*3.1. Methodologies*

In this section, *Z*-number, rough number, DEMATEL method and VIKOR method are briefly introduced. These methods will be used in the proposed risk assessment model.

(1) *Z*-number

*Z*-number, a 2-tuple fuzzy numbers that includes the restriction of the evaluation and the reliability of the judgment, was first introduced by Zadeh in the year of 2011 for overcoming the limitation that fuzzy numbers does not consider the reliability of the information [61]. The idea of a *Z*-number is providing a mode for calculation with numbers that has partial reliability in the evaluation [62]. A *Z*-number can be utilized to express the information of an uncertain judgement in the form of two fuzzy numbers that the first fuzzy number indicates the fuzzy restriction and the second fuzzy number represents an idea of confidence, reliability, and probability. Thus, *Z*-number is more efficient than fuzzy number in describing the knowledge of human judgment since it describes both the restraint and reliability. Due to the powerful ability in modeling uncertain information in real world, *Z*-number has gained attention by some researchers and efforts have been made to apply *Z*-number to various situations such as in computing with words (CWW) [63] and decision making problems [64].

A *Z*-number can be denoted as Z = (A, R) where the first component is the fuzzy restriction for the evaluation of objects and the second component is the reliability of the first component. In *Z*-number, A and R are described in natural language using linguistic terms and presented in a fuzzy number form such as triangular or trapezoidal fuzzy numbers [61]. For example, in risk analysis, the severity of a failure mode is very high, with a confidence of very sure, then the *Z*-number for evaluating the failure mode can be written as Z = (Very high, Very sure).

(2) Rough number

Rough set theory as a mathematical tool for dealing with the imprecision, uncertainty and vagueness knowledge [65] has been extensively applied in the fields of knowledge discovery, data mining, decision analysis and pattern recognition. By using its lower approximation and upper approximation, rough set theory can fully express and describe the ambiguity and randomness of uncertain information and can lessen the information

loss of aggregation process to a certain extent. Based on rough set theory, rough number is developed by Zhai et al. [66] for managing customers' subjective judgments and determining their boundary intervals. By introducing rough number to FMECA, the evaluations of experts in FMECA can be transformed to rough numbers by calculating their lower approximation and upper approximation on the basis of original data without any requirement of auxiliary information. Since it can effectively extract experts' actual opinion and reduce their subjectively in decision-making [67], in this section, rough number is applied to aggregate the evaluations of experts.

#### (3) DEMATEL method

Decision-making trial and evaluation laboratory (DEMATEL) method was first proposed in 1973 to solve the fragmented and antagonistic issues of world societies [68]. It is a method of system analysis using the structural modeling technique to find the influence relation among complex elements. DEMATEL is a tool of based on the graph theory and matrix, which constructs the direct influence matrix by means of the logical relation among various elements in the system and calculates the effect degree and cause degree of each element to other elements. Because of its ability to pragmatically visualize complicated causal relationships [69], DEMATEL can be used as an effective tool in studying the interdependence among elements in a complex systems and can be well used to identify the dependence among failure modes in FMECA process.

#### (4) VIKOR method

The VIKOR (VIsekriterijumska optimizacija i KOmpromisno Resenje) method was first proposed by Opricovic [70] to rank and select the optimum solution among a set of choices under different units criteria. As one of the MCDM method, VIKOR ranks alternatives based on the multicriteria ranking index by calculating the particular measure of "closeness" to the "ideal" solution [71]. It is an effective method in the field of multicriteria decision making especially in the case where the decision makers may not have enough knowledge to express their preferences at the beginning of system design [72]. Comparing to other MCDM methods, VIKOR helps decision makers reach a feasible decision closest to the ideal by proposing a compromised solution with an advantage rate. Moreover, it is facile to conduct without any parameter settings. Thus, VIKOR has been extensively applied to practical decision making issues.

#### *3.2. The Proposed Risk Assessment Model for FMECA*

In this paper, a new risk assessment model for FMECA by integrating *Z*-number, Rough number, DEMATEL method and VIKOR method is proposed. In the proposed approach, *Z*-number is introduced to express experts' judgements on the evaluation of failure modes, which has a strong ability to describe the knowledge of human beings by using a 2-tuple fuzzy numbers and can be well used in representing vagueness and uncertainty information. Rough number is applied to aggregate different types of evaluations transformed by the given 2-tuple fuzzy numbers of experts and manipulate the subjectivity and vagueness in the evaluation process. Based on its flexible boundary interval, the epistemic uncertainty of evaluations can be generally represented and the different sources of uncertainty can be effectively tackled in aggregation process. DEMATEL method is introduced to calculate the effect degree and cause degree of each failure mode by constructing the direct influence matrix of failure modes, which is a very effective tool to study the relationship among various failure modes in complex systems. Finally, VIKOR method is utilized to determine the risk priorities of failure modes under a compromise way, which can help experts achieving a reasonable ranking results on the basis of maximizing the group utility for the "majority" and minimizing the individual regre<sup>t</sup> for the "opponent".

