## 1. Introduction

In the context of the global energy crisis, wind energy has been attracting increasing attention as a clean and pollution-free renewable energy source, and the installed capacity of wind power has increased significantly. With the increase in the stock of operational equipment, the wind power operation and maintenance market has ushered in development opportunities and space, thereby becoming an important factor that affects the development of the wind power industry. With the high price of wind turbines, the harsh working environment, the remote location, and the high height of equipment, maintenance work is very inconvenient. When a shutdown failure occurs, the wind farm must pay an expensive fee for repairs. Therefore, because the gearbox is one of the components with the highest failure rate in wind turbines, reliability analysis and the optimization of maintenance strategies are necessary [

1,

2].

Preventive maintenance is aimed at preventing the occurrence of failures. Through the inspection and monitoring of equipment, failure symptoms are discovered, and various maintenance activities are carried out before failures occur to maintain the prescribed functional status. This approach is an effective way to ensure that the equipment is in good condition. Nowadays, two main types of preventive maintenance are used. The first is time-based maintenance, which can ensure that the unit meets the desired performance requirements after implementation, equipment maintenance is conducted at fixed time intervals, and sufficient maintenance resources are arranged at corresponding time points [

3,

4,

5]. The second is condition-based maintenance (CBM). This maintenance strategy can determine the abnormality of the device and predict the development trend of the unit status by monitoring and diagnosing the equipment status, and make a maintenance plan in advance, thereby improving equipment utilization and overcoming many defects of traditional fault repair and active maintenance [

6,

7,

8]. Therefore, the CBM is receiving growing attention.

In CBM, according to the different ways of condition detection, it can be divided into continuous monitoring state maintenance and discrete detection state maintenance. Continuous state monitoring means that it can continuously return to the real-time operating status of the system through wireless communication technology after a one-time installation. Zhu [

9] uses a multivariate state estimation technique to estimate the bearing temperature of the wind turbine gearbox, and judges whether the gearbox operates normally through the real-time change trend of the residual error between the estimated value and the actual value of the bearing temperature. Hameed [

10] uses back propagation neural network to model and predict the state of the gearbox and generator, as well as a multi-agent method to comprehensively analyze the diagnosis results of different components and gives the overall operating state of the unit. The one-time installation cost of continuous monitoring is high, and the requirements for data acquisition and signal processing are high. It is suitable for nuclear power plants, aviation components and other systems with high safety requirements. Discrete detection needs to carry specific detection equipment at a specific interval to detect the key components of the system. The cost of single detection is high, and the probability of system failure is high between two successive detections. Anders [

11] considers two types of random failures and aging failures in the system, with maintenance including minor repair and replacement. The unit state is represented by a state probability vector, and the adjacent state probability vectors were linked together by one-step transition probability. A method for determining the maintenance time by using Markov chain theory to estimate the remaining life of the system is proposed. Kang [

12] takes the continuous degradation state as the independent variable and the continuous detection interval as the dependent variable, and derives the functional relationship between the state and the detection interval based on the established ordinary differential equation describing the state transition of the system. Zhu [

13] uses a proportional hazard model to reflect the relationship between gearbox status and failure rate and take reliability as the maintenance goal to predict the maintenance interval of the gearbox. It can be seen that the state of the equipment plays an important role in finding the best detection time. However, when constructing the state transition model of the equipment, most of the studies did not consider the impact and interference of random factors such as daily inspection, maintenance, and repair on the system process behavior, which cannot accurately estimate the state of the equipment. It is worth noting that when the state of the device is different, the same disturbance will affect the state of the device to different degrees.

For equipment maintenance, constructing a state transition model of the equipment is very important. The judgment of equipment status has a direct effect on the selection of maintenance strategy. In light of the aforementioned problems, this study took a wind turbine gearbox as the research object paper, and the stochastic differential equation model of the gearbox state transition was established. Compared with the traditional ordinary differential equation, this model considers that the equipment failure rate is affected by the equipment state in addition to the time. At the same time, it takes into account the influence of the external random disturbance on the equipment state and integrates the relationship between the equipment failure rate and external random interference. This model can more accurately predict the state of the equipment, thereby more accurately predicting the inspection moment.

The remainder of this paper is organized as follows.

Section 2 constructs the state transition model of the gearbox.

Section 3 introduces the constituent indicators and weight distribution of the gearbox state.

Section 4 introduces the construction method and parameter-solving method of the failure rate model and the state volatility model in the state transition model.

Section 5 verifies that this model is more accurate through comparison and analysis with traditional differential equations, and then with reliability taken as the decision goal, the best maintenance time of the gearbox is predicted, and validates the effectiveness of the model through an example analysis.

## 2. Gearbox State Transition Modeling

The change of the gearbox state is formed by the combination of the decline of the gearbox itself and external factors. The decline of the gearbox itself is represented by the failure rate, and the received external interference is random.

