# Integrated Transformer Health Estimation Methodology Based on Markov Chains and Evidential Reasoning

Authors: Srdjan Milosavljevic; Aleksandar Janjic
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

Due to the large number of power transformers (ETs) in the distribution system, there is a need for a relatively simple representation of the status of each unit in order to more easily determine where and how to allocate the budget for preventive and corrective maintenance. In recent years, the concept of the transformer health index (HI) as an integral part of resource management was adopted for the condition assessment and ranking of ETs. HI algorithms take different forms and can be determined based on a large number of specific parameters. However, the main problem in HI methodology or any modern diagnostic technique is the existence of regular measurements and inspections and accurate test results. The paper proposes a solution in the form of the upgraded HI and the novel methodology for ET ranking including the value of available information to describe ET current state. The confidence to the measurement results is calculated using evidential reasoning (ER) algorithm based on Dempster–Shafer theory. The contribution to the ER methodology is the calculation of the initial degrees of belief using Markov chains. The aging process of an ET and transition probabilities from state to state are modelled using the statistical data for the population of 300 ETs and 20 years monitoring data. The proposed methodology is tested on the real data for 110/35 kV transformer, and in the second case, compared to the sample of 30 110/x kV transformers with traditional HI calculation.


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## Body

## 1. Introduction

Determining the health index (HI) of a transformer is a suitable tool that should provide in a reproducible and consistent way information about its condition, the operational correctness, and availability. Transformer indexing by operating condition, with additional risk analysis, enables a better understanding of the availability and reliability of large transformer populations [1]. HI is a tool that combines the results of in-service electrical testing, laboratory (chemical) testing of transformer oil, maintenance data, and work history data to manage basic resources and build priorities when designing maintenance plans using a numerical grade of transformer status. The choice of methodology depends on the technical capabilities of a particular ET in terms of the number and types of diagnostic tests used to evaluate the condition of transformer elements, as well as on the business and organizational arrangements for storing and updating data and expert teams involved in testing and diagnostics of transformer conditions.Certain measurement methods are performed on a regular basis, such as oil diagnostics, insulation resistance tests, turns ratio measurements, and inspection of accessories. They are performed relatively often and in the state of normal trouble-free operation. Some methods such as the measuring of partial discharges, in spite of their great importance for the assessment of the condition, are not usually used in determining the HI. The reason is possible financial losses resulting from the interruption of electricity supply that can be caused by unnecessary inspections. The advanced methods are used to assess the condition of ETs with developing defects or aged units, such as frequency domain spectroscopy (FDS), frequency response analysis (FRA), advanced bushings, and on-load tap changer (OLTC) diagnostics, as well as measurement of partial discharges.With the advance of prognostics and health management applications, it is possible to enhance traditional transformer health monitoring techniques. For instance, in estimating the remaining life of a transformer, besides the data of paper insulation condition, it is necessary to know the aging kinetics of the oil-impregnated cellulose insulation, which depends on the temperature and water content in the paper. AC conductivity of the composite of cellulose, mineral oil, and water nanoparticles was investigated in [2], while the relationships between the paper degree of polymerization and the furan content for dry and wet insulation were developed in [3]. The water content distribution along windings in transformers is given in [4]. To improve the accuracy of these methods, the dependence of the degree of polymerization of Kraft paper to 2-FAL content in the oil, corrected to a reference temperature of 20°C, is developed in [5], and the method converting experimental frequency dependence of the loss angle tangent measured by the FDS to the reference temperature is presented in [6].As stated before, the advanced measurements are required only for aged ETs with developing defects, but for the vast majority of units, a practical condition index is required. This index should overcome the main problems of previous condition assessment approaches: (a) the rational aggregation of different ET components, (b) uncertainties, accuracy, and confidence of the inspection results, and (c) consistent grade assessment and weighting of different ET components. Recently, the concept of the transformer health index (HI) as an integral part of resource management was adopted for the condition assessment and ranking of ETs. HI calculation method that combines the impact of all available data and criteria based on the industry’s common practices and technical standards is presented in [7], while Gorgan et al. [8] calculated HI based on the standard model of twenty-four diagnostic factors expanding the list with three factors (loss factor at very low frequency, conductivity factor, and polarization index). HI concept can be extended to other equipment, like in [9], where HI was determined for a number of around 2,000 secondary substations, each consisting of a MV switchgear, MV/LV transformer, and LV rack. The algorithm based on regularly conducted oil diagnostics and easily available maintenance data to enable estimation and update the device’s health status in short intervals is proposed in [10]. A comprehensive study of previous research related to the transformer health index by using the mathematical equation/algorithm or expert judgment is given in [11], but very few research studies are dealing with the uncertainties, accuracy, and confidence of the inspection results.Using machine learning techniques, the collected data can be used for the HI prediction in the future period. Various classification techniques have been investigated to reduce ET assessment complexities and decrease the number of features by extracting the most influential ones when determining the HI [12, 13]. Markov model has been developed in [14] to predict the state of health of a transformer and suggest a schedule for regular gas filtration. Markov model is also used in [15] for the determination of the future deterioration performance curve of the transformers. The conclusion is as test results are sometimes imprecise and even incomplete, a suitable evidence integration method is required to process the data. Furthermore, the technical condition of distribution equipment is facing uncertainties concerning the ambiental and operational condition in the future period. One of the solution is proposed in [16], where the methodology is based on discrete convolution of criteria probability distribution functions and OWA operators, modelling different types of criteria aggregation. In [17], a new methodology for the multicriteria risk assessment of the distribution network assets, based on influence diagrams and fuzzy probabilities, is proposed.The evidential reasoning (ER) approach is a suitable method for dealing with the aggregation problem. The process developed on the basis of Dempster–Shafer evidence theory [18] and evaluation analysis model [19] can model various types of qualitative and quantitative uncertainties. With the introduction of the concepts of the belief structure [20, 21] and the belief decision matrix, it became possible to model various types of uncertainties in a unified format.The methodology of how to transfer a transformer condition assessment problem into a multicriteria decision solution under an ER framework is presented in [22]. Subjective judgments are produced using various dissolved gas analysis (DGA) methods for all of the alternatives involved. Then, the concept of a preference degree was introduced to quantify these evaluation grades and subjective judgments with uncertainty. An interesting approach of usage of Dempster–Shafer theory for information fusion tasks combining the outcome of source classifiers through evidence combination strategies is presented in [23]. The study [24] presents an ER approach to transformer winding assessment based on frequency response analysis (FRA), but the degree of uncertainty, like in the previous study, relies only on the expert’s judgment.The integrated fuzzy and evidential reasoning model is presented in [25], with previous operation history, results of the latest inspection, and states of the on-load tap changer taken as evidence to assess the working state of the transformer. The fuzzy model is proposed for generating the original basic probability assignments for the second-level model. The testing data of indices are normalized according to the attention value on transformer tests and operation standards. Fuzzy variables are also used in [26], where the evaluation of the health status of a transformer using DGA, oil testing, and evidential reasoning criterion has been used. In the analysis, the dissolved gases and oil testing parameters are first normalized and then transformed into fuzzy variables using trapezoid membership function. Finally, an index assessing system, considering the main body, the bushing, and the accessory components, was established in [27]. A fuzzy evidence fusion method was represented to handle the fuzzy evidence fusion processes, but in this paper, like in the previous one, the complete state of nature is presumed to be known (the complete belief sum of all propositions within the recognition framework is equal to 1).To overcome the problem of subjective treatment of old, dubious, and uncertain data, the hybrid model of ER and Markov chains is proposed in this paper. A basic tree structure necessary for ER assessment is developed based on the complete transformer model and individual HI of every component, and a general, multilevel evaluation process is used for dealing with multicriteria decision problems. The ET condition is represented as a probability distribution over all possible health states using the Markov chain model of component ageing. The confidence to the measurement results is calculated using evidential reasoning (ER) algorithm based on Dempster–Shafer theory. The contribution to the ER methodology is the calculation of the initial degrees of belief using Markov chains. The proposed methodology is tested on the real data for 110/35 kV transformer, and in the second case, compared to the sample of 30 110/x kV transformers with traditional HI calculation. The methodology is tested to the sample of 30 110/x kV transformers and compared with other HI calculation methods.The rest of the paper is organized as follows. Section2 briefly outlines the health index (HI) approach and how it works as a prioritization method. Section 3 explains the evidence reasoning algorithm. Section 4 provides data analysis and discussion, while Section 5 offers conclusion.


