The fuzzy hierarchical analysis method has been widely used as a basis for quantifying evaluation indicators and selecting an optimal solution. It combines a fuzzy consistency matrix with a hierarchical analysis, retaining the advantages of the latter while overcoming its fuzziness in the judgment matrix to ensure consistency and functionality more to imitate the human decision-making.

The fuzzy complementary matrix of hydropower unit evaluation indicators can be established by comparing the evaluation indicators r i and r j to determine and quantitatively express the importance of one factor over the other. The fuzzy relationship affiliation degree between these two indicators, a i j , is quantitatively described on a scale of 0.1–0.9, with the meaning of each interval in this range expressed in Table 1. Thus, the fuzzy complementary judgment matrix of the evaluation indicators for hydropower units is expressed as A = a i j n × n which satisfies 0 ≤ a i j ≤ 1 and a i j + a j i = 1, where: a i i = 0.5 i = 1,2 , … , n   a i j + a j i = 1   i , j = 1,2 , ⋯ , n ;

As shown in Table 1, a i i = 0.5 indicates that each factor is equally important; a i j ∈[0.1,0.5) indicates that factor x j is more important than x i ; and a i j ∈ [0.5,0.9] indicates that factor x i is more important than x j . For the last case, the fuzzy consistency matrix performs a summation operation for each row of the fuzzy judgment matrix as follows: r i = ∑ j = 1 n a i j (1)

The matrix transformation conducted according to Eq. 1 yields the fuzzy judgment matrix E = e i j n × n , which is consistent, as follows. e ij = r i − r j 2 n + 0.5 (2) where e i j is the consistency-processed value of a i j in the i th row and j th column of matrix E and r i is the value from the fuzzy complementary matrix A = a i j n × n . The value of the row-by-row summation of the weight of each indicator i , w i =( w 1 , w 2 , w 3 , … , w n ) is subsequently obtained from the matrix E = e i j n × n as follows: w i = 2 ∑ j = 1 n e i j − 1 n n − 1 (3)

The CRITIC method is an objective weighting method that uses the quantity of information for each indicator to calculate the value of its weight. This represents a superior method compared to entropy weighting or the coefficient of variation when the information describing each indicator to be evaluated is provided by varying intensities and conflicts with that describing other indicators. Indeed, the CRITIC method determines the weights not only by taking the variability of the indicator information as a premise, but also by combining the correlation between indicators, thereby preventing subjective factors from having an outsized effect and causing the results to deviate from objective reality.

Indicator dissimilarity is generally expressed as a standard value that represents the volatility of the indicator; the greater the volatility, the greater the indicator dissimilarity. Conflict between two different indicators is expressed in terms of correlation; the greater the conflict, the smaller the correlation.

Generally, CRITIC method modeling is conducted using the following steps.

(1) Dimensionless processing

Considering the inconsistency in the scale of the data for each indicator, dimensionless processing is first conducted on each indicator to remove the influence of such inconsistency on the evaluation results. When there are many objects to be evaluated, standardization can be used for this task as follows: y i = x i − x 0 / s (4) where x i is the sample mean for each indicator, x 0 is the actual variable value, and s is the standard value.

(2) Calculate indicator dissimilarity

The CRITIC method uses the standard deviation to express the variability between indicators as follows: S j = ∑ i = 1 n x i j − x j ∼ 2 n − 1 (5) where x j ∼ is the mean value of each indicator. The larger the standard deviation, the greater the difference between the indicator values, the more information is provided among them, and the greater their assessment strength, suggesting that they should be given higher weights.

(3) Calculate indicator conflict

Indicator conflict is expressed in terms of the correlation coefficient as follows: E j = ∑ i = 1 n 1 − r i j (6) where r i j denotes the Pearson correlation coefficient (Edelmann et al., 2021) between indicators i and j

(4) Calculate quantity of information

The quantity of information is determined by: V j = S j × E j (7)

(5) Calculate the objective weights

The objective weight of the j indicator W j is given by W j = V j ∑ j = 1 n V j (8) and W = W 1 , W 2 , ⋯ , W n T is the indicator objective weight vector.

