Correlation Investigation of Wind Turbine Multiple Operating Parameters Based on SCADA Data

Abstract

The primary wind turbines’ in-service performance evaluation method is mining and analyzing the SCADA data. However, there are complex mathematical and physical relationships between multiple operating parameters, and so far, there is a lack of systematic understanding. To solve this issue, the distribution of wind turbines’ operating parameters was first analyzed according to the characteristics of the energy flow of wind turbines. Then, the correlation calculation was performed using the Spearman correlation coefficient method based on the minute-level data and second-level data. According to the numerical characteristics of the nacelle vibration acceleration, the data preprocessing technology sliding window maximum (SWM) was proposed during the calculation. In addition, taking temperature correlation as an example, two-dimensional scatter (including single-valued scatter) and three-dimensional scatter features were combined with numerical analysis and physical mechanism analysis to understand the correlation characteristics better. On this basis, a quantitative description model of the temperature characteristics of the gearbox oil pool was constructed. Through this research work, the complex mathematical and physical relationships among the multi-parameters of the wind turbines were comprehensively obtained, which provides data and theoretical support for the design, operation, and maintenance.

1. Introduction

Energy is the basis for human survival and development and the basic driving force for social development. Especially since the Industrial Revolution, the global energy demand has grown rapidly, causing an energy supply crisis. In addition, the massive consumption of fossil energy has brought serious environmental pollution problems to human beings. In this context, the wind power industry has received great attention and developed rapidly [1,2,3,4]. With the development of the wind power industry, wind turbines are becoming bigger and bigger, and the blades are becoming longer and longer. After the large-scale development of wind turbines, the evaluation of their performance, especially the performance of the entire unit, has to rely on the field operating data because the laboratory can support less and less data. For example, for a 5 MW wind turbine, the diameter of its rotor exceeds 100 m, which obviously cannot be tested indoors [5].

Wind turbines will experience a variety of working conditions during operation. How to evaluate whether a wind turbine is in the best state under different working conditions is an important issue to be solved. Due to the coupling of aerodynamic, structural, and electrical, the state evolution is the result of a combination of factors. Currently, large-scale wind turbines are generally equipped with supervisory control and data acquisition (SCADA) systems, and important operating parameters are monitored in real-time and stored in the database [6]. Based on SCADA data, condition monitoring and fault diagnosis of wind turbines can be carried out [6,7,8]. The use of SCADA data to monitor wind turbines has recently attracted increasing interest in the diagnostics community because of the very low cost and the availability of large amounts of data without any additional sensors [9]. Usually, SCADA data is combined with other algorithms, such as the convolutional neural network (CNN), the long-term and short-term memory network (LSTM), the deep neural network (DNN), etc. [10,11,12]. Y. Pang et al. presented a novel Spatio-temporal fusion neural network (STFNN) for the fault diagnosis of wind turbines [13]. H. Chen et al. introduce a method based on LSTM and Autoencoder (AE) neural network to evaluate continuous condition monitoring data of wind turbines [14]. In addition, the use of SCADA data can also identify some physical changes of components on wind power, such as the identification of blade icing [15]. R. K. Pandit et al. present a Gauss Process-based power curve incorporating the blade pitch angle and rotor speed to evaluate their impact on Gauss Process model accuracy [16]. The current focus is also on wind turbine performance evaluation and energy management using SCADA data [17,18,19,20]. For example, offshore wind power forecasting is a challenging business due to the harsh environment. Z. Lin et al. built a deep learning neural network (DLNN) to predict wind power based on high-frequency SCADA data with a sampling rate of 1-s [21]. X. Gao et al. studied the power generation of four wind turbines by combining wake and topography effects with field measurements and SCADA data [22].

In the process of using SCADA state parameters to carry out research, it must involve the analysis of state parameter selection and its correlation. P. Sun et al. analyzed the typical operating parameters of wind turbines recorded in the SCADA system. These conditional parameters can be divided into two categories: Category 1 parameters include various component temperatures, output power, and rotor speed, which are greatly affected by environmental conditions. Type 2 parameters include pitch angle error, hydraulic oil pressure, and yaw angle error [23]. Y. Zhao et al. investigated 58 parameters collected in SCADA systems of wind turbines and divided these parameters into four categories: condition parameters, health parameters, performance parameters, and control parameters [24]. In addition to the clear physical relationship between some parameters in the wind turbine SCADA data, some parameters do not have a clear mutual physical mapping relationship. So far, the global correlation characteristics of operating parameters have not been established. Since the amount of SCADA data is vast, the relationship between parameters is complex, so it is necessary to further mine and analyze the correlation characteristics. To solve this issue, the distribution of operating parameters of wind turbines will be analyzed in the following sections. In Section 2, the structure and operation condition of wind turbines are introduced, which is the basis of the correlation investigation. In Section 3, the operating parameters of the wind turbine are classified and analyzed according to the energy flow characteristics. Then in Section 4, the correlation of operating parameters of the wind turbine is analyzed. A quantitative description of correlation characteristics is presented with a case on the temperature in Section 5, and the conclusions are given in Section 6. The overall research workflow chart is shown in Figure 1 and the key scientific contributions of this paper are as follows.

• SCADA data with multiple sampling frequencies were used to compare and study the correlation characteristics of wind turbines;
• The distribution of operating parameters of wind turbines was first analyzed according to the characteristics of the energy flow of wind turbines;
• The correlation characteristics between multi-state parameters of wind turbines were investigated. The understanding of the correlation characteristics between the state parameters of wind turbines was enriched and deepened;
• The sliding window maximum (SWM) algorithm was proposed for vibration data preprocessing;
• Using the correlation coefficient calculated by statistics, combined with the physical mechanism of the wind turbine, taking the oil temperature parameter of the gearbox as an example, the state parameters related to the oil temperature of the gearbox were screened, and a quantitative description model of the temperature characteristics was constructed.

2. Structure and Operation Condition of Wind Turbines

Wind turbines are complex electromechanical systems composed of blades, hub, transmission system, generator, tower, foundation, and other components, converting wind energy into mechanical energy and electrical energy [25,26]. There are three types of large-scale wind turbines: double-fed, direct-drive, and half-direct-drive. Figure 2a shows the typical structure of doubly-fed wind turbines which is selected to conduct the subsequent research. The parameters of the wind turbine are listed in

Table 1. Technical parameters of the wind turbine.

Parameters	Value
Rated output power (kW)	2100
Cut-in speed (m/s)	3
Cut-out speed (m/s)	20
Rated wind speed (m/s)	9.5
Rated rotor speed (rpm)	13.3
Gearbox transmission ratio	90:1

. The low-speed rotation of the wind rotor (blade and hub) is converted into the high-speed rotation of the generator rotor through the speed-increasing gearbox. The rotating speed of the wind rotor is usually below 20 r/min, and the rotating speed is above 1000 r/min through the speed-increasing gearbox [27]. The structural advantage of doubly-fed wind turbines is that the generator can be smaller. However, the introduction of the speed-increasing gearbox increases the system’s failure rate. By comparison, the structural advantage of direct-drive wind turbines is that it omits the speed-increasing gearbox, and the rotating speed of the wind rotor is that of the generator rotor. As a result, the rotating speed of the generator is reduced, but it must be much larger than that of the doubly-fed generator under the same power condition. Half-direct-drive wind turbines can be considered a combination of the two, with the expected advantages.

Figure 2b shows the wind speed-power relationship of wind turbines, which essentially describes the operation state of wind turbines. When the wind speed is above the cut-in wind speed (generally about 3 m/s), the power of wind turbines increases gradually with the increase of the wind speed. After the wind speed reaches the rated wind speed, the rated power remains unchanged. According to the existing research results, Equation (1) is generally used to describe the relationship between the two [28].

$$P=\frac{1}{2}\rho v^{3}S{C}_P$$

(1)

where, P is the power of wind turbines absorbing wind energy (W), $\rho$ is the air density (kg/m³), $v$ is wind speed (m/s), $S$ is swept area of wind rotor (m²), $C_P$ is the power coefficient.

