# Hypothesis Tests-Based Analysis for Anomaly Detection in Photovoltaic Systems in the Absence of Environmental Parameters

## Abstract

**:**

## 1. Introduction

## 2. Statistical Methodology

_{0}(Equation (4)) that the means of the distributions, ${\mu}_{k}$, are equal:

- (a)
- all the observations are mutually independent;
- (b)
- all the distributions are normally distributed;
- (c)
- all the distributions have equal variance.

_{HDT}. By fixing the significance value α = 0.05, if p-value

_{HDT}< α is satisfied, the null hypothesis of the unimodality is rejected, the distribution is surely not Gaussian, ANOVA cannot be applied and a nonparametric test has to be used.

_{K-W(MM)}that has to be compared with the significance value α = 0.05. If p-value

_{K-W(MM)}< α is satisfied, the null hypothesis is rejected and the arrays have not produced the same energy; otherwise, they have. Instead, if the unimodality is satisfied, other checks are needed, before deciding whether ANOVA can be applied. In fact, it is needed to verify the previous assumptions (b) and (c). Only if both of them are satisfied (blue blocks 2 and 3, respectively), ANOVA can be applied.

_{JBT}. By fixing the significance value α = 0.05, if p-value

_{JBT}< α is satisfied, the null hypothesis is rejected and the distribution is not gaussian, otherwise it is. It results:

_{BT}. The BT is effective for Gaussian distributions; in fact, in the flow-chart of Figure 2 it is used only if the distributions are Gaussian. Also in this case, it is possible to fix the common significance value α = 0.05 and to compare it with the p-value

_{BT}. If the inequality p-value

_{JBT}< α is satisfied, the null hypothesis is rejected and the variances of the distributions of the arrays are different, then the condition (c) is violated, and ANOVA cannot be applied. In this case, it is necessary to use K-W or MM, in accordance with the green block. Otherwise, ANOVA can be applied and it return another p-value

_{AN}that must be compared with the significance level α = 0.05. If the inequality p-value

_{AN}< α = 0.05 is satisfied, then the null hypothesis (${H}_{0}:{\mu}_{1}={\mu}_{2}={\mu}_{3}=\cdots ={\mu}_{A}$) is rejected and the conclusion is that the identical arrays have not produced the same amount of energy; so, a low-intensity anomaly is present and it is located in the array that has the mean value different from the other ones. To detect it, a multi-comparison analysis—one-to-one—between the distributions is done by means of the Tukey’s Test (TT), which is a modified version of the well-known t-test and returns a p-value

_{TT}, which states whether the means between two distributions are equal or not. For a small sample size (about 20 samples) the TT is reliable only for normal distribution, instead, for a lager sample size it is valid also for not normal distributions, because of the central limit theorem. Otherwise, no criticality is present and the dataset can be updated with new data to continue the monitoring of the PV plant. As the energy dataset increases, the monitoring becomes more accurate.

## 3. Description of the PV Plant under Investigation

_{dc}of each inverter; moreover, the number of the operating hours is stored. The default monitoring system of the PV plant uses the power in AC and the voltage V

_{dc}, the daily and total energy produced by each inverter, and the number of the operating hours. It is worth noting that the monitoring system is an internal software of the datalogger. As the operation of the monitoring system occupies the internal memory, for default the internal monitoring system does not utilize all the data available into the datalogger, in order not to occupy the internal memory quickly. This approach allows to monitor the PV plant for a longer time, but only the high-intensity anomalies can be detected. Instead, to detect even the low-intensity anomalies, it is necessary to use the methodology described in Figure 2 and all the data stored into the datalogger. Moreover, even if the measurement system of this PV plant does not measure all the variables mentioned in the Section 2 (the produced energy, other than the voltage and current in both the DC and AC side), nevertheless it acquires the produced energy that is the unique variable necessary for the proposed methodology; so it can be applied. The observation period refers to a full year during which the plant has shown some malfunctions, whereas in the previous years the PV plant has not shown any malfunctions, therefore the results of the previous years are not reported in the paper.

## 4. Cumulative Statistical Analysis

- one-month analysis (January);
- three-months analysis (January–March);
- six-months analysis (January–June);
- one-year analysis (January–December).

_{HDT}of each distribution to test the unimodality; the p-value

_{JBT}of each distribution to test whether each one of them is gaussian; the p-value

_{BT}to test the homoscedasticity among the distributions; the p-value

_{AN}to test whether all the distributions have the same mean value; the p-value

_{K-W(MM)}of the non-parametric test (when ANOVA cannot be applied) to check whether all the distributions have the same median value; the box plot of the ANOVA test or of the non-parametric test; the mean value of each distribution and its spread with respect to the global mean of the PV plant.