The framework of the proposed approach is depicted in Figure 1, which comprises four different stages. The first stage is to evaluate the failure modes by using *Z*-number, the second stage is the aggregation of different experts' evaluations based on rough number, the third stage is to determine the dependency among failure modes on the basis of historical

failure data, and the fourth stage is to rank the failure modes using VIKOR method. The four stages are explained in detail as follows.

**Figure 1.** The framework of the proposed FMECA approach.

Step 1: Identify the objectives of the risk assessment process and determine the analysis level.

Step 2: Establish the FMECA team, list the potential failure modes and describe a finite set of relevant risk factors.

Suppose there are m failure modes in FMECA needed to be ranked according to the evaluations of failure modes and *K* experts are responsible for the evaluation with respect to the risk factors of severity, occurrence, detectability and failure propagation of failure modes.

Step 3: Evaluate the identified failure modes based on *Z*-number

In FMECA, failure modes are usually evaluated using linguistic variables such as very high, high, moderate, low, and very low, these evaluations are usually expressed in a fuzzy and imprecise way. In this section, failure modes are evaluated by using *Z*-number, which can not only express the evaluation of failure modes in a fuzzy and imprecise way, but also consider the confidence and reliability of the evaluations. In our work, failure modes are first evaluated according to Table 2, then the given linguistic terms are converted to fuzzy number according to Table 3. The transferred evaluations for failure mode in the form of 2-tuple fuzzy numbers are expressed as

$$Z = (A, B) = \{ (a\_1, a\_2, a\_3), (\beta\_1, \beta\_2, \beta\_3) \} \tag{1}$$

**Table 2.** Evaluation criterion for *S*, *O*, and *D* and the corresponding linguistic terms.


**Table 3.** The relationship between linguistic terms and fuzzy numbers.


Step 4: Convert *Z*-numbers into crisp number.

For effectively aggregating the evaluations of experts, the *Z*-number form evaluations should be defuzzified to obtain a crisp value. The crisp value of evaluations can be obtained by

$$w = \frac{\int x \mu\_B(x) dx}{\int 10 \mu\_B(x) dx} \cdot \frac{(\alpha\_1 + 4 \times \alpha\_2 + \alpha\_3)}{6},\tag{2}$$

$$\mu\_B(\mathbf{x}) = \begin{cases} 0, \mathbf{x} \in (-\infty, \beta\_1) \\ \frac{\mathbf{x} - \beta\_1}{\beta\_2 - \beta\_1}, \mathbf{x} \in [\beta\_1, \beta\_2] \\ \frac{\beta\_3 - \mathbf{x}}{\beta\_3 - \beta\_2}, \mathbf{x} \in [\beta\_2, \beta\_3] \\ 0, \mathbf{x} \in (\beta\_3, +\infty) \end{cases} \tag{3}$$

where is an algebraic integration, *μB*(*x*) is the membership function of triangular fuzzy number (*β*1, *β*2, *β*3).

Step 5: Aggregate the evaluations of *K* experts for each failure mode by using rough number.

For failure mode *i* (*i* = 1, 2, ··· , *m*) with respect to risk factor *j* (*j* = *S*,*O*, *<sup>D</sup>*), the evaluations is denoted as *Vij* = )*v*1*ij*, *v*2*ij*, ··· , *<sup>v</sup>Kij*. The first step in the aggregation process is to obtain the lower approximation and upper approximation of *vkij*(*<sup>k</sup>* = 1, 2, ··· , *K*) by the following equations:

$$\underline{Apr}(\underline{v\_{ij}^k}) = \cup \left\{ \underline{v\_{ij}^t} \in V\_{ij}/\underline{v\_{ij}^t} \le \underline{v\_{ij}^k} \right\},\tag{4}$$

$$\overline{A\ pr}(v\_{ij}^k) = \cup \left\{ v\_{ij}^t \in V\_{ij} / v\_{ij}^t \ge v\_{ij}^k \right\}. \tag{5}$$

Based on the lower approximation and upper approximation of *vkij*, the lower limit and upper limit of *vkij*are obtained by

$$\underline{\dim}(v\_{ij}^k) = \frac{1}{M\_L} \sum v\_{ij}^t \Big| v\_{ij}^t \in \underline{\underline{\underline{\underline{\underline{\underline{\underline{\underline{\underline{\alpha}}}}}}}}(v\_{ij}^k) \ . \tag{6}$$

$$\overline{Lin}(v\_{ij}^k) = \frac{1}{M\_{lI}} \sum v\_{ij}^t \Big| v\_{ij}^t \in \overline{Apr}(v\_{ij}^k) \tag{7}$$

where *ML* is the number of elements contained in *Apr*(*vkij*), and *MU* is the number of elements contained in *Apr*(*vkij*).

