Wind Speed Prediction Model Based on Improved VMD and Sudden Change of Wind Speed

Abstract

An accurate wind speed prediction system is of great importance prerequisite for realizing wind power grid integration and ensuring the safety of the power system. Quantifying wind speed fluctuations can better provide valuable information for power dispatching. Therefore, this paper proposes a deterministic wind speed prediction system and an interval prediction method based on the Lorentzian disturbance sequence. For deterministic forecasting, a variational modal decomposition algorithm is first used to reduce noise. The preprocessed data are then predicted by a long and short-term neural network, but there is a significant one-step lag in the results. In response to such limitation, a wind speed slope is introduced to revise the preliminary prediction results, and the final deterministic wind speed prediction model is obtained. For interval prediction, on the basis of deterministic prediction, the Lorenz disturbance theory is introduced to describe the dynamic atmospheric system. B-spline interpolation is used to fit the distribution of Lorenz disturbance theory series to obtain interval prediction results. The experimental results show that the model proposed in this paper can achieve higher forecasting accuracy than the benchmark model, and the interval prediction based on the Lorentzian disturbance sequence can achieve a higher ground truth coverage rate when the average diameter is small through B-spline interpolation fitting.

1. Introduction

With the serious problem of climate change and the deepening concept of sustainable development, wind energy, as a clean and renewable energy, has gradually become one of the most important energy consumption channels for future power generation in the world. Wind power generation has the characteristics of randomness and volatility; a large-scale high proportion of access to the grid will bring many challenges, such as power balance, power consumption, power grid security, and stability control. Therefore, improving the accuracy of wind speed prediction and quantifying the risk of wind speed fluctuation is not only the focus and difficulty of current research work but also will promote the construction of clean, low carbon, safe and efficient energy system, which is conducive to actively respond to global climate change caused by carbon emissions.

The existing wind power research models can be divided into four categories, namely, physical model, spatio-temporal prediction model, statistical method, and intelligent prediction technology. Among them, the physical model is the prediction model based on the relationship between the physical environment information and the prediction object [1,2], which provides information support for the prediction model by excavating the internal relationship between the two. The main form of its realization is to construct the numerical weather prediction (NWP) model [3,4]. The spatio-temporal prediction model introduces the spatial correlation between various factors in the wind farm or cluster into the model so that the original time dimension model is expanded to the time–space dimension model, and the prediction accuracy is greatly improved [5,6]. Next is the statistical prediction model based on historical data [7,8,9]. This method is better at mining linear information and is interpretable, and the commonly used method is the autoregressive moving average model (ARMA) [10,11,12], autoregressive integrated moving average model (ARIMA) [13,14] and support vector machine regression (SVM) [15,16]. Intelligent prediction technology is better at solving complex nonlinear mapping problems and has high prediction accuracy. It has been applied to many fields. The commonly used models include extreme learning machine (ELM) [17], back propagation neural network (BP) [18,19,20], long short-term memory (LSTM) neural network [21,22] and hybrid models [23,24,25]. Although the above methods have their own advantages, their limitations also exist:

• Physical methods require a large number of factors and data costs and are suitable for long-term forecasting.
• The construction of spatio-temporal prediction model needs to be based on a large amount of information, and the computational complexity increases. The more space-related sites are included, the lower the prediction accuracy may be.
• Statistical methods have great demand and quality for data, and the nonstationarity of data will limit the improvement of prediction level of some statistical methods.
• Common neural networks sometimes lead to prediction lag, and a single prediction model has limited ability to deal with prediction problems in different occasions, that is, the generalization performance is not strong.

