3.1. Health index based on a weighted sum model

Developed by ^{Li and Song (2014)} and based on WSM, HI_WSM is computed as the result of adding four health subindices, each one of the following criteria: 1) age of the PT, 2) the insulating paper, 3) Dissolved Gas Analysis (DGA) 4) Oil Quality Factor (OQF). The method involves a process where the input parameters are quantified and normalized employing linear functions. The tables concerned with the corresponding relationship to calculate the normalized parameters for each of the four parts and tables for the assigned weights can be consulted in (^{Li & Song, 2014}).

The first health subindex HI_age is concerned with the age and loading of the PT and is given by Equation (1).

Where HI_O, is the initial value, B is the aging coefficient calculated by the equations given in (^{Li & Song, 2014}), T₁ is the year corresponding to HI_O, which is usually the year that the PT was put into operation, T₂ is the year that the PT condition is analyzed, can be the current year or some year into the future.

The insulating paper subindex HI_iso considerers the overall insulation aging characteristics and is constituted by two parts: subindex HI_CO formed by the contents of carbon and oxygen, and subindex HI_fur formed by the Furfural Content (FC). DGA and FC analysis are the tests needed to obtain the input parameters to calculate the HI_iso.

The subindex HI_CO consists of three parameters F_CO(i), the three factors are considered to be equally important and all the weights are set as ω=0.333. The HI_CO is calculated using Equation (2).

HI_fur reflects the FC results and it is given by (3).

The resultant HI_iso can be obtained by adding HI_CO and HI_fur together with their respective weights. In this paper, the weights are set to be 0.3 and 0.7, respectively. The index HI_iso is shown in Equation (4).

The subindex HI_CH based on DGA uses five gases, H2, CH4, C2H6, C2H2 and C2H4, to assess the health state of the PT. The subindex is calculated using Equation (5), which is a function of five hydrocarbon factors F_CH(𝑖).

The subindex based on OQF HI_oil analyses the proprieties of oil and how it is correlated with the overall state of the PT. Considered properties are moisture content, acid value, dielectric loss, and breakdown voltage (BV), each of them is represented by F_oil(𝑖) and are linear functions.

The index HI_oil is calculated using Equation (6) and the weights given (^{Li & Song, 2014}).

To compute the final HI_WSM, Equation (7) is used.

The overall condition of the PT based on the value of HI is presented in Table 1.

3.2. Health index based on fuzzy inference system

Proposed by ^{Cerón et al. (2015)} the second HI employs the FIS methodology to process the input data. It uses six parameters to calculate the final value, HI_FIS, which are: BV, moisture content (humidity), acidity, power factor, FC and DGA. A membership function is designed for each input. The author employs (^{IEEE Std C57.152-2013, 2013}) and (^{IEC 60422, 2013}) to establish the limits to score the input data. The set of membership functions for the input parameters can be consulted in (^{Cerón et al., 2015}).

The membership functions corresponding to the output HI_FIS are presented in Figure 1.

Figure 1 Membership functions for the HIFIS. 

A Mamdani FIS composed by 80 rules is used to integrate the six linguistic inputs with the output. Once the output membership function is obtained, a defuzzification process takes place to convert the linguistic HI into a numerical value. The complete HI fuzzy model can be observed in Figure 2.

Figure 2 HI fuzzy model. 

3.3. Health index based on hybrid WSM-FIS

The third approach to compute a HI was proposed by ^{Mharakurwa and Goboza (2019)}. It combines WSM with four FIS to assess the PT condition. The general approach is shown in Figure 3.

Figure 3 Health index calculation method based on WSM and FIS. 

The WSM component scores results of DGA, OQF and FC analysis, while the four FISs are implemented for different purposes. The first FIS evaluates the degree of polymerization (DP) based on the FC and the carbon dioxide (CO2) and carbon monoxide (CO) ratio. The second one assesses the operational stress of the PT, which depends on the load and the hotspot temperature. The third FIS combines the results of the previous two FISs, i.e., it combines the obtained scores for DP and operational stress. Afterwards, the four and last FIS uses the outputs of the WSM and the third FIS to obtain _HIWSM-FIS finally.

To calculate the scores of DGAF, OQF and FC, tables given in (^{Mharakurwa & Goboza, 2019}) are employed. After that, Equations (8) and (9) are applied in order to obtain the normalized score to be used in the fuzzy logic model.

Where K_i corresponds to the assigned minimum weight in the four conditions, x_i is the current value of the parameter analyzed, a_i and b_i are the lower and upper limits of the conforming cluster of the parameter.

In order to compute the operational stress; hotspot temperature and load are taken into account. To calculate the DP value the furan content and CO₂/CO will be considered. The membership functions for both parameters and the output can be consulted in (^{Mharakurwa & Goboza, 2019}).

