# Risk Assessment Algorithm for Power Transformer Fleets Based on Condition and Strategic Importance
## 2 Materials and Methods: Risk Assessment Algorithm  
### Health Index Calculation  
The health index (HI) is an indicator that quantifies and provides an undemanding understanding of the general condition of a PT. Most of the methodologies developed to calculate the HI employ weighting techniques [21; 22]. However, in recent years new methods based on Markov chains [23], fuzzy logic systems [24] and artificial neural networks [25] have been developed.  
Some authors [26; 27; 28], take into account the uncertainty, vagueness, or impreciseness of the available information. Derived from this approach, they consider the fuzzy inference systems (FIS) to be the most appropriate method to calculate the HI. The use of the FIS methodology requires the fuzzification of all the input data, which is often given in numerical values. The fuzzification process involves the conversion of those numerical values into linguistic functions which is accomplished with the modeling of membership functions.  
The proposed method has its roots in the works in Ref. [29] and Ref. [9]. The original methodology uses six entry criteria to calculate the final value: breakdown voltage, moisture content (humidity), acidity, power factor, furan content and dissolved gas analysis (DGA). The furan content analysis is carried out to assess the degree of degradation of the winding's insulating paper. However, recent studies have proven that the furan content may be inaccurate to properly evaluate the state of the insulating paper [30; 31; 32]. Taking into account the facts stated above, this paper proposes the replacement of the furan content criterion for the _DP_ one. The degree of polymerization (_DP_) value is usually estimated from the proportion of 2-_FAL_ found in the furan content analysis., [PERSON]'s Equation (1) is one of the most widely accepted methods to calculate the _DP_ value.  
\[DP_{value}=\frac{1.51-\log(2\text{-}FAL)}{0.0035} \tag{1}\]  
Established equations for the _DP_ only evaluate the 2-_FAL_ content to calculate its estimates. The imprecise outcomes from considering only the 2-_FAL_ content were suggested in researches conducted by [33; 34; 35]. Moreover, they highlight the importance of the CO\({}_{2}\)/CO ratio to estimate the real degree of degradation of the insulating paper. This work implements a FIS approach to calculate the _DP_ value, and entry parameters for this system are the furan content and the CO\({}_{2}\)/CO ratio. The membership functions of the entry criteria and the output of the FIS process are presented in Figures 1-3.  
According to the authors of [4], for the CO\({}_{2}\)/CO ratio, the respective values of CO\({}_{2}\) and CO should exceed 5000 uL/L (ppm) and 500 uL/L (ppm) in order to improve the certainty factor. Additionally, a normal CO\({}_{2}\)/CO ratio should be around 7. [PERSON] et al. [29] state that a _DP_ value higher than 700 represents an insulating paper with its mechanical properties closer to 100%.  
The set of membership functions for the breakdown voltage, moisture content, acidity, power factor, DGA and the final HI value, were taken from Ref. [9] and are shown in Figures 4-9.  
Using the Mamdani FIS methodology the linguistic entry values are integrated with the linguistic output using fuzzy inference rules. For the \(DP\) value, the methodology developed a set of 20 inference rules. Meanwhile, for the final HI value, a group of 80 rules was implemented. Once the output membership function is obtained, a defuzzification process takes place to convert the linguistic result into a numerical value. One can observe the complete HI fuzzy model in Figure 10.  
Figure 5: Membership functions for moisture content.  
Figure 8: Membership functions for DGA.  
Figure 6: Membership functions for acidity.  
Figure 7: Membership functions for power factor.  
Figure 9: Membership functions for the final HI.