Therefore, this paper constructs a hybrid prediction model to improve the prediction effectiveness: Firstly, the wind power data are denoised and preprocessed, and the prediction results are corrected according to the wind speed climbing identification. It can reduce the one-step lag problem and the prediction accuracy from two aspects of input data preprocessing and output correction of LSTM. The traditional denoising methods include wavelet decomposition [26], wavelet packet decomposition [27,28], empirical mode decomposition (EMD) [29], ensemble empirical mode decomposition (EEMD) [30,31]. However, this paper selects a variational mode decomposition (VMD) signal processing method to decompose and reconstruct wind speed for denoising. VMD can filter out the sequence with frequency characteristics, leaving a cluttered noise part, and this filtering property cannot be achieved by the above commonly used algorithms [32]. The VMD decomposition algorithm reduces the probability of modal aliasing and residual noise. In addition, LSTM with long-term memory features is used to denoise the preprocessed sequences. In order to reduce the time delay problem [33], it is different from the previous methods of directly predicting the error sequence or predicting its decomposition-reconstruction to correct the prediction results [34,35]. Find error changes by identifying wind trends by defining WSR and wind slope ratio (WRR). The error caused by one step lag is larger when there is a significant enhancement or a significant weakening [36]. Correcting the prediction results at significantly changed time points can greatly reduce errors and improve accuracy. These two methods are good complements to each other to guarantee satisfactory predictions. The complete wind speed deterministic forecasting model is named VMD-LSTM-PSOR.

Based on deterministic forecasting, this paper further proposes a wind speed interval forecasting method. Although the deterministic prediction model can provide high-precision wind speed value prediction, it only gives a certain time value, which cannot measure the future fluctuation range of wind speed. Therefore, it has more practical guiding significance for interval prediction of wind speed series. The atmospheric dynamical system is a chaotic system, and the dynamical chaotic system has strong uncertainty. The Lorentz equation describes the motion of air fluid by differential equations. When considering the strong randomness of the wind under the influence of the dynamic atmospheric system, the Lorentz equation was applied to the wind speed sequence [37]. Thus, the Lorentzian perturbation sequence (LDS) is obtained by solving the Lorentzian equation, and then B-spline interpolation is used to fit the distribution of the LDS. The upper and lower bounds of the interval prediction are obtained according to the different confidence intervals of the fitted function. The validity of the proposed method is verified by comparing the average interval diameter of the forecasting interval obtained by B-spline interpolation and kernel density estimation (KDE) fitting with the true wind speed coverage. The contributions are summarized as follows:

• Using VMD to de-noise wind speed, a better data preprocessing effect is obtained and paves the way for subsequent forecasting.
• WSR and WRR are defined. According to WRR, the points in which wind speed changes fast are corrected, which reduces the error caused by a one-step lag of LSTM forecasting.
• A wind speed interval prediction based on the Lorenz theory is proposed. The effect of atmospheric power system on wind speed is innovatively expressed in the form of interval prediction, and its good forecasting results are verified.

The structure of this paper is arranged as follows: Section 2 introduces the theory described in this paper. Section 3 gives the model establishment and flow chart. Section 4 introduces the experiment and shows the results and discussions. Conclusion See Section 5.

2. Materials and Methods

This section will introduce four main theories related to the developed wind speed deterministic prediction and interval prediction, which include the variational mode decomposition (VMD), wind speed ramp (WSR), and Lorenz system.

2.1. Variational Mode Decomposition (VMD)

VMD is a new adaptive signal processing technology; its essence is several adaptive Wiener filter groups. VMD decomposes the wind speed data series $f$ into a discrete number of component signals $u_k$ and assumes that all modes $k$ are close to the central pulsation ${\omega }_k$ determined by the decomposition [38]. The process of VMD is actually a problem of minimizing Equation (4) and using alternate direction method of multipliers (ADMM) to implement VMD by iteration and minimization of $u_k$ and ${\omega }_k$. The algorithm is shown in Algorithm 1.

Algorithm 1: Solving of VMD
Initialize $\left\{{\widehat{u}}_k^1\right\},\left\{{\omega }_k^1\right\},{\widehat{\lambda }}^1,n\leftarrow 0$  Repeat  $n\leftarrow n+1$  for $k=1:K$ do  Update ${\widehat{u}}_k$$\mathrm{for}\mathrm{all}\omega \ge 0$:
$${\widehat{u}}_k^{n+1}(\omega )\leftarrow \left[\widehat{f}(\omega )-{\displaystyle \sum _{i\ne k}{\widehat{u}}_i(\omega )}+\frac{{\widehat{\lambda }}^n(\omega )}{2}\right]/\left[1+2\alpha {(\omega -{\omega }_k^n)}^2\right]$$ (1)
Update ${\omega }_k$: $${\omega }_k^{n+1}\leftarrow {\displaystyle \underset{0}{\overset{\infty }{\int }}\omega {\left|{\widehat{u}}_k^{n+1}(\omega )\right|}^{2}d\omega }/{\displaystyle \underset{0}{\overset{\infty }{\int }}{\left|{\widehat{u}}_k^{n+1}(\omega )\right|}^{2}d\omega }$$ (2)  End for
Dual ascent for all $\omega \ge 0$ $${\widehat{\lambda }}^{n+1}(\omega )\leftarrow {\widehat{\lambda }}^n(\omega )+\tau \left(\widehat{f}(\omega )-{\displaystyle \sum _k{\widehat{u}}_k^{n+1}(\omega )}\right)$$ (3)
Until convergence: ${\displaystyle \sum _k{\Vert {\widehat{u}}_k^{n+1}-{\widehat{u}}_k^n\Vert }_2^2/{\Vert {\widehat{u}}_k^n\Vert }_2^2}<\epsilon$