Game theory provide a mathematical method for investigating how to make decisions and maximize benefits when there are multiple struggling or competing individuals in a group. When used in subjective–objective combination weight calculations, game theory can apply weighting to the combinations between indictors and determine weights that deviate the least from each basic value, thereby ensuring that the calculated values match the actual situation. This “portfolio empowerment” method is conducted as follows.

(1) Employ the game theory concept of outlier minimization to find the optimal weights w k * as follows:

The indicator data from a hydropower unit include random measurement errors and exhibit obvious normal distribution characteristics (Zheng et al., 2017; Zhang S. et al., 2020), as shown in Figure 2. For normally distributed unit monitoring indicators V i = { V i 1 , V i 2 , … , V i t … , V i m , the probability that an indicator value falls in the range μ − 3 σ , μ + 3 σ ], where μ is the optimal value of the indicator, is 99.74% and the probability it falls outside this range is 0.26%. Thus, the interval μ − 3 σ , μ + 3 σ ] can be considered the normal operation limit of an indicator and the interval μ − 4 σ , μ + 4 σ ] can be considered its overall, where: u = m e a n V i = 1 m ∑ i = 1 m V i t (13) σ = 1 m ∑ i = 1 m V i t − μ 2 (14)

The degradation degree when using indicators for which larger values are better is expressed as: g = 1 x i < a β − x i β − a a < x i < β 0 x i > β (15) where g is the degradation degree, x i is the measured value of the indicator, β is the optimal value of the indicator, a is the lower limit of the indicator, and a < β .

The degradation degree when using indicators for which smaller values are better is given by: g = 0 x i < a x i − β β − a a < x i < β 1 x i > β (16) where β < a .

When establishing the safety evaluation model for a hydropower unit in use, most of the risk evaluation indicators have randomness, fuzziness, and other uncertainties. The fuzzy comprehensive evaluation method has unique advantages for solving such problems. Fuzzy comprehensive evaluation is based on fuzzy mathematics and uses fuzzy relationship synthesis to quantify factors with unknown boundaries that are not easy to quantify, then comprehensively evaluate the system considering multiple factors. This approach is based on correlation theory, which converts a qualitative evaluation into a quantitative evaluation. This study accordingly adopted the fuzzy comprehensive evaluation method to evaluate and analyze the state of hydropower units.

The triangular/semi-trapezoidal membership function has been widely used in fuzzy comprehensive assessments due to its simple distribution, and it provides results comparable to other more complex membership functions used in risk assessments and equipment evaluations. However, as the degradation values used in the traditional triangular/semi-trapezoidal affiliation function for equipment evaluation depend on expert experience, and the correspondence among different levels lacks explanation. To explain the rationality of the relationships corresponding to different evaluation levels, an improved triangular/semi-trapezoidal affiliation function was proposed for use in this study as shown in the diagram in Figure 3.

The diagram shows that the hydropower unit can be classified into four states: I , II , III , and IV , representing good, qualified, attention, and abnormal conditions, respectively, corresponding to degradation degrees of 0.25, 0.50, 0.75, and 1, respectively. These four conditions can intuitively express hydropower unit health as defined in Table 2.

Since the hydropower unit indicators exhibit normal distribution characteristics, as shown in Figure 2, the hydropower unit states in Table 2 were defined using the four indicator value intervals shown in Table 3. These intervals are defined as μ , μ + 2 σ , μ + σ , μ + 2 σ , μ + 2 σ , μ + 3 σ , and μ + 3 σ , μ + 4 σ , where μ , μ + σ , [ μ + 2 σ , μ + 3 σ , and μ + 4 σ are zone boundary values corresponding to degradation degrees of 0, 0.25, 0.5, 0.75, 0.1, respectively, μ , μ + 3 σ denotes the standard limits, and μ , μ + 4 σ denotes the overall limits.