#### 4.1. One-Month Analysis (January)

_{HDT}> α = 0.05 for each distribution, so all the distributions are unimodal. To apply ANOVA, conditions (b) and (c) have to be verified.

_{JBT}; as p-value

_{JBT}> α = 0.05, all the distributions are Gaussian, so condition (b) of ANOVA is satisfied. Condition (c) about the homoscedasticity has to be verified by means of BT (see Figure 2). The p-value

_{BT}= 0.999 in Table 1 (again higher than α = 0.05) says that the homoscedasticity is verified, then all the variances are equal. Therefore, the main conditions of the flow chart in Figure 2 (blocks 1,2,3) are satisfied and ANOVA can be applied. The p-value

_{AN}= 0.999 in Table 1 says that the distributions have the same mean values, so all the arrays have produced the same energy in this month. Figure 4 is the box plot of ANOVA. For each box, the central red mark indicates the median, and the bottom and top edges of the box indicate the 25th and 75th percentiles, respectively. The whiskers extend to the most extreme data points. Figure 4 highlights that the five distributions have produced almost the same energy, both with respect to the median value (in red color) and to the first and third inter-quartiles; moreover, outliers are absent. Therefore, no anomaly is present in the PV plant. Particularly, from PVGIS [31], it results that the estimated average energy of the PV plant in January should be about 3173 kWh, corresponding to a daily average energy for each array of about 3173/(31 × 5) = 20.5 kWh, that is almost equal to the global mean value 19.74 kWh of Table 1. Figure 5 diagrams the mean value and confidence interval at 95% of each distribution; the values are very similar each other, as it results also from Table 2 that reports the one-to-one comparisons of the mean values. In particular, the high p-values confirm that the differences are not significant.

#### 4.2. Three-Months Analysis (January–March)

_{HDT}> α = 0.05 for each distribution, so all the distributions are still unimodal. The p-value

_{JBT}> α = 0.05, so all the distributions are gaussian and the condition (b) of ANOVA is satisfied. The homoscedasticity is also satisfied (p-value

_{BT}> α = 0.05). Therefore, the main conditions of the flow chart in Figure 2 (blocks 1,2,3) are satisfied and ANOVA can be newly applied. The p-value

_{AN}= 0.998 affirms that the distributions have the same mean values, so all the arrays have produced the same energy also in these three months. Figure 6 is the box plot of ANOVA and it highlights that the five distributions have produced almost the same energy, both with respect to the median value (in red color) and to the first and third inter-quartiles; moreover, outliers are not present. Therefore, no anomaly is present in the PV plant in these three months. Particularly, from PVGIS [31], it results that the estimated average energy of the PV plant in the period January–June should be about 11,695 kWh, corresponding to a daily average energy for each array of about 11,695/(90 × 5) = 25.99 kWh, that is almost equal to the global mean value 25.95 kWh of Table 2.

#### 4.3. Six-Months Analysis (January–June)

_{HDT}> α = 0.05 for each distribution, so all the distributions are still unimodal. As p-value

_{JBT}< α = 0.05 for each distribution, then the null hypothesis is rejected and the constraint of the block 2 (corresponding to the condition (b) of ANOVA) is not satisfied: the distributions are not Gaussian. Therefore, it has no sense to verify the homoscedasticity, because it is mandatory to use a nonparametric test. As no outlier is present, it is advisable to use K-W, as suggested by the green block. The p-value

_{K-W}= 0.861 affirms that the distributions have the same median values, so all the arrays have produced the same energy also in these six months, even if the distributions are no longer Gaussian. Figure 8 is the box plot of K-W and it highlights that the five distributions have produced almost the same energy, both with respect to the median value (in red color) and to the first and third inter-quartiles; moreover, it is confirmed that outliers are not present. Therefore, no anomaly is present in the PV plant in these six months. Particularly, from PVGIS [31], it results that the estimated average energy of the PV plant in the period January–June should be about 32,285 kWh, corresponding to a daily average energy for each array of about 32,285/(181 × 5) = 35.67 kWh, that is almost equal to the global mean value 36.23 kWh of Table 3. Figure 9 illustrates the mean value and confidence interval at 95% of each distribution; the values are very similar each other, as it results also from Table 6 that reports the one-to-one comparisons of the mean values. In particular, the high p-values confirm that the differences are not significant.