where n is the number of iterations, the Fourier transform of $f(t)$, $u_k^{n+1}(t)$, $\lambda (t)$, $u_i(t)$ is expressed as $\widehat{f}(\omega )$, ${\widehat{u}}_k^{n+1}(\omega )$, $\widehat{\lambda }(\omega )$, ${\widehat{u}}_i(\omega )$.

$$ℒ(\{u_k\},\{{\omega }_k\},\lambda ):=\alpha {{\displaystyle \sum _k\Vert {\partial }_t[(\delta (t)+\frac{j}{\pi t})\ast u_k(t)]e^{-j{\omega }_{k}t}\Vert }}_2^2+{\Vert f(t)-{\displaystyle \sum _k{u}_k(t)}\Vert }_2^2+\langle \lambda (t),f(t)-{\displaystyle \sum _k{u}_k(t)}\rangle$$

(4)

where the decomposed main signal is represented by $f(t)$, $\left\{u_k\right\}:=\{u_1,\dots ,u_K\}$ and $\left\{{\omega }_k\right\}:=\{{\omega }_1,\dots ,{\omega }_K\}$ represent the set of all decomposition modes and corresponding center frequencies, respectively, δ(t) denotes the Dirac distribution, $\alpha$ is quadratic penalty term, $\lambda$ represents Lagrange multiplier, $\ast$ denotes convolution.

The modal number of VMD needs to be determined beforehand. Each subsequence obtained by VMD decomposition has certain frequency characteristics. The noise information is included in the remaining part of the original signal to remove each subsequence signal.

2.2. Wind Speed Ramp (WSR)

When the prediction results lag, the corresponding error level rises significantly, often accompanied by the occurrence of WSR events [39]. To identify the occurrence time and band of such events, formula (5) defines the ramp.

$$\left|v(t_0+{\Delta }_t)-v(t_0)\right|/{\Delta }_t>v_{val}$$

(5)

Where $v(t_0)$ represents the wind speed value at time $t_0$, ${\Delta }_t$ is the time gap, $v_{val}$ is the threshold of WSR.

The gradient of wind speed is defined as Equation (6).

$$r(i)=(v(i)-v(i-1))/\mathrm{interval}$$

(6)

in the above expression, $v(i)$ represents the $i$th wind speed value and interval is the interval corresponding to the dataset used. If one of the absolute values of the two adjacent gradients is greater than the wind speed climbing threshold $WR{R}_{val}$ as Equation (7), define that the WSR occurs at $i$th point.

$$r(i)=(v(i)-v(i-1))/\mathrm{interval}$$

(7)

The prediction result correction process based on the above definition can be considered to be equivalent to the following parameter optimization process.

• When the wind speed sequence satisfies Equation (7), that means ramp events are happening. When the positive ramp accumulates to a certain extent, Equation (8) can be satisfied. Otherwise, Equation (9) can represent the negative ramp accumulated to a certain extent.

$$\{\begin{array}{c}r(i)>0\cap r(i-1)>0\\ r(i-1)-r(i)<WR{R}_{up}\end{array}$$

(8)

$$\{\begin{array}{c}r(i)<0\cap r(i-1)<0\\ r(i-1)-r(i)<WR{R}_{down}\end{array}$$

(9)

After the slope event is identified, the predicted lag value $\widehat{v}{(i+1)}_{pre}$ can be corrected according to Equation (10) to obtain $\tilde{v}(i+1)$.

$$\widehat{v}(i+1)=[v(i)-\widehat{v}(i)]+\stackrel{⌢}{v}{(i+1)}_{pre}$$

(10)

where $v(·)$ represents the real wind speed, $\widehat{v}(·)$ denotes the predicted wind speed.