Then the rough number of *vkij* is obtained by its corresponding lower limit and upper limit, namely

$$RN(v\_{ij}^k) = [\underline{\underline{Lin}}(v\_{ij}^k), \overline{\underline{Lin}}(v\_{ij}^k)].\tag{8}$$

The interval between *Lim*(*vkij*) and *Lim*(*vkij*) is the rough boundary interval denoted as

$$RBnd(\upsilon\_{ij}^k) = \overline{Lin}(\upsilon\_{ij}^k) - \underline{Lin}(\upsilon\_{ij}^k). \tag{9}$$

With the obtained rough numbers of *vkij*(*<sup>k</sup>* = 1, 2, ··· , *<sup>K</sup>*), the rough sequence *RS*(*Vij*) of *Vij* can be obtained by

$$\text{RS}(V\_{i\bar{j}}) = \left\{ \left[ v\_{i\bar{j}\prime}^{L} v\_{i\bar{j}}^{lI} \right]\_{1'} \left[ v\_{i\bar{j}\prime}^{L} v\_{i\bar{j}}^{lI} \right]\_{2'} \cdot \cdots \cdot \left[ v\_{i\bar{j}\prime}^{L} v\_{i\bar{j}}^{lI} \right]\_{K} \right\}. \tag{10}$$

Thus the rough number of the evaluation for failure mode i with respect to risk factor *j* (*Vij*) is obtained by averaging the rough sequence, that is

$$\text{RN}(V\_{\vec{i}\vec{j}}) = \left[v\_{\vec{i}\vec{j}\prime}^{L}v\_{\vec{i}\vec{j}}^{\text{II}}\right] = \frac{1}{\text{K}}\left(\left[v\_{\vec{i}\vec{j}\prime}^{L}v\_{\vec{i}\vec{j}}^{\text{II}}\right]\_1 + \left[v\_{\vec{i}\vec{j}\prime}^{L}v\_{\vec{i}\vec{j}}^{\text{II}}\right]\_2 + \dots + \left[v\_{\vec{i}\vec{j}\prime}^{L}v\_{\vec{i}\vec{j}}^{\text{II}}\right]\_K\right). \tag{11}$$

Then the aggregated evaluation matrix *EM* for failure modes with respect to *S*, *O* and *D* is given as:

$$EM = \begin{vmatrix} [\boldsymbol{v}\_{1\boldsymbol{\delta}}^{L}, \boldsymbol{v}\_{1\boldsymbol{\delta}}^{\mathrm{II}}] & [\boldsymbol{v}\_{1\boldsymbol{\delta}}^{L}, \boldsymbol{v}\_{1\boldsymbol{\delta}}^{\mathrm{II}}] & [\boldsymbol{v}\_{1\boldsymbol{\delta}}^{L}, \boldsymbol{v}\_{1\boldsymbol{\delta}}^{\mathrm{II}}] \\ \vdots & \vdots & \vdots \\ [\boldsymbol{v}\_{n\boldsymbol{\delta}}^{L}, \boldsymbol{v}\_{n\boldsymbol{\delta}}^{\mathrm{II}}] & [\boldsymbol{v}\_{n\boldsymbol{\delta}}^{L}, \boldsymbol{v}\_{n\boldsymbol{\delta}}^{\mathrm{II}}] & [\boldsymbol{v}\_{n\boldsymbol{\delta}}^{L}, \boldsymbol{v}\_{n\boldsymbol{\delta}}^{\mathrm{II}}] \end{vmatrix} \tag{12}$$

In this step, DEMATEL method is applied to obtain the effect degree (*R*) and the cause degree (*C*) of each failure mode. The first procedure in DEMATEL method is to obtain the direct effect degree between any two failure modes, referred to as *aij* (*i* = 1, 2, ··· , *m*; *j* = 1, 2, ··· , *<sup>m</sup>*), which can be obtained by making statistical analysis for the historical failure data or the expertise and experience of experts. The value of *aij* represents the degree that failure mode *j* influenced by failure mode *i*, and the value of *aji* represents the degree that failure mode *i* influenced by failure mode *j.* In general, *aij* is not equal to *aji*. Specifically, *aij* = 0 if *i* = *j*. For *m* failure modes in FMECA, the direct relation matrix among these failure modes can be expressed as

$$A = \begin{vmatrix} 0 & a\_{1m} & \cdots & a\_{1m} \\ a\_{21} & 0 & \cdots & a\_{2m} \\ \vdots & \vdots & \ddots & \vdots \\ a\_{m1} & a\_{m2} & \cdots & 0 \end{vmatrix}.\tag{13}$$