According to the curve characteristic of wind speed vs. power, the operation state of wind turbines can be divided into two regions: the region below rated power and the region above rated power [29]. Figure 2c shows the relationship between the rotor speed and output power. There is no effective functional expression to describe the relationship between the two. The operation state of wind turbines can be divided into four regions in Figure 2c. In region AB, where zero power output is the norm, wind turbines have yet to capture wind energy. Region BC is the start-up transition region. The relationship between the rotor speed and the output power is linear in this region. The corresponding rotor speed range in this region is small, and the power efficiency will rise to the maximum after the end of this region. In region CD, the rotor speed and power are designed to be non-linear to maintain maximum power efficiency, which is also the region with the most significant variation in rotor speed. Region DE is the transition region of wind energy from maximum capture to limited capture [30]. The rotor speed fluctuates in a small range during this region. The pitch system is activated and the aim is to limit the power of wind turbines to the rated value. That is to say, with the increase of wind speed, the rotating speed is unchanged, while the pitch angle becomes larger, which reduces the power efficiency.

It can be seen that wind turbines will experience many kinds of working conditions in operation. How to evaluate whether a wind turbine is in the best working condition is an important problem to be solved in its operation and maintenance. Due to the coupling effect of aerodynamic, structure, and electric components, the condition evolution of wind turbines is the result of the comprehensive effect of many factors. Therefore, the condition (health) evaluation is a rather complex problem [31]. For example, with the increase of service time, the gradual deterioration of the aerodynamic performance of the blade, the demagnetization of the generator, the frictional wear and deformation of the mechanical structure, the change of the mechanical properties of the structural parts, the aging of electrical components, and the looseness of the connectors will cause the operating state to change. Large-scale wind turbines are generally equipped with the SCADA (Supervisory Control and Data Acquisition) system. The operating parameters are collected and stored through the SCADA system, and their performance can be mined from the SCADA data to evaluate energy efficiency, structural safety, and service life (Figure 3). With the extension of the service time of wind turbines, the data stored in the SCADA system is accumulated year by year, reflecting the current operating status of wind turbines and storing the historical service status. Therefore, fully mining the SCADA data information during the service of wind turbines is an important means to evaluate the status of wind turbines.

3. Wind Turbine Operating Parameter Distribution

There are many kinds of operating parameters in the service of wind turbines. To investigate them better, they are classified and analyzed according to energy flow characteristics. In Figure 4, the energy flow is divided into three parts: (1) the main energy flow part, (2) the energy dissipation part, and (3) the energy supply part. First, the path through which wind energy is converted into mechanical energy and then into electrical energy, and then fed into the grid is defined as the main energy flow path. From energy capture, the electrical energy generated by wind turbines is essentially from wind energy (air kinetic energy). In SCADA systems, parameters related to wind energy such as wind speed, wind direction, and ambient temperature are generally collected. When the airflow flows through the wind rotor, it drives the wind rotor to rotate, and the kinetic energy of the air is converted into mechanical energy. The size of the mechanical energy is related to the rotating speed of the rotor and the generated mechanical torque. Due to mechanical inertia, part of the generated mechanical energy is stored on the wind rotor. Then, the mechanical energy is transmitted to the generator through the main drive system, and its basic form is that the wind rotor drives the generator rotor to rotate. If it is a direct-drive wind turbine, the rotating speed of the wind rotor, the rotating speed of the main shaft, and the rotating speed of the generator rotor are the same. If it is a doubly-fed wind turbine, the rotating speed of the wind rotor will be transmitted to the generator after increasing the speed. That is to say, the rotating speed of the generator rotor is much greater than the rotating speed of the wind rotor. The mechanical energy is converted into electrical energy at the generator through electromagnetic induction. Accordingly, the SCADA system will collect the state parameters such as generator speed, active power, and power generation in real-time. The connection mode between the generator and the grid will vary depending on the type of wind turbine. The stator winding is directly connected to the grid for the doubly-fed asynchronous wind turbines, and the rotor winding is connected to the grid through the converter. The frequency, voltage, amplitude, and phase of the rotor winding power supply are adjusted by the inverter according to the operation requirements, which ensures that the wind turbines can achieve constant frequency power generation at different speeds. Due to the AC excitation, the generator and power system are essentially “flexible connections”. The excitation current can be adjusted according to the grid voltage, current, and generator speed, thereby accurately adjusting the generator output current. For direct drive (semi-direct drive) wind turbines, the generator output end is connected to the machine side of the full power converter. The alternating current generated by the generator is rectified into direct current by the machine-side converter, and then inverted by the grid-side part of the converter into alternating current that meets the requirements of the grid, and is directly connected to the grid after boosting. For different wind turbines, the operating parameters of the converters collected by the SCADA system are different. Figure 4 shows a monitoring scheme for the operating state parameters of the wind turbine converter. The monitoring parameters include the DC bus voltage, current and active power of the converter.

Energy dissipation is inevitable in energy conversion and transfer, and energy dissipation occurs in various parts. Energy dissipation is mainly manifested as heat generation. For example, during the rotation of the wind rotor, the friction heat of the bearing will dissipate energy, and the heat of the yaw and pitch unit will also dissipate energy. At the generator, the heat of the generator winding and the heat of the main bearing will dissipate energy. Likewise, the heat generated by the various components within the converter also dissipates energy. Therefore, the temperature monitoring of wind turbine components is one of the important contents of the SCADA system. Energy dissipation also occurs in other forms. For example, the vibration of wind turbines will also dissipate a certain amount of energy, but the form of dissipation is more complicated. In addition, during the operation of wind turbines, some component units need to obtain energy from wind turbines. Taking the yaw system and pitch system as typical examples, when the yaw system is activated to rotate the nacelle to the wind, it is necessary to supply power to the yaw motor. Similarly, when the pitch system is activated to adjust the blade pitch angle, it is also necessary to provide power. These functions are indispensable functions for the normal operation of wind turbines. Therefore, the operating parameters during the pitch and yaw are also the focus of the SCADA system, typically including pitch current, pitch angle, pitch motor power, pitch speed, yaw speed, yaw motor power, yaw brake pressure, yaw angle, etc.

4. Correlation Analysis of Operating Parameters of Wind Turbines

Many operating parameters of wind turbines are not isolated and unrelated, and some parameters are often related or affect each other. For example, wind speed directly impacts rotating speed, power, etc., and rotating speed and power also affect each other. However, the relationship between parameters is very complex. From the existing knowledge, the relationship between the operating parameters of wind turbines can be divided into definitely related, definitely irrelevant, or potentially related. The relationship will change with the change in the operating conditions of wind turbines. For example, the rotating speed and power relationship is a synchronous rise below the rated wind speed. In contrast, the rotating speed and power remain unchanged above the rated wind speed. Therefore, accurately analyzing the relationship between SCADA state parameters is very complicated.

4.1. Correlation Characteristics Analysis Based on Correlation Calculation

4.1.1. Correlation Calculation Method

In statistics, there are three well-known methods for calculating correlation coefficients: Pearson correlation coefficient, Spearman correlation coefficient, and Kendall correlation coefficient [32,33,34]. The three correlation coefficients reflect the changing trend (direction and degree) between the two variables. A value of 0 indicates that the two variables are not correlated, a positive value indicates a positive correlation, and a negative value indicates a negative correlation. The larger the value, the stronger the correlation. The Pearson correlation coefficient, also known as the Pearson product-moment correlation coefficient, is a linear correlation coefficient used to reflect the statistic of the degree of linear correlation between two variables. When calculating the Pearson correlation coefficient, the variable’s standard deviation cannot be 0. The Spearman correlation coefficient is solved according to the ranking position of the raw data, which is a representation that does not consider the limitations of solving the Pearson correlation coefficient. The Kendall correlation coefficient is calculated for categorical variables, also known as the Kendall rank correlation coefficient.

This paper uses the Spearman correlation coefficient to carry out the correlation analysis of the state parameters in SCADA data. Let $X_1$ and $X_2$ be two sets of parameter data in SCADA data, and the calculation process is as follows.

