# Improved Semi-Supervised Data-Mining-Based Schemes for Fault Detection in a Grid-Connected Photovoltaic System

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## Abstract

**:**

## 1. Introduction

## 2. PV Installation Description

**STC**symbolizes the Standard Test Conditions (

**G**= 1000 W/m

^{2},

**T**= 25 °C, and

_{C}**AM**= 1.5) and

**MPP**symbolizes the Maximum Power Point.

**G**is the received Irradiance by the PV module during the flash test,

**T**is the temperature of the PV Cell, and

_{C}**AM**is the Air Mass.

**I**is the short circuit current,

_{SC}**V**is the open circuit voltage,

_{OC}**I**is the current at MPP,

_{MPP}**V**is the voltage at MPP, and

_{MPP}**P**is the maximum power.

_{M}## 3. Materials and Methods

#### 3.1. PLS (Partial Least Square)

**E**is the residual matrix. Finally, the output variable Y is obtained via the following expression:

#### 3.2. PCR (Principal Component Regression)

**X**using Principal component analysis (PCA). After that, the resulting retained principal components (CPs) are then linked to the response variables via the Ordinary Least Squares (OLS) regression technique [51,52] (Figure 5).

**E**are the approximated and residual matrices, respectively. Here, $\mathrm{T}\in {\mathrm{R}}^{\mathrm{n}\times \mathrm{m}}$ and $\mathrm{W}\in {\mathrm{R}}^{\mathrm{m}\times \mathrm{m}}$ represent the principal components (PCs) and the loading matrices, respectively. Notice that the cumulative percentage variance (CPV) technique is widely employed to decide the number of PCs to be retained in the model [53]. Therefore, PCR constructs the linear regression between the matrix $\widehat{\mathbf{T}}$ of the k retained principal components and the response variable

**y**as the solution for the following optimization problem:

#### 3.3. TEWMA (Triple Exponential Weighted Moving Average)

_{i}is defined as:

_{i}, is expressed as:

_{i}can be calculated as follow:

_{1}, λ

_{2}, and λ

_{3}can either be equal or different. It has been shown that the Double EWMA (DEWMA) maintains mostly the same performance for the same or different smooth parameters [56,57]. The TEWMA charting statistic with the same smoothing parameter is rewritten as follows [54,58]:

_{i}can be written as [54,58]:

_{i}statistic of the present sample can be computed as pondered sum function of all paste samples j [54,56]:

_{i}and TE

_{i}can be written as [54,56]:

_{i}in Equation (13), DE

_{i}can be expressed in terms of x

_{i}as:

_{0}into DE

_{0}to obtain the final pondered form [54]:

_{i}is expressed as [54]:

_{0}into TE

_{0}, to obtain the TE

_{i}statistic as:

_{i}values below the limits indicate that the monitored system operates without anomalies. On the other hand, anomalies can be identified when TE

_{i}values overpass the defined decision limits.

#### 3.4. KDE-TEWMA (Kernel Density Estimation TEWMA)

_{i}is the ith observation. K is the kernel function; the Gaussian kernel is usually used [60]:

#### 3.5. Dataset Analysis

**PCs**can strongly affect the quality of the constructed prediction models. We have applied the cumulative percentage variance (CPV) method to determine the suitable number of CPs. As depicted in Figure 9, the obtained results show that five and four CPs are needed to describe 99.87% of the variability in X for both PLS (Figure 9a) and 99.98 for PCR (Figure 9b) models, respectively.

## 4. The LVR-TEWMA-Based Fault Detection in PV Systems

## 5. Results

#### 5.1. Scenarios with String Faults

#### 5.2. Scenarios with Inverter Disconnections

#### 5.3. Scenario with Circuit Breaker Faults

#### 5.4. Short-Circuit Fault

#### 5.5. Sensor Bias Faults in the Pyranometer

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 12.**General fault detection procedure in PV system using LVR-based KDE triple exponentially smoothing driven monitoring schemes.

**Figure 13.**Detection results of (

**a**) PLS and (

**b**) PCR-based TEWMA charts in the presence of string fault.

**Figure 14.**Detection results of (

**a**) PLS- and (

**b**) PCR-based TEWMA charts in the presence of inverter disconnections.

**Figure 15.**Detection results of (

**a**) PLS- and (

**b**) PCR-based TEWMA charts in the presence of RCCB faults.

**Figure 16.**Detection results of (

**a**) PLS- and (

**b**) PCR-based TEWMA charts in the presence of two short-circuited modules.

**Figure 17.**Detection results of (

**a**) PLS- and (

**b**) PCR-based TEWMA charts in the presence of sensor bias fault in the pyranometer measurements (a bias of 10% of the total variation in solar irradiance measurements).

