Fig. 2 shows the element analysis of a blade based on the blade element momentum theory. The blade structure is like a cantilever beam, so the moment at its root can be regarded as the primary source of structural damage. M_f and M_e of the blade root can be expressed as [26]:

(1)Mf=0.5∫r0RρrlVrel2(Clcos⁡I+Cdsin⁡I)dr (2)Me=0.5∫r0RρrlVrel2(Clsin⁡I−Cdcos⁡I)dr (3)I=α+β

When the yaw deviation angle θ does not equal 0, the relative wind speed can be expressed as: (4)Vrel=(V0cos⁡θ(1−a))2+(ωrr(1+b))2

Based on M_f and M_e, the DEL of the blade in two directions can be further calculated to more accurately characterize the fatigue characteristics of the blade. By definition, DEL is the amplitude of the sinusoidal stress at a constant frequency f over time T that produces the same damage as the original stress signal. Using the Palmgren-Miner rule [27] and the S-N curve [28], the calculation method of DEL [29] can be determined as follows: (5)DEL=(∑j=1MσjmnjTf)1m

In this paper, the calculation of DEL uses the MCrunch code [13], which can be calculated in MATLAB. However, the calculation of DEL requires the use of measurement data of the two torque parameters, M_f and M_e, over a period of time. This requires additional sensors to measure these two torques and continuously record a large amount of data over a specific period, which is not easy to achieve.

The DEL calculation of the blade is based on the two moments of blades, M_f and M_e. According to Eqs. (1)–(3), M_f and M_e of the wind turbine blade are related to multiple variables such as relative wind speed, attack angle, lift coefficient, drag coefficient, and pitch angle. Among them, the lift and drag coefficient can also be described as related functions with the attack angle as a variable. According to Eq. (4), the relative wind speed is related to actual V₀ and ω_r. The attack angle is the angle between the blade chord and relative wind speed, which is related to its operating conditions and can be a function of the P_e.

In summary, the subsequent sections will select the measured V₀, β, ω_r, and P_e as input parameters to establish a blade fatigue load model driven by measurable data.

According to the above analysis, four parameters are selected as the input parameters of the data-driven model. However, these wind turbine operating parameters are recorded as time series. The amount of data is significant, and it is not easy to use all of them directly as input. Therefore, this section will extract features from multiple time series and filter them based on correlation as direct input to the neural network model. For the above time series signals, effective features can usually be extracted from the time and frequency domain [30].

The time domain analysis method obtains the dynamic change law by describing the autocorrelation structure of the time series. This method is relatively simple and easy to understand. The frequency domain analysis method is most commonly used in Fourier transform analysis and wavelet analysis, which are used to quantify the periodicity or the multi-scale time pattern characteristics [31]. The time and frequency domain parameters are exhibited in Tables 1 and 2.

In Table 1, there are 12 sets of characteristic parameters for the time domain, among which the peak value is also called the maximum value, and the peak difference is the difference between the peak and minimum. When a wind turbine usually operates, the peak and minimum values will not fluctuate too much. If there is a significant deviation from the normal value, it can be judged that the wind turbine has a fault. The abnormal maximum and minimum values and peak differences can also screen normal data. The average value and the rectified average value reflect the average parameters of the data of each working condition when the wind turbine is running, which can represent the changing relationship between different working conditions. The variance, standard deviation, and root mean square values can represent the degree of fluctuation of the working condition within the sampling time. Kurtosis and margin factors are more sensitive to impact signals and can characterize whether the working condition has a change within the sampling time. Skewness is used to measure the asymmetry of the time series, that is, the degree of deviation between the more extensive working condition and the more minor working condition relative to the average value.

In Table 2, the centroid frequency is the frequency of the more significant component in the spectrum, describing the distribution of the power spectrum. The frequency variance is the square of the frequency standard deviation, which can quantify the degree of spectral energy dispersion. The mean square frequency is the weighted average of the square of the root mean square frequency.

In the calculation process of the characteristic parameters mentioned above, the calculation period is 5 min, and the sampling interval is 1 s.

Tables 1 and 2 show that some commonly used time domain and frequency domain indicators are included. These indicators can universally analyze the characteristics of time series data, but not all are related to the DEL to be calculated. Therefore, before being used for DEL modeling, an analysis based on Pearson correlation will be performed first. Indicators with correlation coefficients higher than 0.6 can be considered solid correlation parameters [32] and will be screened as the input of the corresponding DEL model in the subsequent modeling process. The formula of the Pearson correlation coefficient is [32]: (6)r=∑i=1Ns(x(i)−x¯)(y(i)−y¯)∑i=1Ns(x(i)−x¯)2∑i=1Ns(y(i)−y¯)2

The prototype of a deep neural network is a discriminant model. Knowing the input variable can infer the value of the output variable. Generally speaking, the internal network structure of a DNN is shown in Fig. 3.