2.

In addition to the above, if the gradient does not change significantly, and $r(i)$ and $r(i-1)$ have opposite symbols, or the ramp event does not occur, the predicted wind speed value $\widehat{v}{(i+1)}_{pre}$ will not be corrected as Equation (11).

$$\widehat{v}(i+1)=[v(i)-\widehat{v}(i)]+\stackrel{⌢}{v}{(i+1)}_{pre}$$

(11)

The above three thresholds $WR{R}_{val},WR{R}_{up},WR{R}_{down}$ are optimized by PSO algorithm based on minimizing RMSE value.

2.3. Lorenz System

As the earliest chaotic motion dissipative system, Lorenz system was proposed by Edward Lorenz in 1963 to solve the aperiodic phenomenon [40], and Equation (12) describes the simplified model.

$$\begin{array}{c}\dot{x}=-\sigma (x-y)\\ \dot{y}=-xz+rx-y\\ \dot{z}=xy-bz\end{array}\}$$

(12)

In the above formula, $x$, $y$ and $z$ respectively represent the amplitude of convection motion, the horizontal temperature difference between the ascending and descending fluid in the convection, and the vertical temperature difference caused by convection deviates from the equilibrium state without convection. Parameters $\sigma$, $r$ and $b$ represent Prandtl number, Rayleigh number, and the parameter related to the container size respectively. Moreover, if $\sigma =10$, $b=8/3$ and $r$ exceeds 24.74, the system can be regarded as chaotic, and its solution will be unstable and sensitive [41].

The wind speed sequence used in this paper is one-dimensional real value sequence, and LDS is obtained by dimension reduction of Lorenz attractor. Therefore, Chebyshev distance is selected for mapping dimension reduction. The Chebyshev distance between $C_n(x_n,y_n,z_n)$ and $C_0(x_0,y_0,z_0)$ is shown as Equation (13), and the Lorenz attractor and LDS under initial condition (0, 1, 1), $\sigma =10$, $b=8/3$ and $r=28$ is shown as Figure 1.

$$d(C_n-C_0)=\max (\left|x_n-x_0\right|,\left|y_n-y_0\right|,\left|z_n-z_0\right|)$$

(13)

3. Proposed Wind Speed Deterministic and Interval Forecasting Models

In this section, the development of deterministic and interval forecasting is described, including the model building process and corresponding flow chart.

• Phase I: Preliminary forecasting of wind speed.

Step one. First, the wind speed sequence is denoised. In this paper, VMD is used to decompose the original wind speed into several IMFs and a noise sequence according to the frequency characteristics, and then these IMF subsequences are reconstructed to obtain the denoised wind speed sequence. The number of decomposed IMFs is determined by the minimum error of the preliminary forecasting results according to the specific experimental data.

Step two. After noise reduction, the wind speed sequence becomes smoother. Then use the LSTM neural network that is good at learning time series to obtain the initial prediction result $\tilde{v}{(·)}_{pre}$.

2.

Phase II: Reducing one-step lag of preliminary forecasting results.

Step three. According to the real wind speed series, the i-th gradient is calculated. When the absolute value of the gradient is greater than the threshold value $WR{R}_{val}$, and the positive gradient of the current time increases or the negative gradient decreases exceed the corresponding threshold $WR{R}_{up}$ or $WR{R}_{down}$, the predicted value of next point is revised by adding the error of the i-th point.

Step four. For the selection of three thresholds in step three, transform it into a multi-objective optimization problem aiming at minimizing RMSE. PSO is used to solve this optimization problem. Because excessive particle swarm size and large maximum iteration will greatly increase the search time, in order to control the time of this part within 3 min, the particle swarm size and the maximum number of iterations are set to 20 and 5, respectively.

3.

Phase III: Interval forecasting based on LDS.

Step five. Fixing the initial value of Lorenz equation to (0, 1, 1), given the parameter values $\sigma =10$, $b=8/3$ and $r=28$, solving Lorenz equation and deviation standardized the result, three-dimensional LDS is obtained, and then reduced it to one-dimensional according to Chebyshev distance, thus LDS is obtained.