The initial direct relation matrix *A* can be normalized by using the following equations [73]

$$D = A \times \mathcal{S}\_{\prime} \tag{14}$$

$$S = \operatorname{Min} \left[ \frac{1}{\max\_{1 \le i \le m} \sum\_{j=1}^{m} a\_{ij}}, \frac{1}{\max\_{1 \le j \le m} \sum\_{i=1}^{m} a\_{ij}} \right] \tag{15}$$

where the value of each element in matrix *D* ranges from 0 to 1.

Then the total relation matrix is obtained by the following equation

$$T = D(I - D)^{-1} = \left[t\_{ij}\right]\_{m \times m} \tag{16}$$

where *I* is the identity matrix.

The sums of rows and of columns in the total relation matrix *T* are the effect degree (*R*) and the cause degree (*C*) of failure modes, respectively, which are obtained by using the following equations

$$R = (r\_1, r\_2, \dots, r\_m) = \left[\sum\_{j=1}^m t\_{ij}\right]\_{m \times 1} \tag{17}$$

$$\mathbb{C} = (\mathbf{c}\_1, \mathbf{c}\_2, \dots, \mathbf{c}\_m) = \left[ \sum\_{i=1}^m t\_{ij} \right]\_{1 \times m} \tag{18}$$

where *ri* in vector *R* is the sum of *i*th row of matrix *T*, which represents both the direct and indirect effects of failure mode *i* acting on the other failure modes, and *cj* in vector *C* is the sum of *j*th column of matrix *T*, which represents both the direct and indirect effects of failure mode *j* caused by other failure modes.

Step 7: Obtain the ultimate decision making matrix for failure modes.

The effect degree (*R*) and the cause degree (*C*) of each failure mode are regarded as the risk factor for assessing the risk priority of failure mode, namely five risk factors as severity, occurrence, detectability, effect degree and cause degree are chosen in the proposed FMECA approach for the prioritization of failure modes. Thus the ultimate decision making matrix for failure modes is given as:

$$DM = \begin{vmatrix} [\boldsymbol{v}\_{1\boldsymbol{s}}^{\mathrm{L}}, \boldsymbol{v}\_{1\boldsymbol{s}}^{\mathrm{II}}] & [\boldsymbol{v}\_{1\boldsymbol{0}}^{\mathrm{L}}, \boldsymbol{v}\_{1\boldsymbol{0}}^{\mathrm{II}}] & [\boldsymbol{v}\_{1\boldsymbol{D}}^{\mathrm{L}}, \boldsymbol{v}\_{1\boldsymbol{D}}^{\mathrm{II}}] & \boldsymbol{v}\_{1\boldsymbol{R}} & \boldsymbol{v}\_{1\boldsymbol{C}} \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ [\boldsymbol{v}\_{m\boldsymbol{s}}^{\mathrm{L}}, \boldsymbol{v}\_{m\boldsymbol{s}}^{\mathrm{II}}] & [\boldsymbol{v}\_{m\boldsymbol{D}}^{\mathrm{L}}, \boldsymbol{v}\_{m\boldsymbol{D}}^{\mathrm{II}}] & [\boldsymbol{v}\_{m\boldsymbol{D}}^{\mathrm{L}}, \boldsymbol{v}\_{m\boldsymbol{D}}^{\mathrm{II}}] & \boldsymbol{v}\_{m\boldsymbol{R}} & \boldsymbol{v}\_{m\boldsymbol{C}} \end{vmatrix} \tag{19}$$

where *viR* = *ri* and *viC* = *ci* are the effect degree and the cause degree of failure mode *i* respectively.

Step 8: Determine the weight of each risk factor.

As similar as the evaluations of failure modes, the relative weights among risk factors need to be assessed by experts and aggregated using rough number, the rough numbers for the weights of risk factors are expressed as

$$\mathcal{W} = \begin{bmatrix} \ \left[ w\_{\mathcal{S}}^{L}, w\_{\mathcal{S}}^{\mathrm{II}} \right] & \left[ w\_{\mathcal{O}\prime}^{L}, w\_{\mathcal{O}}^{\mathrm{II}} \right] & \left[ w\_{\mathcal{D}\prime}^{L}, w\_{\mathcal{D}}^{\mathrm{II}} \right] & \left[ w\_{\mathcal{R}\prime}^{L}, y\_{\mathcal{R}}^{\mathrm{II}} \right] & \left[ w\_{\mathcal{C}\prime}^{L}, w\_{\mathcal{C}}^{\mathrm{II}} \right] \end{bmatrix}. \tag{20}$$

Step 9: Determine the risk rankings of failure modes using VIKOR method.