Unlike traditional perceptrons, the deep neural network also contains backpropagation and forward channels. The back-propagation channel compares the outputs with the true values and propagates the calculated error back to the input layer by layer. In the subsequent process, the parameters in the forward propagation process are adjusted to minimize the impact of the error value on the network. The idea of model construction can be found in our previous work [22]. After determining the structure of the DNN, the network training process is shown in Algorithm 1.

During forward propagation modeling, an activation function is generally required to introduce nonlinear factors in the propagation process in addition to the linear relationship. Commonly used activation functions in deep neural networks include Sigmoid, ReLU, Leaky ReLU, etc. In the experiment, it is necessary to screen the activation function with the best effect through multiple parameter settings.

During backpropagation modeling, a reasonable loss function is critical, which can be used to measure the deviation between the output calculated by the training data and the true output, and update ω and b based on iteration and forward propagation. Common loss functions include mean square error, etc.

ω and b can be updated as Algorithm 1. However, in the training process, as the depth of the DNN increases, the parameters can be gradually deviate from their set range during the update process. To address this problem, this paper uses the Adam algorithm to calculate the exponentially weighted average of the gradient and then uses the gradient value to update ω and b [33].

As for the accuracy evaluation indicators of the model, RMSE and MAPE will be used. RMSE is used in the model training process, and MAPE is used for the final model evaluation. The formulas of RMSE and MAPE are: (7)RMSE=1Ns∑i=1Ns(x(i)−x⌢(i))2 (8)MAPE=1Ns∑i=1Ns|x(i)−x⌢(i)x(i)|

This paper uses simulation to verify the proposed data-driven blade DEL calculation method. The simulation platform is OpenFAST, as mentioned above. Based on the OpenFAST platform in Matlab/Simulink [34], a pseudo-Monte Carlo experiment is designed to realize the collection of all working conditions. The wind turbine selected in the experiment is the NREL 5MW model [35], and the parameters are exhibited in Table 3.

The so-called pseudo-Monte Carlo method was proposed in [36], essentially a uniform experimental design. In the experimental design, when there are many parameters and an extensive range, a specific sampling interval is selected for each parameter and combined into different working conditions to fully collect parameters that can replace the entire working condition range, similar to the Monte Carlo method.

In Table 4, the range of each parameter is selected based on the following: Referring to the theoretical calibration operating conditions of the NREL 5MW wind turbine, it can be started and operated when the wind speed is greater than 3 m/s and less than 25 m/s, but usually when the wind speed is less than 5 m/s, its power and load are maintained at a low level, so 6–24 m/s is selected as the simulated wind speed range. Referring to the international standard IEC 61400-1-2019, the reference turbulence intensity range of different types of wind turbines is between 0.12–0.18, but the actual working conditions usually exceed this range, so we have expanded the range in the standard to a certain extent, setting the simulation turbulence intensity range to 0.06–0.24. As for the active power and power random values, they are set according to the 0.1–1 and 0.01–0.1 p.u. of the rated power of the NREL 5 MW, respectively.

Furthermore, the average V₀ and T_i can be set by Turbsim, with 10 and 6 different values set, respectively, including 60 working conditions, while the active power setting and power change value are set by OpenFAST, with ten different values set, respectively, including 100 working conditions. Therefore, based on the pseudo-Monte Carlo experiment principle, 6000 different working conditions will be completed. Among them, the power change value mentioned above is added as a random number with a variable range to avoid overfitting the training results of the DNN due to the data being too close.

In summary, this paper constructs 6000 different working conditions for training data-driven models through the OpenFAST platform. When collecting experimental data, the time scale of300 s, commonly used in fatigue calculation of wind turbines, is used as the sampling period for each group of working conditions. The corresponding sampling interval is 0.02 s, and 6000 * 6000 record points are sampled. After collecting the required data through the OpenFAST simulation toolbox, the MCrunch code [13] is used to obtained the DELs of the corresponding blades, which are used as the outputs of the DNN when training the data-driven model.

After the data acquisition process described above, four groups of operating parameters of the wind turbine were sampled under multiple simulation conditions. The indicators in Tables 1 and 2 were used for feature extraction and data dimension reduction of the collected operating parameters. A total of 2 (two DELs of blades) * 4 (four wind turbine parameters) * 15 (number of characteristic parameters) * 6000 (different operating conditions set in the pseudo-Monte Carlo experiment) groups of wind turbine characteristic data were obtained for the data analysis below.

To avoid making the DNN modeling input data too complex and redundant, the parameters with high correlation (>0.6) with blade DEL were selected from the 15 groups of characteristic parameters for modeling. The Pearson coefficient diagram shown in Fig. 4 was finally obtained by performing correlation analysis on the characteristic parameters and DEL.

