Research on Multi-Attribute Decision-Making in Condition-Based Maintenance for Power Transformers Based on Cloud and Kernel Vector Space Models

: Decision-making for the condition-based maintenance (CBM) of power transformers is critical to their sustainable operation. Existing research exhibits signiﬁcant shortcomings; neither group decision-making nor maintenance intention is considered, which does not satisfy the needs of smart grids. Thus, a multivariate assessment system, which includes the consideration of technology, cost-e ﬀ ectiveness, and security, should be created, taking into account current research ﬁndings. In order to address the uncertainty of maintenance strategy selection, this paper proposes a maintenance decision-making model composed of cloud and vector space models. The optimal maintenance strategy is selected in a multivariate assessment system. Cloud models allow for the expression of natural language evaluation information and are used to transform qualitative concepts into quantitative expressions. The subjective and objective weights of the evaluation index are derived from the analytic hierarchy process and the grey relational analysis method, respectively. The kernel vector space model is then used to select the best maintenance strategy through the close degree calculation. Finally, an optimal maintenance strategy is determined. A comparison and analysis of three di ﬀ erent representative maintenance strategies resulted in the following ﬁndings: The proposed model is e ﬀ ective; it provides a new decision-making method for power transformer maintenance decision-making; it is simple, practical, and easy to combine with the traditional state assessment method, and thus should play a role in transformer fault diagnosis.


Introduction
The power transformer greatly affects the stability and security of electrical power systems (EPS). In order to enhance maintenance efficiency and lower costs, it is necessary to improve transformer operation and maintenance strategy. Traditional maintenance strategies usually consist of regular maintenance, which takes into account time but ignores the specific state of the equipment, causing the over-repair or lack of repair of the transformer. This affects the cost and also does not satisfy the need of smart grids [1]. The condition-based maintenance (CBM) of power transformers commonly includes condition monitoring, condition assessment, and maintenance decision-making. During condition monitoring, the electrical and chemical parameters are monitored to evaluate the condition of the voltage transformer insulation [2][3][4]. Dissolved gas analysis (DGA) is used in the actual field diagnosis of engineering to assess the condition of a transformer. Classical machine-learning methods related Six indicators are used: c1 to c6 represent the required technical level for maintenance, the maintenance effect, the total cost, the economic impact of the loss of load due to maintenance, the potential security risk, and the impact of maintenance on system security, respectively.

Cloud Model
The cloud model, proposed by Li et al. [19], is a mathematical model of fuzzy exchange between a qualitative concept and its numerical expression; this model combines the uncertainty of fuzzy theory with the randomness of probability theory, the mathematical characteristics of which can be denoted by three indices: Ex, En, and He. Ex is central to the concept of attributes in the discourse domain and represents the concept of the attributes. En is a measurement metric of the blurring level of property concept and represents the numerical region of intervals in which the property concept is applicable. He represents the degree of dispersion of cloud droplets and reveals the association of the concept of natural language properties between randomness and fuzziness. For example, if Ex = 0.5, En = 0.15, and He = 0.01, the quantitative concepts represented by the cloud model can be described as C(0.5, 0.15, 0.01).
There are many qualitative indicators in the comprehensive evaluation index system that need to be converted into quantitative indicators for subsequent analysis and calculation. The qualitative indicators c1 to c6 are classified in accordance with the following levels: very good, good, average, bad, and very bad. These levels can be represented by five classes under the golden section method. Assuming that C0(E 0 x ,E 0 n ,H 0 e ) is the middle cloud, with the neighboring clouds being C-1(E -1 x ,E -1 n ,H -1 e ), C+1(E +1 x ,E +1 n ,H +1 e ), C-2(E -2 x ,E -2 n ,H -2 e ), and C+2(E +2 x ,E +2 n ,H +2 e ), these five clouds can be defined by the golden section method in Formulas (1)-(3): Six indicators are used: c 1 to c 6 represent the required technical level for maintenance, the maintenance effect, the total cost, the economic impact of the loss of load due to maintenance, the potential security risk, and the impact of maintenance on system security, respectively.

