Analysis of Vibration Characteristics of Horizontal Axis Wind Turbine Tower under Random Wind Action

As the main supporting structure of wind turbines, the service quality of towers directly affects the operational safety of wind turbines. Researching tower vibration characteristics, evaluating its service quality and early warning of damage is an important means to improve the operational safety of wind turbines. Firstly, the external load conditions of the wind turbine tower were determined through wind load calculation, and the load and dynamic theory of the variable pitch torque wind turbine tower were summarized. Secondly, based on the Euler-Lagrange energy equation, the Lagrange equation of the simplified dynamic model of the wind turbine tower is established, and the vibration characteristics of the tower under different wind speeds are analyzed. The research results can provide a theoretical basis for the research on the characteristics of the wind turbine tower.


Introduction
As the main equipment of wind power, wind turbines have received particular attention, and the rapid development of the wind power industry is accompanied by many problems and difficulties that need to be solved.In the industrial field, buildings such as power transmission towers, petrochemical stations, navigation towers, and lighthouses are indispensable for human normal life.These buildings have representative characteristics such as high, large, and small footprint.When many natural disasters such as earthquakes and typhoons occur, on the one hand, due to the large ratio of their own horizontal and vertical dimensions, the overall structure has weak seismic and bending resistance, and on the other hand, due to the strong external load force, the structure is prone to displacement and vibration changes.Wind turbines and the aforementioned high-rise buildings will experience fracture, collapse, and other damage phenomena, as shown in Figure 1.The tower is a key component of wind turbines.Due to the time-varying characteristics of external loads and the harsh working environment, it is inconvenient for real-time monitoring and detection.In addition, due to the time-varying internal and external excitation, the wind turbine tower generates vibration during operation, so it is particularly important to study the dynamic characteristics of the tower more accurately under wind load [1].Load analysis is the most critical foundational work in the design of wind turbines, and it is also the foundation for all subsequent wind turbine analysis and design work [2].The most important and basic load that wind power equipment is subjected to is wind load.Due to the influence of numerous factors such as temperature, air pressure, terrain, and geographical location, wind speed has significant random fluctuations, making wind power generation also characterized by intermittency, random fluctuations, low energy density, and uncontrollability [3].This has a serious impact on the dynamic characteristics, stability, and reliability of wind turbines, which makes it impossible to effectively predict complex wind speed models and wind loads, Making it difficult to calculate and analyze the dynamic and vibration characteristics of wind turbines.

External Load Analysis of Tower
The external load on the wind turbine tower is mainly wind load.Currently, there are two main types of research on wind load.One is to simulate wind speed changes in a short period of seconds or minutes, and the other is to make long-term predictions of wind speed within hours or days.However, most studies on wind turbines use short-term simulation models.The main model methods include combined wind speed method, Auto Recursive Moving Average (ARMA), Kalman filter filter algorithm, duration method, Artificial Neural Network, Fuzzy logic, Spatial, wind speed probability distribution parameter method, least squares support vector method, etc. [5].This topic uses combined wind speed method to simulate wind speed, which can reflect the real characteristics of wind speed.