## 2. Transformer Modelling

### 2.1. Health Index Definition

In recent years, the numerical assessment (indexing) of the current state of an ET and other high-voltage equipment in plants assigning a health index (HI) emerges as a tool that could effectively provide a transition to condition-based maintenance. HI is a numerical value that can be used to estimate the overall condition of an ET. By individually evaluating the most representative key factors that are vital to the reliable operation of transformers and mathematically aggregating them into a quantitative index, this value provides information on the “health” of ET.With this index, it will be possible to evaluate the state of a large population of distribution transformers and group them according to the state. Introducing this concept will increase availability and reliability while reducing maintenance costs. The assessment of the condition of the ET should include an assessment of the condition of the key parts: magnetic core and coil, solid insulation and insulating oil, bushings and voltage regulators, cooling system, transformer tank, expansion tank, and auxiliary equipment. The assessment is based on the results obtained by applying appropriate test methods in the field of chemical and electrical testing and visual inspection as well as evaluation of load histories [28]. For each of these parts, the health indices as well as the health index ET must be determined.Given that the assessment of the condition of an ET is based on the following [29]:(i)
Results of electrical and chemical tests(ii)
Maintenance information(iii)
Work history—exploitation events(iv)
Condition of equipment: isolators, cooling system, transformer tank, expansion tank, and auxiliary equipment(v)
The estimated condition of the paper insulation(vi)
Expert opinionHI represents the sum of these estimates. It is very important to view the health index as a variable parameter because by performing a multiparameter analysis of the condition, it changes over the life of the ET [30].


### 2.2. Weighting Factors of Examination Methods

Based on the previous analysis, the calculation of the transformer HI in the proposed methodology includes an assessment of the condition of its key parts listed in Table1. Each part of the ET is assigned a weight factor Wd based on the impact it has on the overall condition of the ET. The impact of part of the ET was also estimated according to the current statistics of the place of occurrence of failure in the ET [31]. Weighting factors are given based on experience and can take the integer value from 1 to 5, as shown in Table 1. The source of weighting factor values for transformer components was the industry practice, established mostly by the experts of the Electrotechnical institute, “Nikola Tesla.” The condition monitoring and assessment is performed for the long time period (more than 40 years in Electric Power Industry of Serbia), and the factors are the result of accumulated practice and experience, reflecting the local market characteristics and operating practice.Table 1
Weighting factors for different ET components.
NoET componentWeighting factor (Wd)1Magnetic core32Geometry end electric contacts of windings43Insulation44Bushings55On-line tap changer56DGA analysis for the active part57Transformer oil48Transformer tank and auxiliary equipment29Work history3Different test methods are used to evaluate the condition of each part of the ET mentioned in Table1. Some parts are joined by a group of appropriate test methods, each corresponding to a weight factor Wm = (1–5), depending on how accurately the results of that method can describe the state of part ET (Table 2).Table 2
Weighting factors of different inspection methods.
ET componentInspection methodWeighting factor (Wm)Magnetic coreOpen-circuit test/SFRA5Geometry end electric contacts of windingsResistance testing5Leakage inductance test/SFRA5InsulationInsulation resistivity/tgδ and capacitance test5PDC/RVM/FDS/water content in oil4Furan derivative analysis3Bushingstgδ and capacitance4On-line tap changerStatic/dynamic resistance testing5DGA analysis for the active partDissolved gas analysis (DGA)4Transformer oilPhysical and chemical oil characteristics5Content of water in oil4Transformer tank and auxiliary equipmentTesting of cooling system and auxiliary equipment2Visual inspection/leakage control2Work historyLoading and operation history3Since the DGA analysis of the transformer oil sample may indicate a problem of overheating or the occurrence of particles, but it cannot reliably define the location of the resulting fault, it is singled out as special. This limited its impact on the value of total HI but not on specific components, such as windings or cores.


### 2.3. Overall Health Index

The overall health index of a transformer can be calculated using the following expression:(1)HI=∑inOdi⋅Wdi∑inWdi.In expression (1), n corresponds to the number of test methods for which there are applicable results and which assess the state of a given system. Odi is a grade for each individual i-th ET part in the range 0 ≤ Odi ≤ 3 calculated in the following:(2)Od=∑i=1nOmi⋅Wmi∑1nWmi.The estimation of theOm method is given by an expert on the basis of the results of the last and previous tests, experience, and specificity of individual ETs and using the criteria given in the applicable standards and technical recommendations. The possible range is 0 ≤ Om ≤ 3. Alternatively, HI can be calculated with expression (3) using different weighting factors for the on-load tap changer or including the transformer loading history [7]:(3)HI=A1⋅∑i=1n=3ci⋅DIi∑i=1n=34⋅ci+A2∑i=1n=3ci⋅DIi∑i=1n=34⋅ci.A1 and A2 represent the weighting factors for the transformer and load tap changer, respectively, while DIi represents the diagnostic index of the ith ET component. The state estimates for electrical measurements are given in descriptive terms: “good condition,” “moderately good,” “moderately bad,” and “poor.” The corresponding estimates for the health index calculation are 3, 2, 1, and 0, as shown in Table 3.Table 3
Comparison of electrical and chemical test scores with appropriate numerical estimates for health index calculations.
Test resultsHIGood condition3Moderately good2 ≤ HI < 3Moderately bad1 ≤ HI < 2Poor<1The “moderately good” rating indicates dubious results but without major changes over time, e.g., comparing the last two to three trials and continuing the follow-up with more frequent testing. On the contrary, the rating “moderately bad” indicates a growing trend of deterioration of the transformer state, and it tightens control by more frequent testing, recommends additional testing, or emphasizes the need to plan for a specific intervention in the coming period.Because of the irregular inspection period, it is hard to perform accurate yearly ET condition assessment. Some data may be several years old, and the main problem in interpretation is the lack of confidence of testing results. The validity of results can be treated by the similar grading system (from 0 for results older than the maximal inspection period and 3 for actual measurement results). The validity of thei-th transformer component VEi with n inspection methods is given in the following:(4)VEi=∑j=1nVMjn.Then, the HI value composed ofm components, calculated from equation (3), can be corrected with the validity of results (5):(5)HIv=HI⋅∑i=1mVEim.Because this treatment of old measurements is very simplified and not based on the complete transformer aging model, the ET condition must be represented as a probability distribution over all possible health states. In this paper, this distribution is determined using the Markov chain model of component ageing, and evidential reasoning is used for the quantification of different parameters. The integrated methodology is presented in the sequel.


