© X. Yin et al., Published by EDP Sciences 2025

1 Introduction

To achieve the “Carbon Neutrality by 2060” objective, the national investment in renewable energy sources has been substantially increased. Specifically, by 2019, the installed capacity of onshore wind power in China reached 37% of the global total [1]. During the operational phase of wind turbines, numerous instances of gearbox failures have resulted in both human casualties and economic losses [2]. The challenging operational environments of wind turbines contribute to a relatively high failure rate in their gear transmission systems [3]. In reference [4], Ribrant and colleagues analyzed failure data from wind turbine units in Finland, Germany, and Sweden, focusing on the period from 2000 to 2004. Their analysis of the failure rates of various components within these wind turbine units revealed that the gearbox failure rates were significantly higher than those of other components, greatly impacting the stable operation of the wind turbines. Studies show that failures in the gear transmission system cause the longest downtime during wind turbine operations [5–8]. As the scale of wind power installation has expanded, the individual power output of wind turbines has increased, leading to higher gearbox failure rates and increased maintenance costs, thereby progressively raising the failure rates of the turbines and affecting the economic benefits of wind farms [9–14].

Quantitative analyses by Fazlollahtabar et al. calculated the system's failure rates and assessed the severity of failures within the system fault tree [15,16]. Bhardwaj et al. analyzed the failure modes of wind turbine gearboxes, determining the probability and causes of failures among different gearbox components [17]. Cui et al. utilized a nonlinear vibration model to analyze the vibrational responses of gearbox bearings under different operational conditions, thus determining the bearings' failure rates [18]. Yang et al. studied the planetary gear systems within gearboxes, assessing gear engagement states to ascertain the extent of gearbox failures [19]. Ma et al. proposed a new method combining statistical analysis with practical engineering issues, using time-domain signals of turbine gearbox vibrations to diagnose gearbox failures [20]. Cao et al. introduced a frequency analysis method by extracting vibration signals from gearbox fault points and converting them into corresponding frequency signals to analyze the state of gearbox failures [21]. However, this method struggles with detecting low-noise fault states, limiting its effectiveness in characterizing gearbox failures. Azim et al., through multiple experiments using a GMDH predictive model, determined the probability of bearing failures in gearboxes [22]. Sun et al. employed a fuzzy assessment method to develop an artificial intelligence model aimed at improving fault prediction accuracy in wind turbines [23]. Sonawane et al. demonstrated the vibrational characteristics of single-stage spur gearboxes under pitting conditions through empirical methods [24]. Initially introduced by W.H. Watson and D.F. Hansl in 1961, fault tree analysis has been widely applied in various sectors following numerous revisions and enhancements. Traditional fault tree methods, however, struggle with accurately known failure rates, prompting Tanaka Hideo to incorporate fuzzy mathematical theory into fault tree analysis to address this issue. On this basis, Lin CT and others have established a new type of fuzzy gate using logical expressions 0 and 1 to represent inter-event logical relationships. As engineering problems have become increasingly complex and uncertainties have grown, these methods have been unable to meet analytical needs. Therefore, Song Hua and others proposed the T-S fuzzy fault tree analysis method, which has been effectively expanded through integration with Bayesian networks, covering sectors such as construction, aviation, and industrial manufacturing [25–30].

These studies indicate that currently, system reliability is primarily assessed by analyzing event failure data. However, given the complexity of wind turbine gear transmission systems, using the aforementioned methods would inevitably increase research costs and consume extensive resources. Therefore, the introduction of fuzzy mathematical theory into fault tree analysis simplifies the modeling process, offers greater adaptability [31], and provides a more accurate depiction of fault propagation within systems.

2 T-S fuzzy fault tree theory and methods

2.1 Fundamentals of T-S fuzzy modeling

2.1.1 T-S fuzzy algorithm

The Takagi-Sugeno (T-S) fuzzy model is an advanced fuzzy modeling framework specifically developed for representing uncertain systems [32], capable of approximating nonlinear systems with any desired level of precision. This model integrates a fuzzy inference system with a linear time-invariant system, effectively addressing uncertainty within systems and significantly contributing to advancements in the field of reliability [33]. To enhance the resolution of polymorphic issues in complex systems, the T-S gate rule l (l = 1, 2,…, m) is specifically introduced. For instance, assuming the failure severity of a base event x = (x₁, x₂,…, x_n) is quantified as $\left(x_1^1,x_1^2,\dots ,x_1^{k1}\right),\dots ,\left(x_n^1,x_n^2,\dots ,x_n^{kn}\right)$, and the failure severity of a higher-level event y is quantified as (y¹,y²,…,y^kn), the relationship can be expressed as follows:

$$\{\begin{array}{l}0\le x_1^1<x_1^2<\cdots <x_1^{k_1}\le 1,\hfill \\ 0\le x_2^1<x_2^2<\cdots <x_2^{k_2}\le 1,\hfill \\ \cdots \hfill \\ 0\le x_n^1<x_n^2<\cdots <x_n^{k_n}\le 1,\hfill \\ 0\le y^1<y^2<\cdots <y^{k_y}\le 1.\hfill \end{array}$$(1)

Utilizing the concept of normalization, the degree of execution for rule l is calculated according to the following equation:

$${\beta }_l^{*}\left(x^{\prime }\right)={\displaystyle \prod _{j=1}^n{\mu }_{x_j^{ij}}}\left(x_j^{\prime }\right)/{\displaystyle \sum _{l=1}^m{\displaystyle \prod _{j=1}^n{\mu }_{x_j^{ij}}}}(x_j^{\prime }.$$(2)

In the above equation, the ${\mu }_{x_j^{ij}}\left(x_j^{\prime }\right)$ denotes the rule l in x'_(j) the affiliation degree of the membership fuzzy set; β*_(l) then denotes the affiliation degree of the membership degree of the T-S fuzzy fault level to the basic event in rule l.