Step six. Using Normal distribution, Gamma distribution, Rayleigh distribution, and Weibull distribution to fit the frequency histogram converted from LDS. The KS test, which is a test method of fitting effect, is carried out. The results showed that all of them failed. Then the KDE, which is a non-parametric fitting method, and B-spline interpolation were used to fit the frequency histogram, the fitting results were obtained, and the upper and lower quantiles of 90% and 98% confidence intervals were calculated, respectively.

Step seven. The upper and lower quantiles (5%, 95%, 1%, and 99%) of the 90% and 98% confidence intervals are taken as the maximum disturbance values of the wind speed forecasting intervals, and according to Equation (14), the wind speed prediction intervals at each point are obtained.

$$\{\begin{array}{c}{\tilde{v}}_{upper}=\tilde{v}(i)+F_{(1-\alpha /2)}\\ {\tilde{v}}_{lower}=\tilde{v}(i)-F_{(\alpha /2)}\\ {\tilde{v}}_{interval}=[{\tilde{v}}_{upper},{\tilde{v}}_{lower}]\end{array}$$

(14)

Where $F$ is the distribution function of LDS fitted by KDE or B-spline interpolation, $\alpha$ is the significance level, ${\tilde{v}}_{upper}$ denotes the upper bound of the forecasting interval, ${\tilde{v}}_{lower}$ denotes the lower bound of the forecasting interval, ${\tilde{v}}_{interval}$ denotes the wind speed forecasting interval based on LDS.

The flow chart for our proposed VMD-LSTM-PSOR and LDS-based interval forecasting is shown in Figure 2 and the blue box represents the first phase, the light red box represents phase II, and the green box denotes phase III.

4. Experimental Results and Discussions

4.1. Dataset

This paper established a forecasting model based on two datasets of a seaside wind farm located in Spain. The first is from 1 September 2018 00:00 to 7 September 2018 22:40. The second is from 9 March 2019 80:00 to 16 March 2019 06:30. They represent wind speeds in different seasons. The time interval is 10 min, and there are 1000 wind speeds in dataset one and 996 wind speeds in dataset two. Since there are four scattered vacancies in dataset two, use linear interpolation shown in Equation (15) to complete them.

$$y=\frac{y_{t+i}-y_t}{x_{t+i}-x_t}(x-x_t)+y_t,t,i\in N^{*}$$

(15)

where $x_t$ denotes $t$ the moment, and $y_t$ is the corresponding wind speed value at that moment.$x$ is an arbitrary moment between $x_t$ and $x_{t+i}$, and $y$ is the corresponding unknown wind speed value.

For each dataset, the first 900 data (90% of the whole data) are used as a training set, and the next 100 are used as a test set. In other words, the forecast period is 16 h and 30 min. All experiments were conducted in MATLAB R2016 b.

4.2. Metrics

For deterministic forecasting, this paper chooses the most commonly used mean square error (MSE), mean absolute error (MAE), root mean square error (RMSE) and mean absolute percentage error (MAPE) to evaluate.

$$\{\begin{array}{c}{N}_t=\{\begin{array}{c}0,v_{interval}(t)\notin y_t\\ 1,v_{interval}(t)\in y_t\end{array}\\ R_{cover}={\displaystyle {\sum }_{t=1}^n{N}_t/n}\end{array}$$

(16)

$$d_{average}={\displaystyle {\sum }_{t=1}^n({\widehat{y}}_{upper}-{\widehat{y}}_{lower})/n}$$

(17)

where $N_t$ is whether the $t$th interval covers the $t$th observed wind speed, and if it is covered, $N_t$ equals 1, otherwise $N_t$ equals 0. $R_{cover}$ is the interval coverage ratio. $d_{average}$ is the average interval diameter. Obviously, the higher the coverage ratio and the lower the average diameter, the better the effect of interval prediction, which shows that the prediction accuracy is still high when the prediction interval is small.

For interval prediction, both the interval coverage ratio (the ratio of the number of intervals containing real wind speed) and the average interval diameter should be considered. Thus this paper defines Equation (16) and Equation (17), respectively.

4.3. Deterministic Forecasting Results and Discussions

Firstly, the prediction results of six single wind speed prediction models, including ARMA, SVM, GBDT, XGBoost, BP neural network, and LSTM neural network, are compared, and their errors on two datasets as shown in

Table 1. Single model forecasting errors on dataset 1.