#### 4.4. One-Year Analysis (January–December)

_{HDT}> α = 0.05 for each distribution, except for the distribution n. 4, for which p-value

_{HDT}(4) = 0.006 < α = 0.05; therefore, the condition of the block 1 about the unimodality is not satisfied for all the distributions and ANOVA cannot be applied. Consequently, it is mandatory to use a nonparametric test. As no outlier is present, it is advisable to apply K-W, as suggested by the green block. As p-value

_{K-W}= 0.009 < α = 0.05, the null hypothesis is rejected, so the distributions have different median values. This implies that the arrays have not produced the same energy in the complete year, even if they had produced the same energy for the first six months. Figure 10 is the box plot of K-W and it highlights that the median value of the distribution n. 4 is significantly different from the others. It is also confirmed that outliers are not present. Particularly, from PVGIS [31], it results that the estimated average energy of the PV plant in the period January–December should be about 64,724 kWh, corresponding to a daily average energy for each array of about 64,724/(365 × 5) = 35.46 kWh, that is almost equal to the global mean value 35.13 kWh of Table 4. Therefore, high-intensity anomaly is not present, but a low-intensity anomaly is detected in the array n. 4, as confirmed also by the spreads of the mean values reported in Table 4. It can be observed that the array n. 4 produced 6.54% less than the average energy of the PV plant. Figure 11 shows the mean value and confidence interval at 95% of each distribution. It can be observed that the array n. 4 is very different from the other ones, as it results also from Table 8 that reports the one-to-one comparisons of the mean values. In particular, the p-value $0.043<\alpha =0.05$ rejects the hypothesis that the distribution 4 and 5 have the same mean value.

## 5. Conclusions

## Conflicts of Interest

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**Figure 6.**Box plot of K-W test of the five distributions for the three-months analysis (January–March).

**Figure 7.**Mean value of the energy produced by each array for the three-months analysis (January–March).

**Figure 8.**Box plot of K-W test of the five distributions for the six-months analysis (January–June).

**Figure 9.**Mean value of the energy produced by each array for the three-months analysis (January–June).

**Table 1.**p-Value of HDT, JBT, BT, ANOVA for the energy distribution of the arrays, and mean in kWh and spread with respect to the global mean for one-month analysis (January).

Array Number | 1 | 2 | 3 | 4 | 5 | |
---|---|---|---|---|---|---|

p-value_{HDT} | 0.628 | 0.364 | 0.674 | 0.658 | 0.670 | |

JBT | JB | 0.699 | 0.676 | 0.674 | 0.718 | 0.638 |

p-value_{JBT} | 0.500 | 0.500 | 0.500 | 0.500 | 0.500 | |

p-value_{BT} | 0.999 | |||||

p-value_{AN} | 0.999 | |||||

Mean (kWh) | 19.59 | 19.90 | 19.91 | 19.46 | 19.84 | |

Global mean | 19.74 | |||||

Spread % | −0.76 | 0.80 | 0.85 | −1.40 | 0.51 |

Comparison between Samples | LowerBound | DifferenceEstimate | UpperBound | p-Value_{TT} | |
---|---|---|---|---|---|

1 | 2 | −8.77 | −0.31 | 8.16 | 0.999 |

1 | 3 | −8.78 | −0.32 | 8.15 | 0.999 |

1 | 4 | −8.34 | 0.12 | 8.59 | 0.999 |

1 | 5 | −8.71 | −0.25 | 8.21 | 0.999 |

2 | 3 | −8.47 | −0.01 | 8.45 | 1 |

2 | 4 | −8.03 | 0.43 | 8.90 | 0.999 |

2 | 5 | −8.41 | 0.06 | 8.52 | 1 |

3 | 4 | −8.02 | 0.44 | 8.91 | 0.999 |

3 | 5 | −8.40 | 0.06 | 8.53 | 1 |

4 | 5 | −8.84 | −0.38 | 8.09 | 0.999 |

**Table 3.**p-Value of HDT, JBT, BT, ANOVA for the energy distribution of the arrays, and mean in kWh and spread with respect to the global mean for three-month analysis (January–March).

Array Number | 1 | 2 | 3 | 4 | 5 | |
---|---|---|---|---|---|---|

p-value_{HDT} | 0.776 | 0.892 | 0.818 | 0.856 | 0.830 | |

JBT | JB | 5.127 | 5.198 | 5.114 | 5.177 | 5.090 |

p-value_{JBT} | 0.054 | 0.053 | 0.054 | 0.053 | 0.055 | |

p-value_{BT} | 0.999 | |||||

p-value_{AN} | 0.998 | |||||

Mean (kWh) | 25.84 | 25.82 | 26.23 | 25.62 | 26.39 | |

Global mean | 25.95 | |||||

Spread % | −0.54 | −0.62 | 0.98 | −1.36 | 1.58 |

Comparison between Samples | LowerBound | DifferenceEstimate | UpperBound | p-Value_{TT} | |
---|---|---|---|---|---|

1 | 2 | −6.68 | 0.02 | 6.73 | 1 |

1 | 3 | −7.10 | −0.39 | 6.31 | 0.999 |

1 | 4 | −6.49 | 0.21 | 6.92 | 0.999 |

1 | 5 | −7.25 | −0.55 | 6.15 | 0.999 |

2 | 3 | −7.12 | −0.41 | 6.29 | 0.999 |

2 | 4 | −6.51 | 0.19 | 6.90 | 0.999 |

2 | 5 | −7.27 | −0.57 | 6.13 | 0.999 |

3 | 4 | −6.09 | 0.60 | 7.31 | 0.999 |

3 | 5 | −6.86 | −0.15 | 6.55 | 0.999 |

4 | 5 | −7.47 | −0.76 | 5.94 | 0.997 |

**Table 5.**p-Value of HDT, JBT, BT, (KW) for the energy distribution of the arrays, and mean in kWh and spread with respect to the global mean for six-months analysis (January–June).