In this fuzzy system, assuming that the degree of failure of the basic event x' =(x'₁, x'₂,…,x'_n), then the fuzzy likelihood of calculating the degree of failure of the superior event according to the T-S model is:

$$\{\begin{array}{l}P\left(y^1\right)={\displaystyle {\sum }_{l=1}^m{\beta }_l^{*}}\left(x^{\prime }\right)P^l\left(y^1\right),\hfill \\ P\left(y^2\right)={\displaystyle {\sum }_{l=1}^m{\beta }_l^{*}}\left(x^{\prime }\right)P^l\left(y^2\right),\hfill \\ \hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\cdots \hfill \\ P\left(y^n\right)={\displaystyle {\sum }_{l=1}^m{\beta }_l^{*}}\left(x^{\prime }\right)P^l\left(y^n\right).\hfill \end{array}$$(3)

Rule l [34] possibility of implementation $P_0^l$ for:

$$P_0^l=P\left(x_1^{i_1}\right)P\left(x_2^{i_2}\right)\cdots P\left(x_n^{i_n}\right).$$(4)

In the above equation, the $P\left(x_1^{i_1}\right)$ denotes the fuzzy likelihood of various failure levels in the basic event, where:i₁=(1,2,…,k₁)

The fuzzy likelihood of the superior event can also be derived from the above equation:

$$\{\begin{array}{l}P\left(y^1\right)={\displaystyle {\sum }_{l=1}^m{P}_0^l}{P}^l\left(y^1\right),\hfill \\ P\left(y^2\right)={\displaystyle {\sum }_{l=1}^m{P}_0^l}{P}^l\left(y^2\right),\hfill \\ \hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\cdots \hfill \\ P\left(y^n\right)={\displaystyle {\sum }_{l=1}^m{P}_0^l}{P}^l\left(y^n\right).\hfill \end{array}$$(5)

2.1.2 Node state assessment

Fault tree analysis is a qualitative and quantitative tool used for assessing system reliability, depicting paths and probabilities of system failures through a combination of fault trees and logical gates. Interactions among events are typically represented using “AND” and “OR” logical operators [35]. To address uncertainties and ambiguities, Takagi-Sugeno (T-S) fuzzy theory is integrated into traditional fault tree analysis. The T-S fuzzy gate, a universal approximator [35], consists of a series of IF-THEN rules and typically utilizes fuzzy numbers to describe the probability of failures occurring.

Membership functions are frequently employed to express the degree to which a variable (such as probability or frequency) belongs within a specific range, reflecting the fuzzy uncertainty present in failure information. Characteristic functions for these membership functions often include Gaussian, triangular, and trapezoidal shapes. To ensure the generality of the selected membership function, a trapezoidal membership function is adopted [36–38], represented as follows:

$$x\equiv \left(c_0,s_1,f_1,s_2,f_2\right).$$(6)

The affiliation function of its fuzzy numbers is shown in Figure 1.

According to the trapezoidal affiliation function in Figure 1 the segmentation function can be scored as:

See equation (7) below.

$$\text{u}\left(x_i^{a_i}\right)=\{\begin{array}{ll}0,\hfill & 0\le x<c_o-s_1-f_1\hfill \\ \frac{x-\left(c_0-s_1-f_1\right)}{f_1},c_o-s_1-f_1\le x<c_o-s_1\hfill & \hfill \\ 1,\hfill & c_o-s_1\le x<c_o+s_2\hfill \\ \frac{\left(c_0+s_2+f_2\right)-x}{f_2},c_o+s_2\le x<c_o+s_2+f_2\hfill & \hfill \\ 0,\hfill & c_o+s_2+f_2\le x<1\hfill \end{array}$$(7)

In the equation above, f₍₁₎ and f₍₂₎ represent the fuzzy zones of the membership function; s₍₁₎ and s₍₂₎ denote the support radii; c_(o) is the center of the support set of the fuzzy number. The variable $\text{u}\left(x_i^{a_i}\right)$ ranges from 0 to 1, indicating that the component's failure severity can take any value within this interval. When s₍₁₎ = s₍₂₎=0, the trapezoidal membership function transitions into a triangular membership function. Moreover, when f₍₁₎ = f₍₂₎ = 0 and s₍₁₎ = s₍₂₎ = 0, the fuzzy number membership function becomes a deterministic number function.

Fig. 1 Membership functions of fuzzy numbers.

2.2 Importance of T-S fuzzy fault tree

2.2.1 T-S probabilistic significance

The T-S probability importance measures the impact of the failure of a specific event within a system on the overall system failure probability. It is an index used to describe the influence of a basic event on a top event. The definition is formulated as follows:

$$I_{T_q}^{P_T}\left(x_j\right)=\frac{{\displaystyle {\sum }_{i_j}^{k_j^{*}}{I}_{T_q{P}_q}}\left(x_j^{i_j}\right)}{k_{j^{*}}}.$$(8)

In the above equation, the $k_{j^{*}}$ denotes the number of events with non-zero fuzzy failure level of the j event, $I_{T_q}^{P_T}\left(x_j\right)$ represents the T-S probability importance when the top event T of the system equals the fuzzy number T_(q) [39].