Determination of Wind Speed
In research, the combined wind load is generally regarded as the combined effect of average wind (static wind) and fluctuating wind (dynamic wind).According to reference [4], a schematic diagram of natural wind characteristics is presented, in which the average wind can be considered as a stable wind, and its characteristics (such as speed and direction) do not change over time.It can be treated as static load during analysis and belongs to static wind.The other part is called pulsating wind, which is caused by the irregularity of the wind and has no regularity.Its intensity changes randomly over time, and the structure will vibrate under the action of pulsating wind [6].According to Davenport's standard theory, natural wind speed is always the combination of average wind speed and fluctuating wind speed, and the calculation formula is shown in equation (1).
The resistance encountered by near ground winds varies across different frictional landforms on the Earth's surface, resulting in different speeds of wind at different altitudes.In normal terrain, the wind speed increases with the increase of height from the ground, because the wind speed is affected by friction, and the magnitude of friction decreases with the increase of height.Therefore, when the distance from the ground reaches a certain height, the wind speed basically does not change with the height.The height reached is called the gradient wind height G z , and the wind speed at the gradient wind height is called the gradient wind speed In the formula, z is the height at which it is located; m () vz is the average wind speed; g ( , ) v z t is a fluctuating wind speed with time-varying characteristics.
The average wind speed profile from the vicinity of the surface to the height of the gradient wind can usually be described by an index, and the specific expression is shown in equation (2).
In the formula, m () vz represents the average wind speed at any height; Z represents any height; G v represents gradient wind speed; G z represents gradient wind height;  is the ground roughness coefficient.
It can be seen from the exponential distribution (2) that the average wind speed profile of various ground roughness categories is equal at their gradient wind height.Therefore, when the average wind speed of the standard ground roughness category at the standard reference height is known, the average wind speed at any height can be determined by equation ( 2) [8] .
According to Chinese regulations, ground roughness is divided into four categories.Table 1 provides the corresponding gradient wind height and ground roughness coefficient values for each category [7].z for this study is 350m, and the ground roughness coefficient  is 0.16.In order to more realistically reflect the wind force that wind turbines may be subjected to, the average value of strong wind speed in the wind power rating table is used as the average wind speed at the standard height for research.In addition, since China generally collects ground meteorological information through meteorological observation towers with a height of 10m, the reference standard height G z is 10m.
According to the working height Z of the wind turbine generator is set as 90m, and further referring to Table 2, the maximum and minimum wind speeds of the possible average wind speeds selected in this paper are calculated as shown in Equations ( 3) and ( 4), and the average wind speed profile is shown in Figure 2.

Establishment of Wind Load Model
Fluctuating wind is a stationary Gaussian Stochastic process with zero mean value, which is usually analysed by random vibration theory.Davenport conducted extensive wind tunnel tests and provided a widely used wind spectrum that depends on average wind speed and terrain roughness.The Davenport spectrum can be expressed as shown in equation ( 5). Where, f is the gust frequency, and k is the surface Drag coefficient (.005  k  0.015).The fluctuating wind speed can be obtained by inputting the white noise signal of unit standard deviation into the approximate filter of the Davenport spectrum.The filter used in this article is a third-order filter, which is adjusted to be within the same bandwidth range as the Davenport spectrum ( 4[10 ,10] − Hz).Select an approximate filter, whose transfer function can be described as shown in equation (6).
On the basis of wind speed, combined with the calculation of average wind speed and wind force, the wind force calculation expression considering the influence of fluctuating wind can be obtained, as shown in equation (7).The combined wind speed was simulated for 100 seconds using MATLAB/Simulink.According to the measured data of a wind farm in Ningxia, VM=12m/s, the gust start time and gust period are set within 10 minutes by the relevant wind farm data.The wind load speed is randomly generated by a random function within 0-100 seconds, and ODE45 is used to solve it.The simulation results of the 100 second combined wind speed are shown in Figure 2.

