Ultra-short-term wind speed prediction based on variational mode decomposition and optimized extreme learning machine

Abstract

The main purpose of this paper is to improve the prediction accuracy of ultra-short-term wind speed. It is difficult to predict the ultra-short-term wind speed because of its unstable, non-stationary and non-linear. Aiming at the unstable and non-stationary characteristics of ultra-short-term wind speed, the variational mode decomposition algorithm is introduced to decompose the ultra-short-term wind speed data, and a series of stable and stationary components with different frequencies are obtained. The extreme learning machine with good prediction performance and real-time performance is selected as the prediction model of decomposed components. In order to solve the problem of random setting of input weights and bias of extreme learning machine, whale optimization algorithm is used to optimize extreme learning machine to improve the regression performance. The performance of the developed prediciton model is verified by real ultra-short-term wind speed sample data. Five prediction models are selected as the comparison model. Through the comparison between the predicted value and the actual value, the prediction error and its histogram distribution, eight performance indicators, and Pearson’s test correlation coefficient, the results show that the proposed prediction model has high prediction accuracy.

Keywords

Ultra-short-term wind speed prediction variational mode decomposition extreme learning machine whale optimization algorithm

Introduction

Background

As a pollution-free, renewable and clean energy, wind energy has attracted the attention of all countries in the world. However, with the continuous improvement of wind energy industry technology, it also faces some challenges. Due to the complexity, randomness and uncertainty of wind speed, these characteristics will lead to the instability of wind power (He et al., 2021). When a large number of unstable wind turbines are connected to the grid, it will bring great difficulties to the good operation of the whole power grid and seriously affect the safe operation of the power system. Therefore, it is of great significance to predict wind speed through a reliable prediction model (Tian, 2020; Wang and Yang, 2021).

Generally, according to different time scales, wind speed prediction can be divided into ultra-short-term prediction, short-term prediction, medium-term prediction, and long-term prediction. Ultra-short-term prediction is to forecast the future wind speed several minutes to 30 minutes in advance, which is mainly used for wind turbine control and power quality evaluation. Short-term prediction is to forecast wind speed 30–72 minutes in advance, which is mainly used for reasonable dispatching of power grid and maintenance of power quality. Medium-term wind speed prediction refers to the forecast of wind speed several days, weeks and months in advance. Long-term prediction is to forecast the future wind speed several years in advance. Due to the randomness and uncertainty of wind speed, the prediction of ultra-short-term wind speed has more theoretical and practical value (Liu et al., 2019). The research object in this paper is ultra-short-term wind speed.

Literature review

In recent years, scholars have proposed some wind speed prediction schemes. At present, the main models of wind speed prediction can be classified into physical prediction model, statistical prediction model, intelligent prediction model and combination prediction model.

Physical prediction model

Other environmental factors, such as atmospheric pressure, temperature and humidity, should be considered in physical prediction model. At the same time, the weather prediction model is used to simulate the wind speed trend to accurately predict the future wind speed. Among them, the most commonly used physical forecasting model is numerical weather prediction (NWP) model, and some research results have been obtained (Bochenek et al., 2021; Chen et al., 2021). NWP model is generally suitable for medium-term and long-term wind speed prediction. At the same time, NWP usually uses a more complex physical model, which needs to be completed with the help of relevant large-scale meteorological equipment. In addition, some countries or regions do not have corresponding meteorological data detection equipment, which makes it difficult to obtain meteorological data (Al-Yahyai et al., 2010). Therefore, the prediction model based on physical method has some limitations in practice.

Statistical prediction model

Through certain statistical model, the quantitative prediction of wind speed is carried out. Through certain analysis, the corresponding relationship between the original data and the actual predicted value is described. Compared with the physical prediction model, the statistical prediction model is relatively simple, does not need large monitoring equipment, and reduces the cost. Many prediction models based on statistical models predict wind speed or wind power. Statistical prediction model include time series-based model (Do et al., 2016; Eymen and Koylu, 2019; Sim et al., 2019), support vector machine (He and Xu, 2019; Ranganayaki and Deepa, 2019; Tian et al., 2018), neural network model (Duan et al., 2021; Guo et al., 2020a; Navas et al., 2020), etc. These models also have their own shortcomings. The model based on time series is suitable for linear time series, but it is not ideal for wind speed prediction with nonlinear characteristics. Support vector machine and its improved model are sensitive to parameters. Unreasonable parameters will have a great impact on the prediction results of the support vector machine. The traditional neural network can approach any nonlinear process, but the network structure is difficult to determine, the network training process is easy to fall into local optimum, and there is a certain requirement for the number of samples.