The correlation significance shown in the figure is significant at the bilateral 0.01 level. The correlations between different characteristics of different parameters and DEL are significantly different. Taking wind speed as an example, its average value, variance, and standard deviation all have a correlation with DEL_Mf higher than 0.9, which shows that the average level and fluctuation of wind speed are the key factors leading to blade flapwise damage. Higher average wind speed and stronger turbulence will bring more significant blade flapwise damage. As for the damage caused by edgewise bending moment, the most significant related features of the pitch angle are the mean and standard deviation, which shows that frequent blade pitch changes or maintaining a large pitch angle will increase the damage caused by edgewise bending moment. In summary, the results of Fig. 4 are consistent with the laws of physics.

By setting the correlation coefficient to 0.6 as the threshold, 19 sets of feature parameters for DEL_Mf and 25 sets of feature parameters for DEL_Me can be selected as input data for training the deep neural network, and the corresponding DEL as output data of the neural network. The specific parameter selection is shown in Table 5.

In Table 5, the DELs of the blade in both dimensions has obvious correlations with multiple features of V₀ and β. The features of ω_r and P_e have more influence on M_e and have a lower correlation with M_f. The model is trained next based on the features selected in Table 5.

During the DNN model training process, all 6000 simulation scenarios constructed in Section 4.1 were shuffled, 5800 working conditions were the training set, and the remaining 200 working conditions were used for test. After multiple attempts at parameters such as hidden layers, hidden layer nodes, and the activation function, the final structure selected is the DNN model in Table 6. The computer configuration used in the DNN training is as follows: i7-9700 CPU @ 3.00 GHz, 16.0 GB RAM. The training time is 32.7 s.

After training the DNN network based on the structure shown in Table 6, the accuracy was tested using 200 sets of test data, and the results are exhibited in Figs. 5 and 6.

The results in Figs. 5 and 6 show that the constructed DNN model has reached a stable level after 200 training iterations. The trained DNN model has a good prediction ability for the DEL of the test group data, and the calculation accuracy of DEL_Mf and DEL_Me reaches 98.8% and 91.6%, respectively. Moreover, when testing based on the trained model, it takes less than 0.02 s to calculate the blade DEL of the test scenario, which fully meets the requirements of real-time control and optimization.

The results in Figs. 5 and 6 show that the constructed DNN model has a good prediction ability for the DEL of the in-sample working conditions. However, more out-of-sample working conditions may appear in the actual operation. Therefore, it is necessary to conduct further tests on out-of-sample working conditions to clarify the generalization ability of the proposed method. In addition, two other data-driven schemes are provided for comparison. C1 is designed based on [15] and uses the standard deviation of wind speed to construct a linear lookup table of two DELs of the blade. C2 is based on [17], using the average and standard deviation of V₀, etc., as input for deep neural network training.

In this experiment, the average V₀ is 10.85 m/s, the T_i is 13.27%, and the P_e is 1.165 MW, an out-of-sample condition inconsistent with the 6000 training and test conditions in Section 4.1. Under this condition, the DEL comparison results calculated by the proposed model, C1, and C2, are shown in Fig. 7. In the DEL_Mf part, the results calculated by the three schemes are close to TCV, and the MAPE is above 98%. However, in the DEL_Me part, the calculation results of different schemes are significantly different, and the MAPE of the three schemes is 62%, 79%, and 83%, respectively. The proposed scheme also has a good calculation accuracy for out-of-sample working conditions.

For the calculation results in Fig. 7, the following discussion can be carried out:

First, the prediction accuracy of DEL_Mf and DEL_Me for the same algorithm is inconsistent, and the prediction accuracy of DEL_Mf is higher than DEL_Me. This is because the proposed scheme and the two schemes in the control group have more appropriate input selection for DEL_Mf modeling. According to Fig. 4, the features used for DEL_Mf calculation include the variance of wind speed, etc., and its correlation with DEL_Mf is as high as more than 96%. In comparison, the correlation of modeling features for DEL_Me is less than 90%. Therefore, better feature selection will lead to better model accuracy.

Second, compared with the control method, the prediction accuracy of DEL_Me is higher in the proposed method. In particular, compared with C2, the same deep neural network modeling method is used, but the calculation accuracy of out-of-sample data is improved due to the improved input feature extraction method. Therefore, better feature selection will also lead to better model generalization ability.

Therefore, the deep neural network modeling method proposed in this paper has better description ability for nonlinear coupling relationships than the simple table lookup method, and the feature extraction process further improves model accuracy and generalization ability. If more appropriate features of parameters other than wind speed can be selected in future studies, the accuracy of the constructed model could be further improved.