Cloud Model
The cloud model, proposed by Li et al. [19], is a mathematical model of fuzzy exchange between a qualitative concept and its numerical expression; this model combines the uncertainty of fuzzy theory with the randomness of probability theory, the mathematical characteristics of which can be denoted by three indices: E x , E n , and H e . E x is central to the concept of attributes in the discourse domain and represents the concept of the attributes. E n is a measurement metric of the blurring level of property concept and represents the numerical region of intervals in which the property concept is applicable. H e represents the degree of dispersion of cloud droplets and reveals the association of the concept of natural language properties between randomness and fuzziness. For example, if E x = 0.5, E n = 0.15, and H e = 0.01, the quantitative concepts represented by the cloud model can be described as C(0.5, 0.15, 0.01).
There are many qualitative indicators in the comprehensive evaluation index system that need to be converted into quantitative indicators for subsequent analysis and calculation. The qualitative indicators c 1 to c 6 are classified in accordance with the following levels: very good, good, average, bad, and very bad. These levels can be represented by five classes under the golden section method. Assuming that C 0 (E 0 x ,E 0 n ,H 0 e ) is the middle cloud, with the neighboring clouds being x ,E +1 n ,H +1 e ), C −2 (E −2 x ,E −2 n ,H −2 e ), and C +2 (E +2 x ,E +2 n ,H +2 e ), these five clouds can be defined by the golden section method in Formulas (1)-(3): Energies 2020, 13, 5948 With H e set as constant 0.006 and a valid discourse domain defined as [0, 1], the calculation outcomes are listed in Table 1. The corresponding cloud model image is shown in Figure 2. With He set as constant 0.006 and a valid discourse domain defined as [0,1], the calculation outcomes are listed in Table 1. The corresponding cloud model image is shown in Figure 2.      Assuming that the qualitative indicators are evaluated by h experts, the language-based evaluation information given by each expert has a corresponding cloud model. The individual cloud models of each expert are combined as an integrated cloud model in Formulas (4) and (5).

Grey Correlation Analysis
The weight coefficient of each assessment indicator in the comprehensive assessment index system of maintenance decision-making greatly affects the final evaluation results. It is important to establish an objective and comprehensive decision-making process. The grey relational method, originating from the grey system theory, can resolve issues related to fuzzy information, less data, and data shortage [30][31][32][33]. Grey correlation analysis can determine different information between sequences and can calculate the degree of correlation by establishing the different information intervals. The comprehensive evaluation system of the CBM consists of numerous evaluation indicators. The grey correlation analysis method can be used to objectively determine the principal contradiction of the system. It is therefore used to calculate the objective weight of evaluation indicators.
Assuming that G i (i = 1, 2, . . . , m) is the selected maintenance strategy, and G 0 = (g 01 , g 02 , . . . , g 0n ) is the best reference strategy, G ij and G 0j denote the quantified evaluation indicators of the maintenance strategy and the corresponding indicator value of the best reference strategy, respectively. Thus, the selection criteria of the benefit indicators can be calculated using Formula (6): Moreover, the selection criteria of the cost indicators can be calculated using Formula (7): The correlation coefficient matrix (ζ ij ) m×n between the maintenance strategy G i and the best strategy G 0 can be calculated using Formulas (8) and (9): where ∆ ij is the absolute difference between g ij and g 0j , ∆(min) is the minimum difference between the two levels, ∆(max) is the maximum difference between the two levels, and ρ is the resolution coefficient, which was set to 0.5 in this paper as an optimal value. The objective weight vector w j is calculated by Formulas (10) and (11) based on the correlation matrix (ζ ij ) m×n and the correlation coefficient r j :

Determination of the Comprehensive Index Weight
The analytic hierarchy process (AHP) is combined with grey correlation analysis, and the comprehensive weight of the assessment index is calculated by the principle of additive integration. The AHP introduces the subjective prior knowledge of experts as well as their preference information to reach the subjective weight, while grey correlation analysis reflects the inherent correlation of maintenance decision-making for power transformers in an objective and comprehensive manner. Assuming that W S represents the subjective weight vector and W O represents the objective weight vector, according to the additive integration principle, the comprehensive weight can be expressed as specified in Formulas (12) and (13): where P i is the value of subjective weight sorted from small to large, and m is the whole number of indicators.