In recent years, the deep learning model based on the traditional neural network has also been applied to the prediction of wind speed. These deep learning forecasting models include causal convolutional network and gated recurrent unit (Ding et al., 2019), long short term memory (Shahid et al., 2021), convolutional neural network (Zhao et al., 2020), etc. However, the deep learning prediction model needs a lot of sample data, and the training process needs a lot of computing resources. It is not suitable for wind speed forecasting places with insufficient sample data or high real-time requirements.

Intelligent prediction model

In recent years, there are many intelligent optimization algorithms, which are also used in the field of wind speed prediction. These optimization algorithms optimize the parameters or structure of the wind speed prediction model to further improve the prediction accuracy. These optimization algorithms include artificial bee colony algorithm (Tian et al., 2019), backtracking search optimization algorithm (Tian et al., 2020b), particle swarm optimization algorithm (Ma et al., 2019), Jaya optimization algorithm (Liu et al., 2020), and so on. Although the prediction model optimized by optimization algorithm can improve the prediction performance, the optimization algorithm itself is easy to fall into local optimum, and the randomness of optimization algorithm sometimes cannot guarantee the stability of prediction performance. In addition, the maximum number of iterations, the number of population, and some special parameters of these algorithms has great influence on the optimization results.

Combination prediction model

A large number of studies have found that it is not appropriate to use a single model to predict wind speed. Because of the limited ability of single model prediction, the combination prediction model has attracted more and more attention of researchers. The combination prediction model makes full use of the advantages of each model, improves the prediction performance and reduces the difficulty of model selection. The first type of combination prediction model is linear and non-linear combination. For example, autoregressive moving average compensates echo state network (Tian et al., 2020c). The second type is weighted combination model. Different models predict the same wind speed, each model has a weighted coefficient, and the final predicted value is obtained by multiplying the predicted value of all models with the weighted coefficient (Guo et al., 2020b; Wang et al., 2020). The third type is combination model based on signal decomposition. The complex wind speed sequence is decomposed into several components with simple frequency by signal decomposition algorithm. Then, each component is modeled and predicted. The final wind speed prediction value is obtained by adding the predicted values of each component. These decomposition algorithms include empirical mode decomposition (EMD) (Liu et al., 2021), ensemble empirical mode decomposition (EEMD) (Tian et al., 2020a; Yu, 2020) , local mean decomposition (LMD) (Jiang et al., 2019), and variational mode decomposition (Hu et al., 2021), etc. From the current research point of view, the combination prediction model based on decomposition algorithm is a very worthy of in-depth study.

Innovations

In this study, we propose a combination prediction model. This prediction model adopts the variational mode decomposition (VMD) and whale optimization algorithm (WOA) optimized extreme learning machine (ELM). VMD algorithm can solve the end effect and mode aliasing problem of EMD algorithm, and greatly reduce the randomness and non-stationary of the original ultra-short-term wind speed data. As a novel neural network, ELM only needs to set input weights and bias randomly, and its process does not need iteration, so it improves the training speed and generalization ability of neural network. WOA is used to optimize the optimal input weights and the bias of the hidden layer of ELM to further improve the prediction accuracy. Finally, the effectiveness of the prediction model is verified by the real ultra-short-term wind speed data. In short, the innovation of this paper is as follows.

(a)VMD algorithm is used to decompose the ultra-short-term wind speed data to obtain a series of relatively stable components with different frequencies, which reduce the difficulty of the original data modeling.

(b)ELM is chosen as the prediction model of each component after VMD decomposition. At the same time, WOA is used to optimize the input weights and the bias of the hidden layer of ELM.