In this step, VIKOR method is applied to determine the risk rankings of failure modes. Firstly the weights of risk factors in rough number form need to be converted to crisp value by the following equation:

$$w\_{\dot{j}} = \lambda (1 - \frac{w\_{\dot{j}}^{\rm II} - w\_{\dot{j}}^{\rm L}}{2(\beta - a)}) + (1 - \lambda) \frac{w\_{\dot{j}}^{\rm II} + w\_{\dot{j}}^{\rm L}}{2(\beta - a)}, \dot{j} = S, O, D, R, C \tag{21}$$

where *wj* is weight of risk factor *j*, *wUj* and *wLj* are the lower limit and upper limit of the rough number of risk factor *j*, *β* = max*j wUj* ,*α* = min*j wLj λ*, is a discount factor, expressing the effect degree of rough boundary interval imposing on the weight of risk factor. 0 ≤ *λ* ≤ 1, and the greater the value of *λ*, the more effect is imposing on the weight of risk factor, here suppose *λ* = 0.5.

The normalized weight of each risk factor is obtained by using the following equation:

$$w\_j' = \frac{w\_j}{\frac{5}{1}}, j = S\_\prime O\_\prime D\_\prime R\_\prime \mathbb{C}.\tag{22}$$

In VIKOR method, the first step is to determine the optimal and the worst value of each risk factor in *DM*, which is determined by

$$
v\_{j}^{\*} = \left\{ \begin{array}{c} \max\limits\_{i} v\_{ij}^{II} \; , j = \mathcal{S}, O, D \\\max\limits\_{i} v\_{ij} \; , j = R, \mathcal{C} \end{array} \right. \tag{23}$$

$$\boldsymbol{v}\_{j}^{-} = \begin{cases} \min\_{\boldsymbol{i}} \boldsymbol{v}\_{ij}^{L} \; \boldsymbol{j} = \boldsymbol{S} \text{,} \boldsymbol{O} \; \boldsymbol{D} \\\min\_{\boldsymbol{i}} \boldsymbol{v}\_{ij} \; \boldsymbol{j} = \boldsymbol{R} \text{,} \boldsymbol{\mathcal{C}} \end{cases} \tag{24}$$

Based on *v*∗*j* and *v*−*j* , the values of *Si* and *Ri* can be calculated by the following relations

$$S\_i = \sum\_{j=1}^3 w\_j' \frac{\left\{ \left( v\_{ij}^L - v\_j^\* \right)^2 + \left( v\_{ij}^{II} - v\_j^\* \right)^2 \right\}^{1/2}}{\left| v\_j^\* - v\_j^- \right|} + \sum\_{j=4}^5 w\_j' \frac{\left| v\_{ij} - v\_j^\* \right|}{\left| v\_j^\* - v\_j^- \right|}, j = S, O, D, R, C, \tag{25}$$

$$R\_i = \max\_j \left( w\_j' \frac{\left\{ \left( v\_{ij}^L - v\_j^\* \right)^2 + \left( v\_{ij}^{II} - v\_j^\* \right)^2 \right\}^{1/2}}{\left| v\_j^\* - v\_j^- \right|}, w\_j' \frac{\left| v\_{ij} - v\_j^\* \right|}{\left| v\_j^\* - v\_j^- \right|} \right), j = S, O, D, R, C. \tag{26}$$

Then the values of *Qi* (*i* = 1, 2, ··· , *m*) are determined by

$$Q\_i = v \frac{S\_i - S^\*}{S^- - S^\*} + (1 - v) \frac{R\_i - R^\*}{R^- - R^\*} \tag{27}$$

where *S*∗ is the minimum value of *Si* and *S*− is the maximum value of *Si*, *R*∗ is the minimum value of *Ri* and *R*− is the maximum value of *Ri*, *v* is the weight of the strategy of "the majority of criteria" (or "the maximum group utility"), whereas 1 − *v* is the weight of the individual regret. Here suppose *v* = 0.5.

