A Method of Power Supply Health State Estimation Based on Grey Clustering and Fuzzy Comprehensive Evaluation

Health state estimation can evaluate the current degradation state of equipment, and the evaluation results can provide a basis for formulating equipment maintenance strategies. At present, in the research on health state estimation, only one level of indicators of the equipment is usually considered, meaning incomplete estimation results. In addition, the current evaluation methods rarely consider the impact of data noise on the evaluation results, which can easily lead to abnormal evaluation results. To solve the above two problems, this paper first introduces the state indicators such as ripple voltage and output voltage, as well as the ratio of theoretical use time into the evaluation indicators. Furthermore, a health state estimation method combining grey clustering and fuzzy comprehensive evaluation methods is established. This method can consider the characteristics of multiple data groups at the same time, thus the impact of data noise on the evaluation results can be reduced. During the evaluation, the grey clustering method is used to evaluate the clustering coefficient vector of the power supply under a single group of data. After that, the clustering coefficient vector of multiple groups of data is used as the membership vector of fuzzy comprehensive evaluation, and the fuzzy comprehensive evaluation method is used to evaluate the health state of a power supply under multiple groups of data. An example shows that this method is effective in estimating the health state of a normally degraded power supply.


I. INTRODUCTION
With the improvement of equipment integration, the frequency of its failure increases gradually. In order to ensure the smooth operation of the equipment and reduce the maintenance cost during the service life of the equipment, Prognostics health management (PHM) technology came into being [1]. PHM technology represents a change of concept, which turns the maintenance management of equipment from passive maintenance to regular inspection [2].
The associate editor coordinating the review of this manuscript and approving it for publication was Elizete Maria Lourenco .
Health assessment technology is key to PHM [3]. This technology mainly collects the output data of the equipment through various sensors, and processes the data by various methods to comprehensively evaluate the health of the equipment [4]. Based on the health state assessment results, the operators can not only control the equipment state accurately, but also help maintenance personnel formulate maintenance strategies in a timely manner, thus improving the performance and reliability of the equipment [5].
Initially, engineers used the binary function of fault and normal to judge the health state level of equipment [6]. With the development of related technologies, it has been found that this method is insufficient to accurately evaluate the state of equipment. Thereafter, people used multiple levels, such as health, sub-healthy, faulty, etc., to describe the health state of equipment [7], [8].
Due to the different characteristics of different equipment, the health state estimation methods used are usually different. According to the degree of dependence on knowledge, health state estimation methods can be divided into model driven and data driven methods [9], [10]. Among them, the knowledge driven health state estimation methods mainly evaluate health state through knowledge acquisition and knowledge expression [11]. Such methods include markov distance method [12], fusion weight calculation method [13], Euclidean distance method [14] and fuzzy theory method [15]. This kind of method is efficient. However, this kind of method requires expert experience to determine the weight [16], and requires a certain understanding of product characteristics [17].
The data driven methods mainly use the characteristics of data to evaluate the health state based on machine learning [18]. These methods have become a popular choice for health estimation in many areas, e.g. battery, gear, etc [19]. Such methods mainly include linear regression [20], support vector machine [21], neural network, and deep learning [22]. Such methods are highly dependent on data and algorithms, and are usually difficult to interpret at the physical level.
At present, most health state estimation methods still have the following problems to be solved. First, only one level of indicators is considered, which cannot fully reflect the state of equipment. Second, only one set of evaluation indicator data is used in the evaluation process, which cannot avoid the impact of data noise on the evaluation results. In order to solve the above two problems, this paper first introduces the state indicators such as ripple voltage and output voltage, as well as the ratio of theoretical use time into the evaluation indicators. Furthermore, a health state estimation method combining grey clustering and fuzzy comprehensive evaluation methods is established. This method can consider the characteristics of multiple data groups at the same time, thus the impact of data noise on the evaluation results can be reduced.
The framework of this paper is as follows. Section II introduces the relevant theories and estimation steps of health state estimation methods. In section III, the health state estimation results for a power supply are presented. Section IV summarizes the full text and puts forward suggestions for future work.

