A Fuzzy Evaluation Method of Power Transformer Insulation Health State Based on Multi-weight Fusion

During the evaluation of transformer insulation health state, the weight of each evaluation indicator has a significant influence on its evaluation result. To overcome the existing problem of determining the weight of evaluation indicators by a single method, this paper proposes a comprehensive weight fuzzy evaluation method based on multi-weight fusion. Firstly, the proposed method uses principal component analysis (PCA) method to determine the objective weight of each evaluation indicator based on the measured data of the power transformer. Then years of practical experiences from industry experts are introduced to subjectively study the relative importance of the evaluation indicator’s influences on the insulation health states and to build the relative importance of judgment matrix of relative relationship. The subjective weight of each indicator is determined by using the analytic hierarchy process. Next, the objective weight and the subjective weight are fused by using the least square method to obtain the comprehensive weight of the evaluation indicator. And insulation health values and insulation health levels of the power transformers are obtained by using fuzzy comprehensive evaluation. Finally, example analysis and comparison are carried out by using the actual operating data of the transformer. Compared with the single principal component analysis method, the correct rate of the comprehensive weighting method of the evaluation indicator proposed in this paper is improved by 5.9%, which is 10.7% higher than that of the single analytic hierarchy process. The results verify the effectiveness and practicability of the method proposed in this paper. The proposed method provides a scientific basis for the maintenance and overhaul of the transformer. It has a very important practical significance for improving the stability of the power grid and ensuring the normal and orderly progress of the national economy.


Introduction
With the continuous development of China's economy, the demand for power is increasing, and it is becoming more and more important with safe and stable operation for power system. As one of the most important components in the power system, the power transformer plays a vital role in the stable operation of the power system. According to statistics, the forced outage rate of transformers due to faults has been maintained at 0.2 times/100 units per year in the past five years. In 2017 alone, the number of unplanned maintenance due to transformer faults reaches to more than 70 times for power system with voltage levels of no less than 220 kV [1]. By the end of June 2019, the number of transformers with voltage levels of no less than 220 kV is 16,286 in the national power grid. By analyzing transformers fault reasons, it shows that the common faults are mainly caused by leakage current, temperature rising, oil leakage, multi-point connection of iron cores and damp. Besides, faults related to insulation accounted for about 52.37% of the total faults, and it becomes the most common transformers fault [2,3]. Therefore, accurate evaluation of the insulation health states of power transformers has great significances to formulate optimal transformer maintenance strategies, reduce the number of unplanned maintenance and outage time, and ensure the safety and stable operation for the power grid system.
Focusing on the problem of power transformer insulation health evaluation, many methods have been proposed. The main methods used by related scholars are cluster analysis [4], supporting vector machines [5], feature 1 3 quantity diagnosis [6,7]. In [8], in order to determine the fault types of the transformer, the clustering method is used to classify Dissolved Gas Analysis (DGA) at the first step. Then PCA is used to analyze the center distance of the principal components. Because of the different type of transformers, the fault characterizations are also different. Therefore, there are certain limitations in the actual application process for the method. In [9], the paper applied the principles of immune system and clone selection in medicine to the identification of transformer health status. Although it has improved the identification speed, the accuracy has to be improved. In [10], the paper established a two-level and comprehensive evaluation model for transformers based on multi-source information under the basis of the interval matter-element extension principle, cloud model and gray model group difference method. The paper proposed specific schemes for the trend prediction and evaluation of transformer single-factor indicator and the overall state evaluation. However, this method is more complicated. In order to make up for the insufficiency of chromatographic diagnosis of transformer fault oil, reference [11] weighted the two probability indicators of correctness and sensitivity for different diagnosis methods to improve the reliability of the evaluation. But the selection of the weight coefficients used in the article lacked theoretical basis. Reference [12] uses cloud reasoning and unknown theory to determine the weight and evaluate the insulation state of transformer bushings, but there still exists problems in determining the comprehensive weight. Besides, the unknown theory is too cumbersome to determine the weight and the level is not clear. It can be seen from the above references that the weights of evaluation indicators play a vital role in the evaluation of the insulation health of power transformers.
The actual evaluation of transformer insulation states is affected by many factors and it is difficult to quantify accurately. It cannot objectively assign weights to evaluation indicators by using traditional evaluation methods because they are highly subjective. Besides, there is a certain degree of ambiguity in the boundary conditions that lead to the evaluation indicator state level of power transformers. However, the fuzzy algorithm can transform the fuzzy boundary evaluation indicator state levels of the transformer into quantitative evaluations.
Based on this, the objective weights of the transformer insulation health evaluation are determined based on the PCA analysis method in the paper firstly. Then the subjective weights are determined based on the analytic hierarchy process. Next, the least squares method is used to fuse the two weights to obtain the comprehensive weight. Finally, the insulation health evaluation level of the power transformer is obtained by combining the fuzzy algorithm.