## 2.1. Health Index Definition

In recent years, the numerical assessment (indexing) of the current state of an ET and other high-voltage equipment in plants assigning a health index (HI) emerges as a tool that could effectively provide a transition to condition-based maintenance. HI is a numerical value that can be used to estimate the overall condition of an ET. By individually evaluating the most representative key factors that are vital to the reliable operation of transformers and mathematically aggregating them into a quantitative index, this value provides information on the “health” of ET.With this index, it will be possible to evaluate the state of a large population of distribution transformers and group them according to the state. Introducing this concept will increase availability and reliability while reducing maintenance costs. The assessment of the condition of the ET should include an assessment of the condition of the key parts: magnetic core and coil, solid insulation and insulating oil, bushings and voltage regulators, cooling system, transformer tank, expansion tank, and auxiliary equipment. The assessment is based on the results obtained by applying appropriate test methods in the field of chemical and electrical testing and visual inspection as well as evaluation of load histories [28]. For each of these parts, the health indices as well as the health index ET must be determined.Given that the assessment of the condition of an ET is based on the following [29]:(i)
Results of electrical and chemical tests(ii)
Maintenance information(iii)
Work history—exploitation events(iv)
Condition of equipment: isolators, cooling system, transformer tank, expansion tank, and auxiliary equipment(v)
The estimated condition of the paper insulation(vi)
Expert opinionHI represents the sum of these estimates. It is very important to view the health index as a variable parameter because by performing a multiparameter analysis of the condition, it changes over the life of the ET [30].


## 2.2. Weighting Factors of Examination Methods

Based on the previous analysis, the calculation of the transformer HI in the proposed methodology includes an assessment of the condition of its key parts listed in Table1. Each part of the ET is assigned a weight factor Wd based on the impact it has on the overall condition of the ET. The impact of part of the ET was also estimated according to the current statistics of the place of occurrence of failure in the ET [31]. Weighting factors are given based on experience and can take the integer value from 1 to 5, as shown in Table 1. The source of weighting factor values for transformer components was the industry practice, established mostly by the experts of the Electrotechnical institute, “Nikola Tesla.” The condition monitoring and assessment is performed for the long time period (more than 40 years in Electric Power Industry of Serbia), and the factors are the result of accumulated practice and experience, reflecting the local market characteristics and operating practice.Table 1
Weighting factors for different ET components.
NoET componentWeighting factor (Wd)1Magnetic core32Geometry end electric contacts of windings43Insulation44Bushings55On-line tap changer56DGA analysis for the active part57Transformer oil48Transformer tank and auxiliary equipment29Work history3Different test methods are used to evaluate the condition of each part of the ET mentioned in Table1. Some parts are joined by a group of appropriate test methods, each corresponding to a weight factor Wm = (1–5), depending on how accurately the results of that method can describe the state of part ET (Table 2).Table 2
Weighting factors of different inspection methods.
ET componentInspection methodWeighting factor (Wm)Magnetic coreOpen-circuit test/SFRA5Geometry end electric contacts of windingsResistance testing5Leakage inductance test/SFRA5InsulationInsulation resistivity/tgδ and capacitance test5PDC/RVM/FDS/water content in oil4Furan derivative analysis3Bushingstgδ and capacitance4On-line tap changerStatic/dynamic resistance testing5DGA analysis for the active partDissolved gas analysis (DGA)4Transformer oilPhysical and chemical oil characteristics5Content of water in oil4Transformer tank and auxiliary equipmentTesting of cooling system and auxiliary equipment2Visual inspection/leakage control2Work historyLoading and operation history3Since the DGA analysis of the transformer oil sample may indicate a problem of overheating or the occurrence of particles, but it cannot reliably define the location of the resulting fault, it is singled out as special. This limited its impact on the value of total HI but not on specific components, such as windings or cores.


## 2.3. Overall Health Index

The overall health index of a transformer can be calculated using the following expression:(1)HI=∑inOdi⋅Wdi∑inWdi.In expression (1), n corresponds to the number of test methods for which there are applicable results and which assess the state of a given system. Odi is a grade for each individual i-th ET part in the range 0 ≤ Odi ≤ 3 calculated in the following:(2)Od=∑i=1nOmi⋅Wmi∑1nWmi.The estimation of theOm method is given by an expert on the basis of the results of the last and previous tests, experience, and specificity of individual ETs and using the criteria given in the applicable standards and technical recommendations. The possible range is 0 ≤ Om ≤ 3. Alternatively, HI can be calculated with expression (3) using different weighting factors for the on-load tap changer or including the transformer loading history [7]:(3)HI=A1⋅∑i=1n=3ci⋅DIi∑i=1n=34⋅ci+A2∑i=1n=3ci⋅DIi∑i=1n=34⋅ci.A1 and A2 represent the weighting factors for the transformer and load tap changer, respectively, while DIi represents the diagnostic index of the ith ET component. The state estimates for electrical measurements are given in descriptive terms: “good condition,” “moderately good,” “moderately bad,” and “poor.” The corresponding estimates for the health index calculation are 3, 2, 1, and 0, as shown in Table 3.Table 3
Comparison of electrical and chemical test scores with appropriate numerical estimates for health index calculations.
Test resultsHIGood condition3Moderately good2 ≤ HI < 3Moderately bad1 ≤ HI < 2Poor<1The “moderately good” rating indicates dubious results but without major changes over time, e.g., comparing the last two to three trials and continuing the follow-up with more frequent testing. On the contrary, the rating “moderately bad” indicates a growing trend of deterioration of the transformer state, and it tightens control by more frequent testing, recommends additional testing, or emphasizes the need to plan for a specific intervention in the coming period.Because of the irregular inspection period, it is hard to perform accurate yearly ET condition assessment. Some data may be several years old, and the main problem in interpretation is the lack of confidence of testing results. The validity of results can be treated by the similar grading system (from 0 for results older than the maximal inspection period and 3 for actual measurement results). The validity of thei-th transformer component VEi with n inspection methods is given in the following:(4)VEi=∑j=1nVMjn.Then, the HI value composed ofm components, calculated from equation (3), can be corrected with the validity of results (5):(5)HIv=HI⋅∑i=1mVEim.Because this treatment of old measurements is very simplified and not based on the complete transformer aging model, the ET condition must be represented as a probability distribution over all possible health states. In this paper, this distribution is determined using the Markov chain model of component ageing, and evidential reasoning is used for the quantification of different parameters. The integrated methodology is presented in the sequel.