(c)A variety of performance indicators are used to verify the effectiveness of the prediction model for the real collected ultra-short-term wind speed data.

The structure of the paper

The rest content of the paper is arranged as follows. Section 2 describes the knowledge of VMD algorithm; Section 3 introduces the knowledge of WOA optimized ELM prediction model. Section 4 gives the implementation process of the developed prediction model. The case studies are given in Section 5. The conclusions and future work are summarized in Section 6.

VMD algorithm

The essence of VMD algorithm is to obtain a variational mode model, determine different central frequency and bandwidth, and decompose the original signal into a series of intrinsic mode function components (IMFs) with different frequencies. In VMD algorithm, each mode has its limited bandwidth and central frequency. VMD algorithm has good robustness, high computational efficiency, and good performance of modal separation and signal reconstruction (Chouksey and Jha, 2021). It is suitable for the decomposition of ultra-short-term wind speed with nonlinear and non-stationary characteristics. The variational model is as follows.

\min_{{u_{k}}, {ω_{k})} {\sum_{k} | | \partial_{t} [(δ (t) + \frac{j}{π t}) u_{k} (t)] e^{- i ω_{k} t} | |_{2}^{2}}, s . t . \sum_{k} u_{k} = f

(1)

where, ${u} = {u_{1}, \dots, u_{K}}$ is the decomposed K IMFs, ${ω} = {ω_{1}, \dots, ω_{K}}$ is the frequency center of each component, $k = 1, 2, \dots, K$ . The quadratic penalty factor $α$ and Lagrangian multiplier operator $λ (t)$ are used into the variational problem. The constrained variational problem is transformed into an unconstrained variational problem. The extended Lagrangian expression is as follows.

L ({u_{k}}, {ω_{k}}, λ) = α \sum_{k} | | \partial_{t} [(δ (t) + \frac{j}{π t}) u_{k} (t)] e^{- j ω_{k} t} | |_{2}^{2} + | | f (t) - \sum_{k} u_{k} (t) | |_{2}^{2} + 〈 λ (t), f (t) - \sum_{k} u_{k} (t) 〉

(2)

The variational problem is solved by alternating direction method of multiplier algorithm. The original signal is decomposed into K IMFs, and the specific implementation process is as follows.

Step 1 Initialize ${u_{k}^{1}}$ , ${ω_{k}^{1}}$ , n and $λ^{1}$ .

Step 2 Update $u_{k}$ by equation (3), $ω_{k}$ by equation (4).

u_{k}^{n + 1} (ω) = \frac{f (ω) - \sum_{j \neq k} u_{j} (ω) + \frac{λ (ω)}{2}}{1 + 2 α {(ω - ω_{k})}^{2}}

(3)

ω_{k}^{n + 1} = \frac{\int_{0}^{\infty} ω | u_{k} (ω) |^{2} d ω}{\int_{0}^{\infty} | u_{k} (ω) |^{2} d ω}

(4)

Step 3 Update $λ$ by equation (5). If $\sum_{k} | | u_{k}^{n + 1} - u_{k}^{n} | |_{2}^{2} / | | u_{k}^{n} | |_{2}^{2} < e$ , stop iteration. If the conditions are not met, return to Step 2 to continue the iteration until the accuracy requirement is satisfied.

λ^{n + 1} (ω) = λ^{n} (ω) + τ [f (ω) - \sum_{k} u_{k}^{n + 1} (ω)]

(5)

WOA optimized ELM model

In this section, the basic theories of WOA and ELM are introduced, and the WOA optimized ELM prediction model of each component after VMD decomposition is given. ELM is a typical single hidden layer feed-forward neural network (Khatab et al., 2021). The input weights and bias of ELM can be set randomly, so it can greatly reduce the amount of computing, and the connection weights of hidden layer and output layer do not need to be adjusted iteratively, but determined by solving linear equations, so the generalization ability of ELM is greatly improved. For a given training set $D = {(x_{i}, t_{i}) | x_{i} \in R^{n}, t_{i} \in R^{m}, i = 1, \dots, N}$ , activation function $f (x)$ , the number of hidden nodes $L$ . ELM regression model can be expressed as

\begin{matrix} \sum_{i = 1}^{L} β_{i} f (a_{i} x_{1} + b_{i}) = t_{1} \\ \sum_{i = 1}^{L} β_{i} f (a_{i} x_{2} + b_{i}) = t_{2} \\ . \\ . \\ \sum_{i = 1}^{L} β_{i} f (a_{i} x_{k} + b_{i}) = t_{k} \end{matrix}