Comprehensive Evaluation Model of the Insulation Health States of Power Transformer
During the process of evaluating the insulation states of power transformers, it is impossible to use all the factors that affect the insulation states of transformers as evaluation indicators. A transformer insulation states evaluation system must be established before the evaluation is carried out. And during the process of establishing the evaluation indicator system, the principles of systematism, completeness, consistency and independence of the evaluation indicator must be followed to ensure the scientificity and rationality of the established indicator system.

The Evaluation Indicator System of the Insulation Health States of Power Transformer
Focusing on the problems of various types of transformer evaluation indicators and the complex system structure, according to China GB/T7252-2001 'Guidelines for the Analysis and Judgment of Dissolved Gases in Transformer Oil' and DL/596-1996 'Preventive Test Procedures for Power Equipment', oil chromatographic analysis, oil chemical experiment and electrical experiment are used as first-level indicators to construct an evaluation indicator system in this paper. Because power transformers work under multiple environments such as electric field and thermal field, their internal structure and properties exist the phenomenon of ageing. During long-term operation, for example, insulating oil will slowly produce some characteristic gases such as hydrocarbons, carbon monoxide, hydrogen, and carbon dioxide. It is these gases that characterize the deterioration of the insulation state of power transformers [13][14][15][16][17][18]. Therefore, relatively important gas rates, such as C 2 H 2 , H 2 , and relative gas generation rate, such as CO and CO 2 , are selected as the second level indicators for state evaluation. The detection of the insulation states of power transformers can also be carried out through electrical tests and oil chemical tests. Relevant experiences of transformer insulation health evaluation at home and aboard is learned in the paper. And it also selected the main technical parameters such as winding DC resistance, winding dielectric loss, winding absorption ratio, micro water content, oil furfural in electrical test and oil test as the secondary indicators. The specific insulation health evaluation indicator system is shown in Fig. 1.

Comprehensive Evaluation Model of the Insulation Health States of Power Transformer
Power transformer is a complex component. In practice, the insulation health states of power transformer are often a combination of multiple system indicators and factors. Besides, there are many uncertainties and ambiguities. It is difficult to accurately evaluate the insulation health states of power transformers with a single method. It must comprehensively consider the occurrence probability of a certain insulation health states of the transformer and the reliability of the evaluation method. The paper established a fuzzy comprehensive evaluation model of the transformer insulation health states with multi-indicator weight fusion. The details are shown in Fig. 2. Based on the PCA analysis method, the objective data of the transformer measured from the actual working system are processed to determine the objective weight of each indicator of the transformer insulation health states. Taking into account the importance of professional technicians, experts and scholars in evaluating transformer insulation health states based on objective data, the paper constructed a judgment matrix based on the subjective meaning of experts and scholars. And the paper determined the subjective weights of transformer indicators based on the analytic hierarchy process. Finally, the subjective weights and the objective weights are combined with the least square method to determine the comprehensive weight of each index. The method is also combined with fuzzy comprehensive evaluation to obtain the insulation health states of the power transformer. Therefore, the insulation health states level is determined.
In order to clearly and intuitively characterize the health degree of the insulation states of power transformers, the insulation health value HI of power transformers was introduced [19] and HI ∈ [0,1]. The health states of power transformers can be separated into five levels including excellent, good, normal, attention, serious according to the insulation health values of power transformers. The corresponding relationships between health values and state levels are shown in Table 1. Among them, "Excellent" means that the indicators of transformer insulation are in very good condition and it needs not to be maintained; "Good" means that the indicators of transformer insulation are in a mediumto-upper state, and the maintenance time should be appropriately extended; "Normal" means that the indicators of transformer insulation are in a normal state and it needs to be maintained according to plans; "Note" means that a certain indicator of transformer insulation is at a low-medium level and it needs to be maintained in time; "Serious" means that a certain indicator of transformer insulation is in a severely exceeding permitted level state and it needs to be stopped for maintenance immediately.  In Fig. 2, the insulation state health value of the power transformation can be expressed as, where w j is the comprehensive weight of the jth evaluation indicator; ji is the membership value of the jth evaluation indicator corresponding to the ith state level; B ji is the mean value of the upper limit and lower limit of the insulation health states membership level range corresponding to ji . d is the number of evaluation indicators, and it can be seen from Fig. 1 that d = 15 in the paper. According to Table 1, the closer the value of HI is to 1, the better the insulation health degree. Weight and membership value are two variables. The accurate judgment of weight has the most significant impact on the health value of HI. Therefore, the determination of weight is the key to the evaluation model.