## 3. Integrated Assessment Methodology

### 3.1. Evidential Reasoning Algorithm

To evaluate the state of a power transformer, large amount of qualitative and numerical information needs to be interpreted on different hierarchical levels. The ER approach is a suitable method for dealing with the aggregation problem, and the original ER model and algorithm, based on Dempster–Shafer theory [18], are described next.In a two-level hierarchy of attributes with a general attribute at the top level andL basic attributes at the lower level ei (i = 1, …, L), it is possible to define a set of low-level attributes as follows:(6)E=ei,…,ei,…,eL.The weights of the attributes are presented byω = {ω1, …, ωi, …, ωL}, where ωi is the relative weight of the ith lower-level attribute (ei) with value between 0 and 1 (0 ≤ ωi ≤ 1). The evaluation grades are represented by the following:(7)H1,…,Hn,…,HN.It is assumed thatHn+1 is preferred to Hn.The methodology for the evaluation grades for transformer components presented in Table2 is given in [8]. For the sake of illustration, the grading of solid insulation is given in Table 4.Table 4
The evaluation grades for solid insulation.
Dielectric lossesInsulation stateHntgδ ≤ 1%Good condition31% < tgδ ≤ 1.5%Moderately good21.5% < tgδ ≤ 2%Moderately bad1tgδ > 2%Poor0An assessment for theith basic attribute ei may be represented by the following distribution:(8)Sei=Hn,βn,i,n=1,…,N,i=1,…,L,where βn,i denotes the degree of belief and βn,i ≥ 0, ∑n=1Nβn,i≤1. If ∑n=1Nβn,i=1, then assessment S(ei) is complete. In the opposite case, assessment S(ei) is incomplete. Equation (9) denotes a complete lack of information on ei:(9)∑n=1Nβn,i=0.LetHn be a grade to which the general attribute is assessed with certain degree of belief βn. The problem is to generate βn by aggregating the assessments for all associated basic attributes ei. For this purpose, the following algorithm is used.Letmn,i be a basic probability mass representing the degree to which basic ith attribute ei supports judgment that the general attribute y is assessed to the grade Hn, respectively; let mH,i be a remaining probability mass unassigned to any individual grade after all the N grades, concerning the ei attribute, are considered. Next expression explains how basic probability mass is calculated:(10)mn,i=ωiβn,i,n=1,…,N,(11)∑n=1Nωn=1.Remaining probability mass is calculated as(12)mH.i=1−∑n=1Nmn,i=1−ωi∑n=1Nβn,i.Suppose thatEI(i) is a subset of the first i attributes EI(i) = {e1, e2, …, ei}, and according to that, mn,I(i) can be the probability mass defined as the degree to which all the i attributes support the judgment that y is assessed to the grade Hn. Also, mH,I(i) is the remaining probability mass unassigned to individual grades after all the basic attributes in EI(i) have been assessed. Probability masses mn,I(i) and mH,I(i) for EI(i) can be calculated from basic probability masses mn,j and mH,j for all n = 1,…, N, j = 1,…, i. Concerning all the above statements, the original recursive evidential reasoning algorithm can be summarized by the following expressions:(13)mn,Ii+1=KIi+1mn,Iimn,i+1+mn,IimH,i+1+mH,Iimn,i+1,n=1,…,N,(14)mH,Ii+1=KIi+1mH,IimH,i+1,(15)KIi+1=1−∑t=1N∑j=1j≠tNmt,Iimj,i+1−1,i=1,…,L−1,where KI(I+1) is a normalizing factor so that ∑n=1Nmn,Ii+1+mH,Ii+1=1 is ensured. It is important to note that basic attributes in EI(i) are numbered arbitrarily and that initial values are mn,I(1) = mn,1 and mH,I(1) = mH,1. And finally, in the original evidential reasoning algorithm, combined degree of belief for a general attribute βn is given by(16)βn=mn,IL,n=1,…,N,(17)βH=mH,IL=1−∑n=1Nβn,while βH denotes the degree of incompleteness of the assessment.As explained in the introductory section, the hierarchical structure of an ER algorithm proved to be adequate for the condition-based maintenance of the power transformer [18–20]. However, although the process involves decision-making with multiple attribute with uncertainty, very little attention was paid to the determination of degrees of belief βn,i. As the deterioration process from the last inspection can be easily modelled with the appropriate probability distribution, the methodology presented in this paper is using Markov chain modelling of component ageing for the assessment of this parameter. The single grade for the HI can also be calculated using the concept of average expected utility [21]. The concept of expected utility is used to define equivalent numerical values. If u(Hn) is the utility of the grade Hn with u(Hn + 1) > u(Hn) if Hn + 1 is preferred to Hn, then the maximum, minimum, and the average expected utilities on y are given by(18)umaxy=∑n=1N−1βnuHn+βN+βHuHN,uminy=β1+βHuH1+∑n=2NβnuHn,uavgy=umaxy−uminy2.


