Estimation

#### 2.5. Fifth Step: Fault Tree Analysis

#### Advantages of Fault Tree Analysis:

#### Disadvantages of Fault Tree Analysis:

#### 2.5.1. Fifth Step (A): FTA Qualitative Analysis

#### 2.5.2. Fifth Step (B): FTA Quantitative Analysis

#### Reliability Analysis

#### Importance Measures

## 3. Results

#### 3.1. Circuit Model Definitions and Selection of the PDF

#### 3.2. Results of Applying the Fourth Step: Failure Rate Estimation

#### 3.3. Results of Applying the Fifth Step: Fault Tree Analysis

#### 3.3.1. Results of the FTA Qualitative Analysis

#### Minimal Cut-Sets

- {${\mathrm{Battery}}_{\mathrm{Failure}}$};
- {${\mathrm{BMS}}_{\mathrm{Failure}}$};
- {${\mathrm{Inverter}/\mathrm{Charger}}_{\mathrm{Failure}}$};
- {${\mathrm{Microinverter}}_{\mathrm{Failure}}$, ${\mathrm{SolarPanelAC}}_{\mathrm{Failure}}$, ${\mathrm{MPPT}}_{\mathrm{Failure}}$};
- {${\mathrm{Microinverter}}_{\mathrm{Failure}}$, ${\mathrm{SolarPanelAC}}_{\mathrm{Failure}}$, ${\mathrm{SolarPanelDC}}_{\mathrm{Failure}}$}.

#### 3.3.2. Results of the FTA Quantitative Analysis

#### Results of Reliability Analysis

#### Results of Importance Measure

## 4. Final Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 9.**University of Antioquia (UdeA) Microgrid. (

**Left**side): PV panels; (

**right**side): industrial cabinets with power devices.

IGBT | Diode | Capacitor | Inductor | |
---|---|---|---|---|

${\lambda}_{b}$ | 0.00074 | 0.005 | 0.00012 | 0.00003 |

${\pi}_{T}$ | 1.40 | 1.64 | 1.92 | 1.23 |

${\pi}_{S}$ | 0.42 | 0.55 | - | - |

${\pi}_{A}$ | 0.79 | - | - | - |

${\pi}_{R}$ | 8.25 | - | - | - |

${\pi}_{Q}$ | 5.50 | 5.50 | 10 | 3 |

${\pi}_{E}$ | 6 | 6 | 6 | 6 |

${\pi}_{C}$ | - | 1 | 5.87 | - |

${\pi}_{V}$ | - | - | 4.48 | - |

${\lambda}_{\mathit{total}}$ | $9.00\times {10}^{-8}$ | $1.50\times {10}^{-7}$ | $3.60\times {10}^{-7}$ | $6.64\times {10}^{-10}$ |

IGBT | Diode | Capacitor C1 | Cap. C2, C3 | Inductor | |
---|---|---|---|---|---|

${\lambda}_{b}$ | 0.00074 | 0.005 | 0.00012 | 0.00012 | 0.00003 |

${\pi}_{T}$ | 1.40 | 1.64 | 1.92 | 1.92 | 1.23 |

${\pi}_{S}$ | 0.42 | 0.15 | - | - | - |

${\pi}_{A}$ | 0.79 | - | - | - | - |

${\pi}_{R}$ | 16.89 | - | - | - | - |

${\pi}_{Q}$ | 5.50 | 5.50 | 10 | 10 | 3 |

${\pi}_{E}$ | 6 | 6 | 6 | 6 | 6 |

${\pi}_{C}$ | - | 1 | 5.87 | 6.89 | - |

${\pi}_{V}$ | - | - | 1.01 | 1.27 | - |

${\lambda}_{\mathit{total}}$ | $1.90\times {10}^{-7}$ | $4.00\times {10}^{-8}$ | $8.00\times {10}^{-8}$ | $1.20\times {10}^{-7}$ | $6.64\times {10}^{-10}$ |