Comprehensive Weight of Evaluation Indicators
In the fuzzy evaluation model of the insulation health states of power transformers, it has an inseparable relationship between the accurate judgment of the weight of each evaluation indicator and the accuracy of the evaluation results. The weights of the evaluation indicators are determined from the objective and subjective aspects. The objective weight is determined according to the PCA analysis method, which is based on the original data. The subjective weight is determined by determining the importance through comparison and the analytic hierarchy process method according to actual experiences of experts for many years. Then, the least square method is used to fuse the subjective weight and the objective weight to obtain the comprehensive weight.

Objective Weights of Evaluation Indicators Based on PCA
The objective weight is determined by using the principal component analysis method to obtain the initial matrix X. This matrix is the factor coefficient matrix of the original indicator variable, which reflects the correlation between the common factor and the original variable. The larger the value, the stronger the correlation is. The weight of each indicator can be determined according to the principal component analytical model [20]. The steps to determine objective weight are as follows.
1 Constructing a sample matrix. The sample matrix is constructed from the collected n groups of data. Every group of data contains d indicators. Then the sample matrix X can be expressed as following.
The original matrix is standardized by using the deviation method, and the standardized matrix U is obtained. where I is an identity matrix. The characteristic values can be obtained by solving the characteristic equation of (5). Then the characteristic values are arranged as 1 ≥ 2 ≥ ⋯ ≥ d ≥ 0 in the order of magnitude. Besides, calculating the eigenvector e j (j = 1,2,3,…,d) corresponding to every eigenvalue λ j . 5 Calculating the variance contribution rate. The cumulative contribution rate of the first variance and the contribution rate of the variance are expressed by (6) and (7), respectively.
The first m eigenvalues, of whose cumulative contribution rate of variance exceeding 95% and the eigenvalue greater than 1, are selected as the main components.
Determining the objective weight of each indicator. Assuming that the transformed indicator variables are z 1 , z 2 , ⋯ , z m (m ≤ d) , the transformed indicator variables can be expressed as (8) in a manner of linear combination of the original variables.
where ij is the linear combination coefficient of the indi- where e ij is the initial load factor and j is the characteristic value of the jth principal component. Then the objective weight of the indicator is expressed by (10).