### 3.2. Markov Chain Modelling of Component Ageing

The process of transition from state to state can be represented by the diagram given in Figure1. H3 through H0 represent the health status of the transformer according to the established health index. The λij labels indicate the transition rates from state i to state j. If we assume that the transition rates are constant over a period of time, the time of transition from one state to another follows an exponential distribution, and the diagram would represent the Markov process. The same analysis is valid for the individual transformer component according to the analysis in the previous chapter.Figure 1
State transition scheme.The exponential probability distribution (19) is one of the most significant distributions in the reliability analysis:(19)FX=1−exp−λ⋅x,x≥0,0,x<0.The failure intensity of this distribution is a constant value, and the time to failure is independent of the operating time of the equipment being observed. By using Markov transition diagrams, degrees of belief in terms of ER methodology can be modelled in the following way. Let the measurement and inspection results indicate the health state Hi. After the periodt since the last measurement, the probability that the component remained in the same state (Pii) or transitioned to the state j (Pij) can be represented by the following transition matrix:(20)P=P33P32P31P300P22P21P2000P11P100001.Ann-step transition probability P(n) denotes the probability that n time units later the chain will be in state j given it is now in state i (21):(21)Pn=Pn=P×P×⋯×P,n≥1.The intensity of leaving the state is equal to the sum of the transitions from the state to the other states of the system so that(22)λii=−∑k=1k≠inλki.λki is the intensity of transition from state to state k and λii is the intensity of leaving the state and with a negative sign.Exponential distribution has one important feature that allows relatively simple modelling of systems with exponential residence time distributions in characteristic states. The probability that the system that is on state int1 is likely to transition to another state over a period of time (t1, t2) is(23)PT≤t2T>t1=1−exp−∫t1t2λdt=1−exp−λ⋅t2−t1.As can be seen from the previous relation, the considered probability depends solely on the length of time interval for which the probability is calculated and not on the length of stay in the previous state. If the time interval is very short, an approximate replacement of the exponential function is obtained from the previous relation:(24)PT≤t2T>t1≅λ⋅t2−t1.Using the statistical data from the transformer history, the intensities of transition from statei to state j (λij) are calculated based on the following relation:(25)λij=1∑tij/nij,where nij represents the number of transitions from state i to j, while tij represents the average number of years staying in state i before the transition to j.The algorithm for the ET assessment can be presented in the following steps and graphically presented in Figure2.Step 1: define a set ofL inspection methods (basic attributes) influencing the assessment of the ET component state (upper-level attribute)Step 2: for each attribute, determine the transition probability matrix (22) for different health statesStep 3: depending on inspection accuracy and time,εi, and evaluation grade Hn, a degree of belief βn is assigned for each attributeStep 4:mn,i, a basic probability mass, representing the degree to which the ith inspection method εi supports a hypothesis that the health index is assessed to the nth evaluation grade Hn is calculatedStep 5: the combined probability masses are generated by aggregating all the basic probability assignments using the recursive ER algorithmStep 6: calculate the combined degrees of belief for a higher-level propertyFigure 2
State transition scheme.The proposed methodology will be illustrated on individual transformer health estimation and on comparative estimation of 30 transformers operating in EPS (Electric Power Industry of Serbia).


## 3.1. Evidential Reasoning Algorithm

To evaluate the state of a power transformer, large amount of qualitative and numerical information needs to be interpreted on different hierarchical levels. The ER approach is a suitable method for dealing with the aggregation problem, and the original ER model and algorithm, based on Dempster–Shafer theory [18], are described next.In a two-level hierarchy of attributes with a general attribute at the top level andL basic attributes at the lower level ei (i = 1, …, L), it is possible to define a set of low-level attributes as follows:(6)E=ei,…,ei,…,eL.The weights of the attributes are presented byω = {ω1, …, ωi, …, ωL}, where ωi is the relative weight of the ith lower-level attribute (ei) with value between 0 and 1 (0 ≤ ωi ≤ 1). The evaluation grades are represented by the following:(7)H1,…,Hn,…,HN.It is assumed thatHn+1 is preferred to Hn.The methodology for the evaluation grades for transformer components presented in Table2 is given in [8]. For the sake of illustration, the grading of solid insulation is given in Table 4.Table 4
The evaluation grades for solid insulation.
Dielectric lossesInsulation stateHntgδ ≤ 1%Good condition31% < tgδ ≤ 1.5%Moderately good21.5% < tgδ ≤ 2%Moderately bad1tgδ > 2%Poor0An assessment for theith basic attribute ei may be represented by the following distribution:(8)Sei=Hn,βn,i,n=1,…,N,i=1,…,L,where βn,i denotes the degree of belief and βn,i ≥ 0, ∑n=1Nβn,i≤1. If ∑n=1Nβn,i=1, then assessment S(ei) is complete. In the opposite case, assessment S(ei) is incomplete. Equation (9) denotes a complete lack of information on ei:(9)∑n=1Nβn,i=0.LetHn be a grade to which the general attribute is assessed with certain degree of belief βn. The problem is to generate βn by aggregating the assessments for all associated basic attributes ei. For this purpose, the following algorithm is used.Letmn,i be a basic probability mass representing the degree to which basic ith attribute ei supports judgment that the general attribute y is assessed to the grade Hn, respectively; let mH,i be a remaining probability mass unassigned to any individual grade after all the N grades, concerning the ei attribute, are considered. Next expression explains how basic probability mass is calculated:(10)mn,i=ωiβn,i,n=1,…,N,(11)∑n=1Nωn=1.Remaining probability mass is calculated as(12)mH.i=1−∑n=1Nmn,i=1−ωi∑n=1Nβn,i.Suppose thatEI(i) is a subset of the first i attributes EI(i) = {e1, e2, …, ei}, and according to that, mn,I(i) can be the probability mass defined as the degree to which all the i attributes support the judgment that y is assessed to the grade Hn. Also, mH,I(i) is the remaining probability mass unassigned to individual grades after all the basic attributes in EI(i) have been assessed. Probability masses mn,I(i) and mH,I(i) for EI(i) can be calculated from basic probability masses mn,j and mH,j for all n = 1,…, N, j = 1,…, i. Concerning all the above statements, the original recursive evidential reasoning algorithm can be summarized by the following expressions:(13)mn,Ii+1=KIi+1mn,Iimn,i+1+mn,IimH,i+1+mH,Iimn,i+1,n=1,…,N,(14)mH,Ii+1=KIi+1mH,IimH,i+1,(15)KIi+1=1−∑t=1N∑j=1j≠tNmt,Iimj,i+1−1,i=1,…,L−1,where KI(I+1) is a normalizing factor so that ∑n=1Nmn,Ii+1+mH,Ii+1=1 is ensured. It is important to note that basic attributes in EI(i) are numbered arbitrarily and that initial values are mn,I(1) = mn,1 and mH,I(1) = mH,1. And finally, in the original evidential reasoning algorithm, combined degree of belief for a general attribute βn is given by(16)βn=mn,IL,n=1,…,N,(17)βH=mH,IL=1−∑n=1Nβn,while βH denotes the degree of incompleteness of the assessment.As explained in the introductory section, the hierarchical structure of an ER algorithm proved to be adequate for the condition-based maintenance of the power transformer [18–20]. However, although the process involves decision-making with multiple attribute with uncertainty, very little attention was paid to the determination of degrees of belief βn,i. As the deterioration process from the last inspection can be easily modelled with the appropriate probability distribution, the methodology presented in this paper is using Markov chain modelling of component ageing for the assessment of this parameter. The single grade for the HI can also be calculated using the concept of average expected utility [21]. The concept of expected utility is used to define equivalent numerical values. If u(Hn) is the utility of the grade Hn with u(Hn + 1) > u(Hn) if Hn + 1 is preferred to Hn, then the maximum, minimum, and the average expected utilities on y are given by(18)umaxy=∑n=1N−1βnuHn+βN+βHuHN,uminy=β1+βHuH1+∑n=2NβnuHn,uavgy=umaxy−uminy2.