II. RELEVANT THEORIES AND THE STEPS OF HEALTH STATE ESTIMATION
The grey clustering method is used to obtain the clustering coefficient vector of the evaluation object under a single group of data, which is used as the input of fuzzy comprehensive evaluation [23]. The fuzzy comprehensive evaluation method is used to obtain the health state of the evaluation object under multiple groups of data [24]. The specific evaluation process is shown in Fig. 1.

A. ESTABLISH EVALUATION INDICATOR SET
According to the working principle and structure composition of the evaluation object, combined with the characteristics of its output parameters, the health state estimation indicator set is constructed, as shown in Fig. 2.
In addition, the maintenance time of various maintenance behaviors cannot be ignored. It is generally set according to the specific maintenance conditions of each piece of equipment. Generally, it can be assumed that the maintenance time is fixed or not fixed [14], and can follow uniform distribution, exponential distribution and Weibull distribution, etc. [15].
Assume that the evaluation object has m evaluation indicators and n data groups. The characteristic value of evaluation indicator j of the ith data set is x ij . Because the dimension of characteristic value x ij may be different, it is not convenient for subsequent calculation. Therefore, each characteristic value is normalized, compressed between [0, 1], and changed from dimensional data to dimensionless data [25]. For the larger and better evaluation indicator, its standardized processing formula is [26] x , ij = For the smaller optimal evaluation indicator, its standardized formula is VOLUME 11, 2023 where, x ij ′ is the normalized value of the j-th index x ij of the ith dataset; x max j and x min j are respectively the theoretical minimum value and theoretical maximum value of the j-th index of the ith data group.
The characteristic set composed of normalized characteristic values of the evaluation object is taken as the evaluation indicator set X , which can be expressed as

B. CREATING A HEALTH STATE SET
A health state set can be regarded as a set of possible health state types of the evaluation object. The health state set is represented as follows: where, y k is the evaluation results of various health states of the evaluation object, k is the serial number of health states, and s is the total number of health states.

C. ACQUISITION OF CLUSTERING COEFFICIENT VECTOR BASED ON GREY CLUSTERING 1) BASIC PRINCIPLE OF GREY CLUSTERING
The grey clustering method is to use the grey correlation degree matrix to construct the whitening weight function, according to which the assessment indicators are clustered and analyzed to obtain the grey class of the health state of the assessment indicators. If the i-th data group of the evaluation object belongs to the k-th grey class, then a is the clustering coefficient of the i-th data group of the evaluation object belonging to the k-th grey class [27]: In the formula, f k j (·) is the whitening weight function of the k-th grey class, and w j is the weight of the evaluation indicator j.
By normalizing the clustering coefficient σ k i , it can be obtained that the normalized clustering coefficient δ k i of the ith data group of the evaluation object belongs to the k grey class is Then the expression of the normalized clustering coefficient vector δ i of the health state of the ith data group of the evaluation object is If max 1≤k≤s {δ i } = δ k * i , the health state of the i-th data group of the evaluation object belongs to the k * grey class. The grey clustering method needs to determine whitening weight function and index weight, which will be introduced below.

2) INTRODUCTION TO THE BASIC FORM OF WHITENING WEIGHT FUNCTION
Whitening weight function is a quantitative description of the degree to which each data point belongs to a grey hazy set according to the known information [28], which can reflect the preference for different values within the range of the hazy set.
The whitening weight function is generally a piecewise linear function depending on the turning point. Common whitening weight functions include typical whitening weight function, upper limit measure whitening weight function, lower limit measure whitening weight function and moderate measure whitening weight function [29], as shown in Fig. 3.

a: COLOR/GRAYSCALE FIGURES
The typical whitening weight function of grey class k of , and its formula is as follows: where, The evaluation indicator j of this function belongs to the critical value λ k j of grey class k, and the calculation formula is