Depending on the application scenario and domain, its T-S probability importance can also be written as:

$$I_{T_q}^{P_T}\left(x_j^{i_j}\right)=P\left[T_q,P\left(x_j^{i_j}\right)=1\right]-P\left[T_q,P\left(x_j^{i_j}\right)=0\right].$$(9)

In the equation, $x_j^{i_j}$ denotes the failure severity of the event x_j, $P\left(x_j^{i_j}\right)$ represents the fuzzy possibility of failure severity$x_j^{i_j}$, and $P\left[T_q,P\left(x_j^{i_j}\right)=1\right]$ indicates the fuzzy possibility that the system's top event T equals a fuzzy number when$P\left(x_j^{i_j}\right)=1$ . Furthermore, $P\left[T_q,P\left(x_j^{i_j}\right)=0\right]$ expresses the fuzzy possibility that the system's top event T equals the fuzzy number T_(q) when$P\left(x_j^{i_j}\right)=0$.

2.2.2 T-S fuzzy importance level

T-S fuzzy importance is measured by calculating the affiliation of each state in the T-S fuzzy set to measure its importance, which is defined by the following equation [40]:

$$\begin{array}{c}{I}_{T_q}^{F_u}\left(x_j^{i_j}\right)=E\left[P\left(T_q,{\tilde{P}}_{x_j}^{i_j}=1\right)-P\left(T_q,{\tilde{P}}_{x_j}^{i_j}=0\right)\right]=\\ \frac{{\displaystyle {\int }_0^{1}x}{\mu }_{{\tilde{P}}_j^{ij},Tq} \text{d}x}{{\displaystyle {\int }_0^1{\mu }_{{\tilde{P}}_j^{ij},Tq}} \text{d}x}-\frac{{\displaystyle {\int }_0^{1}x}{\mu }_{{\tilde{P}}_j^{ij},0} \text{d}x}{{\displaystyle {\int }_0^1{\mu }_{{\tilde{P}}_j^{ij},0}} \text{d}x}\end{array}$$(10)

In the equation, $I_{T_q}^{F_u}\left(x_j^{i_j}\right)$ represents the T-S fuzzy importance, $x_j^{i_j}$ indicates the failure severity of the event x_j, and ${\tilde{P}}_{x_j}^{i_j}$ denotes the fuzzy operator for the possibility of failure when the failure severity of the event x_j is$x_j^{i_j}$. ${\mu }_{{\tilde{P}}_j^{ij}}$ corresponds to the membership function associated with the fuzzy operator${\tilde{P}}_{x_j}^{i_j}$. $P\left(T_q,{\tilde{P}}_{x_j}^{i_j}=1\right)$ signifies the fuzzy subset of the failure possibility when the fuzzy operator ${\tilde{P}}_{x_j}^{i_j}$ for the failure possibility of the event x_j at severity $x_j^{i_j}$ is 1, leading to the system's top event T equaling the fuzzy number T_(q). The condition $P=\left(T_q,{\tilde{P}}_{x_j}^{i_j}=0\right)$ refers to the fuzzy subset of the failure possibility when the fuzzy operator ${\tilde{P}}_{x_j}^{i_j}$ for the failure possibility of the event x_j at severity $x_j^{i_j}$ is 0, resulting in the system's top event T equaling the fuzzy number T_(q) [41].

Furthermore. $I_{T_q}^{\text{Fu}}\left(x_j\right)$ can also reflect the average impact of each fault state on the system fault state during the evolution of the system fault state from 0 to 1, which is defined by the following equation:

$$I_{T_q}^{\text{Fu}}\left(x_j\right)=\frac{i_j^{i_j^{\prime }}{I}_{T_q}^{\text{Fu}}\left(x_j^{i_j}\right)}{k_j^{\prime }}.$$(11)

In the above equation, the k'_(j) denotes the number of the j event fuzzy failure degree non-zero.

2.2.3 T-S key importance level

In the T-S fuzzy fault tree algorithm, the so-called critical importance is the ratio of the rate of change of the probability of occurrence of the bottom event to the rate of change of the probability of occurrence of the top event, which is defined by the following equation:

$$I_{T_q}^{\text{Cr}}\left(x_j^{\left(i_j\right)}\right)=\frac{P\left(x_j^{\left(i_j\right)}\right)I_T^{\text{Pr}}\left(x_j^{\left(i_j\right)}\right)}{P\left(T=T_q\right)}.$$(12)

In the above equation, the $x_j^{\left(i_j\right)}$ denotes the bottom event x_j is the degree of failure of the bottom event, and $P\left(x_j^{\left(i_j\right)}\right)$ is the bottom event x_j the fuzzy possibility of the bottom event, where: (i_j=1,2,…,k_j), $I_{T_q}^{\text{Cr}}\left(x_j^{\left(i_j\right)}\right)$ denotes the critical importance of the event, and P(T = T_(q)) denotes the top event T is equal to the fuzzy number T_(q) which is the probability of the top event T [42].