Based on the values of *S*, *R* and *Q*, the failure modes can be prioritized with three ranking lists. Moreover, VIKOR method proposes a compromise solution, the failure mode (*FM*(1)), which is the best ranked by the measure *Q* (Minimum) if the following two conditions are satisfied:

C1. Acceptable advantage: *Q*(*FM*(2)) − *Q*(*FM*(1)) ≥ 1/(*m* − <sup>1</sup>), where *FM*(2) is the failure mode with second position in the ranking list by *Q*.

C2. Acceptable stability in decision making: The failure mode *FM*(1) must also be the best ranked by *S* or/and *R*. This compromise solution is stable within a decision making process, which could be: "voting by majority rule" (when *v* > 0.5 is needed), or "by consensus" *v* ≈ 0.5, or "with veto" (*v* < 0.5).

If one of the conditions is not satisfied, then a set of compromise solutions is proposed, which consists of:


*FM*(*M*) is determined by the relation: *Q*(*FM*(*M*)) − *Q*(*FM*(1)) < 1/(*m* − 1) for maximum *M.*

#### **4. Case Study: Application to the Risk Analysis of the Failure Modes in Offshore Wind Turbine Pitch System**

In this section, the proposed approach is used to the risk prioritization of the failure modes in offshore wind turbine pitch system. There are seven main malfunctions of the pitch system, namely, pitch bearing failure, pitch gearbox failure, pitch motor failure, pitch actuator failure, backup power and charger failure, encoder and limit switch failure, and control module failure. Each of the malfunctions could cause the pitch system failure and eventually result in the turbine shutdown. In order to ensure the operation quality and safety of the pitch system, it is necessary to analyze the malfunctions, excavate the potential failure reasons, and identify the weak links and dangerous source of the system.

In this case, four experts with different backgrounds and professional knowledge were invited to identify and evaluate the potential failure modes of pitch system. They are from wind turbine manufacturer, pitch system manufacturer, wind farm and the operation and maintenance enterprise for wind turbine respectively, and all of them have rich experience and knowledge about the fault analysis and diagnosis of pitch system. Based on the analysis of the historical data of pitch system in a wind farm subordinate to Huaneng Group and the experts 'experience knowledge, twenty-four failure modes which are able to cause the seven kinds of malfunctions were identified, and the four experts are responsible for evaluating the severity, occurrence, detectability and failure propagation of these failure modes.

For identifying the weak links and dangerous source of the system, the identified failure modes should to be prioritized based on the values of their severity, occurrence, detectability, effect degree, and cause degree. The twenty-four failure modes and the corresponding code are given in Table 4, and the propagation relationship among different failure modes are provided in Figure 2, which reveals the dependency of the failure modes participated in the failure propagation.


**Figure 2.** The propagation relationship among different failure modes.

The direct relation matrix among the failure modes obtained based on the historical failure data is given as follows:

 (MH, RS)

 (RL, NS)

 (RL, RS)

> (L, U)

 (L, RS)

 (M, NS)

FM6

FM7

FM8

FM9

FM10

FM11

 (RH, U)

 (M, NS)

 (MH, NS)

 (M, NS)

 (RL, NS)

 (M, U)  (L, RS)

 (RL, S)

 (RL, S)

 (L, RS)

 (RL, NS)

 (M, NS)


According to Table 2, the twenty-four potential failure modes were evaluated with respect to severity, occurrence and detectability, and the evaluations for these failure modes were transformed to *Z*-numbers according to Table 3. The evaluations given by expert 1 and the corresponding *Z*-numbers for these evaluations are presented in Tables 5 and 6, respectively. For the sake of space, the other three experts' evaluation information are provided in Appendix A. It is necessary to mention that in our work, the weights of importance of experts are considered as equal. Since each of them has his/her good points, it is difficult to assign a subjective weight to each expert. After converting the *Z*-numbers into the crisp values, the evaluations (in the form of crisp value) given by the four experts were aggregated by using rough number. The aggregation results are presented in Table 7.


 FM18

 FM19

 FM20

 FM21

 FM22

 FM23

**Table 5.** The assessment on *S*, *O* and *D* of the twenty-four potential failure modes given by expert 1.

 (VL, U)

 (RL, RS)

 (RL, NS)

 (L, NS)

 (L, RS)

 (VL, RS)  (L, NS)

 (L, RS)

 (M, U)

 (L, NS)

 (RL, RS)

 (VL, RS)  (RL, U)

 (L, RS)

 (L, RS)

 (L, NS)

 (L, RS)

 (L, U)

According to the direct relation matrix among the failure modes, the effect degree and the cause degree of each failure mode were obtained by using DEMATEL method. In this paper, the effect degree and the cause degree are considered as two risk factors, which reveal the correlation strength between each failure mode and the other failure modes. The greater the effect degree of a failure mode, the more likely the failure mode will lead to other failures/faults to happen, meaning it has a higher severity. The greater the cause degree of a failure mode, the more likely the failure mode can be caused by other failure modes, meaning it has a higher probability of occurrence. The effect degrees and the cause degrees of failure modes are presented in Table 7, thus the ultimate decision matrix for the twenty-four potential failure modes with respect to five risk factors is formed. Similarly, the evaluations of the weights of risk factors were aggregated and presented in Table 8.