: WHITENED WEIGHT FUNCTION OF LOWER BOUND MEASURE
Whitening weight function of k grey lower limit measure , and the formula is as follows: This function cannot specifically determine the first and second turning points x k j (1), x k j (2) and the critical value λ k j is calculated as

c: WHITENING WEIGHT FUNCTION OF MODERATE MEASURE
Whitening weight function of k grey class moderate measure , and its formula is as follows: In this function, the second and third turning points

d: WHITENING WEIGHT FUNCTION OF UPPER BOUND MEASURE
The upper bound measure whitening weight function of k grey class of evaluation indicator j is f k This function has no third and fourth turning points x k j (3), x k j (4), then the formula for calculating the critical value λ k j is

D. DETERMINATION OF WHITENING WEIGHT FUNCTION 1) SHAPE DETERMINATION OF WHITENING WEIGHT FUNCTION
The determination of whitening weight function usually refers to the determination of the shape, starting point, turning point and end point of the function. The lower limit measure whitening weight function is used as the first grey class whitening weight function. The whitening weight function of moderate measure is taken as the whitening weight function of the k-th (k = 2, 3, · · · , s-1) grey class. The whitening weight function of the upper bound measure is taken as the whitening weight function of the last grey class, i.e., the s grey class.

2) DETERMINATION OF STARTING POINT, TURNING POINT AND END POINT OF WHITENING WEIGHT FUNCTION
Since the health state level is s, the number of grey classes is s. In addition, since all evaluation indicators have been standardized, the value of each evaluation indicator will be between [0, 1]. The value range [0, 1] of the evaluation indicator is evenly divided into s intervals, and the size of each interval is 1/s (Fig. 4). According to the size of the evaluation indicator value interval and the vertical coordinate range of the whitening weight function, the coordinates of the starting point, turning point and end point of the whitening weight function of s grey types are determined, as shown in Table 1.

E. INDEX WEIGHT CALCULATION BASED ON INFORMATION ENTROPY
The information entropy method is used to determine the weight of each evaluation indicator in the grey clustering [30].
In the evaluation indicator x , ij of the evaluation object, the information entropy of the j-th evaluation indicator can be expressed as Then the entropy weight w j of the j-th evaluation indicator is Then the weight W of each index can be expressed as

F. DETERMINATION OF HEALTH STATE BASED ON FUZZY COMPREHENSIVE EVALUATION
Fuzzy comprehensive evaluation can determine the subordination degree of multiple data groups to the health state through the evaluation matrix [31], and adjust the the subordination degree through generalized fuzzy operation, so as to obtain the best health state evaluation result. The above has evaluated the i-th data group x , ij in the evaluation indicator set X , and obtained that the clustering coefficient vector of the ith data group for each element in the health state set is b, that is, the membership vector is δ i .  Here, taking the membership vectors of the n data groups as rows, the fuzzy evaluation matrix δ between the evaluation indicator set X and the health state set Y can be obtained: Line i of the matrix δ reflects the impact of the i-th data group on the elements in the health state set of the evaluation object. The k-th column reflects the extent to which the assessment index affects the assessment object to take the kth health state set element.
When the fuzzy evaluation matrix δ and the weightẆ θ of the data group are determined, the fuzzy comprehensive evaluation set Q on the health state set Y can be obtained through the fuzzy comprehensive evaluation model: In the formula, * is the generalized fuzzy composition operation.
According to the principle of maximum membership, the evaluation set element y k corresponding to the maximum value in the fuzzy comprehensive evaluation set Q is taken as the health state evaluation result. Fuzzy comprehensive evaluation method needs to determine the fuzzy evaluation matrix δ and the weight of the data group. The fuzzy evaluation matrix δ has been obtained in the previous section, and the method for obtaining the weight of the data group will be given below.

G. WEIGHT CALCULATION OF MULTIPLE DATA GROUPS BASED ON AN ANALYTIC HIERARCHY PROCESS
An analytic hierarchy process is used to determine the importance of each data group to the evaluation object [32]. The steps of the analytic hierarchy process are as follows. First, establish the analytic hierarchy process structure model, then construct the judgment matrix, after that, calculate the weight coefficient vector of the judgment matrix, then, conduct a consistency check, and finally, conduct the overall ranking and consistency check of the hierarchy. The process is shown in Fig. 5.