Step 1: Convert the elements $X_{1i}$, $X_{2i}$ in $X_1$ and $X_2$ into the ranks in their respective vectors, denoted as $R(X_{1i})$, $R(X_{2i})$;

Step 2: Calculate the grade difference $d_i$ between $R(X_{1i})$ and $R(X_{2i})$ according to Equation (2);

$$d_i=R(X_{1i})-R(X_{2i})$$

(2)

Step 3: Calculate the correlation between $X_1$ and $X_2$ according to Equation (3);

$${\rho }_j=1-\frac{6{\displaystyle \sum _{i=1}^n{{\displaystyle d}}_i^2}}{n(n^2-1)}$$

(3)

where, ${\rho }_j$ is the correlation coefficient, $n$ is the sample data volume.

Then, external ambient temperature, gearbox oil pool temperature, gearbox high-speed bearing temperature, gearbox low-speed bearing temperature, active power, instantaneous wind speed, rotor speed, yaw angle, pitch angle, generator speed, generator torque feedback, outlet active power, acceleration in the front and rear direction of the nacelle, acceleration in the left and right direction of the nacelle are selected to calculate the correlation coefficient. The original SCADA data should be preprocessed first during the analysis, which mainly includes two aspects.

(1) The null values in the SCADA data are first eliminated, that is, the external interference in the sensing test process is eliminated. At the same time, the normal operation data in the SCADA data is extracted, that is, the data in the start-up process and the shutdown process of wind turbines are eliminated. According to the characteristics of SCADA data, the designed elimination algorithm is as follows:

$$\{\begin{array}{l}\mathrm{Delete} X_i, \mathrm{if} X_i= \mathrm{Null}\\ \mathrm{Delete} X_i, \mathrm{if} X_i= \mathrm{zero}\\ \mathrm{Delete} X_i, \mathrm{if} X_{i\mathrm{P}}\le 20\mathrm{kW}\\ \mathrm{Delete} X_i, \mathrm{if} X_{i\beta }\ge {30}^{\circ }\end{array}$$

(4)

where, “$X_i= \mathrm{Null}$” indicates that all data recorded by the SCADA system at a certain time are null values; “$X_i= \mathrm{zero}$” indicates that all data recorded by the SCADA system at a certain time are zero values; $X_{i\mathrm{P}}$ indicates the power data recorded by the SCADA system at a certain time; $X_{i\beta }$ indicates the pitch angle data recorded by the SCADA system at a certain time.

It needs to be further explained that, from the perspective of SCDA data characteristics, when the power is less than 20 kW, wind turbines are mostly in the start-up stage; during normal operation, the pitch angle is generally not greater than 30 degrees, and such data usually occurs during the shutdown process.

(2) The sliding window maximum (SWM) processing is performed on the nacelle vibration acceleration data. In SCADA data, the vibration acceleration of the nacelle has positive and negative values. This data characteristic makes it difficult to correlate with other parameters. It is better to use the SWM (Figure 5).

Assuming that the vibration sample of the nacelle is A = $\left\{A_{ti},A_{ti+1}, \cdots , A_{ti+n}\right\}$, one has

$$A_{ti}=max\left\{A_{ti},A_{ti+1}, \cdots , A_{ti+h}\right\}$$

(5)

where, $h$ is the sliding window width.

4.1.2. Correlation Analysis Based on Minute-Level Data

During the analysis, the SCADA data of a wind turbine in a wind farm from January to March 2021 is extracted (using a 1-min average). After eliminating the null values and the data in the start-up and shutdown processes, the remaining 77,524 sets of data are obtained.

Table 2. Correlation coefficients of operating parameters under all power conditions (minute-level data).

	Te	To	Th	Tl	Pel	v	na	γ	β	ngen	Tor	Pm	Aa	A1
Te	1.00	−0.04	−0.10	−0.15	−0.27	−0.23	−0.24	0.30	0.01	−0.24	−0.27	−0.28	−0.15	−0.20
To	−0.04	1.00	0.92	0.83	0.52	0.52	0.53	0.04	−0.15	0.53	0.52	0.52	0.29	0.38
Th	−0.10	0.92	1.00	0.95	0.76	0.75	0.77	0.02	−0.13	0.77	0.76	0.76	0.42	0.55
Tl	−0.15	0.83	0.95	1.00	0.87	0.85	0.86	0.02	−0.06	0.86	0.87	0.86	0.48	0.61
Pel	−0.27	0.52	0.76	0.87	1.00	0.99	0.97	−0.03	0.03	0.98	1.00	1.00	0.54	0.70
v	−0.23	0.52	0.75	0.85	0.99	1.00	0.96	−0.05	0.05	0.96	0.99	0.99	0.53	0.69
na	−0.24	0.53	0.77	0.86	0.97	0.96	1.00	−0.02	−0.07	0.99	0.97	0.97	0.52	0.68
γ	0.30	0.04	0.02	0.02	−0.03	−0.05	−0.02	1.00	0.04	−0.02	−0.04	−0.04	−0.10	−0.09
β	0.01	−0.15	−0.13	−0.06	0.03	0.05	−0.07	0.04	1.00	−0.07	0.03	0.03	0.07	0.03
ngen	−0.24	0.53	0.77	0.86	0.98	0.96	0.99	−0.02	−0.07	1.00	0.97	0.97	0.51	0.68
Tor	−0.27	0.52	0.76	0.87	1.00	0.99	0.97	−0.04	0.03	0.97	1.00	1.00	0.54	0.70
Pm	−0.28	0.52	0.76	0.86	1.00	0.99	0.97	−0.04	0.03	0.97	1.00	1.00	0.54	0.70
Aa	−0.15	0.29	0.42	0.48	0.54	0.53	0.52	−0.10	0.07	0.51	0.54	0.54	1.00	0.66
A1	−0.20	0.38	0.55	0.61	0.70	0.69	0.68	−0.09	0.03	0.68	0.70	0.70	0.66	1.00

(Note: T_e is the external ambient temperature, (°C); T_o is the gearbox oil pool temperature, (°C); T_h is the gearbox high-speed bearing temperature, (°C); T_l is the gearbox low-speed bearing temperature, (°C); P_el is the generator active power, (kW); v is the instantaneous wind speed at the nacelle, (m/s); n_a is the rotor speed, (rpm); γ is the yaw angle, (°); β is the pitch angle, (°); n_gen is the generator speed, (rpm); T_or is the generator torque feedback, (N·m); P_m is the outlet active power, (kW); A_a is the acceleration in the front and rear direction of the nacelle, (m/s²); A₁ is the acceleration in the left and right direction of the nacelle, (m/s²)).

shows the calculation results of the correlation between the operating parameters of wind turbines. The data selection range is under all power conditions. The purpose is to analyze the correlation characteristics between the operating parameters in the entire operating area. From the calculation results, the correlation coefficient between the external ambient temperature and the temperature of the wind turbine components is relatively low, which means that from the statistical data, the impact of the external ambient temperature on the temperature of the wind turbine components is small. However, from the perspective of the physical mechanism, the external ambient temperature must impact the temperature of the wind turbine components. The correlation coefficient is also related to the following two factors. (1) It is related to the change in the external ambient temperature. If the external ambient temperature is relatively constant, the calculation result of the correlation coefficient must be small; (2) It is related to the mutual influence of other factors. If two factors are highly correlated, their correlation with other factors may be masked. This also shows that when investigating the correlation characteristics between the operating parameters of wind turbines, the calculation results must be combined with the physical mechanism to have a more in-depth and correct understanding. It is difficult to accurately grasp the interaction between parameters only from the mathematical calculation results or the physical mechanism. The correlation coefficients between the gearbox oil pool temperature and the gearbox high-speed bearing temperature, and the gearbox low-speed bearing temperature are relatively large, reaching 0.92 and 0.83, respectively. The calculated correlation coefficients between gearbox oil pool temperature and generator active power, generator torque feedback, and outlet active power are all 0.52. The correlation coefficients between gearbox oil pool temperature, rotor speed, and generator speed are 0.53. This shows that the gearbox oil pool temperature is related to its rotating speed and the transmitted power, which is consistent with its physical mechanism. The greater the motion and power transmitted by the gearbox, the more significant the heat will be. In addition, the gearbox oil pool temperature also has a certain correlation with the wind speed (the correlation coefficient is 0.52). This is because below the rated wind speed, the greater the wind speed, the higher the rotating speed, and the greater the power, so the gearbox oil pool temperature will also be correlated with the wind speed. The correlation coefficients between gearbox oil pool temperature and yaw angle, pitch angle, and nacelle vibration acceleration are all small, indicating that the gearbox oil pool temperature has little correlation with them. The gearbox’s high-speed bearing temperature and low-speed bearing temperature are related to the gearbox oil pool temperature and related to parameters such as generator active power, wind speed, rotor speed, generator torque feedback, and outlet active power.

