## 2 Overview of the FBN
### Construction of the BN for Transformation to the FBN  
For constructing the BN, CPTs are the foundation for representing the conditional relationships between variables. CPTs describe the probability of outcomes for a child node, given the states of its parent nodes. Traditional BNs derive these probabilities using historical data or expert opinions when data are scarce or incomplete. However, this conventional approach struggles to capture real-world data's inherent vagueness and uncertainty, particularly in complex systems such as power grids [39]. To address these challenges, FBNs extend BNs by integrating fuzzy logic, which provides a robust mechanism to handle vague or uncertain data [40]. The following steps outline the process of constructing a BN and transforming it into an FBN:  
1. Development of the BN. The initial step involves creating the BN by identifying critical variables influencing the system, such as environmental factors, external interferences, and operational parameters. The second step establishes dependencies among nodes based on the literature and expert knowledge. The last step is constructing CPTs to represent conditional probabilities derived from historical data and expert elicitation. The general structure of the BN is shown in Figure 1.
2. Integration of fuzzy logic into BNs. The transition to an FBN begins by integrating fuzzy logic principles to handle the uncertainties in node relationships. The variables in the BN are transformed into fuzzy variables characterized by linguistic terms such as \"Low\", \"Medium\", or \"High\". These terms are represented using fuzzy sets with defined membership functions (e.g., triangular or trapezoidal functions) [41]. For all nodes without CPTs through historical data, traditional CPTs are replaced with FPr due to the unattainability of CPr. These values reflect the degree of possibility for each outcome under uncertain conditions.
3. Aggregation of expert opinions using fuzzy logic. As discussed, expert opinions are critical in constructing CPTs when historical data are scarce. To address inconsistencies or subjective biases among experts, each input is assigned a weighting factor based on their expertise level and the relevance of their knowledge to the variable in question. Expert opinions, provided as linguistic terms, are aggregated using fuzzy operations such as weighted averages or fuzzy intersections. This ensures a complete and unbiased representation of the underlying uncertainties.
4. Probabilistic inference in FBNs. Inference in FBNs is carried out through probabilistic and fuzzy reasoning. The process involves propagating evidence across the network and updating node probabilities as fuzzy values, allowing dynamic model adjustment in response to new information. When precise numerical values are required for decision-making, defuzzification techniques are applied to convert fuzzy results into crisp outputs.
5. Validation and Sensitivity Analysis. The last steps are validation and sensitivity analyses to ensure the reliability of the FBN. The impact of variations in input fuzzy sets on output probabilities is assessed, identifying the critical factors influencing system behavior. The model is tested using real-world scenarios to confirm its predictive accuracy and robustness.