Where the top event T of the system is equal to the fuzzy number T_(q), The T-S critical importance at time can also be expressed as:

$$I_{cr}{}^{Tq}\left(X_i\right)=\frac{{\displaystyle {\sum }_{a_i=2}^{k_i}{I}_{cr}}(Tq)\left(X_i=S_i{}^{\left(a_i\right)}\right)}{k_i-1}.$$(13)

2.3 Basic Bayesian Theory

2.3.1 Bayesian principle and transformation process

A Bayesian network is a probabilistic graphical model, denoted by the symbol B(G,P) [43]. G represents the Bayesian network topology, and P denotes the conditional probability, also known as a belief network, because it has been mainly used to deal with ambiguity and uncertainty problems. The network consists of nodes representing different events in the system. Then, the logical relationships between the other events are represented by directed arrows, which generally point in the direction of the "parent" to the "child" nodes. In the study, the Bayesian network nodes x_(i), x_(j) (i≠j,1⩽i⩽n,1⩽j⩽m) are first defined to be called the parent of x_(j) if there exists a directional edge pointing to the node x_(j) from node x_(i), which can be represented by the set E(x_(i)). The full probabilistic representation of the Bayesian network is shown in the following equation:

$$P\left(x_j\right)={\displaystyle \sum _{i=1}^{n}P}\left[x_j\mid E\left(x_i\right)\right]P\left[E\left(x_i\right)\right].$$(14)

In the above equation, P(x_(i)) denotes the edge probability of the parent node x_(i), and P[x_(j)|E(x_(i))] denotes the conditional probability distribution of the child node x_(j).

To describe the multi-state events in the system, this paper maps the fuzzy fault tree model to a Bayesian network model [44], whose mapping rule is shown in Figure 2. In the mapping process, the fault tree model's events correspond to the Bayesian model's nodes one by one [45], which can be derived by calculating the failure probability of a node and, thus, the failure probability of the event itself.

In the transformation of fault trees to Bayesian networks, logic gates are usually transformed into directed edges and conditional probabilities of Bayesian networks [46], as shown in Figure 3 below. In the graph, y₂ and y₃ denote child nodes, and y₁ denotes the parent node; P(y₁=1│y₂=1,y₃=1)=1 means that when child nodes y₂ and y₃ occur simultaneously, the parent node y₁ also occurs.

Fig. 2 Mapping relationship of fault tree to Bayesian network.

Fig. 3 Fault tree logic gate transformation diagram

2.3.2 Node importance assessment

1) Birnbaum importance (Birnbaum measure, BM)

Birnbaum's importance is often used to indicate the value of the increase in the probability of the top event occurring, given that the bottom event occurs [47], whose formula is shown in equation (15) below:

$$I_{X_i}^B=P\left(T=1\mid X_i=1\right)-P\left(T=1\mid X_i=0\right).$$(15)

In the above equation, the P(T=1|X_(i)=1) denotes the probability of occurrence of the top event conditional on the occurrence of the bottom event; P(T=1|X_(i)=0) denotes the probability of occurrence of the top event conditional on the bottom event not occurring.

2) Risk Achievement Worth (RAW)

Hazard importance is primarily used to determine what potential impact the likelihood of a bottom event will have on the occurrence of a top event [48], whose formula is shown in equation (16) below:

$$I_{X_i}^C=\left[P\left(T=1\mid X_i=1\right)-P\left(T=1\mid X_i=0\right)\right]\frac{P\left(X_i\right)}{P(T=1)}.$$(16)

In the above equation, the P(T=1) denotes the probability of the top event occurring.

3) Fussel-Vesely Importance (FV)

Forsythe-Wesley importance, also known as cut-set importance, is primarily used to evaluate the contribution of bottom-event failures to system failures [49], whose formula is shown in equation (17) below:

$$I_{X_i}^{FV}=\frac{P(T=1)-P\left(T=1\mid X_i=0\right)}{P(T=1)}.$$(17)

To facilitate the calculation, several values of importance are normalized by the formula shown in the following equation (18):

$$R_{E,I}=\frac{I_E}{{\displaystyle {\sum }_{i=1}^n{I}_i}}\times 100.$$(18)

In the above equation, the R_(E,I) denotes the bottom event importance; I_(E) denotes the un-normalized value corresponding to the node at a certain importance degree.

The normalized values corresponding to each importance degree for the bottom event are summed to obtain the importance score as shown in equation (19):

$$R_E=R_{E,B}+R_{E,C}+R_{E,FV}.$$(19)

In the above equation, R_(E,B), R_(E,C) and R_(E,FV) denote Birnbaum importance, hazard importance, and Forsythe-Wesley importance, respectively; R_(E) denotes the importance score.

3 Structural model of wind turbine gear train

3.1 Gearbox model selection

The research object selected in this paper is a 5MW doubly-fed wind turbine, whose detailed parameters are shown in Table 1, which is widely used in this category and has a high failure rate of the gearing system, so it is used as the research object.

To more comprehensively analyze the reliability of the wind turbine gear transmission system, the wind turbine transmission system structure is established, the principle of which is shown in Figure 4. The gearbox structure in the figure below has three stages, with the first stage being a high-torque planetary gear and the last two stages being parallel shaft cylindrical gears. In the planetary wheel system, the sun wheel is located in the center of the wheel system and engages with the gear ring through 3 planetary wheels, thus driving the high-speed shaft to rotate. The three-stage transmission of this system can convert the low-speed rotation of the impeller to the high-speed rotation of the generator, thus improving the efficiency of the generator to maintain a stable power output.

Fig. 4 Schematic diagram of wind turbine gear box.