In Table 2, generator active power has a great correlation with wind speed, rotor speed, and generator speed, and the correlation coefficient is in the range of 0.97–0.99. The generator active power has a correlation coefficient of 1 with generator torque feedback and outlet active power. Wind turbine speed change and power change are essentially the influence of wind speed, so from the calculation results, wind speed has a correlation coefficient close to 1 with rotor speed, generator speed, and wind turbine power, respectively. Wind turbines use “speed-power” curve control as the control strategy, and the rotor speed has a correlation coefficient close to 1 with generator speed, generator torque feedback, and generator power. In all power conditions, the yaw angle has a small correlation coefficient with the pitch angle and other parameters. It cannot be considered that there is no correlation between them based on the numerical results because the large-scale wind turbines use yaw control to limit the wind direction deviation within a small range; the pitch angle will only change with the wind speed or power when it is above the rated power. Also, the characteristics of the data used for the calculation need to be considered. The parameters in Table 2 use one-minute mean values, which are acceptable for slow variables. However, once the pitch angle is commanded to change, it will act quickly, and the minute-level data can hardly reflect the essential characteristics of its change. Acceleration in the front and rear direction of the nacelle and generator active power, wind speed, rotor speed, generator speed, generator torque feedback, and outlet active power all have correlation coefficient calculation results of more than 0.5. It should be noted here that the vibration acceleration in Table 2 still adopts the one-minute mean value and is processed according to Equation (5). The acceleration in the left and right direction of the nacelle has a correlation coefficient of over 0.6 for these parameters.

Table 3. Correlation coefficients of operating parameters below the rated power (minute-level data).

	Te	To	Th	Tl	Pel	v	na	γ	β	ngen	Tor	Pm	Aa	A1
Te	1.00	−0.04	-0.08	−0.13	−0.24	−0.20	−0.22	0.29	0.12	−0.21	−0.25	−0.24	−0.10	−0.15
To	−0.04	1.00	0.93	0.87	0.56	0.55	0.56	0.03	−0.32	0.57	0.56	0.56	0.27	0.38
Th	−0.08	0.93	1.00	0.97	0.78	0.76	0.78	0.02	−0.41	0.78	0.78	0.78	0.38	0.52
Tl	−0.13	0.87	0.97	1.00	0.85	0.83	0.85	0.01	−0.44	0.85	0.85	0.85	0.41	0.56
Pel	−0.24	0.56	0.78	0.85	1.00	0.98	0.98	−0.04	−0.46	0.99	1.00	1.00	0.46	0.65
v	−0.20	0.55	0.76	0.83	0.98	1.00	0.97	−0.06	−0.43	0.97	0.98	0.98	0.44	0.63
na	−0.22	0.56	0.78	0.85	0.98	0.97	1.00	−0.02	−0.48	0.99	0.98	0.98	0.45	0.64
γ	0.29	0.03	0.02	0.01	−0.04	−0.06	−0.02	1.00	0.05	−0.02	−0.05	−0.04	−0.11	−0.11
β	0.12	−0.32	−0.41	−0.44	−0.46	−0.43	−0.48	0.05	1.00	−0.48	−0.46	−0.46	−0.18	−0.28
ngen	−0.21	0.57	0.78	0.85	0.99	0.97	0.99	−0.02	−0.48	1.00	0.98	0.99	0.45	0.64
Tor	−0.25	0.56	0.78	0.85	1.00	0.98	0.98	−0.05	−0.46	0.98	1.00	1.00	0.46	0.65
Pm	−0.24	0.56	0.78	0.85	1.00	0.98	0.98	−0.04	−0.46	0.99	1.00	1.00	0.46	0.65
Aa	−0.10	0.27	0.38	0.41	0.46	0.44	0.45	−0.11	−0.18	0.45	0.46	0.46	1.00	0.62
A1	−0.15	0.38	0.52	0.56	0.65	0.63	0.64	−0.11	−0.28	0.64	0.65	0.65	0.62	1.00

shows the calculation results of the correlation coefficient between the operating parameters under the rated power. Compared with the calculation results in Table 2, the correlation coefficients among the operating parameters do not change much except for the pitch angle. For example, under all power conditions, the correlation coefficient between generator speed and wind speed is 0.96, and below the rated power, the correlation coefficient is 0.97. The correlation coefficient between the pitch angle and each operating parameter varies greatly. For instance, under all power conditions, the correlation coefficient between pitch angle and generator active power is 0.03, and below the rated power, the correlation coefficient is −0.46. This does not mean that the two are more correlated under the rated power, and further analysis is required in conjunction with the physical mechanism [35]. Theoretically, when the wind turbines are below the rated wind speed, the pitch angle remains unchanged to capture the maximum wind energy. However, from the actual SCADA data, the initial value of blade pitch angle is not the same in different periods [29]. Therefore, it can be considered that this calculation result is not consistent with its physical nature and needs to be paid extra attention.

Table 4. Correlation coefficients of operating parameters above rated power (minute-level data).

	Te	To	Th	Tl	Pel	v	na	γ	β	ngen	Tor	Pm	Aa	A1
Te	1.00	0.24	0.17	0.43	−0.08	0.02	0.02	0.41	0.06	0.04	0.03	−0.36	−0.07	−0.10
To	0.24	1.00	0.93	0.80	−0.01	0.20	0.02	0.08	0.23	0.04	0.02	−0.11	0.04	0.05
Th	0.17	0.93	1.00	0.71	0.00	0.14	0.02	0.03	0.16	0.03	0.03	−0.07	0.01	0.03
Tl	0.43	0.80	0.71	1.00	0.00	0.21	0.03	0.17	0.25	0.06	0.04	−0.16	−0.02	0.00
Pel	−0.08	−0.01	0.00	0.00	1.00	0.32	0.83	−0.04	0.35	0.88	0.35	0.88	−0.02	−0.03
v	0.02	0.20	0.14	0.21	0.32	1.00	0.26	−0.20	0.90	0.28	0.24	0.23	0.15	0.09
na	0.02	0.02	0.02	0.03	0.83	0.26	1.00	−0.01	0.27	0.93	0.16	0.70	−0.03	−0.04
γ	0.41	0.08	0.03	0.17	−0.04	−0.20	−0.01	1.00	0.05	−0.01	0.06	−0.13	−0.02	−0.02
β	0.06	0.23	0.16	0.25	0.35	0.90	0.27	0.05	1.00	0.29	0.31	0.26	0.21	0.16
ngen	0.04	0.04	0.03	0.06	0.88	0.28	0.93	−0.01	0.29	1.00	0.18	0.71	−0.03	−0.04
Tor	0.03	0.02	0.03	0.04	0.35	0.24	0.16	0.06	0.31	0.18	1.00	0.32	0.05	0.04
Pm	−0.36	−0.11	−0.07	−0.16	0.88	0.23	0.70	−0.13	0.26	0.71	0.32	1.00	0.01	0.02
Aa	−0.07	0.04	0.01	−0.02	−0.02	0.15	−0.03	−0.02	0.21	−0.03	0.05	0.01	1.00	0.45
A1	−0.10	0.05	0.03	0.00	−0.03	0.09	−0.04	−0.02	0.16	−0.04	0.04	0.02	0.45	1.00