Subjective Weights of Evaluation Indicators Based on Improved Analytic Hierarchy Process
The Analytic Hierarchy Process (AHP) is a multi-objective decision-making analysis method proposed by Professor Satty in the early 1970. It is widely used in the field of multi-eigenvector weight calculation. The method integrates qualitative and quantitative problems by a judgment matrix. It divides the various factors of complex issues into interconnected and orderly levels and expresses subjective consciousness in a concrete form. Besides, it makes abstract subjective consciousness more methodical and scientific. However, during the process of evaluating the insulation health states of power transformers, a single decision maker may have problems such as incomplete consideration, high subjectivity, and low credibility. Focusing on the problems, the idea of expert group decision-making is introduced in the paper. It can effectively reduce the possibility of decision errors caused by personal preference of the decision maker.
As a result, it can improve the accuracy of decision-making. The specific steps are as follows. Building a hierarchical model. According to the evaluation indicator system for the insulation state of power transformers, the problems that need decision-making are divided into three levels, of which are the target level, the decision level and the indicator level, respectively. The relationship between every two levels is ownership. Figure 3 shows the hierarchical model of the subjective weight of the power transformer insulation state evaluation indicator.
Constructing an expert judgment matrix. It determines the importance of any two indicators at the same level by asking relevant experts for their opinions, and then it obtains a judgment matrix A k = (a k ij ) n×n by comparison in pairs according to the principle of importance. A k is the judgment matrix of the kth expert, i and j represent two different evaluation indicators, and a k ij represents the relative importance value of the kth expert's opinion of the indicator i to the indicator j. The numbers 1 to 9 and its reciprocal are usually used as a measure of relative importance.
Generating a unified judgment matrix. The judgment value a k ij generated by different experts is processed by using the K-Means clustering algorithm, then the unified judgment matrix A = (a ij ) n×n is obtained. a ij represents the relative importance value of the indicator i that is not related to the expert to the indicator j.
Calculating the weight. It determines the maximum eigenvalue λ max of the judgment matrix A and its corresponding eigenvector B = (b 1 , b 2 , ⋯ , b n ) , AB = max B and the weight vector W c is obtained by normalization of B.
Consistency checking. Since the judgment matrix is obtained by subjective decision-making, the weight vector obtained by the previous method may not meet the requirements, and the consistency check of the judgment matrix is needed. The judgment matrix A is considered to satisfy the requirements only when the consistency coefficient Y R is less than 0.1 after the consistency checking. The judgment matrix that does not meet the requirements needs to be revised until the consistency checking requirements are met. The Y R can be calculated by (11)  where RI is a stochastic indicator, which is shown in Table 3. CI represents a consistency indicator, and it can be calculated by (12).
where max is the maximum eigenvalue of the judgment matrix, and n represents the order of the judgment matrix Table 2.

Weight Fusion Based on Least Squares Method
After analyzing the affections of various factors having on the insulation health of power transformers by using PCA and analytic hierarchy process, it can obtain the objective weight of objective data and the subjective weight of subjective evaluation, respectively. Each weighting method has its advantages and disadvantages and independence. The two analysis methods are complementary in a certain sense. Therefore, it merges the subjective and objective weights of the two methods to obtain a comprehensive weight, which will make the evaluation conclusion more convincing. The subjective and objective comprehensive weights can be obtained by establishing a weighted least squares fusion model [21][22][23][24]. Specific steps are as follows.
S u p p o s i n g t h e o p t i m a l we i g h t ve c t o r i s W = w 1 , w 2 , ⋯ w d . p and q are the number of subjective weighting and objective weighting schemes, respectively. The sum of the weight vectors for each weighting is 1. The optimal weight is obtained by solving the nonlinear (13).
where α and β are subjective weight set coefficient and objective weight set coefficient, respectively.

Fuzzy Comprehensive Evaluation Model
Fuzzy theory is a new theory invented by foreign scholar Zadeh after discovering that precise numerical values are difficult to be applied in some fields [25]. Later, Professor Wang Peizhuang developed a fuzzy comprehensive evaluation method based on fuzzy theory. The boundary conditions of power transformer evaluation indicator state levels are ambiguous, and they are affected by many factors which are difficult to quantify. The fuzzy evaluation method can transform the transformer evaluation indicator state level with fuzzy boundaries into quantitative evaluation. Therefore, fuzzy theory is introduced to obtain the degree of membership in the paper. By introducing the fuzzy theory, the corresponding membership function values can be calculated from the indicators. The state levels corresponding to the state indicators can also be quantitatively analyzed. As a result, the accuracy of the evaluation model can be improved.
It needs to take into account that the units and dimensions of the power transformer insulation health evaluation indicators are different. If the data is not preprocessed, it will cause larger numerical indicators to swallow the errors of smaller numerical indicators, which will seriously affect the overall health states evaluation. In order to unify the data standard, the data is preprocessed according to (16).
where v j represents the standard value of the jth indicator of the insulation health states of the power transformer. If v j < 0, then v j = 0; if v j > 1, then v j = 1. x yj is the threshold value of the j indicator. x cj is the factory value of the j indicator, and x j is the measured value of the indicator. During the evaluation of the insulation health states of power transformers, the characteristics of each evaluation indicator are different, and the classification of the state level is also different, so the fuzzy membership functions are not completely the same. The following analysis takes the hydrogen content as an example. The state level of the hydrogen content evaluation indicator is the same as the classification of the insulation state level of power transformers, which are five state levels including excellent, good, normal, attention, and serious. Figure 4 shows the functional relationship between the standard value of hydrogen content and its state level determined according to the data. If the hydrogen content is the jth indicator, the expressions of the membership functions corresponding to the five state levels of Excellent, Good, Normal, Attention, and Serious are shown in (17,18,19,20,21), respectively.
Then the membership vector µ j of the standard value of hydrogen content can be expressed as (22).
The membership degree vectors of the standard values of other evaluation indicators can also be obtained in the same manner. Then, the fuzzy membership degree matrix µ of the fuzzy comprehensive evaluation model is, From (2), the membership value of the jth evaluation indicator can be obtained. The insulation health value of the power transformer can be obtained by substituting the weights and membership scores of each indicator into (1). And it can also determine its insulation health level.

