1. Introduction
As a key piece of equipment in power systems, the power transformer comprises up to 60% of the total investment in substations and affects the safety and stability of power supply [
1,
2]. With the rapid expansion of the power system network, sudden failures of transformers will affect the security of life and property more seriously than before [
3]. Therefore, grasping the health condition of transformers accurately is of significant importance, which involves transformers’ operation and maintenance [
4]. Health diagnosis methods provide feasibility for changing the maintenance strategy and, accordingly, maximizing the practicable operating efficiency and optimum life, while minimizing the risk of premature failure [
5,
6].
In past years, many techniques, such as neural network [
7], support vector machine [
8] and fuzzy logic [
9], were applied to transformer fault diagnosis. These approaches usually focused on a single factor (e.g., DGA analysis, thermal modeling, winding fault analysis, etc.). Results indicated that these research works could evaluate the transformer fault condition effectively to a certain extent. Nevertheless, these attempts were not sufficient to obtain an overall and precise health condition of the transformers [
5,
6]. They usually gave a qualitative description of transformers whether in good or bad condition. In fact, the transformer health condition is affected by many factors, which reflect its condition from different aspects, degrees and levels. The health condition of power transformers is often somewhere between good and bad. For example, some indices may have deviated from their permissible thresholds, but the overall condition is still acceptable. On the other hand, some indices may be below the thresholds, but the overall condition is bad, since timely maintenance is required. Therefore, it is difficult for power utilities to obtain accurate evaluation results due to varied information sources from transformers, which can be regarded as an MADM problem [
5,
6].
To address such MADM problems, some researchers have attempted to integrate the merits of AHP and evidence theory to evaluate the electric primary devices including diverse condition information [
5,
10]. The AHP method, established by Satty, which has been successfully employed under many actual decision-making situations [
11,
12,
13,
14], is a popular approach for determining the weights of alternatives in MADM problems involving qualitative data [
15,
16]. In addition, the Dempster–Shafer theory (also called evidence theory), initially presented by Dempster [
17] and then developed by Shafer [
18], is applied to handle the uncertain information in MADM problems [
5,
6]. The kernel of evidence theory is the combination rule, which can be adopted to obtain an evaluation result considering various kinds of condition information [
19].
However, these previous studies are still one-sided and unsystematic because of several challenges or drawbacks remaining.
For the classical AHP approach, the scale of pair-wise comparison judgment, derived from experts, is confined to crisp numbers [
20]. However, in many practical applications, such as condition assessment of transformers, expert objective predilection may be fuzzy [
21], and the experts may not be willing to provide exact values for pair comparisons [
22].
For the calculation of weights, determining a suitable weight is a very important step in the decision process. However, both objective and subjective weight have limitations. The objective weightneglects the decision maker’s knowledge and actual situation. On the contrary, the subjective weight is heavily influenced by expert experiences and prejudices, resulting in high subjectivity [
23].
For the traditional evidence theory, it is strongly confined to the definition of exclusiveness hypothesis and the completeness constraint [
24]. Therefore, this limits the actual application of evidence theory, especially the application in the health condition assessment of transformers, including five intersection grades (health, sub-health, minor defect, major defect and critical defect) based on human judgment [
25,
26]. Unfortunately, little attention has been paid to the rigorous mathematical definition of evidence theory.
To effectively overcome these shortcomings of existing methods, several techniques, such as fuzzy extended AHP, game theory and D numbers, have been developed. The fuzzy extended AHP, extending the classical AHP by using a triangular fuzzy number [
27], has become an outstandingly effective tool to determine the weights of evaluation criteria in an actual complex system. Recently, it has been successfully applied in many fields, like green product designs [
28], ship selection [
29] and teaching performance evaluation [
30]. Game theory, a strategic bargaining behavior [
31], has been developed and employed for various fields from economics to engineering [
32]. It can play a better role when it comes to dealing with the conflicts among two or more participants [
23]. Similarly, subjective and objective weight can be considered as two participants of a game, and the comprehensive weight is the result of the ‘weight’ game [
23]. D numbers, a novel theory initially proposed by Deng [
24], has become a powerful method to deal with the uncertainty in actual engineering applications due to its capacity of avoiding the mutually-exclusive hypothesis of the frame of discernment [
25,
26,
33,
34].