a: ESTABLISH THE HIERARCHICAL ANALYSIS STRUCTURE MODEL
According to the internal relationship, mutual influence among data groups, a structure analysis model of orderly arrangement of multiple levels is established, as shown in Fig. 6. The top layer of the model is the target layer, which represents the problems that need to be solved. The middle layer is the criteria layer, which represents the intermediate links used to achieve the overall goal. This layer places the criteria and indicators for evaluating and measuring the overall goal. At the bottom is the solution layer, which represents the measures and methods used to solve the problem.
The symbol A represents the target layer, which has 1 element. The symbol B represents the criterion layer, and B p represents each element in the criterion layer. The criterion layer has r elements, that is, B p is B 1 , B 2 ,. . . , B r . The symbol C represents the scheme layer, and C q represents each element in the scheme. The criterion layer has u elements, that is, C q is C 1 , C 2 , . . . , C u .

b: CONSTRUCT JUDGMENT MATRIX
The judgment matrix reflects the importance of each element of the lower level in the hierarchy model to a related element of the upper level. By comparing the elements of each layer with the elements of the previous layer in the hierarchical analysis structure model, the judgment matrix of the importance of different elements of the same layer with respect to the corresponding elements of the previous layer is obtained. For element A, the criterion level judgment matrix B is B = b pl , p = 1, 2, · · ·, r, l = 1, 2, · · ·, r wherein b pl represents the relative importance of element A, element B p and element B l . The judgment matrix has the following characteristics: The T.L. Sataty1-9 scaling method is used to determine the b pl using 1∼9 and its reciprocal 1, 1/2,. . . , 1/9, a total of 17 numbers as the scale [33]. The meanings of each scale are shown in Table 2.
According to the above principles and hierarchical structure model, the importance of indicators at all levels relative to the superior indicators is compared in pairs, and finally the judgment matrix between elements at each level can be obtained.

c: CALCULATE THE WEIGHT COEFFICIENT VECTOR OF THE JUDGMENT MATRIX
Calculate the relative weights of the elements in the judgment matrix. The sum method is used to calculate the relative weight coefficient. This method is simple and easy to implement. The specific calculation process is shown below. First, normalize each column vector in the judgment matrix to obtainḃ pq Step 2, sumḃ pq by line to getb p = (b p ) T , where b p = u q=1ḃ pq , p = 1, 2, · · ·, r Normalize all elements inb p Then for the total objective element A, the weight coefficient vector of each element in the criterion level judgment matrix isẆ

d: CONDUCT CONSISTENCY INSPECTION
The main reason for consistency inspection is that when the elements to be compared are fuzzy and complex, it is difficult to ensure the consistency of the determined judgment matrix. For a judgment matrix whose consistency cannot be satisfied, its maximum eigenvalue will be greater than the order of the matrix, and there are other non-zero eigenvalues. At this point, each element in the matrix needs to be adjusted.  The formula for solving the maximum eigenvalue λ max of judgment matrix B is as follows: where, (B ·Ẇ θ ) p represents the p-th component of B ·Ẇ θ . The solution formula of consistency deviation degree index C.I. of judgment matrix B is as follows: R. I . is the average random consistency index of judgment matrix B, and its solution is taken The formula to solve the random consistency ratio C.R. of judgment matrix B is where, R.I. is the average random consistency index of judgment matrix B. This index can modify C.I. to eliminate the influence of matrix order. Conduct consistency inspection [34]. In the case of r ≥ 3, when 0< C.R.<0. 10, it indicates that the deviation degree of judgment matrix consistency is small, and the weight value assigned to each index is reasonable. Otherwise, it indicates that there may be some unreasonable contradictions in the process of pairwise comparison or judgment of elements. At this time, each element of the judgment matrix needs to be adjusted. The adjusted matrix repeats the above steps to solve the new C.R. again. Until the C.R. is less than 0.1, the adjustment is finished.
The weight calculated the last time is the single level sorting of elements in this level. VOLUME 11, 2023 e: CALCULATION LEVEL TOTAL SORTING Compared with the target layer, the weight data of single level sorting is required for the overall hierarchical sorting. The total ranking weight of element B p (p = 1, 2, · · ·, r) in the standard layer to target layer A is W p .
The single ranking weight of factor C q (q = 1, 2, · · ·, u) in the scheme layer C of the next layer relative to the upper layer B p is w p q . Then the ranking weight of the lower layer C q relative to the overall goal isW cq = r p=1 W p · w p q , q = 1, 2, · · ·, u . The comprehensive weight coefficient of each element can be solved according to Table 3.