Calculation of Tower Load under Wind Load
For variable pitch variable speed wind turbines, when the wind speed is higher than the rated value, the speed of the blades is kept constant by changing the pitch angle of the blades.At this time, the output power of the impeller is equal to the rated power; When the wind speed is lower than the rated value, controlling the pitch angle of the impeller can maintain the optimal tip speed ratio of the impeller, keeping its wind energy utilization coefficient at the local maximum value at any wind speed, thereby significantly increasing the power of the wind turbine.Compared with the constant speed operation, the variable-speed operation of wind turbines has greater Mechanical efficiency.Some data show that variable-speed operation can increase the annual power generation of wind turbines by about 5% [8].Based on the above content, the natural wind speed can be divided into four regions [9], as shown in Figure 4: (1) When the wind speed is lower than the cut in wind speed cut in v of the wind turbine, the wind speed is too small to make the wind turbine blades rotate, the blades remain stationary, and the input power of the gearbox is zero.
(2) When the wind speed is higher than the cut in wind speed cut in v and lower than the rated wind speed rate v , by controlling the pitch angle to maintain the optimal tip speed ratio of the impeller, the maximum wind energy utilization coefficient is obtained, enabling the wind turbine to achieve the maximum input power; (3) When the wind speed is higher than the rated wind speed and lower than the cut out wind speed cut off v , the blade speed is kept constant by changing the pitch angle, and the output power of the impeller is equal to the rated power; 6 (4) When the wind speed is higher than the cut out wind speed cut off v , excessive torque will damage the wind turbine, a forced shutdown is required, the blades remain stationary, and the input power of the gearbox is zero.According to the data provided by the Wind farm as shown in Table 3, the piecewise function relationship between the input torque and wind speed of the 1.5MW wind turbine booster is shown in Equation ( 8), the piecewise function relationship between the input speed and wind speed is shown in Equation ( 9), and the curve of the input torque and input speed of the variable pitch and variable wind speed wind turbine with the change of wind speed is shown in Figure 5, At the same time, the corresponding input torque and speed curves of the 1.5MW wind turbine booster planetary gear system and external time-varying wind speed within 100s can be obtained, as shown in Figure 6.

Simulation Analysis of Wind Turbine Tower Vibration
Consider the wind turbine platform as a rigid body, the tower as an elastic body, and the springs and dampers connected to the platform represent the stiffness and damping of the platform.The simplified model is shown in Figure 7.
In the formula, T is the total kinetic energy of the system; V is the total potential energy of the system; L is the Lagrange multiplier; Q is non influential for easy analysis.Using the small angle assumption, the vibration equation of the fan can be simplified as equation (13).
In the formula, M, C, and K are the mass, damping coefficient, and stiffness coefficient of the fan, respectively, and F is the wind load.XX 、 , and X are the acceleration, velocity, and displacement vectors of the state variables, respectively.The system modeling parameters are shown in Table 4.

Conclusion
This article establishes a wind load wind speed model and a wind turbine tower dynamics model, and analyses the tower vibration dynamics characteristics under wind load.The following conclusions were drawn: (1) Based on the actual measurement of wind speed in a certain wind power plant, the wind force calculation expression considering the influence of pulsating wind is obtained.We simulated the combined wind speed using MATLAB/Simulink.Solve it using ODE45 and obtain a 100s combined wind speed simulation curve.(2) Based on the Euler Lagrange energy equation, the Lagrange equation for establishing a simplified dynamic model of wind turbines can be expressed as an equation to calculate the tower oscillation of wind turbines under different wind speeds.
Considering that excessive swing angle under wind induced vibration can cause instability of the tower and affect the normal operation of the generator, in order to ensure the safety of wind turbine production, a safe wind speed of no more than 12m/s is set.(3) Through simulation calculations, it can be concluded that under wind load, the displacement response at the top of the wind turbine tower is closely related to the response to external loads, and has a similar variation pattern to time-varying torque.The larger the external excitation, the greater the vibration displacement.At a wind speed of about 10m/s, the tower sways left and right under the influence of wind loads.When the wind speed exceeds 10m/s, the tower sways to one side and the swaying angle is too large; Considering that excessive swing angle under wind induced vibration can cause instability of the tower and affect the normal operation of the generator, in order to ensure the safety of wind turbine production, a safe wind speed of no more than 12m/s is set.

Figure 1 .
Figure 1.Site of a wind turbine tower collapse accident.

Figure 3 .
Figure 3. Wind speed time history curve.

Figure 4 .
Figure 4. Operation area of variable speed wind turbine.

Table 4 .Figure 8 .
Modeling parameters of the system.Wind vibration swing of inspection robot at wind speed of 5m/s.

Figure 9 .
Figure 9. Wind vibration swing of inspection robot at wind speed of 15m/s.

Table 1 .
Ground roughness index table.
Based on the operating environment of the wind turbine, it can be determined that the reference standard height G

Table 3 .
Technical parameters of wind.