Reliability analysis of wind power gearbox based on fault tree and monte carlo simulation

The gearbox is an important speed-up component in wind turbine, and the environment in which wind turbine operates is generally harsh. For maintenance and replacement of the gearbox and its difficult operation, it is self-evident that it is important to ensure its reliable operation. Through collecting and analyzing the structural and functional principle of the gearbox and data of common failure modes, the fault tree of the gearbox is established, and failure rate data and corresponding distribution are obtained by manual and data inquiry for each bottom event. Combining Monte Carlo simulation, the reliability curves of each bottom event mode of the gearbox, the gearbox, and important components are solved, and the assumptions of improving the important bottom events are made. By comparing the curves with the previous ones, it is proved that the improvement of important components can effectively improve the system’s reliability.


Introduction
According to the 2023 Global Wind Energy Report from the Global Wind Energy Council, the global wind power capacity has reached 906 GW as of 2022, showing tremendous potential for wind power development.Compared to 2021, it has grown by 9%.Currently, China's newly installed wind power generation units rank first in the world.Among such a large-scale wind power generation unit, the gearbox is an important speed-increasing component susceptible to internal component failures after long-term operation.Relevant data also shows that the failure rate of gearboxes in wind power generation units is relatively high, which leads to a long downtime [1].Therefore, reliability analysis of the gearbox is very necessary to improve the overall service life of wind power generation units [2].
This article focuses on the failure analysis of two important transmission systems and lubrication systems in the gearbox of a certain type of 2 WM doubly fed wind turbine generator [3].A detailed analysis and hierarchical division are conducted from common component failure modes to their failure mechanisms in order to prepare for the development of fault tree models.After the fault tree model is established, the distribution function of each bottom event is determined to ensure that the failure distribution of each bottom event is known.Based on this, a Monte Carlo simulation is conducted to obtain the required analysis results, i.e., the importance of each bottom event [4].Improvements are then hypothetically made to the components with higher importance to effectively improve the reliability of the gearbox system.

Fault Tree Analysis
Fault Tree Analysis (FTA) is very logical and can intuitively show the connection between faults and faults.Through the overall grasp of the structural and functional failure of the system, with the most undesirable event in the system as the top event, all the cause events that cause the event to occur are found.Then a detailed analysis is made of each intermediate event to find out all the next-level events that cause the intermediate event to occur until the lowest cause event is found and regarded as the bottom event [5].Upper and lower events are linked together by means of logic gates, forming a pyramid-like inverted tree structure.

Fault Tree Establishment and Bottom Event Distribution Determination of Wind Power Gearbox 2.2.1 Gearbox Fault Tree Model
When researching the fault problem, the fault tree can start from the most serious fault problem that causes the electric gearbox system, trace back and reasoning the fault layers until the root cause of the problem is found, and then solve the problem more pertinently.To study the reliability of wind power gearboxes, we need to analyze the faults that will occur in the gearbox system and put forward the method to improve the faults, then feedback to the design and manufacturing stage to form a closedloop system, which fundamentally improves the system's reliability.A graphic deduction is used to analyze the causes of various faults leading to system failure and the logical relationship between them from the top down.Qualitative analysis and numerical calculation are used to identify the most affected parts of the system [6].The fault tree diagram of a certain type of gearbox is shown in figure 1: T represents the fault of the wind power gearbox as the top event, M series is an intermediate event, M1 is the transmission system fault, M2 is a lubrication system fault, M11 is gear fault, M12 is shaft fault, M3 is bearing fault, M21 is lubrication oil deterioration, M22 is oil leakage, M23 is oil temperature anomaly; M111 is broken teeth, M112 is pitting, M113 is gluing, M114 is plastic deformation, M115 is wearing teeth, M121 is a misaligned shaft, M122 is a bent shaft, M123 is a loose shaft, M131 is bearing stall fault, M132 is bearing accuracy problem [7].