According to the intervals defined in Table 3 and based on the triangular/semi-trapezoidal affiliation function in Figure 3, the expressions for the affiliation function were constructed as follows: f 1 = 1 g < 0.125 − 4 g + 1.5 0.125 < g < 0.375 0 g > 0.375 (17) f 2 = 0 g < 0.125 4 g − 0.5 0.125 < g < 0.375 − 4 g + 2.5 0.375 < g < 0.625 0 g > 0.625 (18) f 3 = 0 g < 0.375 4 g − 1.5 0.375 < g < 0.625 − 4 g + 3.5 0.625 < g < 0.875 0 g > 0.875 (19) f 4 = 0 g < 0.625 4 g − 2.5 0.625 < g < 0.875 1 g > 0.875 (20)

The following sequence was used to conduct the hydropower unit case study in this example (a flow chart of this process is provided in Figure 4).

1) According to the actual situation of the case study hydropower plant unit structure and the arrangement of measurement points, a hierarchical analysis system was constructed by dividing the unit into a target layer, project layer, and indicator layer. Historical health data describing the oscillation, vibration, pressure, and temperature indicators—included in the project layer—were obtained under the same working conditions used in the example calculation.

The subjective project level weights included vibration, oscillation, pressure pulsation, and temperature indicators. The hierarchical analysis method was used to determine the weight of each indicator type. As the vibration, oscillation, and pressure pulsation indicators changed faster and were more sensitive to the state of the unit, the indictor importance was defined as vibration fault = oscillation fault > pressure pulsation fault > temperature fault. According to the 0.1–0.9 scale, the fuzzy judgment matrix for the project level was determined as follows: C = C i j = 0.5 0.5 0.7 0.9 0.5 0.5 0.7 0.9 0.3 0.3 0.5 0.7 0.1 0.1 0.3 0.5

And the fuzzy consistency matrix was obtained as E i j = 0.5 0.5 0.6 0.7 0.5 0.5 0.6 0.7 0.4 0.4 0.5 0.6 0.3 0.3 0.4 0.5 and subjective weights at the project level were (0.3, 0.3, 0.233, 0.167).

The objective and subjective weights at the indicator level and the combination weights were subsequently obtained as shown in Table 4.

According to the case study unit model and its parameters, including operating head and rated speed, and other parameters, the oscillation, vibration, pressure pulsation, and temperature indicator limits were obtained by consulting international industry regulations, as well as power station regulations and guidelines. Then, the historical health indicator data and Gaussian threshold method has been applied to obtain the final weights of the oscillation, vibration, and pressure pulsation, and temperature indicators as shown in Figure 5.

As shown in Table 5, the axial vibration class indicators exhibited the largest variation. Therefore, the axial vibration class degradation graph was obtained as shown in Figure 6, which identifies five moments representing the degradation trend over time. Note that all monitoring indicators considered in this example were minimum optimal (smaller is better) type indicators.

During the evaluation of hydropower units, the serious deviation of a certain indicator or indicator type from its normal value often means that the feature of this indicator is abnormal and must be paid careful attention, or the system should be shut down for maintenance (Xu et al., 2016; Lin et al., 2020). However, as the weight value of a problematic indicator could be quite small in practice, the state of the response may not be reflected in the results for the entire hydropower unit. Therefore, an adaptive penalty factor algorithm was introduced to adjust the weight of each indicator according to its operating state as follows: w i = ∂ k w i ∑ j = 1 n ∂ k w i (21) where ∂ k = {1,2,3,4} is the penalty, defined according to the principle of maximum affiliation and the dynamics of the indicator state assessment results. In state 1, ∂ k = ∂ 1 = 1, in state 2 , ∂ k = ∂ 2 = 2 , etc. The project layer fuzzy evaluation matrix was constructed according to the variable weight theory and combined with the subjective weights of the project layer to obtain the overall fuzzy evaluation matrix.

The values of the indicators in Table 5 were substituted into Eq. 16 to obtain the deterioration degree for each indicator. These deterioration degrees were substituted into Eqs 17–20 to obtain the fuzzy evaluation matrix for each item level indicator as follows: M B = 0.623 0.377 0 0 0.588 0.412 0 0 0.313 0.687 0 0 0 0 0.548 0.452 0 0 0.652 0.348 0 0 0.631 0.369

As shown above, the weight of each indicator was reallocated using the variable weight theory. According to the principle of maximum subordination, the top vibrations in the X- and Y-directions are in state 1 (corresponding to penalty factor ∂ 1 ), the top vibration in the Z-direction is in state 2 (corresponding to penalty factor ∂ 2 ), and the axial vibrations A, B, and C are in state 3 (corresponding to penalty factor ∂ 3 ). The vibration class indicator weights were combined into w B and substituted into Eq. 4 to obtain the new vibration class weight vector as follows: w ′ = 0.075,0.061,0.124,0.252,0.250,0.238