(6)

where, $a_{i}, i = 1, \dots, L$ is the output weights, $b_{i}, i = 1, \dots, L$ is bias, k is the numbers of the sample. Equation (6) is rewritten as

T_{k} = H_{k} β_{k}

(7)

where $H_{k}$ is a neuron matrix and can be represented as

H_{k} = [\begin{matrix} f (a_{1} x_{1} + b_{1}) & \dots & f (a_{L} x_{1} + b_{L}) \\ f (a_{1} x_{2} + b_{1}) & \dots & f (a_{L} x_{2} + b_{L}) \\ \dots & \dots & \dots \\ f (a_{1} x_{k} + b_{1}) & \dots & f (a_{L} x_{k} + b_{L}) \end{matrix}]

(8)

$β_{k}$ is the output weights and can be expressed as

β_{k} = {[\begin{matrix} β_{1} & β_{2} & \dots & β_{k} \end{matrix}]}^{T}

(9)

$T_{k}$ is the output weights and can be expressed as

T_{k} = {[\begin{matrix} t_{1} & t_{2} & \dots & t_{k} \end{matrix}]}^{T}

(10)

The output weights can be obtained by solving equation (7).

β_{k} = {(H_{k}^{T} H_{k})}^{- 1} H_{k}^{T} T_{k}

(11)

Therefore, the prediction model based on ELM can be obtained after training,

t = \sum_{i = 1}^{L} β_{i} f (a_{i} x + b_{i})

(12)

The random selection of input weights and bias of ELM can lead to the poor prediction effect of the model (Alaba et al., 2019). Therefore, we use the WOA to optimize the parameters of the ELM to improve the prediction accuracy.

WOA algorithm

WOA is a population intelligent optimization algorithm proposed in 2016, which is derived from the predatory strategy of whales in the ocean. In WOA, the position of each whale can be seen as a potential feasible solution. In the process of whale’s hunting, each whale has two behaviors. First, surround the prey. The whale’s current position will move to other prey. Another behavior is through the bubble net, the whale in accordance with their own way through the cycle of swimming, so as to achieve the effect of driving prey. In each behavior, the whale will randomly produce a kind of hunting behavior. WOA uses random learning strategy to realize the global search of the algorithm, and the convergence speed is faster in the early stage of the search. At the same time, shrinking encirclement, and spiral predator-prey strategy make it have stronger development ability in the later stage, so it is a better global optimization algorithm. It has the advantages of simple process, fast convergence speed, excellent performance in solving optimization problems, and has a wide range of applications. The authors tested the WOA algorithm through six structural optimization problems and 29 mathematical optimization problems. The experimental results show that the WOA algorithm has good global search ability and convergence speed compared with particle swarm optimization algorithm, genetic algorithm and cuckoo search algorithm (Mirjalili and Lewis, 2016). Other related research results also show that WOA has better optimization performance (Chen, 2020; Kaveh and Ghazaan, 2017).

Surround the prey

The whale in its current position will swim in the direction of the whale closest to its prey in all populations, slowly gathering the entire population and eventually capturing its prey. The update formula of whale position is as follows:

{\begin{matrix} X_{i}^{t + 1} = X_{best}^{t} - A * D \\ D = | C * X_{best}^{t} - X_{i}^{t} | \end{matrix}

(13)

where $X_{best}^{t}$ is the optimal position of the whole whale population, $X_{i}^{t}$ is the current position of the whale. A and C are coefficient matrices, as shown below.