**Table 6.** The transformed *Z*-numbers of the twenty-four potential failure modes given by expert 1.

**Table 7.** Decision matrix for the twenty-four failure modes.



**Table 8.** The evaluations and weights for risk factors.

After the aggregation process, VIKOR method was applied to sort the risks of the failure modes based on the decision matrix. The risk priorities of the twenty-four failure modes were determined by calculating the measure of closeness to the weighted vectors of positive ideal point. In the stage of VIKOR method, the optimal and the worst value of each risk factor were determined by Equations (22) and (23), and the values of *S*, *R* and *Q* for all failure modes were computed by using Equations (24)–(26) and presented in Table 9. A failure mode would be closer to the optimal values as the corresponding measure values approaches to zero. Thus, the failure modes can be prioritized or ranked according as the values of *S*, *R*, and *Q* in descending order. In order to make the ranking results better accepted by decision-makers, VIKOR method provides a compromise solution as illustrated in Section 3.2.


**Table 9.** The values and rankings of *S*, *R* and *Q* for all failure modes.

By comparing the risk rankings of the twenty-four failure modes, we see that in the pitch system the weakest link from a reliability standpoint is the pitch bearing, whose failure modes are ranked first and second in all failure modes identified in pitch system, and followed by the pitch gearbox and pitch motor. The failure of pitch bearing may lead to blade pitch to be out of sync or cannot pitch, causing impeller aerodynamic imbalance and fan speeding, which can result in the failure of safely starting and stopping the turbine, and bring about the blade rupture and other accidents. The pitch gearbox is also the

key component that affects the reliability of the pitch system, whose failures of gear and bearing in the gearbox are ranked third and fifth in all failure modes. Through statistical analysis of historical fault data of pitch system, we found that the failure of pitch bearing and pitch gear accounts for 71% of the failure of the whole pitch system, which reveals that attention should be paid to these failure modes, and necessary measures and controls should be taken to lessen the possibility of their occurrence. There are many types of failure of pitch motor, among which the most serious failure modes are short circuit and open circuit of motor winding and motor brake failure, ranked fourth and sixth in all failure modes, respectively. It can also be seen that the failures of bearings, gear and other mechanical components have higher rankings, while the failures of switch, line and other electrical components have relatively lower rankings. This is because mechanical failures are difficult to be detected in time, and electrical failures are easy to be detected according to the abnormal current and voltage signals. Thus, in the reliability design of pitch system, higher reliability should be allocated to mechanical components, and in order to identify the failures of the mechanical components such as bearings and gears early before accidents to ensure the reliable operation of wind turbine, it is necessary to study the on-line monitoring technology for the mechanical failures of pitch system.

From the above analysis, we see that the ranking results of the twenty-four potential failure modes are in accordance with the practical engineering background, which proves the effectiveness of the proposed approach in practical application.

#### **5. Comparison and Discussion**

To further demonstrate the validity and availability of the proposed approach, three comparable method of traditional FMECA, fuzzy TOPSIS and combination weightingbased fuzzy VIKOR were also applied in the case study. The ranking results of the three methods are given in Table 10 and compared with that of the proposed FMECA approach. Based on the rankings in Table 10, it can be seen that the four approaches have a certain degree of similarity on the overall ranking trends of the twenty-four failure modes. For example, FM1 is recognized as the most critical failure mode in the four approaches since it has the highest or second-highest risk ranking. In each of approach, the top four ranked failure modes all contain FM1, FM2, and FM3, and the lowest ranked failure mode is all FM22. Moreover, failure mode FM15, FM17 and FM21 have very similar rankings in the four approaches. However, there are also some failure modes whose rankings of are very different in the four approaches, such as FM4, FM5, FM9, FM11, FM13, FM14, FM16, FM18, FM19, FM20 and FM24. The reasons contributing to the different rankings are analyzed as follows.