Herein, a novel MADM model, which integrates the merits of fuzzy set theory, game theory and modified evidence combination extended by D numbers, is adopted to evaluate the health condition of transformers in this paper. Three factors, (i) dissolved gas analysis (DGA) date; (ii) electrical testing; (iii) oil testing and seventeen sub-indices are involved in the evaluate framework. The followings have been investigated in this paper: (i) adopting the fuzzy set theory to generate the original basic probability assignments for all indices; (ii) applying the game theory to obtain a comprehensive weight based on the subjective and objective weight, which are calculated by fuzzy extended AHP and the entropy weight, respectively; (iii) employing the distance of D numbers to modify original basic probability assignments and obtain final assessment results for transformers. The proposed model is verified by evaluating a realistic transformer and compared to a typical method. The results indicate that the model can evaluate the transformer health condition effectively.
This paper unfolds in the following fashion.
Section 2 presents the framework for transformer condition assessment.
Section 3 demonstrates the detailed procedures of the condition assessment model, including the fuzzy set theory, game theory and modified evidence combination extended by D numbers.
Section 4 takes two cases for example to show the efficiency of the model, and final conclusions are illustrated in
Section 5.
2. Framework for Transformer Condition Assessment
During the whole service period of the power transformer, various subsystems of the power transformer are aging gradually. Although the health condition of the power transformer cannot be observed directly, it can be reflected by all kinds of condition information [
5,
6]. Thus, diverse evaluation indices are acquired to evaluate the health condition of the transformer, which is regarded as an MADM problem. The selected evaluation indices should be typical and reasonable, so as to the reflect health condition of the transformer. Based on the aging mechanisms and fault properties of the transformer, three factors, DGA data, electrical testing and oil testing, are chosen in the evaluation framework.
The evaluation framework, a four-layer structure, is established as shown in
Figure 1. Level 1, the objective level, represents the final condition evaluation result of the power transformer. Level 2, the factor level, describes the condition information of the transformer from three aspects. Level 3, the sub-factors’ level, involves corresponding indices’ information. For example,
= {
,
,
,
,
} represents the DGA data with five indices. Level 4, the assessment result level, indicates the evaluation grades of each index.
Based on previous research [
6,
21] and experts’ experiences, the evaluation grades, relating to maintenance purposes, can be divided into five grades (health, sub-health, minor defect, major defect and critical defect) and are given by:
The relationship between the assessment grades and maintenance strategy is described in
Table 1.
3. Methodology
A novel hybrid MADM model, which integrates the merits of fuzzy set theory, game theory and modified evidence combination extended by D numbers, has been proposed in this paper. The assessment process consists of three key steps. First, the original basic probability assignments for all indices are obtained by the fuzzy set theory. Second, based on game theory, the subjective and objective weights of indices, which are calculated by fuzzy AHP and the entropy weight, respectively, are integrated to generate the comprehensive weights. Third, the modified evidence combination extended by D numbers is proposed to obtain the final assessing result.
3.1. Fuzzy Set Theory
Due to different dimensions or magnitudes, various indices need to be first normalized so as to obtain the membership grades for quantitative indices. Let
be the
j-th index of the
i-th factor, and the normalization mapping
:
f→ [0,1] is given as follows [
6].
If the indices are benefit attributes, the standardization process is:
If the indices are cost attributes, the standardization process is:
where
is the standardized value.
The membership function is adopted widely in the condition assessment of electrical equipment. Nonetheless, there is no unified standard within fuzzy theory for constructing suitable membership functions [
10]. Recently, a trapezoidal membership function is usually employed in the health diagnosis of transformers [
35,
36], and the trapezoidal model is also in accordance with the health condition of transformers [
21]. Hence, the trapezoidal model is adopted to obtain the assessing grades in this paper. The design of the membership function is shown in
Figure 2 [
21] and can be described as follows.
After extensive field testing and validations, the interval values are given as: = 0.05, = 0.25, = 0.3, = 0.45, = 0.5, = 0.75, = 0.8, = 0.95, respectively.
By using Equations (
2)–(
5), the fuzzy membership matrix is then:
where
stands for the index membership matrix of the evaluation level of the
i-th factor.
3.2. Comprehensive Weights Based on Game Theory
3.2.1. Fuzzy Extended AHP
Several fuzzy AHP methods have been developed to determine the weights of alternatives [
37]. Among these methods, the fuzzy extended AHP, proposed by Chang [
27], is employed widely in different application areas due to its lower computation complexity than the other methods [
30]. In this paper, the fuzzy extended AHP is adopted to calculate the weights of alternatives based on experts’ opinions.