is the correlation coefficient between the operating parameters above the rated power. Compared with the results of the correlation coefficients of the operating parameters under all power conditions in Table 2, the correlation coefficients between the external ambient temperature and the parameters such as gearbox high-speed bearing temperature, gearbox low-speed bearing temperature, and gearbox oil pool temperature have certain changes. In particular, the change of the correlation coefficient between the external ambient temperature and the gearbox low-speed bearing temperature is obvious. Under all power conditions, its value is −0.15, and above rated power, its value is 0.43. In theory, the external ambient temperature influences the temperature of each component inside the wind turbines, but under the combined effect of multiple factors, the influence of one factor may be covered by the influence of other factors. In the working condition above the rated power, the influence of the external ambient temperature on the gearbox low-speed bearing temperature is more obvious. It needs to be further explained that the bearing temperature is essentially caused by frictional heat. Above the rated power, the rotor speed and power are constant (ignoring the influence of the bearing lubrication state change), which means that the bearing friction state is stable. In this case, the influence of the external ambient temperature is naturally more apparent. Another notable phenomenon is that under all power conditions, the correlation coefficients between gearbox oil pool temperature and parameters such as generator active power, wind speed, rotor speed, generator speed, generator torque feedback, and outlet active power are all greater than 0.5. Above the rated power, these correlation coefficients are close to 0. This is because, under this working condition, the wind turbines’ rotor speed and power are unchanged. Although the wind speed changes, the friction state of the bearing does not change much, and the change in the gearbox oil pool temperature is caused by the heating and heat transfer of the bearings, gears, and other components. Similar phenomena can also be seen from the correlation coefficients between gearbox high-speed bearing temperature, gearbox low-speed bearing temperature, and these operating parameters, which further shows that the preceding analysis is correct.

Above the rated power, the correlation coefficient between generator active power and wind speed is significantly lower than that under all power conditions. This is mainly because below the rated wind speed, the generator power increases with the increase of the wind speed, while above the rated wind speed, the generator power remains near the rated value and does not increase with the increase of the wind speed. For the same reason, the correlation coefficient between generator active power and rotor speed decreases above the rated power. However, the correlation coefficient between generator active power and pitch angle has increased. Under all power conditions, the correlation coefficient between the two is 0.03, and above the rated power, the correlation coefficient is 0.35. Since the table uses minute-level data, more accurate analysis needs to be further in-depth. Likewise, the correlation coefficient between wind speed and pitch angle reaches 0.9 above rated power (0.05 under all power conditions).

An interesting phenomenon worth noting is that the correlation coefficient between the yaw angle and each parameter is very small in any working condition. In addition, under all power conditions, acceleration in the front and rear direction of the nacelle and acceleration in the left and right direction of the nacelle have correlation coefficients greater than 0.5 with generator active power, wind speed, rotor speed, generator speed, generator torque feedback, and outlet active power. Above rated power, these values are all close to 0. Vibration is inherently caused by varying loads (constant loads do not induce vibration), and the correlation calculations also show that vibration analysis is a complex problem that deserves further exploration.

4.1.3. Correlation Analysis Based on Second-Level Data

To better analyze the correlation characteristics between various operating parameters of wind turbines, based on minute-level data analysis, some second-level data (a total of 772,302 groups) in the same period were extracted for analysis (

Table 5. Correlation coefficients of operating parameters below the rated power (second-level data).

	Pel	v	na	γ	β	ngen	Tor	Pm	Aab	A1b	Aa	A1
Pel	1.00	0.88	0.92	−0.11	−0.40	0.93	1.00	1.00	0.00	0.00	0.28	0.63
v	0.88	1.00	0.83	−0.13	−0.36	0.83	0.88	0.88	0.01	0.00	0.22	0.54
na	0.92	0.83	1.00	−0.10	−0.46	0.96	0.92	0.92	0.00	−0.01	0.26	0.61
γ	−0.11	−0.13	−0.10	1.00	0.00	−0.10	−0.11	−0.11	0.01	0.00	0.02	−0.04
β	−0.40	−0.36	−0.46	0.00	1.00	−0.47	−0.40	−0.40	0.00	0.00	−0.10	−0.28
ngen	0.93	0.83	0.96	−0.10	−0.47	1.00	0.93	0.92	0.01	0.01	0.26	0.61
Tor	1.00	0.88	0.92	−0.11	−0.40	0.93	1.00	1.00	0.00	0.00	0.28	0.63
Pm	1.00	0.88	0.92	−0.11	−0.40	0.92	1.00	1.00	0.00	0.00	0.28	0.63
Aab	0.00	0.01	0.00	0.01	0.00	0.01	0.00	0.00	1.00	0.27	−0.01	0.00
A1b	0.00	0.00	−0.01	0.00	0.00	0.01	0.00	0.00	0.27	1.00	0.00	−0.01
Aa	0.28	0.22	0.26	0.02	−0.10	0.26	0.28	0.28	−0.01	0.00	1.00	0.51
A1	0.63	0.54	0.61	−0.04	−0.28	0.61	0.63	0.63	0.00	−0.01	0.51	1.00

(Note: A_ab is acceleration in the front and rear direction of the nacelle (m/s²); A_1b is acceleration in the left and right direction of the nacelle (m/s²); A_a is acceleration in the front and rear direction of the nacelle after the SWM (m/s²); A₁ is acceleration in the left and right direction of the nacelle after the SWM (m/s²)).

and

Table 6. Correlation coefficients of operating parameters above the rated power (second-level data).

	Pel	v	na	γ	β	ngen	Tor	Pm	Aab	A1b	Aa	A1
Pel	1.00	0.17	0.60	−0.03	0.25	0.81	0.31	0.71	0.04	0.01	−0.07	−0.05
v	0.17	1.00	0.17	−0.10	0.67	0.16	0.05	0.15	0.01	0.00	−0.02	−0.04
na	0.60	0.17	1.00	−0.04	0.19	0.69	0.01	0.59	0.02	−0.03	−0.07	−0.05
γ	−0.03	−0.10	-0.04	1.00	0.03	−0.02	−0.01	−0.04	0.00	0.01	0.05	0.03
β	0.25	0.67	0.19	0.03	1.00	0.24	0.10	0.24	0.01	0.00	0.02	−0.02
ngen	0.81	0.16	0.69	−0.02	0.24	1.00	0.10	0.71	0.04	0.02	−0.08	−0.06
Tor	0.31	0.05	0.01	−0.01	0.10	0.10	1.00	0.08	0.02	0.03	0.00	0.00
Pm	0.71	0.15	0.59	−0.04	0.24	0.71	0.08	1.00	0.05	0.00	−0.07	−0.03
Aab	0.04	0.01	0.02	0.00	0.01	0.04	0.02	0.05	1.00	0.32	−0.01	0.00
A1b	0.01	0.00	−0.03	0.01	0.00	0.02	0.03	0.00	0.32	1.00	0.00	−0.01
Aa	−0.07	−0.02	−0.07	0.05	0.02	−0.08	0.00	−0.07	−0.01	0.00	1.00	0.40
A1	−0.05	−0.04	−0.05	0.03	−0.02	−0.06	0.00	−0.03	0.00	−0.01	0.40	1.00

). The uses of second-level data are mainly for two reasons.

(1) In the minute-level data, the vibration data is also the 1-min average value, and the results of the analysis need to be further demonstrated; the second-level data can better reflect the dynamic characteristics of vibration.

(2) For the wind turbines used in the study, the yaw angle (wind direction deviation) in the SCADA data is recorded from 0 degrees to 360 degrees (sometimes greater than 360 degrees). Therefore, any value in the range of 0 degrees to 360 degrees occurs abundantly in the minute-level yaw angle. In the actual service process of wind turbines, the yaw angle is usually controlled within −20 degrees to 20 degrees. Therefore, it is necessary to use second-level data for data preprocessing and further analyze the correlation between the yaw angle and other operating parameters. The designed data preprocessing algorithm is as follows.

$$\{\begin{array}{l}{X}_{i\gamma }=X_{i\gamma }-{360}^{\circ },{ \mathrm{if} 180}^{\circ }\le X_{i\gamma }\le {360}^{\circ } \\ X_{i\gamma }={360}^{\circ }-X_{i\gamma }, \mathrm{if} X_{i\gamma }\ge {360}^{\circ }\\ \mathrm{Delete} X_{i\gamma }, \mathrm{if} X_{i\gamma }\ge { 20}^{\circ }\mathrm{or} X_{i\gamma }\le -{20}^{\circ }\\ X_{i\gamma }=\left|X_{i\gamma }\right|\end{array}$$

(6)

where, $X_{i\gamma }$ is the yaw angle (wind direction deviation) data in a data record of the SCADA system.

