## 2 Materials and Methods
### Methodology
Statistical metrics that are used for results comparison are the root mean square error (RMSE), the normalized root mean square error (nRMSE), the mean absolute error (MAE), the mean absolute percentage error (MAPE) [35], the mean bias error (MBE) [36] and the Pearson correlation coefficient [37]:  
\[\text{RMSE}=\frac{1}{\text{N}}\sqrt{\sum_{i=1}^{\text{N}}(\text{P}f_{i}-\text {P}m_{i})^{2}} \tag{3}\]  
\[\text{nRMSE}=\frac{\sqrt{\sum_{i=1}^{\text{N}}(\text{P}f_{i}-\text{P}m_{i})^{2} }}{\sqrt{\sum_{i=1}^{\text{N}}(\text{P}m_{i})^{2}}} \tag{4}\]  
\[\text{MAE}=\frac{\sum_{i=1}^{\text{N}}\lvert\text{P}f_{i}-\text{P}m_{i}\rvert}{ \text{N}} \tag{5}\]  
\[\text{MAPE}=\frac{100\%}{\text{N}}\sum_{i=1}^{\text{N}}\frac{\lvert\text{P}f_ {i}-\text{P}m_{i}\rvert}{\text{P}m_{i}} \tag{6}\] \begin{table}
\begin{tabular}{l c} \hline \hline
**ANN Type** & **FF** \\ \hline ANN Dimensions & 1 \\ Inputs & 2 \\ Layers & 1 \\ Outputs & 0 \\ Layer Delays & 0 \\ Weight Elements & 81 \\ ANN connections: & \\ Bias Connections: & [1; 1] \\ Input Connections: & [1; 0] \\ Layers Connections: & [0 0; 1 0] \\ Output Connections: & [0 1] \\ ANN Training hyper-parameters & \\ Maximum Epochs & 1000 \\ Maximum Training Time & Inf \\ Performance Goal & 0 \\ Minimum Gradient & \(1.00\times 10^{-7}\) \\ Maximum Validation Checks & 6 \\ \(\mu_{k}\) & 0.001 \\ \(\mu_{k}\) decrease ratio & 0.1 \\ \(\mu_{k}\) increase ratio & 10 \\ Maximum \(\mu_{k}\) & 1.00 \(\times\) 10\({}^{10}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Design parameters of the specific type of FF ANN applied, along with the hyper-parameter values related to the training procedure. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Network** & **PV Plant** & **Inverter** & **Input Parameters** & **Training Period** \\ \hline \multirow{3}{*}{n1} & \multirow{3}{*}{PVS1} & \multirow{3}{*}{INV4} & Irradiance & \multirow{3}{*}{2013} \\  & & & Back Panel Surface & \\  & & & Temperature & \\ \multirow{3}{*}{n2} & \multirow{3}{*}{PVS2} & \multirow{3}{*}{INV3} & Back Panel Surface & \multirow{3}{*}{2013} \\  & & & Temperature & \\ \multirow{3}{*}{n3} & \multirow{3}{*}{PVS3} & \multirow{3}{*}{INV6} & Irradiance & \multirow{3}{*}{2013} \\  & & & Ambient Temperature & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Characteristics of the three FF ANNs and their respective PV systems. \[\text{MBE}=\frac{\sum_{i=1}^{\text{N}}(\text{Pf}_{i}-\text{P}m_{i})}{\text{N}} \tag{7}\]  
\[\text{PBIAS}=100\%\frac{\sum_{i=1}^{\text{N}}(\text{Pf}_{i}-\text{P}m_{i})}{ \sum_{i=1}^{\text{N}}(\text{P}m_{i})} \tag{8}\]  
\[\text{COR}=\frac{\text{COV}(\text{P}m_{i},\text{Pf}_{i})}{\sqrt{\text{V}(\text {P}m_{i})}\sqrt{\text{V}(\text{P}f_{i})}}100\% \tag{9}\]  
The next step is the correlation of statistical metrics with already known faults in each PV system. These may concern a fault in one of the four strings connected to each inverter (see details in Appendix A), faults in a single PV panel, shading during specific days of year close to the winter solstice. It is important to understand first the networks performance in normal operation, by observing the behavior of statistical metrics in order to establish a performance baseline. Furthermore, the application of ANN to different inverters and different PV systems allows the clear establishment of specific thresholds in the statistical performance metrics that points to the detection of specific types of faults. Different scenarios are employed in this work in order to demonstrate the use of the fault detection methodology. Another important metric that is computed and employed in the methodology is the performance ratio, as described in Appendix B.