In Figures 4 to 6, the membership functions for the final HI value which combines the DGAF+OQF+FF and the DP- value+Operational stress are introduced.

Figure 4 Membership functions for DGAF+OQF+FF. 

Figure 5 Membership functions for DP value+Operational stress. 

Figure 6 Membership functions for the final HI. 

4.1. Description of the performed comparison

Figure 7 presents a flowchart with the major key points of the methodology used to compare the different HI techniques.

Figure 7 Flowchart of the proposed comparison methodology. 

The three approaches presented in the previous section were tested in a PT fleet composed of thirty units. Table 2 summarizes the main results of the diagnostic tests for the complete fleet.

According to ^{Hughes (2003)} and ^{Jürgensen et al. (2017)}, HI is an indicator that enables an effective prioritization/ranking of the assets, in order to support maintenance and replacement decisions.

Therefore, by using the computed HIs, the transformer fleet will be categorized into groups based on level of priority following the guidelines presented in Table 3.

Then, the results of the three methods are combined using the well-known K-means clustering technique to obtain a final categorization ranking and performing a comparison with the rest of the rankings previously obtained.

Finally, in a second stage, to assess how the HIs evolve with time, an extrapolation for five years is performed for the PT diagnosis test. For this purpose, the normal limits for oil condition given by IEEE Std C57.106^TM-2002, (2003) are employed and normalized for a yearly increase according to the equations suggested by ^{Irungu et al. (2017)}. The change rate per year for the oil condition is calculated based on a normally expected life span for a PT, which is 40 years. To extrapolate FC, acidity, moisture, and power factor equation (10) is used. The normal rate of change for BV and interfacial tension is calculated based on (11).

The normal rate of increment for the dissolved gases in parts per million per year and per day can be found in ^{IEC 60599, (2015)} and ^{Gray (2009)} respectively. The consequent values for normal increment for both oil condition and DGA are presented in Table 4.

4.2. Results and discussions

Table 5 shows the HI ranking obtained from each approach for the assessed fleet. From HI results, it is noted that the three methods produce different rankings. Moreover, from Table 5, it can be inferred that also different decisions will be made depending on the approach selected by the asset manager.

There is a level of coincidence between methods 2 and 3, but there are cases where important differences between them can be found. Method 1 returned the lowest HI values for all PTs and always maintained a margin of difference in relation to methods 2 and 3. For the fleet of thirty PTs, the three methods only agreed in the ranking in four cases, i.e., PTs: T2, T22, T24, and T30.

As mentioned above, a clustering technique was applied to combine HI results. Figure 8 shows the clusters representing the three methods and the centroids of the k-means technique.

Figure 8 Cluster for the categorization groups of PT conditions. 

Table 6 presents the categorization groups for the three methods and the inclusion of the k-means merged results. Clear differences can be observed in how the methods categorized the PT fleet.

The first method ranks almost all units in good condition and only one of them receives a bad score, method 2 and 3 showed different results as well. The degree of coincidence between method 1 and 2 is around forty percent as for method 3 the value increases to sixty percent; for method 2 the level of agreement with method 3 is eighty percent. Thus, the overall coincidence of the three methods was around forty percent. The k-means method presented a ranking similar to the one proposed in method 3, the degree of coincidence with method 2 and 3 is greater than eighty percent and for method 1 is placed around sixty percent.

On the other hand, in Figure 9, an individual analysis for each method is developed to simulate how each type of HI varies with time, by using the extrapolated values for the diagnosis tests.

Figure 9 a) to c). Individual comparison for the three methods for the five-year period. a) Health index based on a weighted sum model. b) Health index based on fuzzy inference system. c) Health index based on hybrid WSM-FIS. 

For method 1, the HI rose every year, the increment rate shaped a quasi-linear function as the sample ages. The growth can be explained due to the weight assigned to the age of the PT (56%), in contrast, DGA and oil condition were given much lower weights (26% and 9%). Method 2 presented a different perspective during the five-year period, only fifty percent of the samples presented changes, the rest remained at the same value, this can be attributable to the nature of the trapezoidal fuzzy membership functions (TFMF), where they maintain the same value unless the input value moves to an interception zone between membership functions or another function at all.

It is important to note that this method does not consider, age, load or CO₂/CO relation but shows good performance Method 3 was a particular case, it showed similar results with method 2, but during the five years, there were some samples whose HI experimented a decrease in value (samples 2,9 and 19). The reason for that decrement was the change in the CO₂/CO relation, which has an essential role in the final value in the degradation level of the insulating paper. Also, it can be observed a stiffer rate of change than in method 2, this behavior responds to the nature of the TFMF that is amplified by the multi-FIS scheme employed in the method.