In addition, since the second-level data does not contain temperature data, the correlation coefficient calculation between temperature and other parameters is canceled in the table.

Table 5 shows the correlation coefficients of operating parameters below rated power obtained by using second-level data (sampling frequency: 1 Hz). Compared with the results obtained using minute-level data in Table 3, they are relatively close, and the correlation coefficients between some parameters have some changes. For example, the correlation coefficient between generator active power and wind speed calculated using minute-level data is 0.98, and that calculated using second-level data is 0.88. The correlation coefficient between wind speed and rotor speed calculated using minute-level data is 0.97, and that calculated using second-level data is 0.83. Whether using minute-level data or second-level data, the calculated results of the correlation coefficients between the yaw angle and other parameters are close to 0.

From the results in Table 5, it can also be found that if the nacelle vibration data is not processed with SWM, the correlation coefficients of acceleration in the front and rear direction of the nacelle and acceleration in the left and right direction of the nacelle with other operating parameters are close to 0. After the SWM processing, the correlation coefficients between them and other operating parameters have increased significantly, which also shows that the SWM processing is necessary. It can also be seen from the results that compared with acceleration in the front and rear direction of the nacelle, acceleration in the left and right direction of the nacelle has a greater correlation coefficient with other parameters.

Table 6 shows the correlation coefficients of the operating parameters above the rated power obtained by using second-level data (sampling frequency: 1 Hz). Under this working condition, the correlation coefficient between generator active power and wind speed is small, the result calculated using second-level data is only 0.17, and the correlation coefficient with pitch angle is 0.25. The correlation coefficient between wind speed and rotor speed is 0.17 because, at this stage, the rotation speed will not change with the change of wind speed, but the pitch angle will adjust accordingly. It can also be seen from the table that the correlation coefficient between wind speed and pitch angle is 0.67 (the calculation result using minute-level data is 0.90). Compared with the influence of other parameters on the pitch angle, the influence of wind speed is the most important at this stage. Whether using minute-level data or second-level data, at this stage, the correlation coefficients between other parameters and nacelle vibration are relatively small. It is understood from the physical essence that the fluctuation of wind speed will be an important factor affecting the vibration of the nacelle, whether it is a working condition above the rated power or a working condition below the rated power. It should be noted that in the SCADA data, fluctuations in wind speed are not recorded and quantified.

4.2. Correlation Characteristic Analysis Based on Scatter Distribution—A Case on Temperature Correlation Analysis

4.2.1. Analysis of Temperature Correlation Based on Two-Dimensional Scatter

By using SCADA data to plot the scatter distribution among the relevant operating parameters, the distribution characteristics of the operating parameters under multiple operating conditions can be visualized. Combined with the results of correlation calculation, the correlation properties between operating parameters can be better understood. For a long time, there have been a lot of research reports on the performance analysis of wind turbines based on scatter characteristics of operating parameters, especially on the relationship between wind speed and power, rotor speed and power, wind speed, and rotor speed [18,21,36]. The relationship analysis of basic operating parameters such as wind speed, rotor speed, power, etc., has been relatively sufficient. Due to space limitations, this section mainly analyzes the temperature characteristics, that is, analyzes the scatter relationship between temperature parameters and other operating parameters. In the SCADA system, there are multiple sets of data recorded about temperature, and multiple components have records of temperature parameters. In this section, the oil pool temperature of the main drive system (gearbox) is selected as the observation parameter for analysis.

From the calculation results in Table 2, it can be seen that the parameters whose correlation with the oil pool temperature is more than 0.5 include gearbox high-speed bearing temperature, gearbox low-speed bearing temperature, generator active power, instantaneous wind speed, rotor speed, generator speed, generator torque feedback, outlet active power. The gearbox oil pool temperature, gearbox high-speed bearing temperature, and gearbox low-speed bearing temperature all reflect the heating characteristics of the gearbox in essence. To simplify the analysis, the oil pool temperature is chosen to represent the heating characteristics of the gearbox. Both generator active power and outlet active power essentially reflect the energy output characteristics, so one of them can be selected for analysis, for example, generator active power. The generator speed is higher than or equal to the rotor speed (there is a difference between double-fed wind turbines and direct-drive wind turbines). It is essentially the result of the rotor speed passing through the main drive system so that it can be analyzed with only one of them as a representative. In addition, generator torque feedback and generator power are indirect characterization parameters, and it is feasible to choose only one of the two. Therefore, the operating parameters selected for the correlation analysis of gearbox oil pool temperature mainly include three parameters: wind speed, rotor speed, and power. From the calculation results in Table 2, although the correlation between the external ambient temperature and the gearbox oil pool temperature is small, it can be known from the perspective of the physical mechanism that the external ambient temperature will affect the gearbox oil pool temperature. Therefore, the external ambient temperature is also considered an influencing parameter.

Figure 6 shows the two-dimensional scatter between the gearbox oil pool temperature and other operating parameters of wind turbines. Figure 6a–c shows the scatter relationship between power and gearbox oil pool temperature, which are three working conditions: all power conditions, above-rated power, and below-rated power. From the scatter distribution, below the rated power, the distribution range of gearbox oil pool temperature is wider, especially when the power is low, the distribution of gearbox oil pool temperature is the widest. Above the rated power, the gearbox oil pool temperature distribution range is relatively uniform. The wind turbines are in the initial stage of operation in the start-up stage, and the gearbox oil pool temperature is greatly affected by the ambient temperature. After running for a certain period to reach thermal equilibrium, its value will fluctuate within a certain range. It can also be seen from the figure that the upper contour line of the scatter distribution is almost straight. It can provide a good criterion for operating state judgment based on normal service data.

Figure 6d–f shows the scatter relationship between wind speed and oil pool temperature, where the wind speed is the data collected by the nacelle anemometer. It can be seen that the scatter relationship between wind speed and temperature is similar to the scatter relationship between power and temperature. When the wind speed is low, the distribution of oil pool temperature is wider, while under the condition of high wind speed, the distribution of oil pool temperature is more concentrated. Essentially, the power characteristics of wind turbines depend on the characteristics of wind speed. When the wind speed is low, the wind turbines are in the initial stage of the startup, and the oil pool temperature is greatly affected by the ambient temperature, and it is gradually increasing. Under all power conditions, the oil pool temperature has a correlation coefficient of 0.52 with power and wind speed (Table 2). Below the rated power, the oil pool temperature has correlation coefficients of 0.56 and 0.55 with power and wind speed, respectively (Table 3). The two are also very close. Above the rated power, the oil pool temperature has correlation coefficients of −0.01 and 0.2 with power and wind speed, respectively (Table 4). From the overall distribution trend of scatter, the temperature is relatively stable above the rated power, whether with the increase of power or the increase in wind speed. However, it needs to be further explained that the wind speed has a relatively large increase above the rated power, but the power increase space is limited, and it is controlled near the rated power, or it can be said that the power is kept constant. In this way, no matter how the wind speed rises, the energy transmitted by the transmission system is the same, so there is a phenomenon that the correlation coefficient is close to 0.

Figure 6g–i shows the scatter relationship between the rotor speed and the oil pool temperature. In all power conditions, with the increase of the rotor speed, the oil pool temperature shows a certain upward trend as a whole, and the correlation coefficient between the two is 0.53 (Table 2). Above the rated power, the scatter distribution is concentrated in a certain range, and the correlation coefficient between the oil pool temperature and the rotor speed is 0.02 (Table 4). In fact, above the rated power, the rotor speed is controlled within a certain range or approximately kept constant, so the speed change at this stage is not the main factor affecting the temperature. Figure 6j–l shows the multi-condition scatter relationship between the external ambient temperature and the oil pool temperature. From the scatter distribution, there is no obvious trend feature. In all power conditions, the correlation coefficient between the two is −0.04. Obviously, from the point of view of physical mechanism, the change of ambient temperature will inevitably affect the change of gearbox oil pool temperature. However, neither the calculated results of the correlation coefficient nor the scatter distribution reflect this effect. This is because the change of the external ambient temperature is a relatively slow variable, and under the influence of factors such as wind speed, power, and rotor speed, the influence of the ambient temperature change is masked.