The vibration class fuzzy evaluation matrix was subsequently obtained based on the new vibration indicator weight vector as follows: W B = w B ’ · M B = 0.121,0.138,0.451,0.289

The fuzzy evaluation matrices for the other indicator types have been obtained similarly, then the project level fuzzy evaluation matrix has been determined as follows: M = 0.081 0.531 0.231 0.243 0.121 0.138 0.451 0.289 0.521 0.260 0.219 0 0.494 0.306 0.2 0

The new project layer weight vector has been obtained after the subjective weights of the project layer were weighted as w ′ =(0.316.0.474,0.123,0.087), and the final fuzzy evaluation matrix was given by w ′ · M =(0.190,0.266,0.331,0.213).

According to the calculated fuzzy evaluation matrix, the weights of each indicator were reallocated using variable weight theory, giving the results shown in Table 6.

Furthermore, the comprehensive deterioration degree of the hydropower unit is plotted in Figure 7, which shows that the deterioration of the hydroelectric unit increased obviously from August 26, with abnormalities at C, D, and E. At the end of August, the vibration of the case study unit was reported to have been quite violent. After an accident probe of the site personnel, it was determined that the runner chamber steel plate fell off during this time period, leading to a hydraulic imbalance that would obviously change the vibration pendulum signal. Thus, the calculated deterioration degree is consistent with the reported reality.

This method is compared with the fuzzy evaluation method for hydropower units, which is based on traditional monitoring quantities and regulations. According to the unit type, working head, rated speed and other parameters, the upper and lower limit values of each measurement point of the hydraulic turbine system under steady state operation are determined by consulting the national standards and regulations and the guidelines of power plant regulations, as shown in Table 7.

It can be seen that all the monitored quantities are of the smaller and better type, therefore the degree of deterioration is calculated according to Eq. 16, as shown in the traditional plot of the membership function in Figure 8:

Where g₁,g₂,g₃,g₄, correspond to (0.2, 0.4, 0.6, 0.8), respectively, as a comparison to evaluate the operating status of the turbine system A-E at each moment, corresponding to the actual monitoring values shown in Table 5, According to Table 5, the final state evaluation table under the traditional fuzzy evaluation method can be obtained as shown in Table 8:

Table 8 shows that the turbine system status and vibration indexes at C, D, and E time are still assessed as qualified or attention, but at this time, through the comprehensive deterioration diagram of the unit, it is obvious to see that there are abnormalities in the unit, and the evaluation results obviously cannot reflect the real state of the unit, in contrast, the evaluation results based on the combination of the assignment and the Gaussian threshold method of the state of the unit are more in line with the actual situation, and can be more real and effective. Reflecting the actual operating status of the unit.

According to the real-time evaluation status of each component in the unit hierarchical analysis system, the penalty factor is introduced to dynamically adjust the original weights of each component, so as to propose an adaptive variable weighting method. Compare it with the evaluation method using fixed constant weights to verify the advantages of introducing the variable weight theory. When using constant weights, the penalty factors in Eq. 21 are all 1, and the results are shown in Table 9.

Under the constant weighting mode, the evaluation status of the turbine system of the unit is qualified at two moments B and C. At this time, as shown in Figure 5, the vibration deterioration increases significantly and the evaluation results of vibration indexes at the two moments in Table 7 are attention and abnormality, respectively. In the case of a single category of indicators abnormal, but still the overall evaluation of the system as a qualified obviously does not match the actual situation. Figure 9 shows the difference between the evaluation results of the two methods more intuitively, and it can be seen that, Compared with the fixed constant weights, and conventional weights can not objectively reflect the serious deviation of indicators from the normal situation, so the introduction of variable weights theory can be based on the actual state of the unit to dynamically change the ratio of each component, highlighting the hidden equipment, variable weights mode of the unit’s condition assessment results and the actual operating state of the unit is more in line with the actual operating state of the unit.