f: CONSISTENCY TEST FOR THE OVERALL HIERARCHICAL RANKING
The consistency check of the overall ranking of the hierarchy is calculated from top to bottom, and the final result is the overall ranking of the lowest scheme level elements to the top target level elements. The consistency check is calculated using the following formula: If the hierarchical structure can pass the formula test, it is the optimal scheme. Otherwise, it is necessary to adjust each element of the judgment matrix and repeat the above steps until a satisfactory result is obtained. Fig. 7 shows the health state estimation indicator set of a power supply. According to this evaluation indicator set, the characteristic vector X of grey clustering evaluation is established. X includes three characteristic parameters, namely, X ij ={x i1 , x i2 , x i3 }={ripple voltage, output voltage, and the proportion of theoretical service time}.

III. HEALTH STATE ESTIMATION IN CASE OF POWER SUPPLY DEGRADATION A. ESTABLISH EVALUATION INDICATOR SET
The method of combining simulation and theoretical calculation is used to obtain the evaluation indicator data for a continuous period of time for subsequent power supply health state evaluation [35]. First, 8 groups of ripple voltage and average voltage data near the time of power source degradation are obtained through simulation. Then, according to the theoretical working hours of the power supply and the working hours near the actual degradation, the theoretical service time proportion is obtained. The specific data are shown in Table 4. These data are obtained near the time of power supply failure. At this time, the theoretical health state of the power supply is ''Scrap''. Therefore, the ''Scrap'' can be used as the standard health state to verify the accuracy of evaluation results subsequently.  The theoretical minimum value, theoretical maximum value and threshold value of each evaluation indicator are listed in Table 5. Among them, the threshold value is the value of the evaluation indicator in case of power supply failure. Its value is generally conservative, slightly less than the theoretical maximum value.
The standardized values of the evaluation indicators obtained from the standardized processing of the original data of the evaluation indicators are shown in Table 6.

B. CREATE HEALTH STATE
According to the actual working state and health information characteristics of the power supply, its health state can be divided into five grades: healthy, good, sub-healthy, faulty and scrap. The description of each grade is shown in Table 7.
The health state set Y of the power supply is Y = y 1 = Healthy, y 2 = Good, y 3 = Sub-healthy, The health state of the assessment object has been divided into 5 levels (as shown in Table 7 Table 8.
The starting point, turning point and end point of the five grey whitening weight functions are respectively brought into the corresponding whitening weight formula to obtain the unknown parameters in the whitening weight function.
The images of the whitening weight function under each health grey category and their corresponding expressions are listed in Fig. 8 and Table 9.
The evaluation indicator feature vector data is brought into the formula in the table above, to obtain the whitening weight function values of each index under five different grey categories. The whitened weight function values obtained are 5 matrices with 8 rows and 3 columns.

3) GET THE CLUSTERING COEFFICIENT VECTOR
Take the whitening weight function value and weight into (5) to (7), and calculate the normalized clustering coefficient VOLUME 11, 2023  vector δ of the data group. Later, this vector is used as the comprehensive evaluation matrix of the fuzzy comprehensive evaluation method.

D. HEALTH STATE ESTIMATION BASED ON FUZZY COMPREHENSIVE EVALUATION 1) ESTABLISH THE HIERARCHICAL ANALYSIS STRUCTURE MODEL
The hierarchical structure model is established as shown in Fig. 9, which is used to obtain the weight of 8 groups of data points. The two elements included in the criteria layer are the time distance from the latest measurement data point and the relative health of the evaluation indicator data point. The 8 elements included in the scheme layer are mainly 8 groups of evaluation indicator data.