Since the hierarchical structure is constructed, the triangular fuzzy comparison matrix [
27], based on expert judgments, is given by:
where:
The triangular fuzzy numbers and corresponding linguistic description are illustrated in
Table 2. The linguistic description should be converted into fuzzy scales, which aims to be convenient for mathematical operation. The steps of fuzzy extended AHP are demonstrated as follows [
27].
Step 1: Sum up each row of the fuzzy comparison matrix
A, then normalize the row sums. The fuzzy synthetic extent values of the
i-th object are:
Step 2: Compare the degree of possibility
. Thus:
where
and
d is the intersection point between
and
(
Figure 3).
Step 3: Compute the minimum degree of possibility. We have:
Then, the weight vector is:
where
are
n design alternatives.
Step 4: Normalize the weight vectors. The final weight vector is given by:
where
are non-fuzzy numbers.
3.2.2. Entropy Weight
The information entropy theory was first set forth from thermodynamics to information systems by Shannon [
38]. Based on the information entropy theory, the entropy weight can reflect the useful quantitative information of evaluation indices [
39]. Assume that there are
m evaluation objects and
n indices for decision-making problems. The procedures are demonstrated as follows [
40].
Step 1: Calculation of the entropy. Information entropy of index
j is:
where
is the normalization value of the quantitative index.
Step 2: Acquisition of the weight. The weight acquired from information entropy is:
where
.
3.2.3. Game Theory
As discussed previously, there are certain limitations to consider a single weighting method under many situations. The objective weight neglects the decision-maker’s knowledge and actual situation. On the contrary, the subjective weight is heavily influenced by expert experiences and some prejudices, resulting in high subjectivity. Therefore, the comprehensive weight, combining the subjective and objective weight with an effective algorithm, is more reasonable in the decision-making process.
Game theory, the research of strategic interaction, is an important branch of modern mathematics. Specifically, game theory is adopted to obtain the optimum equilibrium solution among two or more conflicts. In game theory, a decision is made either individually or collectively. Additionally, the decision can maximize the utility payoffs out of participants’ expectations. Thus, a decision of either a consensus or compromise is suggested. Analogously, the comprehensive weight, which reaches a compromise between the subjective weight and the objective weight, can be regarded as an optimum equilibrium solution. The calculation steps of comprehensive weight based on game theory are described as below [
23].
Step 1: Generate
m weights using
m kinds of weighting approaches. Then, establish a basic weight vector set
. Thus, a possible weight set is formed by
m vectors with the form of an arbitrary linear combination, expressed as:
where
is a possible weight vector in set
and
is the weight coefficient.
Step 2: Calculate the optimum equilibrium weight vector
of the possible weight vector sets based on game theory, indicating that a consensus is reached among
m weights. Such a consensus can be taken as the optimization of the weight coefficient
, which is a linear combination. The purpose of the optimization is to minimize the deviation between
and
using the following formula.
Based on the differentiation property of the matrix, the condition of the optimal first-order derivative in Equation (
17) is determined as:
Step 3: Compute the weight coefficient
by using Equation (
19), then normalize them using the following equation.
Step 4: Obtain the final comprehensive weight with the following formula:
3.3. Modified Evidence Combination Based on D Numbers
Although the evidence theory is widely applied to solve MADM problems, many issues are still unsolved in some situations. Among these problems, the definition of mutually-exclusive and conflict evidence have attracted more attention. Recently, two methods, D numbers and weighted average combination, are proposed by Deng et al. [
24,
41] to address the mentioned problems effectively. Inspired by the two methods, a modified evidence combination extended by D numbers is formulated as follows.
3.3.1. Dempster–Shafer Theory
Dempster–Shafer (D-S), also named evidence theory, is mainly introduced to solve the MADM problems with uncertainty. In the evidence theory, a sample set Θ that is collectively exhaustive and mutually exclusive, called a frame of discernment, is defined as [
6]:
The power set of Θ is described as
, namely:
If
,
A is called a proposition. The combination rule is one of the most important performances in evidence theory. Suppose there are two pieces of evidence indicated by
and
on the same discernment framework Θ, and the combination rule is performed [
6], with the following signs:
where:
In (25), k is a conflict coefficient, which reflects the conflict degree between the two pieces of evidence and .