IGBT | Diode | Capacitor | Inductor | |
---|---|---|---|---|

${\lambda}_{b}$ | 0.00074 | 0.005 | 0.00012 | 0.00003 |

${\pi}_{T}$ | 1.40 | 1.64 | 1.92 | 1.23 |

${\pi}_{S}$ | 0.42 | 0.15 | - | - |

${\pi}_{A}$ | 0.79 | - | - | - |

${\pi}_{R}$ | 12.88 | - | - | - |

${\pi}_{Q}$ | 5.50 | 5.50 | 10 | 3 |

${\pi}_{E}$ | 6 | 6 | 6 | 6 |

${\pi}_{C}$ | - | 1 | 5.87 | - |

${\pi}_{V}$ | - | - | 4.48 | - |

${\lambda}_{\mathit{total}}$ | $1.50\times {10}^{-7}$ | $4.00\times {10}^{-8}$ | $3.60\times {10}^{-7}$ | $6.64\times {10}^{-10}$ |

Diode | |
---|---|

${\lambda}_{b}$ | 0.004 |

${\pi}_{T}$ | 1.57 |

${\pi}_{Q}$ | 5.50 |

${\pi}_{E}$ | 1 |

${\lambda}_{\mathit{total}}$ | $3.00\times {10}^{-8}$ |

IGBT | Diode | Capacitor | Inductor | |
---|---|---|---|---|

${\lambda}_{b}$ | 0.00074 | 0.005 | 0.00012 | 0.00003 |

${\pi}_{T}$ | 1.40 | 1.19 | 1.92 | 1.23 |

${\pi}_{S}$ | 0.42 | 0.12 | - | - |

${\pi}_{A}$ | 0.79 | - | - | - |

${\pi}_{R}$ | 20.59 | - | - | - |

${\pi}_{Q}$ | 5.50 | 0.70 | 10 | 3 |

${\pi}_{E}$ | 6 | 6 | 6 | 6 |

${\pi}_{C}$ | - | 1 | 8.03 | - |

${\pi}_{V}$ | - | - | 5.21 | - |

${\lambda}_{\mathit{total}}$ | $2.30\times {10}^{-7}$ | $3.00\times {10}^{-9}$ | $5.80\times {10}^{-7}$ | $6.64\times {10}^{-10}$ |

Device | Failure Rate [Failure/Hour] | MTTF [Years/Failure] |
---|---|---|

Microinverter | $1.71\times {10}^{-6}$ | 66.73 |

Inverter/Charger | $2.16\times {10}^{-6}$ | 52.83 |

MPPT | $7.41\times {10}^{-7}$ | 154.13 |

Solar Panel | $1.80\times {10}^{-6}$ | 63.42 |

BMS | $8.37\times {10}^{-6}$ | 13.63 |

Battery | $1.289\times {10}^{-5}$ [28] | 8.86 |

Probability of Failure | Year | ||||||||
---|---|---|---|---|---|---|---|---|---|

5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | |

AC Bus | 0.630 | 0.863 | 0.950 | 0.982 | 0.993 | 0.997 | 0.999 | 1.000 | 1.000 |

Battery and BMS | 0.602 | 0.841 | 0.937 | 0.976 | 0.991 | 0.996 | 0.998 | 0.999 | 1.000 |

K of N , AC solar module | 0.000 | 0.004 | 0.079 | 0.327 | 0.641 | 0.856 | 0.954 | 0.988 | 0.997 |

DC solar module | 0.032 | 0.065 | 0.101 | 0.138 | 0.178 | 0.219 | 0.262 | 0.305 | 0.349 |

Battery power | 0.000 | 0.000 | 0.008 | 0.045 | 0.114 | 0.188 | 0.250 | 0.302 | 0.348 |

Time (hrs) | Unavailability | Conditional Failure Intensity | Failure Frequency | Mean Unavailability |
---|---|---|---|---|

0.000 | 0.000000 | 23.420000 | 23.420000 | 0.000000 |

100.000 | 0.002339 | 23.420000 | 23.365215 | 0.001170 |

200.000 | 0.004673 | 23.420000 | 23.310558 | 0.002338 |

300.000 | 0.007001 | 23.420001 | 23.256028 | 0.003505 |

400.000 | 0.009324 | 23.420001 | 23.201627 | 0.004669 |

500.000 | 0.011642 | 23.420002 | 23.147353 | 0.005832 |

600.000 | 0.013954 | 23.420002 | 23.093206 | 0.006993 |

700.000 | 0.016260 | 23.420003 | 23.039186 | 0.008152 |

800.000 | 0.018562 | 23.420004 | 22.985292 | 0.009310 |

900.000 | 0.020857 | 23.420006 | 22.931525 | 0.010465 |

1000.000 | 0.023148 | 23.420007 | 22.877883 | 0.011619 |

Probability | Events |
---|---|

0.012807 | ${\mathrm{Battery}}_{\mathrm{Failure}}$ |

0.008335 | ${\mathrm{BMS}}_{\mathrm{Failure}}$ |

0.002158 | ${\mathrm{Inverter}/\mathrm{Charger}}_{\mathrm{Failure}}$ |

$2.275956\times {10}^{-9}$ | ${\mathrm{Microinverter}}_{\mathrm{Failure}}$, ${\mathrm{SolarPanelAC}}_{\mathrm{Failure}}$, ${\mathrm{MPPT}}_{\mathrm{Failure}}$ |