3.2 Failure data sources

To ensure the reliability of the analysis results, the operation data and fault data recorded in the SCADA (Supervisory Control and Data Acquisition) system of a doubly-fed wind turbine at a wind farm in Inner Mongolia are used as the primary data source. These data are collected These data are collected at intervals of 10 minutes and encompass all parameters pertinent to the turbine's operation, such as the angle of the paddle blade, wind speed, temperature, gearbox oil temperature, and any fault reports. The detailed data collection and analysis methods can be referenced in Tables 2 and 3.

4 T-S fuzzy fault tree modeling of wind turbine gear train system

4.1 Gearbox system fault tree

According to the relevant data of FD5500Z gearbox in a wind turbine SCADA system acquired, to establish the gearbox system fault tree as shown in Figure 5. The model gearbox fault tree model includes four parts: mechanical components, lubrication subsystem, cooling subsystem, and monitoring and protection subsystem, which will be introduced in this section. In this fault model, the gearbox fault y₁ is designated as the top event, representing an undesired event in the system. The bottom event is denoted by x. The nomenclature corresponding to each event is enumerated in Table 4.

Fig. 5 Fault tree of gearbox system.

4.2 Lubrication subsystem fault tree

The lubrication subsystem, as an offshoot of gearbox failure, is primarily responsible for providing a constant, proper flow of lubricant to the moving parts inside the gearbox, such as the gears and bearings, to minimize friction and carry away the heat generated by that friction. In this system, the lubrication system fault y₂ is taken as the top event, and its lubrication subsystem fault tree is shown in Figure 6, and the names of the corresponding events are shown in Table 5.

Fig. 6 Lubrication subsystem fault tree.

4.3 Cooling subsystem fault tree

The cooling subsystem is also an essential part of the gearbox structure. The double-fed unit selected in this paper utilizes air cooling in the form of gearbox cooling to prevent degradation of lubricant performance or damage to mechanical components due to overheating. Taking the cooling system fault y₉ as the top event, its cooling subsystem fault tree is shown in Figure 7, and the names corresponding to each event are shown in Table 6.

Fig. 7 Cooling subsystem fault tree.

4.4 Monitoring and protection subsystem fault tree

The monitoring and protection subsystem, as the protection device of the gearbox, can monitor the temperature, oil quality, vibration level, and other status parameters in the gearbox in real-time, to facilitate the operation and maintenance personnel to discover abnormalities in time and take corresponding measures. In this subsection, the monitoring and protection system fault y₁₀ is taken as the top event, and its monitoring and protection subsystem fault tree is shown in Figure 8, and the names of the corresponding events are shown in Table 7.

Fig. 8 Fault tree of monitoring and protection subsystem.

4.5 Transformation of T-S fuzzy fault tree into Bayesian network

By analyzing the causal relationship between the bottom events and the intermediate events in the fault tree model of the gear transmission system, the mapping algorithm is used to convert the fault tree network model of the gear transmission system into a Bayesian network model. In this conversion, each basic event in the fault tree corresponds to the nodes in the Bayesian network one by one, and the logic gates in the fault tree are represented by CPTs. The Bayesian network model is shown in Figure 9.

Fig. 9 Bayesian model of wind turbine gearing system.

5 Reliability analysis of wind turbine gearing system

5.1 Logic gate judgment of T-S fuzzy fault tree

In the T-S fuzzy fault tree analysis system, fault types are typically categorized based on the system's fault status into no-fault, minor fault, and severe fault, represented in the membership functions as 0, 0.5, and 1, respectively [50]. In this system, gearbox failure is considered the top event, and a top-down analysis approach is used to describe the fault status of intermediate event y={y₁,y₂,…,y₂₈} and basic event x={x₁,x₂,…,x₅₈}.

To calculate the fuzzy probability of the top event's fault severity, assumptions are made based on the established T-S fault tree model. It is assumed that the fault severities of basic event x={x₁,x₂,…,x₅₈} and intermediate event y={y₁,y₂,…,y₂₈} are set to 0, 0.5, and 1. According to reference [51], the parameters for the membership function are set to f₁=f₂=0.3 and s₁=s₂=0.1. Utilizing relevant empirical data, the gate rules for the T-S fuzzy fault tree are detailed in Tables 8 through 12. Due to space limitations, only Tables 8 to 12 are provided as examples.

For ease of analysis, it is stipulated that each row in Tables 8 to 12 represents one fuzzy rule. For example, Rule 1 in Table 8 posits that if the fault severities. x₁, x₂, x₃, and x₄ are all 0, then the likelihood of no fault for y₄ is 1, with the possibility of minor and severe faults being 0 [52].

5.2 Calculation of reliability assessment indicators

5.2.1 Fuzzy likelihood of system failure calculated with known event fault states

Taking the lubrication subsystem failure as an example, assume that the known bottom events x=(x₁₉,x₂₀,…,x₃₂) of the fault state are x'=(x'₁₉, x'₂₀,…,x'₃₂). For x₂₃, let c₀ = 0.3(Taking values from literature 35), its affiliation function is obtained as:

$$\text{u}\left(x_i^{a_i}\right)=\{\begin{array}{ll}0,\hfill & \hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}x\le -0.1\hfill \\ \frac{10x+1}{3},\hfill & -0.1<x\le 0.2\hfill \\ \frac{7-10x}{3}{,}^{}\hfill & \hspace{1em}\hspace{1em}\hspace{1em}0.2<x\le 0.4\hfill \\ 1,\hfill & \hspace{1em}0.4<x\le 0.7\hfill \\ 0,\hfill & \hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0.7<x\le 1\hfill \end{array}$$(20)

The fault severity of x₂₃, when characterized as 0, 0.5, and 1, corresponds to membership values of 1/3, 2/3, and 0, respectively.