3.3.2. D Numbers
As mentioned above, the frame of discernment is a strong hypothesis of being mutually exclusive. However, it is inevitable that linguistic assessments based on human judgment intersect each other, such as “health”, “sub-health”, “minor defect”,”major defect” and “critical defect”. Therefore, it is not reasonable to apply D-S theory under such circumstances. To address this problem, a novel technique, D numbers, was proposed.
Let Θ be a finite nonempty set, and a D number is a mapping defined by [
24,
26]:
with:
where
is an empty set and
A is a subset of Θ.
Since the frame of discernment does not need to be a mutually-exclusive set in D numbers theory, the five grades of transformer health condition from health to critical defect can be regarded as a frame of discernment of D numbers.
3.3.3. Distance between D Numbers
A relative matrix, explaining the relationship between each D number, is described as follows [
26]. Let the number
i and number
j of
m linguistic constants be expressed by
and
, the union region between
and
be
and the intersection region between
and
be
. The nonexclusive degree
is expressed as below.
The relative matrix is established as:
For instance, suppose
m linguistic constants are shown in
Figure 4. The non-exclusive degree
is obtained to stand for the non-exclusive degree between two D numbers based on the region of union
and intersection
between
and
.
Then, an intersection degree of two subsets is described as below.
where
. In the relative matrix
R, the variable
i represents the row number of the first element of set
and the variable
j represents the column number of the first element of set
.
shows the cardinality of
, and
shows the cardinality of
. Note that when
.
Based on the above, the distance between two D numbers
and
is defined as:
where
and
are two
matrices. Their elements are defined as:
3.3.4. Modified Evidence Combination Based on D Numbers
After obtaining the distance of D numbers, we can construct a
matrix as:
Let
be the similarity value between
and
, then the similarity value is given as [
41]:
It is obvious that the bigger the value of the distance is, the smaller the similarity of them will be, and vice versa. The similarity matrix is expressed as [
41]:
The support degree of each evidence is illustrated as [
41]:
To normalize
, the objective weights of evidence are obtained as [
41]:
where
.
Considering the relative importance of different factors in the power transformer, the optimum equilibrium weights of evidence (three factors), by integrating the objective and subjective weights of evidence, are determined based on game theory, described as:
where
are the subjective weights of three factors (DGA date, electrical testing and oil testing).
After obtaining the optimum equilibrium weights of each piece of evidence, the new modified pieces of evidence are defined as [
41]:
In this study, we take the modified pieces of evidence as independent of each other. If there are n pieces of evidence, we can apply the traditional Dempster–Shafer’s combination rule to combine the new modified evidence times.
3.4. Procedures for Transformer Condition Assessment
The detailed procedures of the novel multi-attribute decision-making model for transformer condition assessment are shown in
Figure 5 and can be summarized as the following steps.
Step 1: Construct a framework of transformer condition assessment. Three factors and seventeen indices are involved in the assessment framework. The evaluation grades are divided into 5 grades (health, sub-health, minor defect, major defect, critical defect), defined by using Equation (
1).
Step 2: Establish a fuzzy membership matrix. Due to the fact that various indices have different dimension values, a uniform standard, obtained by the fuzzy membership function, is needed in the assessment framework of the transformer. After determining the fuzzy membership function by using Equations (
4) and (
5), a fuzzy membership matrix for all of the indices is constructed in Equation (
7).
Step 3: Calculate the subjective and the objective weight. The subjective weight is computed based on the fuzzy extended AHP by using Equations (
8)–(
14), and the objective weight is solved based on the entropy weight method by using Equations (
15) and (
16).
Step 4: Compute the comprehensive weight. As the subjective weight and objective weight are obtained, the comprehensive weight is generated based on the game theory by using Equations (
17)–(
22).
Step 5: Determine the original basic probability assignments. The original basic probability assignments are obtained through the additive weighting method, expressed as:
where
can be regarded as the original basic probability assignment of the
i-th factor,
stands for the index membership matrix of the evaluation level of the
i-th factor and
reflects the comprehensive weight of index
.
Step 6: Combine the modified pieces of evidence to generate the evaluation results of the transformer health condition by using Equations (
24) and (
25).
Step7: Judge the final evaluation results based on the decision-making rule. The decision rule is defined as [
42]:
where
ε = 0.04.