Determination of Failure Distribution Function for Bottom Event of Fault Tree
Failure distribution of bottom events of the fault tree can be defined as an exponential distribution [8], i.e., there will be a stable failure rate during the accidental failure period.Considering the principle of product approximation, the failure rate of bottom events [9,10] can be obtained by approximate modification combined with the operating environment of products.After consulting the data, its parameters are shown in the following table 1

Reliability Numerical simulation based on fault tree
In reliability engineering, the fault tree itself can carry out qualitative and quantitative calculations.The main consideration of simulation calculation by fault tree is that traditional calculation can no longer meet the requirements as the tree structure becomes more and more complicated, and even the complex calculation of exponential explosion may occur.Monte Carlo simulation just solves this problem.By using the law of large numbers, the problem is transformed into probability statistics.As the number of simulations increases, the simulation result will approach the true value [11,12].To combine the fault tree with Monte Carlo simulation, it is necessary to inversely transform the key time variables in the failure distribution function of the bottom event, which was previously solved as in equation ( 1): where  is a random number that is evenly distributed on (0,1), i.e.,  ∼U (0, 1); For example, the exponential distribution is calculated as the following equation (2): This means a random sampling process from a uniformly distributed random number to a given distribution time variable random number, i.e., sample value.By using MATLAB, the random numbers subject to various distributions correspond to their time variables one by one.When the distribution parameters of each bottom event are known, the corresponding random numbers of time [13] can be obtained.
Since problems that occur at any position in the gearbox can lead to gearbox failure, the whole structure is in series, and the mapping to the fault tree is all expressed in terms of gates or gates.Assuming that each bottom event does not interfere with each other independently, the structure-function of the fault tree is as the following equation (3): The simulation takes the form of a minimum cut set of fault trees.Since all or gate is the minimum cut set, every bottom event can be known as the minimum cut set.2, it can be seen that the fatigue fracture of gear components is the most important, followed by bearing wear.As high-frequency rotating parts in gear box drive system, gear and bearing will be damaged after a long time of operation, which is also in line with reality.
The reliability function image of the important subsystems of the wind turbine gearbox can also be obtained by simulation as shown in figure 3.As can be seen from the above two diagrams, the reliability of the transmission system is inferior to that of the lubrication system for a specified time, after a long period of operation of the gearbox.That is to say, the system composed of gear, shaft, and bearing components will reach its service life more quickly.Therefore, it is necessary to pay more attention to these components and maintain them in time.
Based on the importance analysis above, it can be seen that the fatigue factor of gear teeth will make the gear reach the service life more quickly.Now it is assumed that the fatigue life of gear teeth will be doubled [14] by improving the lubricant and maintaining in time.The simulated curves are as follows.It can be seen from the figure 4 and figure 5 that the reliability of the gearbox system of the wind turbine can be effectively improved by the assumption of related component improvement.The necessity of maintenance improvement for key components of the system is also confirmed.

Concluding remarks
After collecting and sorting out more comprehensive data on structure function and common fault modes of the wind power gearbox, this paper carries out reliability modeling for two important transmission systems and lubrication system of the gearbox by fault tree analysis method of reliability engineering theory.Through the fault tree structure, we can intuitively and clearly see common fault modes and fault mechanisms that cause gearbox failure.Based on this model, maintenance personnel can be organized to carry out maintenance work; Then, based on the fault tree, the failure distribution functions of each bottom event are explained.The parameters in the distribution functions are approximately corrected by failure rate data of similar products and corresponding environmental factors, and the final failure data are obtained.
Considering the complexity and diversification of fault tree structure, the calculation of exponential explosion will be solved quantitatively.Using the known fault tree structure function and failure distribution function of each event, the numerical reliability simulation of the gearbox fault tree based on the minimum cut set of the fault tree is carried out by Monte Carlo simulation.The simulation can solve the complex quantitative calculation of the fault tree and obtain the required importance of each bottom event mode, thus determining the importance of each bottom event and providing effective data proof for subsequent maintenance of key components.Then, the reliability curves of the two subsystems and gearbox further provide maintenance personnel with time nodes for preventive improvement and carry out effective preventive maintenance on the premise of guaranteeing specific reliability to further increase the service life of the gearbox.Finally, through the maintenance improvement assumption of the most important gear fatigue, its average fatigue life is increased and re-simulated.The obtained gearbox reliability curve is compared with the original simulation curve, which verifies the rationality of the improvement assumption.Also, it shows that the service life of the gear box system can be effectively improved by timely treatment and maintenance of the failure causes of key components.

Figure 2 .
Figure 2. Fault tree Monte Carlo simulation flow chart.

Figure 3 .
Figure 3. Reliability curve of important subsystems of the gearbox.

Figure 5 .
Figure 5.Comparison of reliability curves before and after improvement.

Table 1 .
. Names of bottom events and their failure distributions.

Table 2 .
Importance of bottom event patterns.