Fractional Levy Stable and Maximum Lyapunov Exponent for Wind Speed Prediction

: In this paper, a wind speed prediction method was proposed based on the maximum Lyapunov exponent (Le) and the fractional Levy stable motion (fLsm) iterative prediction model. First, the calculation of the maximum prediction steps was introduced based on the maximum Le. The maximum prediction steps could provide the prediction steps for subsequent prediction models. Secondly, the fLsm iterative prediction model was established by stochastic di ﬀ erential. Meanwhile, the parameters of the fLsm iterative prediction model were obtained by rescaled range analysis and novel characteristic function methods, thereby obtaining a wind speed prediction model. Finally, in order to reduce the error in the parameter estimation of the prediction model, we adopted the method of weighted wind speed data. The wind speed prediction model in this paper was compared with GA-BP neural network and the results of wind speed prediction proved the e ﬀ ectiveness of the method that is proposed in this paper. In particular, fLsm has long-range dependence (LRD) characteristics and identiﬁed LRD by estimating self-similarity index H and characteristic index α . Compared with fractional Brownian motion, fLsm can describe the LRD process more ﬂexibly. However, the two parameters are not independent because the LRD condition relates them by α H > 1.


Introduction
When the penetration of wind power exceeds a certain value, it seriously affects power quality. At present, the error rate of wind speed forecasting of wind farms is about 25%-40%, and the research on wind speed forecasting of wind farms has not reached a satisfactory level [1]. If wind speed and wind power could be accurately predicted, it would be beneficial for the power system dispatching department to adjust the scheduling plan in time, which could effectively reduce the impact of wind power on the power grid [2]. At the same time, the improvement of prediction accuracy could also reduce the operating cost and rotation reserve of power systems [3], increase the limit of wind power penetration, and lay the foundation for wind farms to participate in bidding for power generation [4]. Many researchers have developed several different wind speed prediction methods. The simplest prediction method is the continuous method, which uses the closer wind speed or power observation value as the prediction value for the next point [5]. Other prediction methods include Kalman filters [6], ARMA [7], artificial neural network (ANN) [8], fuzzy logic, and so on. These methods only need the wind speed or power time series of the wind farm to build a model and make predictions. The spatial

Maximum Prediction Steps Based on Lyapunov Exponent
The small-data method [16] is defined as follows. Let {x 1 , x 2 · · · x N }, be a given chaotic time series, then the reconstructed phase space is defined as: where N = M + (m − 1)τ. The embedding dimension m and the time delay τ can be chosen according to the C-C method [17,18]. After the reconstruction of phase space, find the nearest adjacent point of each point on the given orbit, i.e., where p is the average period of the time series, which can be estimated by the inverse of the average frequency of the power spectrum, and the maximum Le can be estimated by the average divergence rate of each point on the basic orbit. For each reference point, calculate the distance to the nearest discrete point after the first discrete time step by The average divergence rate obeys the exponential divergence, i.e.,: Take the logarithm on both sides to get: Obviously, the maximum Le is roughly equivalent to the slope on this set of straight lines. It can be obtained by approximating this set of lines by the method of least squares.
The reciprocal of the maximum Lyapunov exponent is the maximum prediction steps ε when λ 1 > 0.

Parameter Meaning of Levy Stable Motion
Levy stable motion represents a non-Gaussian random process with LRD and high variability, we denote by X ∼ S α (β, δ, µ) the stable distribution with parameters α, β, δ and µ. Its characteristic function form is as follows [26]: where δ > 0, β ∈ [−1, 1], µ ∈ R. The parameter β is called the skewness parameter, while δ is called scale parameter, and µ the location parameter. In this article, we studied symmetric stable distribution, so we make β = 0. The location parameter µ indicates the mean, and the scale parameter δ represents the discrete nature of the distribution. Where α ∈ (0, 2]. The parameter α is the tail parameter and the distribution is Gaussian when α = 2, whereas the tail is exponential. In what follows, we typically supposed 0 < α < 2. When x → ∞ , the probability tails of X satisfy [27]: where C a is a constant. The tail of the distribution with 0 < α < 2 obeys a power law and decreases to zero so slowly that the variance is infinite; the smaller the value of α, the slower the decrease. From the perspective of probability distribution, as the value of α decreases, its tail becomes thicker ( Figure 1).