**Definition** **1.** x(t) represents the state of the gearbox at time t. x(t) = 1 means that the device is in a brand-new state at t, and x(t) = 0 means that the device is in a completely damaged state at t.

**Definition** **2.** The failure rate of the gearbox is related to the state of the equipment and the operating time of the equipment. The strength of the random disturbance received by the gearbox on the equipment state is related to the state of the equipment.

**Hypothesis** **1** **(H1).** The gearbox repair process is completed in an instant, and each repair means that the equipment enters a new life cycle. The change of x(t) represents the state change process of the gearbox in one life cycle.

**Hypothesis** **2** **(H2).** The random disturbance of the gearbox at all times is independent and stable.

**Hypothesis** **3** **(H3).** The expected value of the random disturbance of the gearbox is zero.

The state transition model of the gearbox is constructed as follows:

In (1), the first term on the right side of the equation is the decline of the gearbox itself and $\lambda \left(x\left(t\right),t\right)$ is the failure rate of the gearbox, the second term represents the random disturbance received by the gearbox, $\mu \left(x\left(t\right),t\right)$ is the random disturbance coefficient, which is called the equipment state fluctuation rate, and B(t) is the Brownian motion, which satisfies the Hypotheses 2 and 3.

$\lambda \left(x\left(t\right),t\right)$, $\mu \left(x\left(t\right),t\right)$ is a $B\left[0,T\right]\times {B}^{t}$ measurable function defined on $\left[0,T\right]\times {R}^{t}$.

There is a constant

K, let:

In (2), both

$x\left(t\right)$ and

$\tilde{x\left(t\right)}$ are the solutions of (1), (2) proves the uniqueness of the solution of (1), and (3) proves the existence of the solution of (1). If 1 and 2 are both satisfied, then Equation (1) has a unique solution

x(

t), and

x(

t) is continuous with respect to

t [

14].

## 3. The Determination of the State Characteristic Value of the Gear Box

#### 3.1. Selection of Modeling Indicators

In the condition-monitoring system of wind turbines, the commonly used condition monitoring parameters are wind direction, wind speed, tower torque and bending moment, oil temperature, shaft temperature, and main bearing vibration [

15,

16,

17]. In establishing the gearbox state transformation model, the following variables that are closely related to the gearbox state are selected to form the observation indicators:

Oil temperature (OT): the oil temperature of the gearbox reflects the operation of the gearbox air cooler, lubrication system, and relief valve. If these parts are functioning abnormally, then the gearbox temperature will rise sharply.

Shaft temperature (ST): the increase in the gearbox shaft temperature means that the frictional force increases, which may be caused by gear wear, abnormal denature or bite, or aging and deterioration of lubricating oil.

Amplitude: the rapid increase in gearbox amplitude may be caused by problems with rotors, shafts, gears, and bearings.

#### 3.2. Weights of Modeling Indicators

The oil temperature, shaft temperature, and amplitude are closely related to the state of the gearbox. Therefore, when a state change model is being built, the degree of influence of the three observation indicators on the state needs to be clarified, that is, these indicators must be weighted. In this paper, the entropy method is used to weight the observed indicators.

Construct the data matrix:

In (4), ${x}_{ij}$ is the value of the jth observation index at the ith moment (j = 1, 2, 3).

The measurement units of the various observation indicators are not uniform, which is why they must be normalized before the overall state of the device is calculated. In this paper, a large

x(

t) corresponds to a good device state. The data must be translated to avoid the meaningless logarithm of the entropy value. The specific methods are as follows:

Calculate the weight of the

jth observation indicator at the

ith moment in the indicator:

Calculate the entropy value of the

jth feature observation indicator:

where

k is related to the number of samples n, generally let

k = −1/ln

n.

For the ith moment, a great difference in the ${X}_{ij}$ value corresponds to a great effect on the program evaluation and a small entropy value. Therefore, the difference coefficient of the jth index is calculated as ${g}_{j}=1-{e}_{j}$, a large ${g}_{j}$ index corresponds to its increased importance.

The weight of each observation indicator is:

The weight set constructed to reflect the relative importance of each observation indicator is:

## 6. Results

To address the problem of wind turbine gearbox failure, this paper builds a stochastic differential equation model of gearbox state transition, which can accurately predict the equipment state and maintenance time. In the model, the Weibull model and polynomial approximation were used to construct the failure rate function of the equipment, and Brownian motion was used to simulate the external random disturbance, which integrates the relationship between the equipment’s own degradation and external interference. At the same time, the accuracy of the model were verified by comparing with the ordinary differential model and the gamma distribution model. In the case analysis, the validity of the model was verified by predicting the maintenance time. Hence the state transition model constructed by stochastic differential equations is more accurate and effective. Clearly, the application of this model has a positive effect on the development of CBM.