	MSE	MAE	RMSE	MAPE
ARMA	0.2853	0.4195	0.5341	6.6086
SVM	0.6042	0.5953	0.7773	9.1708
GBDT	0.8502	0.7321	0.9221	13.2483
XGBoost	0.4409	0.5352	0.6640	8.2239
BP	0.2867	0.3958	0.5355	6.2363
LSTM	0.2565	0.3896	0.5064	6.1140

and

Table 2. Single model forecasting errors on dataset 2.

	MSE	MAE	RMSE	MAPE
ARMA	0.2361	0.3754	0.4859	12.8248
SVM	1.3405	0.9185	1.1578	37.4274
GBDT	1.2722	0.8859	1.1279	27.4912
XGBoost	0.5611	0.5565	0.7490	19.2091
BP	0.2943	0.4130	0.5425	14.7883
LSTM	0.2061	0.3574	0.4540	12.1753

. The parameters $p$ and $q$ of ARMA are determined according to the minimum AIC criterion. The most commonly used radial basis function is selected as the kernel function of SVM. The learning rate and the number of training sets for GBDT and XGBoost are obtained by minimizing their MAEs, respectively, and the number of boosting trees of the two models are both set to 100. The structure of BP is three layers, and the number of nodes is 3-30-1; LSTM is also set to three layers, and the number of nodes is 3-5-1. In this paper, the number of hidden layer nodes of LSTM is set to less than BP because too many nodes will increase the computational time, especially for LSTM (More than one minute was consumed in our programming environment when six nodes were used in the operation and this will greatly increase the computational time of the whole model); moreover, even LSTM has less hidden layer nodes, its prediction performance is still better than BP.

As can be seen from Table 1 and Table 2, all error metrics of LSTM neural network are minimal on any dataset. It can be seen from Table 1 that GBDT has the largest error. LSTM’s MSE is reduced by 69.83%, MAE by 46.78%, RMSE by 45.08%, and MAPE by 53.85% compared with GBDT. In other words, LSTM has the highest prediction accuracy in commonly used single prediction models, and Table 2 illustrates that the errors of SVM are the largest. LSTM’s MSE is reduced by 84.63%, MAE by 61.09%, RMSE by 60.79%, and MAPE by 67.47% compared with SVM. That is to say, LSTM has the highest prediction accuracy compared with the benchmark prediction models.

The forecasting results of the six models on two datasets are shown in Figure 3 and Figure 4, respectively.

It is clear from Figure 3 and Figure 4 that the red line, the green line and the cyan line are far away from the black line, whereas the magenta line, the yellow line and the black dotted line are closer to the black solid line. The black dotted line representing the LSTM forecasting result is closest to the real variation trend. This also confirms our conclusion in the error metrics tables.

However, as can be seen from the figures, there is a lag in the predictions of LSTM compared to the actual values, thus the LSTM was improved by denoising the data and modifying the predictions according to the WSR. The prediction errors on the two datasets are shown in

Table 3. Forecasting errors after revising on dataset 1.

	Error	MSE		MAE		RMSE		MAPE
Model		Before PSOR	After PSOR	Before PSOR	After PSOR	Before PSOR	After PSOR	Before PSOR	After PSOR
WD-LSTM (lev = 2)		0.1880	0.1384	0.3183	0.2779	0.4336	0.3720	5.1055	4.2841
WD-LSTM (lev = 3)		0.2103	0.1709	0.3621	0.3300	0.4586	0.4134	5.7926	5.1563
WD-LSTM (lev = 4)		0.2588	0.2071	0.4062	0.3639	0.5087	0.4550	6.4514	5.7514
WD-LSTM (lev = 5)		0.2621	0.2077	0.4091	0.3733	0.5119	0.4557	6.4973	5.8561
VMD-LSTM (IMF = 4)		0.1796	0.1502	0.3459	0.3157	0.4238	0.3876	5.5162	4.8386
VMD-LSTM (IMF = 5)		0.1526	0.1263	0.2965	0.2682	0.3906	0.3554	4.6730	4.0650
VMD-LSTM (IMF = 6)		0.1465	0.1248	0.2899	0.2655	0.3828	0.3532	4.5463	4.0041
VMD-LSTM (IMF = 7)		0.1462	0.1280	0.2872	0.2696	0.3824	0.3578	4.5058	4.0653
VMD-LSTM (IMF = 8)		0.1519	0.1408	0.2947	0.2795	0.3897	0.3753	4.6511	4.2372

and

Table 4. Forecasting errors after revising on dataset 2.