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constant. The tail of the distribution with 0 < < 2 obeys a power law ly that the variance is infinite; the smaller the value of , the slower the d e of probability distribution, as the value of decreases, its tail be

Long-Range Dependence and Self-Similarity Fractional Levy Stable Motion
The model of fLsm [14] is given by the following stochastic integral: where a and b are the arbitrary constants, x H−1/α is the symmetric Levy stable random measure, and H is the self-similarity parameter. The incremental process of fLsm [28] is as follows: where ω α (s) is the Levy stable white noise.
The key parameters α, H of the fLsm model are not independent in some cases, i.e., the fLsm has LRD characteristics for αH > 1 [30]. It is worth noting that the fLsm model has no long memory when 0 < α < 1, therefore, the range of α is limited to (1,2) to ensure that the fLsm model has the LRD characteristic. At the same time, 0.5 < H < 1 is also required.

Iterative Forecasting Model
Let us consider the following Langevin-type stochastic differential equation driven by Levy stable motion [19]: where dL α (t) stands for the increments of Levy α-stable motion L α (t). By replacing L H,α (t) to L α (t), we obtain the Langevin-type stochastic differential equation driven by fractional Levy stable motion: where b(t, X(t)) and δ(t, X(t)) represent the drift and diffusion functions, respectively. The fractional Black-Scholes model [20,21], which was developed by W. DAI et al. [31,32] has expression in the form: where µ indicates the expected return rate and δ is the volatility rate. The Levy stable distribution is the Gaussian distribution when α = 2 so that when α = 2 the fLsm becomes the fractional Brownian motion, µ represents the mean, and δ represents the diffusion coefficient. The parameters b and δ in the Levy stable distribution represent the mean and diffusion coefficient, respectively, in 1 < α ≤ 2. Consequently, Equation (14) can be rewritten as follows: where µ and δ are constants. They are derived from the novel CF method in the Appendix. By using the Maruyama symbol [22], dB t = w(t)(dt) 1/2 , the following equations can be obtained: where 0 < a < 1, and a represents the self-similar parameter of x. The incremental expression of fLsm can be obtained by replacing f (t) with w α (t): Equation (16) can be written the discrete form, which reads as follows: The iterative predictive model was obtained from the identity ∆X(t) = X(t + 1) − X(t):

Parameter Estimation with the Characteristic Function
In the essay of Wang et al. [23][24][25], some methods were introduced and the validity of these methods was compared, including the quantiles method, empirical characteristic function method, logarithmic moment method, Monte Carlo method, etc. It was concluded that the CF accuracy method was better. The parameter estimation methodology can be subdivided into the following steps: Step 1: Let x i | i=1...N be the sampling data for the fLsm, Step 2: δ estimation: The estimated δ has the form: Step 3: Further, we estimate parameter α, Step 4: Parameter µ is estimated by complex domain of the cumulant generating function of fLsm, Step 5: As we know, the fLsm model drive function is symmetricβ = 0.