Array Number | 1 | 2 | 3 | 4 | 5 | |
---|---|---|---|---|---|---|

p-value_{HDT} | 0.794 | 0.722 | 0.782 | 0.808 | 0.842 | |

JBT | JB | 13.95 | 14.02 | 13.92 | 13.94 | 13.85 |

p-value_{JBT} | 0.007 | 0.007 | 0.007 | 0.007 | 0.007 | |

p-value_{BT} | ------------ | |||||

p-value_{K-W} | 0.861 | |||||

Mean (kWh) | 36.01 | 35.78 | 36.60 | 35.74 | 37.04 | |

Global mean | 36.23 | |||||

Spread % | −0.60 | −1.23 | 1.02 | −1.35 | 2.23 |

Comparison between Samples | LowerBound | DifferenceEstimate | UpperBound | p-Value_{TT} | |
---|---|---|---|---|---|

1 | 2 | −5.14 | 0.22 | 5.60 | 0.999 |

1 | 3 | −5.96 | −0.58 | 4.78 | 0.999 |

1 | 4 | −5.10 | 0.27 | 5.64 | 0.999 |

1 | 5 | −6.40 | −1.02 | 4.34 | 0.985 |

2 | 3 | −6.18 | −0.81 | 4.55 | 0.993 |

2 | 4 | −5.33 | 0.04 | 5.41 | 1 |

2 | 5 | −6.62 | −1.25 | 4.11 | 0.968 |

3 | 4 | −4.51 | 0.85 | 6.23 | 0.992 |

3 | 5 | −5.81 | −0.43 | 4.93 | 0.999 |

4 | 5 | −6.67 | −1.29 | 4.07 | 0.964 |

**Table 7.**p-Value of HDT and K-W for the energy distribution of the arrays, and mean in kWh and spread with respect to the global mean for one-year analysis.

Array Number | |||||
---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | |

p-value_{HDT} | 0.820 | 0.846 | 0.898 | 0.006 | 0.636 |

p-value_{JBT} | - | - | - | - | - |

p-value_{BT} | - | - | - | - | - |

p-value_{K-W} | 0.009 | ||||

Mean (hWh) | 35.37 | 35.24 | 35.92 | 32.83 | 36.31 |

Global mean | 35.13 | ||||

Spread % | 0.68 | 0.32 | 2.25 | −6.54 | 3.35 |

Comparison between Samples | LowerBound | DifferenceEstimate | UpperBound | p-Value_{TT} | |
---|---|---|---|---|---|

1 | 2 | −3.76 | 0.12 | 4.01 | 0.999 |

1 | 3 | −4.44 | −0.55 | 3.33 | 0.995 |

1 | 4 | −0.86 | 3.02 | 6.91 | 0.210 |

1 | 5 | −4.83 | −0.94 | 2.95 | 0.964 |

2 | 3 | −4.56 | −0.67 | 3.21 | 0.989 |

2 | 4 | −0.98 | 2.90 | 6.79 | 0.249 |

2 | 5 | −4.95 | −1.06 | 2.82 | 0.945 |

3 | 4 | −0.31 | 3.57 | 7.47 | 0.088 |

3 | 5 | −4.28 | −0.38 | 3.50 | 0.998 |

4 | 5 | −7.85 | −3.96 | −0.07 | 0.043 |

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**MDPI and ACS Style**

Vergura, S.
Hypothesis Tests-Based Analysis for Anomaly Detection in Photovoltaic Systems in the Absence of Environmental Parameters. *Energies* **2018**, *11*, 485.
https://doi.org/10.3390/en11030485

**AMA Style**

Vergura S.
Hypothesis Tests-Based Analysis for Anomaly Detection in Photovoltaic Systems in the Absence of Environmental Parameters. *Energies*. 2018; 11(3):485.
https://doi.org/10.3390/en11030485

**Chicago/Turabian Style**

Vergura, Silvano.
2018. "Hypothesis Tests-Based Analysis for Anomaly Detection in Photovoltaic Systems in the Absence of Environmental Parameters" *Energies* 11, no. 3: 485.
https://doi.org/10.3390/en11030485