	Error	MSE		MAE		RMSE		MAPE
Model		Before PSOR	After PSOR	Before PSOR	After PSOR	Before PSOR	After PSOR	Before PSOR	After PSOR
WD-LSTM (lev = 2)		0.1659	0.1271	0.3227	0.2815	0.4073	0.3565	11.3157	9.0404
WD-LSTM (lev = 3)		0.1827	0.1305	0.3444	0.2865	0.4275	0.3612	10.6325	9.2621
WD-LSTM (lev = 4)		0.1778	0.1383	0.3338	0.2932	0.4216	0.3718	10.6544	9.5665
WD-LSTM (lev = 5)		0.1951	0.1427	0.3527	0.2943	0.4417	0.3777	11.2441	9.7730
VMD-LSTM (IMF = 3)		0.1680	0.1238	0.3223	0.2663	0.4098	0.3518	11.4498	8.6970
VMD-LSTM (IMF = 4)		0.1420	0.1039	0.2828	0.2401	0.3769	0.3223	9.7019	7.6814
VMD-LSTM (IMF = 5)		0.1334	0.1200	0.2855	0.2701	0.3652	0.3464	9.5620	8.1130
VMD-LSTM (IMF = 6)		0.1374	0.1201	0.2921	0.2727	0.3707	0.3466	9.7768	8.2673
VMD-LSTM (IMF = 7)		0.1427	0.1333	0.3778	0.2923	0.3824	0.3651	10.4884	8.8785

, respectively.

In Table 3 and Table 4, the minimum error achieved by WD-LSTM is highlighted for different decomposition levels (lev equals two to five in both tables) and for different IMFs’ numbers (ranging from four to eight in Table 3 and ranging from three to seven in Table 4), the minimum error achieved by VMD-LSTM is shown in bold.

From the two tables, it can see that when the WD level is two, the error of WD-LSTM reaches the minimum. Furthermore, before WSR correction, most VMD-LSTM errors are smaller than the minimum error of WD-LSTM. This fully proves that the effect of denoising by VMD is much better than that by WD. On the two datasets, after correcting the forecasting results by WSR, the errors of each model are significantly reduced. The reduction rate is between 5% and 30%. The minimum error on dataset one is obtained by VMD-LSTM-PSOR (IMF = 6), and the minimum error on dataset two is obtained by VMD-LSTM-PSOR (IMF = 4).

Figure 5 and Figure 6 are forecasting results of LSTM, VMD-LSTM, and VMD-LSTM-PSOR with IMF = 6 on dataset one and IMF = 4 on dataset two, respectively.

As can be observed from these two figures, the solid green line representing the LSTM prediction results has a noticeable one-step lag. After VMD preprocessing, the fluctuation of wind speed series is weakened and the wind speed curve becomes smooth. The hysteresis has been significantly improved, but the fitting effect is poor at the sampling points with large variations. In order to better predict the wind speed, WSR with PSO optimization is used to correct the prediction results, so that the wind is closer to the real curve in detail. It is evident from the figure that the red line representing the VMD-LSTM-PSOR prediction results most closely represents the true black line wind speed.

4.4. Interval Forecasting Results and Discussions

The Lorenz disturbance series are fitted by KDE and B-spline interpolation, respectively. Based on the results of deterministic forecasting results, the interval forecasting of wind speed is obtained according to different confidence intervals. The coverage ratio, average diameter, and the uncovered points are shown in

Table 5. Interval forecasting results analysis.