Kernel Vector Space Model
In this paper, the kernel vector space model derived from the support vector machine theory was applied to map the input data from low-dimensional space to high-dimensional space through kernel function. This model increases the difference and distance between samples to gain more objective and scientific evaluation results.
At present, there is no theory that can explain the selection of kernel functions perfectly, though the Gaussian kernel function is commonly used. The expression of the function is as specified in Formula (14): where σ is the radial basis parameter of the Gaussian kernel function and the parameters x and y are the corresponding space vectors. By calculating the cosine of the angle of the vectors, the proximity between the vectors can be calculated using Formula (15): where R and R 0 are the quantitative indicator and the best quantitative index mentioned in Section 2, respectively. The angle between the space vector R and R 0 is defined as θ.
Considering the difference between the factors and the weight distribution of each evaluation index, the space vector affected by the combined weight should be taken into account, and the comprehensive weight should be added before each space vector, as specified in Formula (16): where cosθ is the weighted cosine of the space vector after the allocation of the index weights in the kernel space. The calculated weighted cosine value can be regarded as the proximity of each decision candidate and the optimal decision. Thus, the maintenance decision evaluation result of a power transformer should be based on a close degree of proximity.

Case Analysis
The case analysis was based on the equipment failure of a 110 kV transformer. Monitoring of the transformer showed that the total amount of hydrocarbon oil exceeded the alert value.
Three kinds of maintenance strategies were developed according to the results of the fault diagnosis and production planning arrangement, denoted as M 1 to M 3 . M 1 overhauls in advance; maintenance items are established by the relevant guidelines. M 2 uses targeted overhauling; the maintenance Energies 2020, 13, 5948 7 of 11 schedule is arranged by the fault diagnosis and the trend forecasting results. M 3 tracks and monitors the transformer continuously; it does not arrange the overhauling until the overhauling cycle.
After analysis and comparison, four experts published their own viewpoint of the alternative strategy according to the indices mentioned in Section 2. The evaluation of the qualitative indices is shown in Table 2, with each row of M 1 , M 2 , and M 3 corresponding to the evaluation of the qualitative indicators of one expert. The determination process of the optimal maintenance strategy is described as follows: The natural language evaluation information of the experts should first be pre-processed. The quantification of the of technical, economic, and security qualitative indices should be calculated using Equations (1)- (5), and the initial decision matrix (D) of the three strategies should be composed of six indicators, as follows: According to the criterion of the best program as established in Section 2, the best quantitative sequence was constructed as follows: (1) Based on the influence of maintenance in power transformers, the expected results of qualitative index c 2 were calculated using Equation (6), as follows: (2) The expected results of qualitative indicators c 1 and c 3 to c 6 were calculated using Equation (7), as follows: Additionally, the augmented matrix (M z ) can be expressed as: According to the process mentioned in Section 2, the final weight coefficients are shown in Table 3, and a bar diagram of the comprehensive weight is shown in Figure 3.
According to the process mentioned in Section 2, the final weight coefficients are shown in Table 3, and a bar diagram of the comprehensive weight is shown in Figure 3. Table 3. Weight of the evaluation indices. The weighted proximity (Q) between decision vector R and R0, as calculated by Equations (14)-(16), is listed in Table 4, with σ set to 1.12. The weighted proximity (Q) between decision vector R and R 0 , as calculated by Equations (14)- (16), is listed in Table 4, with σ set to 1.12. The optimal membership degrees of each maintenance strategy were sorted, and the result was Q 3 > Q 2 > Q 1 . The third maintenance scheme, Q 3 , was the relative optimum strategy, which is the same as the results of [34,35].