Example Data
The data used in the paper are obtained from a power transformer of a substation with a model of SFPSZ9-120,000/220 in a certain place. The power transformer is used outdoor. The transformer was put into use in 1998 and it was designed for 30 years. The evaluation model was tested with the data measured on June 20, 2020. The relative gas production rate of carbon monoxide and carbon dioxide was calculated from the data measured on May 20, 2020. Table 3 showed the experimental data of evaluation indicators of power transformer insulation health.

Evaluation Indicator Objective Weight Calculation
The weight of each indicator can be obtained by analyzing the evaluation indicators of the transformer. Then the paper calculated the eigenvalues of principal components, the cumulative contribution rate of variance, and the contribution rate of variance on the 100 sets of original data samples collected according to (2,3,4,5,6,7). The calculation results are shown in Table 4. It can be seen from Table 4 that the eigenvalues of the first 4 principal components are all greater than 1. The Fig. 4 Relationship between hydrogen content and health states contribution rates of the four principal components to the whole are 40.153%, 25.675%, 17.895% and 13.105%, respectively. The cumulative contribution rate of the variance of the four principal components reaches 96.828% (The rate of the variance is required to be greater than 95% generally). It indicates that the 4 principal components can replace the original data. Therefore, it can calculate the linear combination coefficients of the first 4 principal components z i (i = 1, 2, 3, 4) and the original indicator variable x j (j = 1, 2, ⋯ , d) according to (9). The results are shown in Table 5.
It can be seen from Table 5 that the first principal component contains 40.153% of the original data information. And the main quantities included are the relative gas production rate of CO, the relative gas production rate of CO 2 , DC resistance imbalance coefficient, casing dielectric loss, leakage current, oil dielectric loss, furfural content, micro water content and iron core grounding current. The second principal component contains 25.675% of the original data information. And the main components are C X H X , absorption ratio, oil dielectric loss, breakdown voltage, etc. The third principal component contains 17.895% of the original  (10)

Subjective Weight Calculation of Evaluation indicators
According to Fig. 2, the paper established a hierarchical analysis and evaluation system with the evaluation of the insulation states of power transformers as the target layer, oil chromatographic analysis, electrical test, and oil chemical test as the decision-making layer, and the specific indicators of each analysis method as the indicator layer. It has invited five experts from different directions to evaluate the five indicators of H 2 content, C 2 H 2 content, C X H X content, CO relative gas production rate, and CO 2 relative gas production rate based on the principle of importance, and construct the judgment matrix of the five experts as 1 1 , 2 1 , 3 1 , 4 1 , 5 1 , and then the unified judgment matrix A 1 is obtained by the K-Means clustering algorithm.
In the same way, it can be evaluated the six indicators of DC resistance unbalance coefficient, absorption ratio, winding dielectric loss, casing dielectric loss, partial discharge, and core ground current, and the judgment matrix A 2 can be obtained then. Finally, the judgment matrix A 3 is obtained by evaluating the four indicators of micro-water content, oil dielectric loss, breakdown voltage, and furfural content.  According to (11) and (12), .0163 can be calculated respectively, and the values are all less than 0.1. The consistency test is passed and the inspection meets the requirements.
Then the judgment matrix B can be obtained after evaluating the decision-making level of oil chromatographic analysis, electrical test, and oil chemical test.