2) CONSTRUCT THE JUDGMENT MATRIX
The reference specific gravity of element B1 in the determination criteria layer is 3 times that of element B2. Then for element A, the criterion level judgment matrix B is wherein, b 11 represents the relative importance of element B1 and element B2 for element A. There are two judgment matrices in the scheme layer: the judgment matrix is C 1 and the judgment matrix is C 2 . Wherein, the judgment matrix C 1 represents the importance of elements B1 in the criteria layer and C1, C2, C3, . . . , C8 in the scheme layer. For element B1, element C1 is the earliest sampling point, and element C8 is the latest sampling point.
The rule for setting the importance is that the earlier the points are collected, the lower the importance is, and the newer the points are collected, the higher the importance is. The lower the importance of the points collected earlier, the higher the importance of the newly collected points. That is to say, the importance of element C1 is the lowest and that  of element C8 is the highest. The importance of element C8 is set to be 8 times that of element C1, and the importance of element C7 is set to be 7 times that of element C1. The importance of other elements compared with element C1 is also set according to the above rule. Then the judgment matrix C1 is The judgment matrix C 2 represents the importance of elements C1, C2, C3 and C8 in the scheme layer for element B2. For element B2, set element C1 as the earliest sampling point and element C8 as the latest sampling point. The earlier the point is set, the lower its health is, and the more newly collected the point is, the higher its health is. That is, element C1 is the least important and element C8 is the most important. All the sampling points are divided into four areas. The area with the earliest sampling is low in relative health, while the area with the latest sampling is high in relative health. Set the importance of elements C7 and C8 to be 4 times that of elements C1 and C2, the importance of elements C5 and C6 to be 3 times that of elements C1 and C2, and the importance of elements C3 and C4 to be 2 times that of elements C1 and C2. Then the judgment matrix C 2 is

3) CALCULATE THE WEIGHT VECTOR OF THE JUDGMENT MATRIX
Use the sum method to calculate the weight vectors of judgment matrices B, C 1 and C 2 , which are respectively:

4) CONSISTENCY INSPECTION
The consistency of judgment matrices of two adjacent levels is checked, and the calculation results are shown in Table 10.
The results show that the element C.R. is between 0 and 0.10. That is, the weight value assigned to each index is reasonable, and the judgment matrix of two adjacent layers of all elements meet the consistency requirements.

5) CALCULATE THE TOTAL RANKING OF LEVELS AND CHECK THE CONSISTENCY
Obtain the weight vector of fuzzy comprehensive evaluation of 8 data groups: Continue iteration according to (31) to calculate the consistency check ratio of the whole analytic hierarchy model. Since C.R. is 0.0019 and less than 0.1, it indicates that the judgment matrix of the whole hierarchy model meets the consistency requirements, and it also indicates that the ranking is reasonable and effective.

6) HEALTH STATE DETERMINATION BASED ON FUZZY EVALUATION
The membership vector of grey clustering is used as the fuzzy evaluation matrix δ. The weights of 8 data groups obtained by analytic hierarchy process are taken as the weights in fuzzy comprehensive evaluation. By introducing the data into the formula of fuzzy evaluation matrix, we can get According to the principle of maximum membership, the overall health state of the power supply can be determined as scrap.

IV. CONCLUSION
In this paper, the ripple voltage, output voltage and other state indicators, and the ratio of theoretical service time are introduced into the evaluation indicators of the model. In addition, based on grey clustering and fuzzy comprehensive evaluation, a health state estimation method considering multiple data groups is established to evaluate multiple groups of data at the same time, so as to reduce the impact of data noise on the evaluation results.
Firstly, the clustering coefficient vector of the power supply in a single group of data is evaluated by using the grey clustering method. Then, the clustering coefficient vector under multiple single data points is synthesized as the membership vector of fuzzy comprehensive evaluation. The fuzzy comprehensive evaluation method is used to evaluate the health state of power supplies under multiple groups of data. An example shows that this method can be used to estimate the health state of a normally degraded power supply.
It is worth noting that the health state estimation method proposed in this paper only takes the power supply under specific working conditions as the analysis object. The applicability of this method to severe working conditions and other research objects has not been discussed. In the future work, the applicability of this method for equipment operating under severe working conditions can be further explored.