*3.2. Development of Numerical Method*

There are few studies related to the use of mathematical theories to improve the effectiveness of maintenance strategy. An investigation in the study [11] applied Markov prediction model (MPM) to assess transformer condition based on available CM parameters under various pre-determined maintenance repair rates, and to ascertain the corrective action for maintenance, repair, and replacement. The MPM provides a simulation of the asset's deterioration using health index % (HI) to assess the condition as good, fair, poor, etc. HI is defined in [11,19] as a method that uses historical condition data in a single measurable index in order to provide a comprehensive assessment of the current state of transformers. In another study [19], a statistical distribution model (SDM) was utilized to predict the HI of transformers using the data of CM-parameters. The main outcome of applying SDM was establishing a modelling framework for future health index of transformers with even limited historical CM-parameters data. MPM is also applied in an investigation [20] based on CM-parameters to model the future deterioration in the transformers. However, all the mentioned models provided relevant information about the overall current condition status of transformers, and predicted the condition in the future, but none of them integrated a method to detect early faults.

In this study, a novel numerical method was created in order to create a model for early fault diagnosis. The numerical method aims to calculate relative alarm threshold (RAT) and relative fault detection value (RFDV) of a measurable variable before a measured value exceeds its caution limit (CL) in order to detect faults in the initial stage. In addition, the daily trend (DT%) of a measured value was calculated to investigate the increased amount percentage of this value per day after a period.

1. Relative alarm threshold (RAT): RAT is estimated as a critical threshold for any measurable variable that can be used in the indication of the probability of having a fault. According to the international standard IEC 60599 [13], the probability of having a failure may increase significantly at values much above typical concentration levels. The situation is then considered critical, for even though a fault may never occur at these high levels, the risk of having one is high. To calculate the RAT, three limits are required. The first limit is the caution limit (CL), that can be recorded from standard guidelines such as IEC 60599 [13], IEC 60422 [15], IEEE C57.106 [44], ASTM D3487 [45], CIGRE TB 771 [46], as well as limits reported by experimental investigations [2]. Selecting one of these standard guidelines is based on the organization's maintenance plan. Table 2 displays the CLs and standard guidelines used in the PSMD power plant. The second limit is the warning limit (WL), which was estimated here by supposing the starting of a fault's progression can occur when a measured value exceeds a typical value such as 50% of the caution limit (CL) value.

$$\text{WL} = \text{CL} \times 0.50 \tag{1}$$

The third limit is alarm limit (AL) based on 80% of the CL. Estimation of 80% is based on an experimental investigation carried out by [47], which revealed that the relative error in the oil analytical method could be until 20% of the measured value. The sources of the error can be from oil sampling procedure, instrument inaccuracy, result deviation, human error, etc.

$$\text{AL} = \text{CL} \times 0.80 \tag{2}$$

So, the relative alarm threshold (RAT) can be calculated based on the difference between AL and WL relative to WL (see Equation (3)).

$$\text{RAT} = (\text{AL} - \text{WL}) / \text{WL} \tag{3}$$

As seen in Table 2, the RAT value for any measurable variable was found to be 0.60 at different CLs.


**Table 2.** Relative Alarm Threshold (RAT) of measurable variables.

1,2 according to standard guideline IEC 60599 [13], <sup>3</sup> according to standard guideline IEC 60422 for power transformer HV <sup>≥</sup> 170 kV [15], 4,5 according to experimental investigation [2].

2. Relative fault detection value (RFDV): RFDV is calculated based on the difference between the first measured value (*w*1) of a measurable variable and WL relative to the WL, as demonstrated in Equation (4):

$$\text{RFD}\text{V} = (w\,1-\,\text{WL})/\text{WL} \tag{4}$$

where *w*1 is the first measured value of a measurable variable.

If the value of the RFDV < RAT (0.60), the transformer is still in a good condition. Whereas, if REDV ≥0.60, that is an indication of the probability of having a fault. Hence, a new oil sample from the same transformer should be analyzed after a period, i.e., one month as recommended in [13] to record the second measured value (*w*2).

3. Daily trend (DT%): DT is calculated according to the following equation:

$$\text{Daily Trend, DT\%}=((w2 - w1)/w1) \times 100\text{)/n}\tag{5}$$

where *w*1 is the first measured value, *w*2 is the second measured value after a period, and nd is the number of days between *w*1 and *w*2.

If the DT ≥ 0.33%, there is a strong indication of starting a fault in the transformer. Accordingly, it is recommended to immediately carry out a corrective action to prevent the progression of the fault to a risky level. The factor 0.33% is the critical percentage of increasing the rate of measured value per day according to the standard IEC 60599 [13], which considered the increase in the measured value more than 10% per one month (~0.33% per day) as an indication of an active fault. In contrast, if DT < 0.33% this is an indication of not having a significant trend in the measured values. Accordingly, the transformer can be considered still in a safe condition.