$1.787111\times {10}^{-14}$ | ${\mathrm{Microinverter}}_{\mathrm{Failure}}$, ${\mathrm{SolarPanelAC}}_{\mathrm{Failure}}$, ${\mathrm{SolarPanelDC}}_{\mathrm{Failure}}$ |

Time (Years) | Unavailability | Conditional Failure Intensity | Failure Frequency | Mean Unavailability |
---|---|---|---|---|

1 year | 0.183191 | 23.420499 | 19.130069 | 0.094682 |

2 years | 0.332829 | 23.421953 | 15.626446 | 0.177607 |

3 years | 0.455063 | 23.424307 | 12.764783 | 0.250406 |

4 years | 0.554912 | 23.427516 | 10.427304 | 0.314472 |

5 years | 0.636477 | 23.431544 | 8.517894 | 0.370991 |

Probability | Events |
---|---|

0.426988 | ${\mathrm{Battery}}_{\mathrm{Failure}}$ |

0.303428 | ${\mathrm{BMS}}_{\mathrm{Failure}}$ |

0.089091 | ${\mathrm{Inverter}/\mathrm{Charger}}_{\mathrm{Failure}}$ |

$1.678368\times {10}^{-4}$ | ${\mathrm{Microinverter}}_{\mathrm{Failure}}$, ${\mathrm{SolarPanelAC}}_{\mathrm{Failure}}$, ${\mathrm{MPPT}}_{\mathrm{Failure}}$ |

$2.230790\times {10}^{-6}$ | ${\mathrm{Microinverter}}_{\mathrm{Failure}}$, ${\mathrm{SolarPanelAC}}_{\mathrm{Failure}}$, ${\mathrm{SolarPanelDC}}_{\mathrm{Failure}}$ |

Event | Marginal | Criticality | Diagnostic | Risk Achievement Worth | Risk Reduction Worth |
---|---|---|---|---|---|

Battery | 0.634406 | 0.425598 | 0.670861 | 1.571148 | 1.740941 |

BMS | 0.521874 | 0.248793 | 0.476730 | 1.571148 | 1.331191 |

Inverter/Charger | 0.399077 | 0.055861 | 0.139975 | 1.571148 | 1.059166 |

Microinverter | $8.679799\times {10}^{-4}$ | $9.711004\times {10}^{-5}$ | 0.071300 | 1.001267 | 1.000097 |

MPPT | 0.001936 | 9.583571 × 10^{−5} | 0.031597 | 1.002946 | 1.000096 |

Solar Panel | $1.049981\times {10}^{-5}$ | $1.234180\times {10}^{-6}$ | 0.074815 | 1.000015 | 1.000001 |

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## Share and Cite

**MDPI and ACS Style**

Patiño-Álvarez, G.A.; Arias-Pérez, J.S.; Muñoz-Galeano, N.
A Step-by-Step Methodology for Obtaining the Reliability of Building Microgrids Using Fault TreeAnalysis. *Computers* **2024**, *13*, 131.
https://doi.org/10.3390/computers13060131

**AMA Style**

Patiño-Álvarez GA, Arias-Pérez JS, Muñoz-Galeano N.
A Step-by-Step Methodology for Obtaining the Reliability of Building Microgrids Using Fault TreeAnalysis. *Computers*. 2024; 13(6):131.
https://doi.org/10.3390/computers13060131

**Chicago/Turabian Style**

Patiño-Álvarez, Gustavo A., Johan S. Arias-Pérez, and Nicolás Muñoz-Galeano.
2024. "A Step-by-Step Methodology for Obtaining the Reliability of Building Microgrids Using Fault TreeAnalysis" *Computers* 13, no. 6: 131.
https://doi.org/10.3390/computers13060131