A = 2 a * r_{1} - a

(14)

C = 2 * r_{2}

(15)

where $r_{1}$ and $r_{2}$ are random numbers generated by uniform distribution in the range of [0, 1], $a$ is the convergence factor, which is shown in equation (16). It decreases linearly from 2 to 0 with the number of iterations.

a = 2 - 2 t / t_{max}

(16)

where $t$ is the current number of iterations and $t_{max}$ is the maximum number of iterations.

The whale in the current position randomly selects a whale and swims its direction, thus increasing the search range of the whale population. The location update method is

X_{i}^{t + 1} = X_{rand}^{t} - A * | C * X_{rand}^{t} - X_{i}^{t} |

(17)

where $X_{rand}^{t}$ is a random position for whales. At the same time, the value of coefficient matrix $A$ determines the target that the whale moves toward. When $| A | \geq 1$ , the whale in the current position moves toward a random target, while when $| A | < 1$ , the whale moves toward the individual with the optimal position. In WOA algorithm, the larger convergence factor has better global search ability to avoid the algorithm falling into local optimum, while the smaller convergence factor has stronger local search ability to accelerate the convergence speed of the algorithm.

Bubble net

The whale uses bubble net to drive away the prey, reduces the encirclement by spiral search, and finally makes all the individual whales gradually approach the best individual position. The whale swims in a spiral path around its prey. In order to simulate this synchronous behavior, we assume that the selection probability of the shrink encircling mechanism and the spiral update position are both 0.5. The expression for generating the new individual is as follows.

X_{i}^{t + 1} = {\begin{matrix} X_{gbest}^{t} - A * | C * X_{gbest}^{t} - X_{i}^{t} |, p < 0.5 \\ X_{gbest}^{t} + D * e^{bl} * \cos (2 π l), p \geq 0.5 \end{matrix}

(18)

where b is a constant that defines the logarithmic shape, l is the random number between [−1, 1], and p is the random number generated by uniform distribution in the range of [0, 1].

WOA-ELM model

In this section, we introduce WOA to optimize the weights and the bias of ELM model. The corresponding sample data is collected, and the training set of the generated sample data is input into the ELM model, and then the input weights and bias are optimized by the WOA. After obtaining the best parameters, the validity of the model is verified by using the test set. The specific implementation process of the WOA optimized ELM model is as following:

Step 1 The data is pre-processed, and the new data after pre-processing is divided into training set and test set. Generally, the first 80% is the training set, and the last 20% is the test set.

Step 2 The data of training set and test set are normalized. In order to reduce the training error, the values with large difference are transformed into values between [0, 1]. The formula of data normalization is as follows.

x = \frac{x - x_{min}}{x_{max} - x_{min}}

(19)

Step 3 The range of the input weights and bias are determined as [0, 1]. The weight $a_{i}$ and bias b are taken as the objects to be optimized. The number of the two parameters is the number of hidden layer nodes L.

Step 4 The individual fitness of the whale population is calculated, and the root mean square error (RMSE) between the actual value and the predicted value of ELM model is taken as the fitness value of WOA. The expression of fitness is as follows.

fitness = \sqrt{\frac{1}{N} \sum_{i = 1}^{N} {(y_{i} - \bar{y_{i}})}^{2}}

(20)

where $y_{i}$ is the actual value, $\bar{y_{i}}$ is the predictive value, N is the length of sample data.

Step 5 Judge whether the termination conditions are met. If the condition is hold, the optimal weights and bias are output. Otherwise, go back to iterative optimization.

The process of WOA optimized ELM model is shown in Figure 1.

Figure 1.

Flow chart of WOA optimized ELM model.

Proposed prediction model

In this section, we combine the WOA optimized ELM with VMD algorithm to propose a combination prediction model. Firstly, the ultra-short-term wind speed time series is decomposed by VMD algorithm to form a number of relatively stable IMF components with obvious local characteristics, which are predicted by ELM model. Meanwhile, the input weights and bias of ELM are optimized by WOA algorithm. Finally, the predicted value of each ELM model is superimposed to get the final predicted value. The framework of the proposed ultra-short-term wind speed prediction model is shown in Figure 2. In this study, the single-step prediction iterative method is used to achieve multi-step prediction. As can be seen from Figure 2, the implementation steps of the proposed prediction model are as follows.

