# High-impedance Fault Detection in DC Microgrid Lines Using Open-Set Recognition
## 4 Case Study
### Results  
The FFNN is used because its architecture is simple and the training procedure is relatively straightforward. The FFNN is first trained to achieve the best possible accuracy (defined as ratio of correctly classified samples and total number of samples) on the test dataset. The second FFNN, with the same architecture, is trained but with improvements  
Figure 5: Representatives of dataset classes (steady state, load change) and HIF in the time domain. The stationary current is flat with ripples, while the load change current transitions to another steady state after a transient period. Notably, the arcing phenomenon is clearly visible during the HIF. Figure 6: Representatives of dataset classes (steady state, load change) and HIF in the frequency domain. Amplitudes of high-frequency components of the current signal are relatively low during the stationary state and load changes. However, during the HIF event, these amplitudes significantly increase, indicating a distinct pattern. that aim to increase its open-set performance. The FFNN is set to take 500 points as inputs, as this is the size of the transform of the time sequence that describes an event, as explained before. The input layer is followed by three linear layers with sizes 2000, 2000, and 400, each of which has a Rectified Linear Unit (ReLU) nonlinearity at its output. The number of nodes in the first hidden layer is selected to increase the dimensionality of the input. The second hidden layer retains the same number of nodes, as iterative parameter selection indicated that further increases in dimensionality did not enhance the neural network's performance. The third hidden layer reduces dimensionality before the final reduction in the output layer. The number of nodes in this layer is reduced by a factor of five compared to the previous hidden layer, as the iterative parameter selection process demonstrated that this achieves an optimal balance between the neural network's complexity and accuracy. The final linear layer reduces the size from 400 to a vector of size 2, with each element representing a class. The ReLU nonlinear function is applied at the output of each hidden layer, as it is computationally efficient and avoids the vanishing gradient problem encountered with other nonlinear functions, such as the sigmoid function. A batch size of 32 samples is used, as it provides a good balance between convergence speed and the neural network's generalization ability. This same rationale guided the selection of the learning rate and number of epochs. Additionally, RMSprop is employed as the optimizer due to its ability to ensure stable convergence, even in cases with sparse gradients. Both the learning rate warmup and label smoothing were determined iteratively in alignment with the OSR framework. The architectures for both classifiers, as well as other parameters and the optimizer used, can be found in Table 2.  
As can be seen from the table, the only difference between the classifiers is in the improvements related to the OSR performance. Instead of a fixed value, the cosine learning rate with warmup was used, which means that the learning rate is increased from \(1\times 10^{-4}\) to \(1\times 10^{-3}\) during the first 10% of the total number of epochs. After it reaches its maximum, it is reduced for the remaining number of epochs according to the cosine law. The accuracy of the classifier is not reduced after applying all OSR rules, and the learning process converges as shown in Figure 7.  
The reason for using MLS becomes clearer when dealing with unknown samples that are not part of the dataset, in this case HIFs. The standard approach is that for each input sample, the classifier produces a vector of size equal to the number of classes. The vector elements are normalized using the softmax function to obtain probabilities of the input belonging to each class; i.e., each vector element is in the range [0, 1] and their sum is equal to 1. However, this approach does not take into account the vector amplitude information,