Similarly, the membership values for the remaining basic events are presented in Table 13.

Calculated from equation (2), the rule enforcement degree of T-S fuzzy gate 10 is shown in Table 14.

Combining equation (3) yields the failure probability of the intermediate event y₍₁₃₎:

$$\begin{array}{l}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}P\left(y_{13}=0\right)=1\times {\beta }_1^{*}+0.1\times {\beta }_4^{*}=0.4\hfill \\ P\left(y_{13}=0.5\right)=0.2\times {\beta }_4^{*}=0.133\hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}P\left(y_{13}=1\right)=0.7\times {\beta }_4^{*}=0.467\hfill \end{array}$$

Replacing the affiliation of the intermediate event with its fuzzy likelihood, the fuzzy likelihood of the top event is finally obtained as:

$$P\left(y_1=0\right)=0.203,P\left(y_1=0.5\right)=0.055,P\left(y_1=1\right)=0.742$$

This can be derived from the above calculations:

When the bottom event x₂₃ and x₂₄ are faulted to different degrees, the impact on the intermediate event y₍₁₃₎ is larger, which leads to the possibility of failure of its top event increased, up to 74.2%, which is consistent with the actual operating conditions of the wind turbine, so the application of T-S fuzzy fault tree methodology in the process of reliability analysis of the wind turbine is completely feasible.

5.2.2 Fuzzy likelihood of known event failures and calculation of fuzzy likelihood of system failures

In this paper, eight experts in the field of wind power generation were selected to form an assessment team, and the members of the team were involved in scientific researchers, front-line workers, etc., to make a professional assessment of the reliability of the gear drive system [53]. When integrating the assessment opinions of different experts on the same basic event, the concept of expert weight is introduced to improve the credibility of expert assessment probability [54]. The experts' basic information and weight distribution are shown in Table 15.

The fuzzy numbers in the assessment opinions of different experts were combined through the following equation (21):

$$P_j={\displaystyle \sum _{i=1}^m{w}_j}\otimes A_{ij},j=1,2,\cdots ,n.$$(21)

In the above equation, P_(j) denotes the fuzzy failure possibility of event j, w_(j) denotes the weight factor of expert j, A_(ij), and the description of event i by expert j.

To facilitate the calculation, the center of mass index method (centroid-index) [55] is used to denazify the fuzzy number of the probability of the occurrence of the event assessed by the experts above and the fuzzy number is expressed by the fuzzy possibility score (FPS), which is calculated as shown in the following equation (22):

$$\begin{array}{l}FPS=\frac{{\displaystyle \int \mu }(x)x \text{d}x}{{\displaystyle \int \mu }(x)\text{d}x}=\frac{{\displaystyle {\int }_{a_1}^{a_2}\frac{x-a_1}{a_2-a_1}}x \text{d}x+{\displaystyle {\int }_{a_2}^{a_3}x}dx+{\displaystyle {\int }_{a_3}^{a_4}\frac{a_4-x}{a_4-a_3}}x \text{d}x}{{\displaystyle {\int }_{a_1}^{a_2}\frac{x-a_1}{a_2-a_1}} \text{d}x+{\displaystyle {\int }_{a_2}^{a_3}d}x+{\displaystyle {\int }_{a_3}^{a_4}\frac{a_4-x}{a_4-a_3}} \text{d}x}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\frac{1}{3}\frac{{\left(a_4+a_3\right)}^2-a_4{a}_3-{\left(a_1+a_2\right)}^2+a_1{a}_2}{a_4+a_3-a_1-a_2}\hfill \end{array}$$(22)

After obtaining the fuzzy likelihood score, it is usually transformed into fuzzy probability (FFR) to ensure that the fuzzy probability of all events is harmonized with the true likelihood, and the transformation process is shown in equation (23) below:

$$FFR=\{\begin{array}{ll}1/{10}^K,\hspace{1em}\hspace{1em}FPS\ne 0\hfill & \hfill \\ 0,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}FPS=0\hfill & \hfill \\ K=2.301\times {\left[\frac{1-FPS}{FPS}\right]}^{\frac{1}{3}}\hfill & \hfill \end{array}.$$(23)

In the above equation, the FPS denotes the fuzzy likelihood score; the K denotes a fixed constant.

To enhance the precision of the prediction of the probability of failure of the bottom event of the gearing system by the selected experts, this paper utilizes the seven-level linguistic terms “VL,” “L,” “FL,” and “M,” as well as “FH,” “H,” and “VH.” The estimation of the probability of failure of the system bottom event by seven levels of linguistic terms is presented herein [56-57]. Using fuzzy numbers to quantify the expert language is not only in line with human thinking habits but can also reduce computational complexity and improve computational efficiency [58]. The correspondence between trapezoidal fuzzy numbers and expert language is shown in Table 16 below.

In the statistical process of gear transmission system failure data, there are such as x₇,x₈,x₁₀,x₂₉,x₃₄,x₃₅ and other bottom events that can not be retrieved from the SCADA system of the probability of failure, i.e., through the members of the group of experts following the terminology in the above Table 16 on the bottom of the event to put forward the corresponding comments, the results of the results are shown in the following Table 17.

According to the above equation (23), the fuzzy probability of the bottom event (FFR) is shown in Table 18.

The probability of occurrence of bottoming events in the wind turbine gearing system is shown in Table 19 [59–67].