## 3.2. Markov Chain Modelling of Component Ageing

The process of transition from state to state can be represented by the diagram given in Figure1. H3 through H0 represent the health status of the transformer according to the established health index. The λij labels indicate the transition rates from state i to state j. If we assume that the transition rates are constant over a period of time, the time of transition from one state to another follows an exponential distribution, and the diagram would represent the Markov process. The same analysis is valid for the individual transformer component according to the analysis in the previous chapter.Figure 1
State transition scheme.The exponential probability distribution (19) is one of the most significant distributions in the reliability analysis:(19)FX=1−exp−λ⋅x,x≥0,0,x<0.The failure intensity of this distribution is a constant value, and the time to failure is independent of the operating time of the equipment being observed. By using Markov transition diagrams, degrees of belief in terms of ER methodology can be modelled in the following way. Let the measurement and inspection results indicate the health state Hi. After the periodt since the last measurement, the probability that the component remained in the same state (Pii) or transitioned to the state j (Pij) can be represented by the following transition matrix:(20)P=P33P32P31P300P22P21P2000P11P100001.Ann-step transition probability P(n) denotes the probability that n time units later the chain will be in state j given it is now in state i (21):(21)Pn=Pn=P×P×⋯×P,n≥1.The intensity of leaving the state is equal to the sum of the transitions from the state to the other states of the system so that(22)λii=−∑k=1k≠inλki.λki is the intensity of transition from state to state k and λii is the intensity of leaving the state and with a negative sign.Exponential distribution has one important feature that allows relatively simple modelling of systems with exponential residence time distributions in characteristic states. The probability that the system that is on state int1 is likely to transition to another state over a period of time (t1, t2) is(23)PT≤t2T>t1=1−exp−∫t1t2λdt=1−exp−λ⋅t2−t1.As can be seen from the previous relation, the considered probability depends solely on the length of time interval for which the probability is calculated and not on the length of stay in the previous state. If the time interval is very short, an approximate replacement of the exponential function is obtained from the previous relation:(24)PT≤t2T>t1≅λ⋅t2−t1.Using the statistical data from the transformer history, the intensities of transition from statei to state j (λij) are calculated based on the following relation:(25)λij=1∑tij/nij,where nij represents the number of transitions from state i to j, while tij represents the average number of years staying in state i before the transition to j.The algorithm for the ET assessment can be presented in the following steps and graphically presented in Figure2.Step 1: define a set ofL inspection methods (basic attributes) influencing the assessment of the ET component state (upper-level attribute)Step 2: for each attribute, determine the transition probability matrix (22) for different health statesStep 3: depending on inspection accuracy and time,εi, and evaluation grade Hn, a degree of belief βn is assigned for each attributeStep 4:mn,i, a basic probability mass, representing the degree to which the ith inspection method εi supports a hypothesis that the health index is assessed to the nth evaluation grade Hn is calculatedStep 5: the combined probability masses are generated by aggregating all the basic probability assignments using the recursive ER algorithmStep 6: calculate the combined degrees of belief for a higher-level propertyFigure 2
State transition scheme.The proposed methodology will be illustrated on individual transformer health estimation and on comparative estimation of 30 transformers operating in EPS (Electric Power Industry of Serbia).


## 4. Case Studies

The fleet of 344 transformers of 110 kV primary voltage operating in EPS was monitored during 10 years period (2008–2018). Transition probabilities are derived from the measurement databases consisting of more than 10,000 measurements of 18 main transformer components. The proposed methodology will be illustrated on the individual transformer health assessment (Case 1) and the ranking of a sample of 30 transformers (Case 2).


### 4.1. Case 1

The methodology for the condition assessment will be applied to the existing transformer 110/35/10 kV, 20/20/10MVA. Starting from a complete model presented in Tables2 and 3, a reduced model concerning only the main transformer parts without the on-line tap changer is presented in Figure 3.Figure 3
Hierarchical scheme for transformer HI assessment.Initial data with the available measurements, together with weighting factors for ET component (Wi) and testing method (Wm), are presented in Table 5. Starting from values in Tables 2 and 3, factors are normalized to fulfill condition (11). Because of different dates of inspection methods, different degrees of belief are presented in the table. The degree of belief denotes the source’s level of confidence when assessing the level of fulfillment of a certain property. For instance, due to the lack of frequency domain spectroscopy (FDS) test, all belief values equal to zero. For the sake of illustration, the transition intensity matrix for the physical/chemical oil characteristics λP,C (1/year) obtained from the 10-year period is given in the following:(26)λP,C=0,750,140,080,0200,820,170,01000,80,20001.Table 5
Initial data for the degrees of belief calculation.
OilInsulationActive partWindingsWd0.240.240.280.24Wm0.550.450.410.340.250.50.5HiPhys, chem (βi, 1)H2O (βi,2)tgδ (βi,1)FDS (βi,2)Furan (βi,3)DGA (βi)R (βi,1)L (βi,2)30.750000.800.50.520.140.80000.90.30.310.080.20.80000000.020000000The physical/chemical characteristics have been inspected two years ago, and the state of oil has been graded as “good.” Using expressions (23) and (24), the probabilities that the oil characteristics are in the states 3, 2, 1, and 0 are 0.75, 0.145, 0.08, and 0.02, respectively.Recursively using equations (13)–(17) for the aggregation of probability masses for individual inspection method, probability masses for individual ET component are obtained and represented in Table 6. For instance, assessment of transformer oil for the grade H3 = “good,” H2 = “moderately good,” H1 = “moderately bad,” and H0 = “poor,” equals 0.17, 0.44, 0.045, and 0, respectively.Table 6
The degrees of belief of the main transformer components.
3210mH,iOil0.170.440.04500.345Insulation0.06200.1400.8DGA00.252000.748Windings0.1530.07000.777With the values calculated in step 3, we get the combined degrees of belief forH3 = “good,” H2 = “moderately good,” and H1 = “moderately bad” which equal to 0.32, 0.175, and 0.08, respectively. The average HI, obtained from (19)–(21) is HIavg = 2.03.Using the traditional HI calculation method, using the data from Table7 and equation (2), the grade Od for oil, insulation, active part, and windings equals 2.56, 1.75, 2, and 3, respectively. Using equation (3), the value of the HI is given in the following:(27)HI=∑14Odi⋅Wdi∑14Wdi=4⋅2,56+4⋅1,75+5⋅2+4⋅317=2,3.Table 7
Transformer assessment using the traditional HI.
OilInsulationActive partWindingsWd4454Wm5454355Phys, chemH2OtgδFDSFuranDGARLOm321—3233