Index Subjective Weights Objective Weights Combined Weights
The above example takes the technical, economic, and security aspects into consideration. It can also be concluded that M 1 , M 3 , and M 2 are the relative optimum strategies for the technical, economic, and security aspects, respectively, where aspects are considered individually. Decision-makers should consider various evaluation indicators and the potential risk of loss comprehensively, synthetically, and systematically before making the final decision to avoid unnecessary economic loss. The model proposed in this paper offers a good interpretation of the psychology of power enterprise decision-makers, and thus reflects expected human behavior in the selection of the maintenance strategies. Table 5 and Figure 4 show comparisons of the results obtained based on the strategies in [34,35] and those obtained by the proposed strategy in this paper.  The optimal membership degrees of each maintenance strategy were sorted, and the result was Q3 > Q2 > Q1. The third maintenance scheme, Q3, was the relative optimum strategy, which is the same as the results of [34,35].
The above example takes the technical, economic, and security aspects into consideration. It can also be concluded that M1, M3, and M2 are the relative optimum strategies for the technical, economic, and security aspects, respectively, where aspects are considered individually. Decision-makers should consider various evaluation indicators and the potential risk of loss comprehensively, synthetically, and systematically before making the final decision to avoid unnecessary economic loss. The model proposed in this paper offers a good interpretation of the psychology of power enterprise decision-makers, and thus reflects expected human behavior in the selection of the maintenance strategies. Table 5 and Figure 4 show comparisons of the results obtained based on the strategies in [34,35] and those obtained by the proposed strategy in this paper.  According to the results of Table 5 and Figure 4, although in both [34,35] the third maintenance strategy was used as a relatively optimal scheme, the degree of uncertainty in [34] was still relatively high and the determination of the strategy in [35] was too absolute, which resulted in a deviation from reality. In comparison to the conclusions in [34,35], the model put forward in this paper is more suitable for practical engineering applications. The results obtained by the proposed model reflect the psychology of decision-makers in power companies when choosing maintenance strategies. More importantly, the implementation of the model is achieved with a solid mathematical foundation. According to the results of Table 5 and Figure 4, although in both [34,35] the third maintenance strategy was used as a relatively optimal scheme, the degree of uncertainty in [34] was still relatively high and the determination of the strategy in [35] was too absolute, which resulted in a deviation from reality. In comparison to the conclusions in [34,35], the model put forward in this paper is more suitable for practical engineering applications. The results obtained by the proposed model reflect the psychology of decision-makers in power companies when choosing maintenance strategies. More importantly, the implementation of the model is achieved with a solid mathematical foundation.

Conclusions
This paper proposed an integrated evaluation model for decision-making for power transformers that includes the cloud and kernel vector space models. It also suggested a comprehensive evaluation system based on technical, economic, and security indices. The cloud model allows for the quantitative expression of the qualitative language assessment index. The subjective weight of the assessment index is calculated by the analytic hierarchy process, while the objective weight is calculated by the grey relational analysis method. In terms of the principle of the additive synthesis method, the comprehensive weight, including the influence of subjective judgment and objective information, can be obtained.
The results of the case analysis show that the proposed strategy is applicable. It is characterized by a simple model and practical method, and it is convenient to combine with traditional condition assessment method to play a greater role in the CBM of power transformers. The constructed comprehensive evaluation system is able to reflect the complexity of the maintenance decisions in the field and provides scientific and reasonable evaluation results. The model outlined above is considered an efficient model for maintenance decision-making of the power transformer and offers a new means of CBM decision-making for power enterprises.

Conflicts of Interest:
The authors declare no conflict of interest.