Assuming that the probability data of the bottom events x₁ to x₅₈ with a fault degree of 0.5 is equal to the probability data of the fault degree of 1, the fuzzy likelihood of the intermediate events is known based on Tables 9–11 and equations (4) and (5):

$$\begin{array}{l}P\left(y_{12}=0.5\right)={\displaystyle \sum _{l=1}^{81}{P}_0^l}{P}^l\left(y_{12}=0.5\right)=p_0^2{p}^2\left(y_{12}=0.5\right)+p_0^6{p}^6\left(y_{12}=0.5\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=0.9P_0^2+P_0^6=1.714\times {10}^{-5}\hfill \end{array}$$

$$\begin{array}{l}P\left(y_{12}=1\right)={\displaystyle \sum _{l=1}^{81}{P}_0^l}{P}^l\left(y_{12}=1\right)=p_0^2{p}^2\left(y_{12}=1\right)+p_0^3{p}^3\left(y_{12}=1\right)+\cdots +p_0^{81}{p}^{81}\left(y_{12}=1\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=0.1P_0^2+P_0^3+\cdots +P_0^{81}=6.784\times {10}^{-6}\hfill \end{array}$$

$$\begin{array}{l}P\left(y_{13}=0.5\right)={\displaystyle \sum _{l=1}^9{P}_0^l}{P}^l\left(y_{13}=0.5\right)=p_0^2{p}^2\left(y_{13}=0.5\right)+p_0^4{p}^4\left(y_{13}=0.5\right)+p_0^5{p}^5\left(y_{13}=0.5\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=0.4P_0^2+0.2P_0^4+0.8P_0^5=5.557\times {10}^{-6}\hfill \end{array}$$

$$P\left(y_{16}=0.5\right)={\displaystyle \sum _{l=1}^9{P}_0^l}{P}^l\left(y_{16}=0.5\right)=0.3P_0^2+0.4P_0^4+0.8P_0^5=5.502\times {10}^{-6}$$

$$\begin{array}{l}P\left(y_{16}=1\right)={\displaystyle \sum _{l=1}^9{P}_0^l}{P}^l\left(y_{16}=1\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}=0.5P_0^2+P_0^3+0.4P_0^4+0.2P_0^5+P_0^6+P_0^7+P_0^8+P_0^9\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}=2.432\times {10}^{-5}\hfill \end{array}$$

Similarly, the fuzzy failure rates for the remaining intermediate events y₂ to y₁₁, y₁₄, y₁₅, y₁₇ to y₂₈ were calculated using MATLAB software and the results are shown in Table 20.

Therefore, the fuzzy probability of the top event y₁ gearbox failure is:

$$\begin{array}{c}P\left(y_1=0.5\right)={\displaystyle \sum _{l=1}^{81}{P}_0^l}{P}^l\left(y_1=0.5\right)=0.3P_0^2+0.4P_0^4+0.8P_0^5=16.371\times {10}^{-6}\\ P\left(y_1=1\right)={\displaystyle \sum _{l=1}^{81}{P}_0^l}{P}^l\left(y_1=1\right)=0.3P_0^2+P_0^3+\cdots +P_0^{81}=19.425\times {10}^{-6}\end{array}.$$

5.2.3 Bayesian network inference analysis

According to the above formulas (15) to (17), the node importance index of the gearbox fault y₍₁₎ can be calculated, and then the results are normalized using formula (18) and formula (19), and the results are shown in Table 21.

From the results calculated above, it is evident that the fuzzy possibility of event failures is of the same order of magnitude as that of gearbox failures in wind turbine systems. The fuzzy probabilities of intermediate events listed in Table 20 indicate that within the subsystems. y₂, y₃, y₉, and y₁₀ of the gearbox system y₁ in wind turbines, the failure rates rank from highest to lowest as follows: y₁₀ > y₍₂₎> y₉ > y₃. The harsh natural environments of wind farms, combined with the location of gearboxes at the tops of towers, which are often several tens or even over a hundred meters high, mean that electronic components such as sensors in the nacelle are constantly subjected to factors like vibration and humidity. These conditions lead to increased failures or errors in electronic components, making the monitoring and protection system y₁₀ more prone to failure. In contrast, the gearbox housing, made of cast iron, has a more stable internal structure, resulting in a relatively lower probability of failure. The failure rates of the various gearbox components are depicted in Figure 10.

The lubrication system y₂ and the cooling system y₉ are significantly affected by environmental conditions. Due to the substantial diurnal temperature variations within the nacelle, the viscosity of the lubricant in the lubrication system can change, which diminishes its lubricating performance and can lead to damaged seals and oil leakage. Similarly, the wind turbine's cooling system y₉ is also susceptible to environmental impacts, which can lead to blockages in the cooling radiators and subsequent system failures. These computational results are consistent with the actual operational experiences of wind turbine units and provide valuable insights for the reliability analysis of gear transmission systems.

Fig. 10 Failure rate of various components in the gearbox.

6 Conclusion

• By integrating T-S fuzzy theory with fault tree analysis, a comprehensive assessment of the reliability of wind turbine gear transmission systems is performed. This method effectively addresses the complexity involved in constructing traditional fault trees by eliminating the need to analyze logical relationships such as “AND,” “OR,” and “NOT,” thereby facilitating the inference of system failure states.