### 4.2. Case 2

The estimation of HI was performed on a sample of 30 distributive energy transformers. The complete transformer model with the elements in Table2 is used. The first estimation was made by experts, who gave an assessment on the basis of existing electrical and chemical tests, i.e., data on maintenance and load history. The results were rechecked during the next two-year period, and their assessment was accepted as the reference value for the benchmarking with two other HI calculation methodologies.Figure4 shows the condition estimation of the mentioned sample of 30 transformers obtained by different mathematical models.Figure 4
HI obtained by different assessment methods.The HI is calculated using the proposed ER methodology (denoted ER in Figure4) and compared with the results obtained using equation (3) and adjusted HI index, taking into account the measurement period (equation (5)). The traditional approach to HI calculation gives the large deviations from the accurate health grades, especially for transformers in poor condition. Because of the lack of certain parameter values, (3) gives far more optimistic values (for transformers 1, 2, 8, and 18). On the contrary, integrated ER methodology gives the results much closer to accurate HI values. Transformers 14 and 15 are rated high with both expert's opinion and equation (3), but the ER methodology gave more pessimistic results, thanks to the lack of accurate data. Equation (5) gives better results than the traditional method, but great variations from the exact values are still present.Deviations of all ER results from the accurate health state are presented in5.Figure 5
Deviations of the real health values from the HI obtained by the ER method.Maximal deviations from the exact values are obtained with equation (3). The problem with this methodology is that the nonexistent measurements are simply not taken into account in the calculation. Transformers 1, 2, and 18 have the high HI values although their real condition is alarming (the complete table with the inspection results is given in Table 8). Equation (5) methodology gives smaller deviations that are equally dispersed for all HI values. However, the deviations are still great (above unity) for greater number of transformers. ER methodology gives best results. Only two transformers (14 and 15) have the deviations from the exact values that are greater than 1 point. These transformers were ranked as “good” by the experts, which means that the ER methodology gives the security offset. The deviations for ETs that are ranked as “bad” are much more lower than for other two methods.Table 8
Inspection grades and assessment results.
Magnetic core 3Geometry end electric contacts of windings, 4Insulation, 4Bushings, 5On-line tap changer, 5Oil, 4DGA, 5Tank, 2HI [0–3]Total HI [0–100%]Validity ažurnostExpertEquation (3)ERE 5Dev equation (3)Dev ERDev equation (5)HIVHIVHIVHIVHIVHIVHIVHIV119833102.51036.670103101.441031000.7424.6783.3402.760.410.6167162.760.410.6167162351.52.536.6700303103103100.782655.2102.490.410.4306382.490.410.4306381819863103102.671001103103103101.7959.6787.501.080.21.566251.080.21.566251719793101.5536.6731030353001.963.3345.840.61.231.210.870960.630.610.2709613195831031021001102103102.56101.655587.50.750.931.471.443750.180.720.69375519693103103103101103103103101.8461.331001.351.781.41.840.430.050.49819723103102.678.333101103103103101.8260.6797.911.352.581.41.7819621.230.050.431962719683102.5102.386.67103103103103101.775983.341.52.111.61.4751180.610.10.024882919622101.25101.421003102103103101.625487.51.52.581.61.41751.080.10.082511197831031031003103103101.22101.9364.3387.51.52.21.91.688750.70.40.1887519197231031031001102.44103102.44101.7558.3387.51.52.151.71.531250.650.20.031252319893101.5536.6731030353001.963.3345.841.52.782.230.870961.280.730.62904419710302.336.67310001.44531000.723.3339.591.652.641.80.277130.990.151.3728712198803536.67310310352002.8193.6745.841.652.782.241.2881041.130.590.36189620201303536.67310310310310310310077.091.652.111.82.31270.460.150.66276196800353501.4410310351.8461.3343.751.82.221.70.8050.420.10.9951619800353103103103103100310068.751.81.231.222.06250.570.580.26252219710100100.586.670310350000.391352.091.82.952.740.2031511.150.941.5968492119653102.5103103102102.56103102.56102.7901001.952.721.92.70.770.050.752419683103102.67103103102.44103103102.8996.331001.952.712.432.890.760.480.94319783103103100310310302.56102.9799752.12.542.22.22750.440.10.12751019823101.552.67103103102.56103103101.866293.752.11.491.71.743750.610.40.356252519803101.5536.672530353001.8260.6739.592.12.432.560.7205380.330.461.3794621419543103101.466.6703102.44103103101.8461.3383.342.251.381.231.5334560.871.020.71654426198631031031031031031031031031001002.252.642.2330.390.020.752719743103102.581031031031021002.7892.6787.52.252.111.562.43250.140.690.1825153102.51033.333103100002.996.6754.1631.531.241.570641.471.761.429362819783103102.581003101.441031001.8461.337532.582.541.380.420.461.622919783103103103103103103103103100100332.74300.26030197431031036.673103101531001.8762.3377.09332.741.44158300.261.558417


## 4.1. Case 1

The methodology for the condition assessment will be applied to the existing transformer 110/35/10 kV, 20/20/10MVA. Starting from a complete model presented in Tables2 and 3, a reduced model concerning only the main transformer parts without the on-line tap changer is presented in Figure 3.Figure 3
Hierarchical scheme for transformer HI assessment.Initial data with the available measurements, together with weighting factors for ET component (Wi) and testing method (Wm), are presented in Table 5. Starting from values in Tables 2 and 3, factors are normalized to fulfill condition (11). Because of different dates of inspection methods, different degrees of belief are presented in the table. The degree of belief denotes the source’s level of confidence when assessing the level of fulfillment of a certain property. For instance, due to the lack of frequency domain spectroscopy (FDS) test, all belief values equal to zero. For the sake of illustration, the transition intensity matrix for the physical/chemical oil characteristics λP,C (1/year) obtained from the 10-year period is given in the following:(26)λP,C=0,750,140,080,0200,820,170,01000,80,20001.Table 5
Initial data for the degrees of belief calculation.
OilInsulationActive partWindingsWd0.240.240.280.24Wm0.550.450.410.340.250.50.5HiPhys, chem (βi, 1)H2O (βi,2)tgδ (βi,1)FDS (βi,2)Furan (βi,3)DGA (βi)R (βi,1)L (βi,2)30.750000.800.50.520.140.80000.90.30.310.080.20.80000000.020000000The physical/chemical characteristics have been inspected two years ago, and the state of oil has been graded as “good.” Using expressions (23) and (24), the probabilities that the oil characteristics are in the states 3, 2, 1, and 0 are 0.75, 0.145, 0.08, and 0.02, respectively.Recursively using equations (13)–(17) for the aggregation of probability masses for individual inspection method, probability masses for individual ET component are obtained and represented in Table 6. For instance, assessment of transformer oil for the grade H3 = “good,” H2 = “moderately good,” H1 = “moderately bad,” and H0 = “poor,” equals 0.17, 0.44, 0.045, and 0, respectively.Table 6
The degrees of belief of the main transformer components.
3210mH,iOil0.170.440.04500.345Insulation0.06200.1400.8DGA00.252000.748Windings0.1530.07000.777With the values calculated in step 3, we get the combined degrees of belief forH3 = “good,” H2 = “moderately good,” and H1 = “moderately bad” which equal to 0.32, 0.175, and 0.08, respectively. The average HI, obtained from (19)–(21) is HIavg = 2.03.Using the traditional HI calculation method, using the data from Table7 and equation (2), the grade Od for oil, insulation, active part, and windings equals 2.56, 1.75, 2, and 3, respectively. Using equation (3), the value of the HI is given in the following:(27)HI=∑14Odi⋅Wdi∑14Wdi=4⋅2,56+4⋅1,75+5⋅2+4⋅317=2,3.Table 7
Transformer assessment using the traditional HI.
OilInsulationActive partWindingsWd4454Wm5454355Phys, chemH2OtgδFDSFuranDGARLOm321—3233