To further analyze the influence of external ambient temperature, Figure 7 shows the multi-condition scatter distribution of power and gearbox oil pool temperature when the external ambient temperature is greater than 10 degrees (Figure 7a–c), and when the external ambient temperature is less than 10 degrees, the multi-condition scatter distribution (Figure 7d–f). To more accurately analyze its distribution characteristics, the scatter distribution is processed by the kernel density estimation method. The processing steps are as follows.

Step 1: Power interval dividing. According to the power region, a number of points (P₁, P₂,..., P_N) are selected as the basic points of single-valued processing.

Step 2: Extracting the oil pool temperature data corresponding to a certain power point. Considering that there is less temperature data corresponding to a certain power point, the temperature data corresponding to a certain power interval is extracted. For example, when extracting the oil pool temperature data corresponding to the power P₂, data T₁, T₂,..., T_n corresponding to the power interval ${[\mathrm{P}}_2-\mathrm{\Delta }\mathrm{P},{\mathrm{P}}_2+\mathrm{\Delta }\mathrm{P}]$ are actually extracted.

Step 3: Calculating the probability distribution density of oil pool temperature based on kernel density estimation. The calculation expression is [37]:

$$f_h(x)=\frac{1}{nh}{\displaystyle \sum _{i=1}^{n}K\left(\frac{x-X_i}{h}\right)}$$

(7)

where, $X_i$ is oil pool temperature, $K\left(\right)$ is the kernel density function, and h is the window width.

For example, after extracting the oil pool temperature data T₁, T₂,..., T_n corresponding to the power interval ${[\mathrm{P}}_2-\mathrm{\Delta }\mathrm{P},{\mathrm{P}}_2+\mathrm{\Delta }\mathrm{P}]$, the probability of each temperature value is calculated according to Equation (7).

Step 4: According to the density distribution curve, the temperature data point with the highest probability is extracted and used as the single-point temperature data corresponding to the power.

Step 5: Repeating Step 2 to Step 4. Then, the single-point oil pool temperature data corresponding to each selected power point are extracted. Subsequently, these points are drawn as a curve, which is the singularization curve of scatter.

Although it is difficult to see the effect of different ambient temperatures on oil pool temperature from the two-dimensional scatter plot comparison, it is clear from the single-valued results that when the ambient temperature is higher, oil pool temperature at the same power is higher overall, especially above rated power.

4.2.2. Analysis of Temperature Correlation Based on Three-Dimensional Scatter

According to the previous analysis, wind speed, rotor speed, and power are the main factors affecting the gearbox oil pool temperature (bearing temperature and oil pool temperature are regarded as different manifestations of gearbox temperature). To better analyze the correlation characteristics of gearbox oil pool temperature with wind speed, rotor speed, and power, Figure 8 shows the three-dimensional scatter analysis of oil pool temperature characteristics.

Figure 8a–c shows the multi-condition scatter relationship between wind speed, rotor speed, and oil pool temperature. It can be seen that the three-dimensional scatter distribution also has obvious spatial distribution characteristics. In the low wind speed stage, the oil pool temperature scatter distribution is broader, and in the high wind speed stage, the oil pool temperature scatter mainly concentrates in the interval (50 °C, 60 °C). In the low wind speed stage, as the wind speed increases, the rotor speed gradually increases. When the wind speed exceeds the rated wind speed, the rotor speed remains unchanged and fluctuates in a small range (around 1210 rpm). Figure 8d–f shows the multi-condition scatter relationship between wind speed, power, and oil pool temperature. As the wind speed increases, the power gradually increases; when the wind speed is above the rated wind speed, the power changes slightly around the rated value of 2000 kW. When the power is small, the oil pool temperature scatter is wider, and when the power increases to 1000 kW (approximate value), the oil pool temperature is more concentrated. This means that the gearbox is basically in thermal equilibrium above this power. Figure 8g–i show the multi-condition scatter relationship between rotor speed, power, and oil pool temperature. Based on the scatter distribution feature, it can be seen that the oil pool temperature changes with power and rotor speed. Using the oil pool temperature feature is also more conducive to judging the relationship between rotor speed and power. Specifically, when the oil pool temperature is at the upper-temperature limit, it means that the wind turbines are in a more stable equilibrium state, and the “speed-power” curve at this time is closer to the actual situation of the wind turbines.

5. Quantitative Description of Correlation Characteristics—A Case on Temperature

To better analyze the correlation characteristics between the operating parameters of the wind turbines, this section takes the gearbox oil pool temperature as an example to establish the relationship model between the gearbox oil pool temperature and its associated parameters. There is a very vast literature about data-driven modeling of wind turbine sub-components’ temperature [38]. Most of the literature focus on fault diagnosis [39,40]. Here, the purpose of modeling is different, mainly to better understand the correlation characteristics between operating parameters. First, the generalized expression is given as follows.

$$T_{\mathrm{O}}=f(T_{\mathrm{e}},T_{\mathrm{h}},T_{\mathrm{l}},p_{\mathrm{el}},v,n_{\mathrm{a}},\gamma ,\beta ,\cdots )$$

(8)

Equation (8) gives a general expression of the relationship between gearbox oil pool temperature and external ambient temperature, gearbox high-speed bearing temperature, gearbox low-speed bearing temperature, generator active power, instantaneous wind speed, rotor speed, yaw angle, pitch angle, etc. According to the calculation results in Table 2, with the correlation coefficient of 0.38 as the critical point, parameters related to the oil pool temperature can be extracted including gearbox high-speed bearing temperature, gearbox low-speed bearing temperature, generator active power, wind speed, rotor speed, generator speed, generator torque feedback and outlet active power, acceleration in the left and right direction of the nacelle. Considering a direct relationship between generator active power and outlet active power, one of the two is selected for quantitative analysis of the correlation. Similarly, there is a direct relationship between rotor speed and generator speed. In addition, from the perspective of the physical mechanism, the external ambient temperature will inevitably affect the gearbox oil pool temperature. Therefore, although the correlation coefficient between the two given in Table 2 is small, the external ambient temperature is still considered a major factor.

Then, the relationship between gearbox oil pool temperature and its associated parameters can be expressed as:

$$T_{\mathrm{O}}=f(T_{\mathrm{e}},T_{\mathrm{h}},T_{\mathrm{l}},p_{\mathrm{el}},v,n_{\mathrm{a}},T_{\mathrm{or}},A_{\mathrm{l}})$$

(9)

to obtain the specific function expression of Equation (9), the quadratic regression analysis equation is introduced as follows:

$$y=b_0+{\displaystyle \sum _{i=1}^8{b}_i{x}_i}+{\displaystyle \sum _{i=1}^7{\displaystyle \sum _{j=i+1}^8{b}_{ij}{x}_i{x}_j}}+{\displaystyle \sum _{i=1}^8{b}_{ii}{x}_i{}^2}$$

(10)

where, $y$ denotes gearbox oil pool temperature T_o; $x_i(i=1,\cdots ,8)$ denotes $T_{\mathrm{e}},T_{\mathrm{h}},T_{\mathrm{l}},p_{\mathrm{el}},v,n_{\mathrm{a}},T_{\mathrm{or}},A_{\mathrm{l}}$, respectively.

In data processing, considering that the magnitude of each state parameter differs greatly, each parameter is subjected to “mean normalization” processing before regression analysis, that is, each parameter variable is dimensionless. The specific processing method is as follows:

$$x_{\mathit{in}}=\frac{x_{\mathit{in}}-\overline{x_i}}{x_{i\max }-x_{i\min }}$$

(11)

where ${\overline{x}}_i$ is the average value of the i-th parameter, $i=1,\cdots ,8$; $n=1,\cdots ,N$ (N is the number of samples).