Fuzzy Comprehensive Evaluation of The Insulation Health States of Power Transformer
According to (16), the data in Table 4 is processed to obtain the membership value of the evaluation indicator as shown in Table 6. According to the membership function of each evaluation indicator, the membership matrix of the evaluation indicator is, By substituting the membership matrix and the comprehensive weight W into (1), it can calculate the insulation health value of the power transformer. The result is HI = 0.3930. Then it can obtain that it belongs to the attention level by looking up the insulation health states of the power transformer in Table 1. In addition, during the onsite maintenance of the transformer, it found that due to the long time operation, its internal insulation was severely damaged, which caused several indicators such as hydrogen content, micro water content, and relative gas production rate of carbon monoxide to approach the threshold value. The transformer was in a state requiring maintenance in time. If the transformer is not maintained in time, the insulation health state of the transformer would drop quickly in a short time. As a result, it would cause a power supply accident and cause greater economic losses. It was consistent with the evaluation results of the proposed method in the paper. The result verified the effectiveness of the proposed method. The paper compared the results with only using a single method (PCA or analytic hierarchy process) to determine the indicator weight. the calculated insulation health values are shown in Table 7.
It can be seen from Table 7 that the insulation health value under the PCA weighting method is relatively low, while the insulation health value under the analytic hierarchy process weighting method is relatively high, and the insulation health value under the comprehensive weighting method is more similar to the on-site inspection results. The factors that can have affection on the health of insulation are complex. Some simple data are often difficult to reflect, and it is difficult to correctly evaluate by using a single objective weight method. The analytic hierarchy process is a decision made by experts based on years of practical experience to evaluate the relationship between indicators and health states. The comprehensive weight of the indicators after the fusion of the least square method can modify the conclusions obtained from purely objective data to make the health value more precise.
The paper used the comprehensive weight of the evaluation indicator and the single weight of PCA and analytic hierarchy process to carry out simulations by using the monitoring 17 sets of transformer data. And it also compared the simulation results with the insulation health level of the transformer obtained from the maintenance site. Table 8 shows the correct group number and correct rate of transformer insulation health evaluation.
It can be seen from Table 8 that the PCA weight method can correctly evaluate 15 sets of transformer data with a correct rate of 88.2%. The analytic hierarchy process weight method can correctly evaluate 14 groups with a correct rate of 83.4%. The comprehensive weighting method of evaluation indicator proposed in this paper correctly evaluates 16 sets of transformer data. Compared with the single principal component analysis method and the single analytic hierarchy process, the correct rate is increased by 5.9% and 10.7% respectively, which verifies the effectiveness of the method proposed in this paper.

Conclusion
In order to ensure the power system a stable and reliable operation system, reduce unnecessary overhauls and unplanned overhauls, and improve the efficiency of power grid operation, it is particularly important to accurately evaluate the insulation health states of power transformers. The paper established a fuzzy assessment model of transformer insulation health state based on the combination of PCA and analytic hierarchy process. Firstly, the paper analyzed the relationship between the factors that affect the insulation health states, and use the PCA analysis method to process the objective data to obtain the objective weights of the transformer health indicators. In addition, the analytic hierarchy process is used to obtain the subjective weights of the transformer indicators. Then the paper fused the weights of various indicators to obtain the comprehensive weight of the evaluation, and combined with the fuzzy comprehensive evaluation to obtain the insulation health value so that it can determine the insulation health level. Finally, by using the actual operating data of the transformer, the results verified the effectiveness of the comprehensive weight evaluation model of the evaluation indicator. The proposed method can overcome the problem of determining the weight of the evaluation indicators by a single method, and provide a scientific basis for the maintenance and reparation of power transformers.
Because the health states of power transformers are composed of many evaluation indicators, the determination of the level boundaries of quantitative indicators is often determined based on actual experience, and there also exists a certain degree of subjectivity, it is needed to be further studied in the future. Big data technology can be used to mine historical information much more accurately. Besides, the accuracy of the overall assessment can also be improved by dividing the equal boundaries of the indicators. In addition, in order to improve the accuracy of the evaluation, it is necessary to take into account of the characteristics of each indicator and the variation trend to select a more consistent function and refine each indicator.