	Parameter	Confidence Interval	${\mathit{R}}_{\mathit{c}\mathit{o}\mathit{v}\mathit{e}\mathit{r}}$	${\mathit{d}}_{\mathit{a}\mathit{v}\mathit{e}\mathit{r}\mathit{a}\mathit{g}\mathit{e}}(\mathbf{m}/\mathbf{s})$	Uncovered Points
Fitting Method
KDE Fitting LDS (on dataset 1)		90%	93%	1.5009	2, 17, 23, 39, 71, 74, 84
		98%	91%	1.718	2, 17, 19, 23, 39, 71, 74, 81, 95
B-spline interpolation fitting LDS (on dataset 1)		90%	95%	1.5075	17, 23, 39, 71, 84
		98%	94%	1.6923	2, 17, 23, 39, 71, 74
KDE Fitting LDS (on dataset 2)		90%	92%	1.5009	6, 11, 25, 59, 69, 90, 96, 98
		98%	88%	1.718	6, 11, 18, 25, 26, 49, 69, 70, 80, 90, 96, 98
B-spline interpolation fitting LDS (on dataset 2)		90%	94%	1.5075	6, 11, 59, 69, 90, 96
		98%	92%	1.6923	6, 11, 25, 69, 70, 90, 96, 98

.

In Table 5, the interval forecasting results obtained by different fitting methods under 90% confidence interval and 98% confidence interval are calculated, respectively. As for dataset one, on the one hand, for a 90% confidence interval, the average diameter of KDE and B-spline interpolation is almost the same (1.5009 m/s and 1.5075 m/s, respectively), but the coverage of B-spline is 2% higher than that of KDE. B-spline interval prediction covers two points, 2 and 74, more than KDE. As for the 98% confidence interval, the situation is similar. The $d_{average}$ is almost the same, but the coverage of B-spline is 3% higher than that of KDE. B-spline interval prediction covers three points, 19, 81, and 95, more than KDE. This shows that when the confidence interval is the same and the average diameter is almost the same, the fitting effect of KDE on LDS is not as good as that of B-spline interpolation, and the confidence interval obtained from the latter can cover more real wind speed points. On the other hand, for the same fitting method, higher confidence intervals lead to a larger average diameter, which is consistent with statistical principles but cannot lead to a higher coverage ratio. Because a higher confidence interval means that the interval prediction may have a higher upper bound and a higher lower bound, the coverage range will also change (on dataset one, for KDE Fitting LDS, 90% confidence interval covers points 19, 81, and 85 while 98% confidence interval not, on the opposite, 98% confidence interval covers point 84 while 90% confidence interval not. The situation is similar in other conditions). It means that the fitting of LDS can obtain a high coverage ratio and good interval prediction results without using very high confidence intervals, and it is effectually to structure forecasting intervals with LDS.

The results of interval prediction are plotted on the two datasets, as shown in Figure 7 and Figure 8, respectively. In Figure 7 and Figure 8, (a) draws the result of interval forecasting under 90% confidence interval, (b) draws the result of interval prediction under 98% confidence interval. Most of the real wind speed points are covered by the results of interval prediction, which shows that the prediction of wind speed interval based on LDS is effective, and the prediction results obtained by fitting LDS with B-spline interpolation are better than those of KDE fitting LDS, which is consistent with the result shown in Table 5.

5. Conclusions

The intelligent development of the wind power industry is inseparable from accurate and effective wind power forecasting. This article presents an innovative deterministic forecasting model named VMD-LSTM-PSOR and a wind speed interval forecasting model based on Lorenz disturbance theory. In the deterministic forecasting stage, the denoising effect of VMD on wind speed series is obviously better than that of the traditional wavelet decomposition denoising method. In view of the forecasting results of one-step lag caused by the LSTM neural network, this paper corrects it by defining WSR and reducing the forecasting error obviously. In the interval forecasting stage, the Lorenz equation is used to describe the effect of the dynamic atmospheric system on wind speed, and the upper and lower bounds of interval prediction are determined by the distribution based on LDS. Comparing the forecasting effects of fitting LDS with KDE and B-spline interpolation, respectively, the effect of B-spline interpolation is better than that of KDE in terms of average interval length and true wind speed coverage.

Compared with the traditional model, the wind speeds deterministic prediction model proposed in this paper improves the prediction accuracy. The Lorentz system is innovatively introduced into wind speed interval forecasting, which effectively quantifies the volatility of wind power generation. In order to further improve the accuracy of wind speed prediction, the optimization algorithm and decomposition algorithm of the model can be improved in the future. Multi-factor input can be selected, and the number of input variables can be increased so as to increase the stability of prediction. Combined with the numerical weather forecast system and real-time wind speed data, an online wind speed prediction platform can be established to realize an ultra-short-term forecast of wind speed.