77,524 groups of data were extracted, and the stepwise regression analysis was carried out. The regression coefficient was obtained by the “least square method”.

Table 7. Quadratic stepwise regression parameter estimation.

Variable	Estimated Value	Standard Error	T-Value	p-Value
constant term	52.548	0.14433	364.07	0
$x_1$	−1.5813	0.026129	−60.521	0
$x_2$	62.813	0.26904	233.47	0
$x_3$	2.7635	0.32131	8.6007	8.0659 × 10−18
$x_4$	34.445	3.6055	9.5532	1.292 × 10−21
$x_5$	3.6202	0.19953	18.144	2.0496 × 10−73
$x_6$	−15.333	1.0086	−15.202	4.0914 × 10−52
$x_7$	−35.432	3.0016	−11.804	3.9559 × 10−32
$x_8$	−0.68524	0.082746	−8.2813	1.2375 × 10−16
$x_{12}$	2.1965	1.1008	1.9954	4.6003 × 10−2
$x_{13}$	−7.3378	1.2131	−6.0487	1.4669 × 10−9
$x_{14}$	−8.5192	1.5624	−5.4526	4.9797 × 10−8
$x_{15}$	−2.2374	0.76858	−2.9111	3.6027 × 10−3
$x_{16}$	0.56283	0.29495	1.9082	5.6366 × 10−2
$x_{17}$	14.55	1.756	8.2863	1.1867 × 10−16
$x_{18}$	1.385	0.36915	3.752	1.7558 × 10−4
$x_{23}$	333.85	10.282	32.471	9.6808 × 10−230
$x_{24}$	245.91	12.77	19.257	1.9076 × 10−82
$x_{26}$	−43.076	2.552	−16.879	8.3262 × 10−64
$x_{27}$	−245.48	13.248	−18.53	1.743 × 10−76
$x_{28}$	8.2258	2.6581	3.0946	1.9714 × 10−3
$x_{34}$	−42.82	13.191	−3.2462	1.1701 × 10−3
$x_{35}$	12.693	2.7292	4.6509	3.3103 × 10−6
$x_{36}$	−5.5556	2.8741	−1.933	5.324 × 10−2
$x_{37}$	64.782	13.768	4.7053	2.5393 × 10−6
$x_{38}$	−13.688	3.1075	−4.4048	1.0602 × 10−5
$x_{45}$	−63.589	11.16	−5.6977	1.2189 × 10−8
$x_{46}$	−28.842	5.2426	−5.5015	3.7774 × 10−8
$x_{47}$	55.003	9.6068	5.7254	1.0355 × 10−8
$x_{48}$	−23.682	5.1202	−4.6252	3.7488 × 10−6
$x_{56}$	−12.878	1.8254	−7.055	1.7402 × 10−12
$x_{57}$	68.116	11.319	6.0177	1.7771 × 10−9
$x_{67}$	11.402	5.8774	1.9399	5.2395 × 10−2
$x_{68}$	6.0297	0.87594	6.8836	5.8791 × 10−12
$x_{78}$	25.56	5.2997	4.8229	1.4176 × 10−6
$x_2^2$	−202.34	4.6432	−43.578	0
$x_3^2$	−97.101	5.8802	−16.513	3.7581 × 10−61
$x_6^2$	7.1562	0.83282	8.5927	8.6524 × 10−18
$x_7^2$	−56.099	9.5619	−5.8669	4.4578 × 10−9
$x_8^2$	0.93965	0.38071	2.4682	1.3583 × 10−2

presents the quadratic stepwise regression’s parameter estimates, T values, and F values.

Since the p values of the F test for items $x_{25}$, $x_{58}$, $x_1^2$, $x_4^2$ and $x_5^2$ are all greater than 0.05 (shown in

Table 8. p-values of excluded variables in stepwise regression.

Variable	F-Value	p-Value
$x_{25}$	0.06862	0.79336
$x_{58}$	0.71827	0.39672
$x_1^2$	0.74045	0.38952
$x_4^2$	0.06862	0.79336
$x_5^2$	0.64201	0.42299

), they are not significant items. These terms were excluded from the stepwise regression analysis. The RMSE obtained by stepwise regression analysis is 1.1, the R-squared and adjusted R-Square are both 0.961, as shown in

Table 9. Stepwise regression analysis of variance.

RMSE	R-Squared	Adjusted R-Square
1.1	0.961	0.961

. Finally, the quadratic regression expression about multiple correlation factors and oil pool temperature is obtained, as shown in Equation (12).

$$\begin{array}{ll}{T}_{\mathrm{O}}& =52.548-1.5813x_1+62.813x_2+2.7635x_3+34.445x_4+3.6202x_5-15.333x_6-35.432x_7\\ & -0.68524x_8+2.1965x_1{x}_2-7.3378x_1{x}_3-8.5192x_1{x}_4-2.2374x_1{x}_5+0.56283x_1{x}_6+14.55x_1{x}_7\\ & +1.385x_1{x}_8+333.85x_2{x}_3+245.91x_2{x}_4-43.076x_2{x}_6-245.48x_2{x}_7+8.2258x_2{x}_8-42.82x_3{x}_4\\ & +12.693x_3{x}_5-5.5556x_3{x}_6+64.782x_3{x}_7-13.688x_3{x}_8-63.589x_4{x}_5-28.842x_4{x}_6+55.003x_4{x}_7\\ & -23.682x_4{x}_8-12.878x_5{x}_6+68.116x_5{x}_7+11.402x_6{x}_7+6.0297x_6{x}_8+25.56x_7{x}_8-202.34x_2{}^2\\ & -97.101x_3^2+7.1562x_6^2-56.099x_7^2+0.93965x_8^2\end{array}$$

(12)

Figure 9 is a fitting curve obtained according to Equation (12). The abscissa in the figure is the observation number of the dependent variable (gearbox oil pool temperature), and the ordinate is the value of the dependent variable. The overall trend of the fitted curve is consistent with the original scatter. However, there will be some differences in the local data, especially in regions where the original temperature changes sharply. In general, the quantitative description equation of oil pool temperature obtained in this paper can better reflect the quantitative relationship between it and related variables and is also beneficial to the design and maintenance of wind turbines.

6. Conclusions

For large wind turbines, the assessment of their status currently relies heavily on SCADA data, which is recorded all the time. A large amount of SCADA data is stored in the database. How to effectively utilize SCADA data is a hot issue at present. Many operating parameters are not isolated, and there is often a certain correlation between them. Finding such correlation characteristics is the premise of using SCADA data efficiently. To gain an understanding of the correlation characteristics of the operating parameters of wind turbines, a combination of mathematical calculation and physical mechanism was adopted. Specifically, the Spearman correlation coefficient was used to calculate the correlation coefficient between multiple operating parameters, and the direct calculation result was obtained. The two-dimensional and three-dimensional scatter distributions of some parameters were also given. The innovation lies in the profound combination of digital analysis, graphical analysis, and physical mechanism, thus forming a new understanding of the global characteristics of wind turbine operating parameters.

This study showed that the correlation coefficient between two operating parameters cannot be judged simply by using the correlation coefficient. For example, under all power conditions, the correlation coefficient between pitch angle and generator active power is 0.03, and below rated power, it is −0.46. This does not mean that the two are more correlated below the rated power, and further analysis is required in conjunction with the physical mechanism. If the nacelle vibration data is not processed with SWM, the correlation coefficient between nacelle acceleration and other operating parameters is relatively small (close to 0). The correlation coefficient was significantly improved after processing. The quantitative description equation of oil pool temperature obtained using quadratic regression can better reflect the quantitative relationship between it and related variables and is also beneficial to the design and maintenance of wind turbines.

To expand this research, further study is needed on the problems revealed in the paper. For example, SCADA data will be further mined to analyze the time evolution characteristics of wind turbine state parameters. SCADA state parameters related to aging will be selected for wind turbine state identification, and a mathematical model (quantitative description) for wind turbine performance aging evaluation will be established.