Wind Speed Forecasting
We used the average daily wind speed data from the 2011 actual historical wind speed of Inner Mongolia. The historical wind speed waveform is shown in Figure 2. When the wind speed is too high, it will seriously affect the power grid, so we focused on accurately predicting the time period when the wind speed is high. It can be seen from Figure 2 that the wind speed data began to fluctuate greatly from the 100th day, which was harmful to the power grid, so we chose to start from the 100th forecast. In terms of selecting the prediction steps, the small-data method of the second part was used to calculate the maximum prediction steps. The calculation results are shown in Table 1. The maximum forecast steps were 43 days, we could set the forecast time period from the 100th day to the 140th day. Before using the fLsm iterative forecasting model, we needed to determine whether the wind speed sequence was LRD. Through parameter estimation, we could get the value of H and α (Table 2), satisfying αH > 1. Finally, the fLsm iterative forecasting model was used to forecast the wind speed sequence, and the forecast result is shown in Figure 3. The specific method flow is shown in Figure 4.          As can be seen from Figure 3, when the prediction steps exceeded nine steps, the prediction error gradually increased, and the prediction data was often larger than the actual data. However, by calculating the maximum prediction steps of 43, its effective prediction steps were much less than the maximum prediction steps.
As fLsm is an infinite variance process and the variance of the wind speed data is not large, if the historical wind speed data is used to estimate the parameters of the fLsm iterative prediction As can be seen from Figure 3, when the prediction steps exceeded nine steps, the prediction error gradually increased, and the prediction data was often larger than the actual data. However, by calculating the maximum prediction steps of 43, its effective prediction steps were much less than the maximum prediction steps.
As fLsm is an infinite variance process and the variance of the wind speed data is not large, if the historical wind speed data is used to estimate the parameters of the fLsm iterative prediction model, a large error will occur. In this section, we used a method of weighting the wind speed data to increase the variance of the data, thereby reducing the error in parameter estimation.
It can be seen from Figures 5 and 6 that the prediction effect of the wind speed weighted data had been significantly improved. Generally speaking, increasing the variance will cause the tail parameter α to decrease. However, it can be seen from Table 2 that the α value of the five-times weighted wind speed data was larger than the α value of the unweighted wind speed data, which indicated that a larger error occurred when modeling the wind speed sequence using the fLsm iterative prediction model. Of course, αH > 1 must be guaranteed when weighting the wind speed data.
Symmetry 2020, 12, x FOR PEER REVIEW 8 of 12 model, a large error will occur. In this section, we used a method of weighting the wind speed data to increase the variance of the data, thereby reducing the error in parameter estimation. It can be seen from Figure 5 and 6 that the prediction effect of the wind speed weighted data had been significantly improved. Generally speaking, increasing the variance will cause the tail parameter α to decrease. However, it can be seen from Table 2 that the α value of the five-times weighted wind speed data was larger than the α value of the unweighted wind speed data, which indicated that a larger error occurred when modeling the wind speed sequence using the fLsm iterative prediction model. Of course, > 1 must be guaranteed when weighting the wind speed data.    In order to prove the extensiveness of the wind speed prediction model in this paper, we forecasted the wind speed data of Inner Mongolia in 2012. As can be seen from Figure 7, the wind speed data on the 70th day began to fluctuate. We then calculated the maximum number of prediction steps to 45 and set the prediction time period to the 70th to 115th days. After weighting the wind speed data 10 times, the fLsm iterative prediction model was used for prediction in Figure 8. In addition, the fLsm iterative prediction model was compared with the GA-BP neural network, which showed that the wind speed prediction model in this paper had better prediction accuracy. In order to prove the extensiveness of the wind speed prediction model in this paper, we forecasted the wind speed data of Inner Mongolia in 2012. As can be seen from Figure 7, the wind speed data on the 70th day began to fluctuate. We then calculated the maximum number of prediction steps to 45 and set the prediction time period to the 70th to 115th days. After weighting the wind speed data 10 times, the fLsm iterative prediction model was used for prediction in Figure 8. In addition, the fLsm iterative prediction model was compared with the GA-BP neural network, which showed that the wind speed prediction model in this paper had better prediction accuracy.  In order to prove the extensiveness of the wind speed prediction model in this paper, we forecasted the wind speed data of Inner Mongolia in 2012. As can be seen from Figure 7, the wind speed data on the 70th day began to fluctuate. We then calculated the maximum number of prediction steps to 45 and set the prediction time period to the 70th to 115th days. After weighting the wind speed data 10 times, the fLsm iterative prediction model was used for prediction in Figure 8. In addition, the fLsm iterative prediction model was compared with the GA-BP neural network, which showed that the wind speed prediction model in this paper had better prediction accuracy.   Table 3 lists the maximum and average percentage errors for the two prediction models. As can be seen from the table, the fLsm iterative prediction model had higher prediction accuracy. At the same time, it can be seen from Figures 9 and 10 that the GA-BP neural network had a poor prediction of the peak value, which will lead to the inability to prevent the impact of excessive wind speed on the grid.  Table 3 lists the maximum and average percentage errors for the two prediction models. As can be seen from the table, the fLsm iterative prediction model had higher prediction accuracy. At the same time, it can be seen from Figures 9 and 10 that the GA-BP neural network had a poor prediction of the peak value, which will lead to the inability to prevent the impact of excessive wind speed on the grid.   Table 3 lists the maximum and average percentage errors for the two prediction models. As can be seen from the table, the fLsm iterative prediction model had higher prediction accuracy. At the same time, it can be seen from Figures 9 and 10 that the GA-BP neural network had a poor prediction of the peak value, which will lead to the inability to prevent the impact of excessive wind speed on the grid.      Table 3 lists the maximum and average percentage errors for the two prediction models. As can be seen from the table, the fLsm iterative prediction model had higher prediction accuracy. At the same time, it can be seen from Figures 9 and 10 that the GA-BP neural network had a poor prediction of the peak value, which will lead to the inability to prevent the impact of excessive wind speed on the grid.