First, the weights of risk factors are different in the four approaches. The traditional FMECA reckons the weights of risk factors as equal, which is not reasonable in actual case. Since under the hypothesis of equal weights, some risk factors may be overestimated and others may be underestimated. In the fuzzy TOPSIS, fuzzy VIKOR and the proposed approach, such equal weight assumption is abandoned by determining the real weights of risk factors based on evaluations of experts. In the three kinds of approaches, the weights of risk factors are evaluated by experts using linguistic items. Meanwhile, the fuzzy TOPSIS determines the weight of risk factors by fuzzy AHP method, in which the weight of risk factors is (*wS* = 0.41, *wO* = 0.31, *wD* = 0.28). The fuzzy VIKOR determines the weight of risk factor based on a combined weighting method integrated by fuzzy AHP and entropy method, in which the weight of risk factors is (*wS* = 0.4, *wO* = 0.38, *wD* = 0.22). The proposed approach determines the weight of risk factors based on rough number and Equations (20) and (21), in which the weight of risk factors is (*wS* = 0.27, *wO* = 0.21, *wD* = 0.19, *wR* = 0.18, *wC* = 0.15). Take the FM14 as an example, although experts evaluate FM14 with respect to occurrence with high value, they put relatively low importance on occurrence. Thus, FM14 gets relatively high ranking in the traditional FMECA compared to the rankings in the other three approaches, since occurrence is

overestimated when regarding the weights of severity, occurrence and detectability as equal in traditional FMECA.


**Table 10.** The values and rankings of *S*, *R* and *Q* for all failure modes.

The second reason is the different representation and aggregation method for experts' evaluation information in the four approaches. As we know, FMECA is a team collaboration behavior which cannot be implemented alone on an individual basis [25]. On one hand, traditional FMECA, fuzzy TOPSIS and fuzzy VIKOR aggregate different experts' evaluations by average method. The aggregation results by this method are largely influenced by expert's opinion with subjectively and uncertainly. In fact, because of the different experience and backgrounds of experts, the evaluations of experts may be different and diverse, and some of which may be vague, imprecise and uncertain. In the proposed approach, the evaluations of different experts were aggregated by rough number, which could effectively aggregate the diversity evaluations and reduce the subjectivity and uncertainly in aggregation process. On the other hand, traditional FMECA, fuzzy TOPSIS and fuzzy VIKOR evaluate failure modes in the form of crisp number or triangular fuzzy number. Although fuzzy numbers are able to deal with the human vagueness evaluation to some extent, it does not consider the reliability of the restricted evaluation. In the proposed approach, the limitations of fuzzy number are overcome by *Z*-number, which describe the evaluations of failure modes by using 2-tuple fuzzy numbers. Compared to fuzzy number, *Z*-number has a stronger ability to express vague and uncertain information.

The third reason is the different ranking mechanism for failure modes in the four approaches. The traditional FMECA ranks the failure modes by multiplying the values of *S*, *O*, and *D*, which is questionable as mentioned in Introduction section. While the fuzzy TOPSIS, fuzzy VIKOR and the proposed approach take the ranking problem of failure modes as a multiple criteria decision-making (MCDM) issue and rank the failure modes by TOPSIS and VIKOR method. One difference between TOPSIS and VIKOR method is the different mechanism of aggregation function for ranking in the two methods. The aggregation function of VIKOR method represents the distance from the optimal values [72]

with the ranking index of aggregating all risk factors, the weights of risk factors, and the balance between group and individual satisfaction. While the aggregation function of TOPSIS method represents the distances from the optimal value and from the worst value, which introduces the ranking index by summing these distances without considering their relative importance. The other difference between these two methods is the different means of normalization. The VIKOR method utilizes linear normalization method for normalizing, while the TOPSIS method uses vector normalization method. Moreover, the VIKOR method proposes a compromise solution with an advantage rate.

The last and most critical reason is that the traditional FMECA, fuzzy TOPSIS and fuzzy TOPSIS do not consider the propagation effect among failure modes, while the proposed approach considers. The failure propagation takes into account how a failure of a component could spread within a system, leading other components to failure. In fact, the practical impact to system reliability of the failure propagation is to increase the severity and occurrence of failure modes which can cause the occurrence of other failure modes or be affected by other failure modes. It can be seen that a very different ranking of failure mode is found in FM19 between the proposed approach and the other three approaches, which is ranked as twentieth in the traditional FMECA and fuzzy TOPSIS, eighteenth in the fuzzy VIKOR and ninth in the proposed approach. The remarkably different ranking for FM19 result from its high effect degree and cause degree. The high effect degree indicates that FM19 has a large possibility of causing other failures, which means its severity can be increased through the failure propagation. The high cause degree indicates that FM19 is more likely to be caused by other failures, which means its occurrence can be increased through the failure propagation. Thus, due to consideration of the failure propagation, the ranking of FM19 has greatly increased in the proposed approach compared to the ranking in the other three approaches, and so does the other failure modes such as FM3, FM4, FM20, etc.