## 4.2. Case 2

The estimation of HI was performed on a sample of 30 distributive energy transformers. The complete transformer model with the elements in Table2 is used. The first estimation was made by experts, who gave an assessment on the basis of existing electrical and chemical tests, i.e., data on maintenance and load history. The results were rechecked during the next two-year period, and their assessment was accepted as the reference value for the benchmarking with two other HI calculation methodologies.Figure4 shows the condition estimation of the mentioned sample of 30 transformers obtained by different mathematical models.Figure 4
HI obtained by different assessment methods.The HI is calculated using the proposed ER methodology (denoted ER in Figure4) and compared with the results obtained using equation (3) and adjusted HI index, taking into account the measurement period (equation (5)). The traditional approach to HI calculation gives the large deviations from the accurate health grades, especially for transformers in poor condition. Because of the lack of certain parameter values, (3) gives far more optimistic values (for transformers 1, 2, 8, and 18). On the contrary, integrated ER methodology gives the results much closer to accurate HI values. Transformers 14 and 15 are rated high with both expert's opinion and equation (3), but the ER methodology gave more pessimistic results, thanks to the lack of accurate data. Equation (5) gives better results than the traditional method, but great variations from the exact values are still present.Deviations of all ER results from the accurate health state are presented in5.Figure 5
Deviations of the real health values from the HI obtained by the ER method.Maximal deviations from the exact values are obtained with equation (3). The problem with this methodology is that the nonexistent measurements are simply not taken into account in the calculation. Transformers 1, 2, and 18 have the high HI values although their real condition is alarming (the complete table with the inspection results is given in Table 8). Equation (5) methodology gives smaller deviations that are equally dispersed for all HI values. However, the deviations are still great (above unity) for greater number of transformers. ER methodology gives best results. Only two transformers (14 and 15) have the deviations from the exact values that are greater than 1 point. These transformers were ranked as “good” by the experts, which means that the ER methodology gives the security offset. The deviations for ETs that are ranked as “bad” are much more lower than for other two methods.Table 8
Inspection grades and assessment results.
Magnetic core 3Geometry end electric contacts of windings, 4Insulation, 4Bushings, 5On-line tap changer, 5Oil, 4DGA, 5Tank, 2HI [0–3]Total HI [0–100%]Validity ažurnostExpertEquation (3)ERE 5Dev equation (3)Dev ERDev equation (5)HIVHIVHIVHIVHIVHIVHIVHIV119833102.51036.670103101.441031000.7424.6783.3402.760.410.6167162.760.410.6167162351.52.536.6700303103103100.782655.2102.490.410.4306382.490.410.4306381819863103102.671001103103103101.7959.6787.501.080.21.566251.080.21.566251719793101.5536.6731030353001.963.3345.840.61.231.210.870960.630.610.2709613195831031021001102103102.56101.655587.50.750.931.471.443750.180.720.69375519693103103103101103103103101.8461.331001.351.781.41.840.430.050.49819723103102.678.333101103103103101.8260.6797.911.352.581.41.7819621.230.050.431962719683102.5102.386.67103103103103101.775983.341.52.111.61.4751180.610.10.024882919622101.25101.421003102103103101.625487.51.52.581.61.41751.080.10.082511197831031031003103103101.22101.9364.3387.51.52.21.91.688750.70.40.1887519197231031031001102.44103102.44101.7558.3387.51.52.151.71.531250.650.20.031252319893101.5536.6731030353001.963.3345.841.52.782.230.870961.280.730.62904419710302.336.67310001.44531000.723.3339.591.652.641.80.277130.990.151.3728712198803536.67310310352002.8193.6745.841.652.782.241.2881041.130.590.36189620201303536.67310310310310310310077.091.652.111.82.31270.460.150.66276196800353501.4410310351.8461.3343.751.82.221.70.8050.420.10.9951619800353103103103103100310068.751.81.231.222.06250.570.580.26252219710100100.586.670310350000.391352.091.82.952.740.2031511.150.941.5968492119653102.5103103102102.56103102.56102.7901001.952.721.92.70.770.050.752419683103102.67103103102.44103103102.8996.331001.952.712.432.890.760.480.94319783103103100310310302.56102.9799752.12.542.22.22750.440.10.12751019823101.552.67103103102.56103103101.866293.752.11.491.71.743750.610.40.356252519803101.5536.672530353001.8260.6739.592.12.432.560.7205380.330.461.3794621419543103101.466.6703102.44103103101.8461.3383.342.251.381.231.5334560.871.020.71654426198631031031031031031031031031001002.252.642.2330.390.020.752719743103102.581031031031021002.7892.6787.52.252.111.562.43250.140.690.1825153102.51033.333103100002.996.6754.1631.531.241.570641.471.761.429362819783103102.581003101.441031001.8461.337532.582.541.380.420.461.622919783103103103103103103103103100100332.74300.26030197431031036.673103101531001.8762.3377.09332.741.44158300.261.558417


## 5. Conclusions

The main challenge in finding the relation between the HI and all the elements of the calculation is the lack of data. The regular, precisely defined dynamics of the full-scale testing is not present at all power distribution companies. Therefore, there is a difficulty in selecting a mathematical tool to calculate objective transformer state estimates in the absence of some parameters. The integration of evidential reasoning methodology with the Markov chain model of component ageing enables the more accurate estimation of transformer health. The proposed methodology takes the probability that the component remained in the same state or transitioned to another state in a quantitative way.By calculating the HI for a fleet of transformers in the system, it is possible to rank them according to their current state or HI value. The traditional approaches to HI calculation give the large deviations from the accurate health grades, especially for transformers in poor condition. If the scope of the tests is small and the last tests were performed a long time ago, it is obvious the reliability of the HI is lower. However, the traditional HI methodologies did not give the quantitative evaluation of this reliability. The proposed methodology gives a clear signal that more accurate or full-scale testing should be performed to more accurately assess the current state of the unit. Further research will be focused on the more precise estimation of utility functions, concerning financial losses resulting from the interruption of electricity supply that can be caused by an ET failure or unnecessary inspections.



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