#### 4.1. Concept of D–S Evidence Theory

D–S evidence theory can fuse information from different data sources. It is an effective method of information fusion. It has achieved good results in many fields.

Suppose Θ is the identification framework and 2^{Θ} is the power set of Θ. If the mapping relationship m: 2^{Θ}→[0, 1] satisfies $m(\phi )=0$, $\sum _{A\subset \Theta}m(A)=1$, then m is called the basic probability assignment (BPA) function and m(A) is the basic reliability of evidence to A. If $\forall A\in \Theta $ and m (A) > 0, then A is named the focal element of BPA.

However, when there is a high conflict between evidences, the Dempster rule has a big defect in the process of synthesis [

39]. The synthesis results may be even completely contrary to the actual situation. In order to solve this problem, this paper introduces weight coefficient to readjust BPA. The specific process is as follows:

- (1)
Calculate the conflict extent

k_{ij} between evidence

E_{i} and

E_{j} (

j = 1, 2, …,

i − 1,

i + 1, …

n) to form the collision vector

K_{i}.

- (2)
Standardize the collision vector

K_{i}.

- (3)
Calculate the entropy value

H_{i} of

K_{i}^{N}- (4)
Calculate the reciprocal of

H_{i}- (5)
Calculate the weight coefficient

w_{i} of

E_{i}.

Through above five steps, we can get the weight coefficients of each evidence, and combine them to obtain the weight vector W_{i} = {w_{1}, w_{2}, …, w_{n}}.

Let the maximum weight be

w_{max} = max{

w_{1},

w_{2}, …,

w_{n}}, and the relative values of weight coefficients can be obtained as:

W* = (

w_{1},

w_{2}, …,

w_{n})/

w_{max}. Discount coefficient is introduced into the basic probability assignment function, shown as:

where

k = 1, 2, …,

d_{i}, and

d_{i} is the number of non-Θ focal elements in the identification framework. Therefore, the value of BPA can be adjusted as

m_{i}^{*}(

A_{k}),

k = 1, 2, …,

d_{i}.

To construct the complete BPA, the concept of uncertainty is defined as:

A complete BPA function can be constructed by Formulas (24) and (25).

#### 4.2. Transformer Fault Diagnosis Based on Improved D–S Evidence Theory

In order to enrich information sources in transformer fault diagnosis and improve the diagnostic efficiency, D–S evidence theory is employed to merge HIA, KELM, and HMSVM together. In this section, a diagnostic fusion model of power transformer fault is constructed based on improved D–S evidence theory. The diagnostic procedure is as follows.

- (1)
Three fault diagnosis models are used to test DGA samples and make a preliminary diagnosis. If a unanimous conclusion is obtained, then stop the operation and output the result directly. Otherwise continue the following steps.

- (2)
Different BPA of three diagnostic approaches are constructed as follows.

a. HIA

In the recognition framework Θ, the Euclidean distances from

A_{g} to all memory antibodies are calculated. Assuming that the minimum Euclidean distance between

A_{g} and

B_{i} is

d_{i}, if

d_{i} >

σ, and then let

x_{i} = 0, otherwise

x_{i} = 1/

d_{i}, where

σ is the affinity threshold,

i = 1, 2, …, 6. In HIA [

40], the BPA of

A_{g} can be described as follows:

b. KELM

There are 6 kinds of transformer fault states in this paper, that is, the number of recognition types of KELM is 6. If

f_{i}(

x) is used to represent the output function for the

ith state of transformer faults, then the output function

f(

x) of KELM is expressed as [

41]:

The numerical output of KELM is transformed into probabilistic output.

For Sample x, the BPA of KELM is defined as:

c. HMSVM

m_{3}(

B_{i}) denotes the BPA of HMSVM that a group of DGA samples belong to fault

A_{i}.

where

P(

A_{i}) can be calculated by Formula (14).

The above three groups of BPAs are satisfied:

- (3)
The weight coefficient is used to adjust the BPA.

- (4)
The synthesized BPA is discriminated according to the following rules.

Let

A_{1} and

A_{2} belong to U, satisfying

where

A_{1} is the judgment result. Further,

ε_{1} and

ε_{2} are preset thresholds, which are set to be 0.2 after repeated comparison.

The fault diagnosis model based on improved evidence theory is shown in

Figure 10.

#### 4.3. Case Analysis and Discussion

In this section, 5 sets of DGA samples are collected from Hebei north electric power company, with actual fault labels attached, shown in

Table 8.

The DGA samples are processed and input into the transformer fault diagnosis model based on HMSVM. The output membership values of five groups of samples are obtained, as shown in

Table 9. The value appears in bold in

Table 9 means the highest membership of each sample.