• The reliability of the wind turbine gear transmission system is calculated using two approaches: one that computes the system's fuzzy possibility from known event failure states, and another from known event fuzzy possibilities. By integrating data from the SCADA system with expert experience, the membership and degree of execution under T-S fuzzy gate rules are calculated, which are then used to determine the fuzzy possibility of the top event occurring. Results from both methods indicate that the probability of failure is highest for the top events, lowest for the basic events, and intermediate for the middle events, consistent with practical observations. This demonstrates the feasibility of the T-S fuzzy fault tree analysis method for wind turbine gear transmission systems.

• The integration of the T-S fuzzy fault tree with the Bayesian network not only addresses the limitations of the conventional Bayesian network, which is incapable of articulating the fuzzy logic relationships between the nodes, but also overcomes the challenges posed by the T-S fuzzy fault tree, which is intricate in nature and lacks bidirectional reasoning capabilities. In the future, the structure of the T-S fuzzy model can be further optimized so that the research method can be adapted to a wider range of engineering fields and a new model can be explored for improving the reliability of various systems.

• This paper employs a polymorphic analysis method to assess the reliability of the wind turbine gear transmission system, overcoming the limitations of bistate analysis methods that struggle to identify inter-event relationships. This approach yields results that more closely align with the actual operational conditions of the gearbox.

Funding

This study was generously supported by the Liaoning Provincial Department of Science and Technology, and we would also like to thank the Shenyang Engineering Institute for partially funding this study under project number: LJKZZ20220139.

Conflicts of interest

We declare that we are not aware of any other private or financial interests that may influence the results and conclusions of this study other than the conflicts of interest mentioned above.

We understand the importance of the conflict of interest statement, which helps the reader to assess the credibility of the study results. We promise to update the statement and inform the journal's editorial board if any new conflict of interest arises.

Data availability statement

We, the author team listed above, hereby submit our research paper titled “Reliability Assessment of Multistate Wind Turbine Gear Train Systems Based on T-S Fuzzy Fault Trees and Bayesian Networks” to your journal along with our Data Availability Statement.

• Data storage: All raw data collected in this study have been stored in China's National Knowledge Infrastructure and Wanfang database, and have been securely encrypted in accordance with relevant data protection regulations. We have ensured that all data have been stored in a manner consistent with current best practices and standards in the field of scientific research.

• Data access: In order to ensure the integrity and confidentiality of the data, we have set certain restrictions on data access. Researchers can access the data by making a formal data access request to the corresponding author or our research organization. We promise to provide the necessary data and support within no more than 24 hours after receiving a reasonable data request. We understand the importance of data availability statements for scientific research, not only to help promote transparency in research, but also to enhance the credibility and utility of the results. We are committed to providing the scientific community with high-quality data resources to support validation, replication, and further research.

Author contribution statement

Xiaowei Yin: Designed the main concept and experimental program of the study. Took care of the data collection, analysis and interpretation. Wrote the first draft of the research paper and revised the final manuscript. Bingbing Jiao: Participated in the development of experimental design and methodology. Responsible for the exact execution of the experiment and data collection. Provided important intellectual input, and reviewed and provided feedback during the manuscript writing process. Jiushen Liu: Provided professional support in data analysis and statistical processing. Participated in the discussion of the results and the formation of the paper's conclusions. Assisted in the preparation of relevant figures and supplementary materials. HuiWen Hu: Provided key laboratory resources and materials for the research project. Played a key role in experimental manipulation and technical support. Reviewed the technical content of the paper at the draft stage.

Ethics approval

We, the author team listed above, hereby solemnly declare that we have strictly followed the ethical guidelines and norms of scientific research in writing and submitting the research paper titled “Reliability Assessment of Polymorphic Wind Turbine Gear Train System Based on T-S Fuzzy Fault Tree and Bayesian Networks” to your journal. The following are the details of our ethical statement:

• research subjects and participants: All human participants involved in our study were fully informed in advance about the purpose, methods, potential risks and benefits of the study and have provided their informed consent.

• Data collection and analysis: We guarantee that the data collected is true and accurate and that the data analysis process was fair and unbiased. We promise that we have not engaged in any form of data falsification or manipulation and that all results are based on actual observations and records.

• Conflict of interest: We declare that we do not have any financial or personal conflicts of interest in the course of this study that could affect the objectivity of the results. If potential conflicts of interest exist, we have clearly identified them in the manuscript and have taken appropriate measures to eliminate or minimize the effects of these conflicts.

• Intellectual property and copyright: We confirm that the work in this paper is original and does not infringe the intellectual property rights or copyright of any third party. We understand and accept that once a manuscript is accepted for publication, the corresponding copyright will be owned by the journal publisher.

• Funding: We have clearly disclosed in the manuscript all the sources of funding, including any commercial or governmental grants, that funded this study. We are committed to the accuracy of the above and we are willing to bear the consequences of any breach of ethical norms. We believe that by following high standards of ethical practices, our research will make a valuable contribution to the scientific community.

References

All Tables

All Figures

	Fig. 1 Membership functions of fuzzy numbers.
In the text

	Fig. 2 Mapping relationship of fault tree to Bayesian network.
In the text

	Fig. 3 Fault tree logic gate transformation diagram
In the text

	Fig. 4 Schematic diagram of wind turbine gear box.
In the text

	Fig. 5 Fault tree of gearbox system.
In the text

	Fig. 6 Lubrication subsystem fault tree.
In the text

	Fig. 7 Cooling subsystem fault tree.
In the text

	Fig. 8 Fault tree of monitoring and protection subsystem.
In the text

	Fig. 9 Bayesian model of wind turbine gearing system.
In the text

	Fig. 10 Failure rate of various components in the gearbox.
In the text