**Figure 18.**AUC values by PLS- and PCR-based TEWMA charts for different bias magnitudes in the pyranometer.

Parameters | I_{SC} (A) | V_{OC} (V) | I_{MPP} (A) | V_{MPP} (V) | P_{M} (W) |
---|---|---|---|---|---|

PV Module | 6.54 | 21.6 | 6.1 | 17.4 | 106 |

PV Sub-Array | 13.08 | 324 | 12.2 | 261 | 3180 |

Parameters | Nominal AC Power (W) | DC Voltage Range (V) | Inverter Efficiency (%) | AC Voltage Range (V) | Frequency Range (Hz) |
---|---|---|---|---|---|

Value | 2500 | 150–400 | 92.7–94.3 | 195–253 | 49.8–50.2 |

Method | TPR | FPR | Accuracy | AUC | EER |
---|---|---|---|---|---|

PLS-TEWMA | 0.98 | 0 | 0.9942 | 0.99 | 0.0034 |

PCR-TEWMA | 0.98 | 0 | 0.9942 | 0.99 | 0.0034 |

Method | TPR | FPR | Accuracy | AUC | EER |
---|---|---|---|---|---|

PLS-TEWMA | 1 | 0.0418 | 0.9583 | 0.9791 | 0.0417 |

PCR-TEWMA | 0.75 | 0.0399 | 0.9593 | 0.8550 | 0.0407 |

Method | TPR | FPR | Accuracy | AUC | EER |
---|---|---|---|---|---|

PLS-TEWMA | 0.9815 | 0.0220 | 0.9782 | 0.9797 | 0.0218 |

PCR-TEWMA | 0.9815 | 0.0210 | 0.9791 | 0.9802 | 0.0209 |

Method | TPR | FPR | Accuracy | AUC | EER |
---|---|---|---|---|---|

PLS-TEWMA | 0.8649 | 0 | 0.9823 | 0.9324 | 0.0095 |

PCR-TEWMA | 0.1351 | 0 | 0.8867 | 0.5676 | 0.0606 |

Bias Sensor (B) | TEWMA Method | TPR | FPR | Accuracy | AUC | EER |
---|---|---|---|---|---|---|

50% | PLS | 0.9780 | 0 | 0.9931 | 0.9890 | 0.0041 |

PCR | 0.9451 | 0 | 0.9828 | 0.9725 | 0.0102 | |

40% | PLS | 0.9670 | 0 | 0.9897 | 0.9835 | 0.0061 |

PCR | 0.9341 | 0 | 0.9794 | 0.9670 | 0.0122 | |

30% | PLS | 0.9670 | 0 | 0.9897 | 0.9835 | 0.0061 |

PCR | 0.9231 | 0 | 0.9759 | 0.9615 | 0.0143 | |

20% | PLS | 0.9560 | 0 | 0.9863 | 0.9780 | 0.0081 |

PCR | 0.9011 | 0 | 0.9691 | 0.9505 | 0.0183 | |

10% | PLS | 0.9451 | 0 | 0.9828 | 0.9725 | 0.0102 |

PCR | 0.8571 | 0 | 0.9553 | 0.9286 | 0.0265 | |

5% | PLS | 0.9231 | 0 | 0.9759 | 0.9615 | 0.0143 |

PCR | 0.7802 | 0 | 0.9313 | 0.8901 | 0.0407 |

Bias Sensor (B) | DEWMA Method | TPR | FPR | Accuracy | AUC | EER |
---|---|---|---|---|---|---|

50% | PLS | 0.9622 | 0 | 0.9776 | 0.9811 | 0.0224 |

PCR | 0.9553 | 0 | 0.9735 | 0.9777 | 0.0265 | |

40% | PLS | 0.9588 | 0 | 0.9756 | 0.9794 | 0.0244 |

PCR | 0.9313 | 0 | 0.9593 | 0.9656 | 0.0407 | |

30% | PLS | 0.9313 | 0 | 0.9593 | 0.9656 | 0.0407 |

PCR | 0.9038 | 0 | 0.9430 | 0.9519 | 0.0570 | |

20% | PLS | 0.9003 | 0 | 0.9409 | 0.9502 | 0.0591 |

PCR | 0.8729 | 0 | 0.9246 | 0.9364 | 0.0754 | |

10% | PLS | 0.7388 | 0 | 0.8452 | 0.8694 | 0.1548 |

PCR | 0.7457 | 0 | 0.8493 | 0.8729 | 0.1507 | |

5% | PLS | 0.7216 | 0 | 0.8350 | 0.8608 | 0.1650 |

PCR | 0.6976 | 0 | 0.8208 | 0.8488 | 0.1792 |

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**MDPI and ACS Style**

Bouyeddou, B.; Harrou, F.; Taghezouit, B.; Sun, Y.; Hadj Arab, A.
Improved Semi-Supervised Data-Mining-Based Schemes for Fault Detection in a Grid-Connected Photovoltaic System. *Energies* **2022**, *15*, 7978.
https://doi.org/10.3390/en15217978

**AMA Style**

Bouyeddou B, Harrou F, Taghezouit B, Sun Y, Hadj Arab A.
Improved Semi-Supervised Data-Mining-Based Schemes for Fault Detection in a Grid-Connected Photovoltaic System. *Energies*. 2022; 15(21):7978.
https://doi.org/10.3390/en15217978

**Chicago/Turabian Style**

Bouyeddou, Benamar, Fouzi Harrou, Bilal Taghezouit, Ying Sun, and Amar Hadj Arab.
2022. "Improved Semi-Supervised Data-Mining-Based Schemes for Fault Detection in a Grid-Connected Photovoltaic System" *Energies* 15, no. 21: 7978.
https://doi.org/10.3390/en15217978