According to the diagnostic principle of HMSVM, if the sample is located inside the hypersphere, the value of membership is greater, otherwise, the value of membership is smaller.

Table 9 shows that except for Sample 3, the maximum membership of DGA samples is much greater than that of other fault states. This is because the four samples belong to only one state hypersphere, so the diagnosis conclusion is clear, and the membership value is usually greater than 0.5. For T

_{12} and T

_{3}, the membership values of Sample 3 are greater than 0.5, and close to each other. This is because the sample is located in the overlapping area of hyperspheres, which is prone to misjudgment. According to HMSVM diagnostic criteria, the diagnosis conclusion of this sample is T

_{3}, which is not consistent with the actual fault.

HIA and KELM are employed to analyze above typical samples. The diagnostic results using three intelligent algorithms are shown in

Figure 11.

As can be seen from

Figure 11, for the five sets of fault samples, the diagnostic results obtained by three intelligent algorithms are quite different. Among them, the HMSVM method has the best diagnostic effect, and only Sample 3 is not recognized correctly. The reason is this sample is located in the overlapping area of hyperspheres, which results in misjudgment. The KELM method has some problems in the diagnosis of Sample 2 and Sample 3, which shows that it is easy to misjudge the overheating fault. The HI method has misjudged Sample 4 and Sample 5, which indicates that this method is not effective for the distinction between high energy discharge and high temperature superheating faults. According to above analysis, except for normal samples, three intelligent methods have great difference in the diagnosis effect of transformer faults. For Sample 3, the diagnostic conclusions given by the three intelligent methods are completely different. Therefore, this paper chooses Sample 3 as the high conflict data for detailed analysis. Afterwards, the evidence theory fusion method is used for comprehensive diagnosis.

In this section, the proposed fusion algorithm is utilized to analyze the high-conflict DGA sample. First of all, the characteristics of the sample are extracted. Then, three different intelligent approaches are employed for preliminary diagnosis. According to the BPA construction method of three different algorithms, the BPA to each fault state is obtained shown in

Table 10. In the table,

E_{1},

E_{2},

E_{3} represent the evidence of KELM, HIA, and HMSVM respectively, and

m denotes the basic probability assignment.

Table 10 shows that the reliability to T

_{3} is the highest for KELM and HMSVM, and the reliability to T

_{12} is the highest for HIA. The conflictive degree of evidences is calculated, defined as

k.

k_{12} denotes the conflictive degree between KELM and HIA.

k_{13} means the conflictive degree between KELM and HMSVM. Lastly,

k_{23} denotes the conflictive degree between HMSVM and HIA. After multiple calculations, the values of

k_{12},

k_{13},

k_{23} are obtained, which are 0.9827, 0.1721, and 0.9700. This indicates that the conflictive degree between HIA and each other evidence is higher.

In order to solve the problem of high conflict between evidences, the weight coefficient is introduced for BPA readjustment. With the degree of conflict

k_{12},

k_{13},

k_{23}, the weight coefficient

w_{i} can be obtained by Formulas (20)–(23) as

w_{i} = (0.3845, 0.2335, 0.3819). Due to the serious conflict between E

_{2} and the other evidence, HIA has smaller weight coefficient. Different weight coefficient values can distinguish the importance degree of each evidence. The uncertainty

m(Θ) is introduced in the process of BPA adjustment. The adjusted BPA based on weight coefficients is obtained in

Table 11.

With the BPA in

Table 11, the diagnostic decision can be made by formula (24) and (25). After calculation,

m(T

_{12}) and

m(T

_{3}) are the first and second maximum of basic probability assignment. The difference between these two values is 0.466, which is greater than

ε_{1}(0.2). In addition, the value of uncertainty

m(Θ) is 0.0201, which is much less than

ε_{2}(0.2). Above all, the final diagnostic result is T

_{12}, which is the same as the actual state.

Using above 253 testing DGA samples, the diagnostic results based on different algorithms are shown in

Table 12.

It can be seen from

Table 12 that, due to the superior performance in processing nonhomogeneous and small samples, the total diagnostic accuracy of HMSVM reaches 90.9%, which is better than KELM and HIA. However, for

D_{1}, HIA performs better than HMSVM. In addition, HMSVM gets worse results for

PD than both HIA and KELM. It indicates that three intelligent algorithms have diverse performances for different transformer fault states. For each transformer fault state, the diagnostic accuracy of the fusion method based on improved D–S evidence theory is no less than 90.0%. In total, the proposed fusion method has the highest diagnostic accuracy, which is obviously better than the other three single approaches. It shows that IDET can fuse the advantages of each single algorithm together and recognize transformer fault states effectively.