Relationship between Wind Speed and Wind Power
Taking a variable-pitch wind turbine with a single unit capacity of 600 kW as an example, the power characteristics are shown in Figure 11. The cut-in wind speed, cut-out wind speed, and rated wind speed were 3, 50, and 25 m/s, respectively. The raw data of wind power time series could be obtained from the original data of wind speed and power characteristic curve of the wind turbines.

Relationship between Wind Speed and Wind Power
Taking a variable-pitch wind turbine with a single unit capacity of 600 kW as an example, the power characteristics are shown in Figure 11. The cut-in wind speed, cut-out wind speed, and rated wind speed were 3, 50, and 25 m/s, respectively. The raw data of wind power time series could be obtained from the original data of wind speed and power characteristic curve of the wind turbines. When the wind speed was less than the cut-in wind speed and greater than the cut-out wind speed, the power generation was zero; when the wind speed was equal to the cut-in wind speed, the rated wind speed, and the cut-out wind speed, the power characteristic curve had a significant turning point. When the wind speed was greater than the rated wind speed and less than the cut-out wind speed when the wind speed was out, the generating power was a certain value. Only when the When the wind speed was less than the cut-in wind speed and greater than the cut-out wind speed, the power generation was zero; when the wind speed was equal to the cut-in wind speed, the rated wind speed, and the cut-out wind speed, the power characteristic curve had a significant turning point. When the wind speed was greater than the rated wind speed and less than the cut-out wind speed when the wind speed was out, the generating power was a certain value. Only when the wind speed was greater than the cut-in wind speed and less than the rated wind speed did the generating power, and the wind speed approximate a linear relationship.
where P(v) is the wind power, P r is the rated power of the generator, v i is the cut-in wind speed, v c is the cut-out wind speed also known as the cut-off wind speed, v r is the rated wind speed, and f p (v) is the output characteristic of the wind speed between v i and v r . Its characteristics can be linear functions, quadratic functions, or cubic functions.

Conclusions
(1) Wind speed prediction is of great significance to the stable operation and operating efficiency of the power system. At the same time, it improves the ability of wind farms to participate in market competition.
(2) The wind speed prediction method based on the maximum Lyapunov exponent and fLsm iterative prediction model was effective. Based on the historical wind speed sequence, this paper calculated the maximum prediction steps, weighted the wind speed data, and established an fLsm iterative prediction model. It can be seen from the MATLAB simulation curve that the model can better predict the wind speed and reflect the change of the sequence, which has certain guiding significance. It can be seen from Section 6 that after the conversion of the power characteristic curve, its regularity was partially destroyed, and the regularity of the obtained wind energy was even weaker, which led to a larger forecast error of wind power. Therefore, the wind speed needs to be predicted first, and then the amount of electricity can be calculated.
(3) In practice, wind speed has strong randomness, and some regions may not have LRD. The wind speed sequence of short-range dependent (SRD) has yet to be studied.