Step 1 Ultra-short-term wind speed data are collected to generate training set and test set. Assume ultra-short-term wind speed time series is $w (i)$ , i is the sampling time.

Step 2 The training set data is decomposed by VMD algorithm to generate S IMF components. The current sampling time is i, the input $x (i), x (i - 1), \dots, x (i - m + 1)$ and the output $x (i + 1)$ of each IMF are generated. That is to say, the number of input layers of ELM is m, and the number of output layers is 1. The ELM model is trained for each IMF component. The input weights and bias of ELM are optimized by WOA. The optimal parameters are determined.

Step 3 Part of the training set data is removed, and the missing data is supplemented by the test set. Keep the input data length unchanged.

Step 4 VMD is adopted to generate S IMF components. The current sampling time is i, $x (i), x (i - 1), \dots, x (i - m + 1)$ after VMD decomposition is chosen as the input of ELM. The input weights and bias are optimal values obtained in Step 2. The predictive value $\bar{x} (i + 1)$ of each component is obtained by the corresponding ELM model. The final forecasting value $\bar{w} (i + 1)$ of wind speed is obtained by adding all output value $\bar{x} (i + 1)$ of each ELM model.

Step 5 The sliding window method is used to replace the oldest wind speed data with $\bar{w} (i + 1)$ , i=i+1, and return to Step 4 until all the test set data are predicted.

Figure 2.

The framework of the proposed ultra-short-term wind speed prediction model.

Case study

The validity of the model is verified by the real ultra-short-term wind speed data. Compared with other models, the comparison results of performance indicators fully show the effectiveness of the model.

Data set

In the case study, the wind speed data of Manjing wind farm in Shangyi County, Zhangjiakou City, Hebei Province of People’s Republic of China in 2012 are used as the research object. The data are sampled every 15 minutes, and 500 groups of ultra-short-term wind speed data with a sampling period of 15 minutes are collected. At the same time, the data set is divided into two parts, the first 80% of 400 groups of data as the training set, and the last 20% of 100 groups of data as the test set. The collected ultra-short-term wind speed sample data is shown in Figure 3.

Figure 3.

The collected ultra-short-term wind speed sample data.

It can be seen from Figure 3 that the ultra-short-term wind speed time series has strong nonlinear, non-stationary and fluctuating characteristics, and there will be great differences in the wind speed values at the two sampling times, which brings great difficulties to its modeling and prediction. On the other hand, the ultra-short-term wind speed time series contains a lot of frequency components, so the original wind speed can be decomposed by VMD algorithm to reduce the complexity of modeling.

Performance indicators

In order to verify the prediction accuracy of the ultra-short-term wind speed prediction model, the following eight performance indicators are introduced to measure the prediction effect of the prediction model (Tian et al., 2020a).

a. Root mean square error (RMSE)

RMSE = \sqrt{\frac{1}{N} \sum_{i = 1}^{N} {(w (i) - \bar{w} (i))}^{2}}

(21)

b. Mean absolute error (MAE)

MAE = \frac{1}{N} \sum_{i = 1}^{N} | w (i) - \bar{w} (i) |

(22)

c. Mean absolute percentile error (MAPE)

MAPE = \frac{1}{N} \sum_{i = 1}^{N} | w (i) - \bar{w} (i) | \times 100 / w (i)

(23)

d. Relative root mean square error (RRMSE)

RRMSE = \sqrt{\frac{1}{N} \sum_{i = 1}^{N} {(\frac{w (i) - \bar{w} (i)}{w (i)})}^{2}}

(24)

e. Square sum error (SSE)

SSE = \sum_{i = 1}^{N} {(w (i) - \bar{w} (i))}^{2}

(25)

f. $R^{2}$ (R Square)

R^{2} = 1 - \frac{\sum_{i = 1}^{N} {(w (i) - \bar{w} (i))}^{2}}{\sum_{i = 1}^{N} {(w (i) - w_{m})}^{2}}

(26)

g. Theil inequality coefficient (TIC)

TIC = \frac{\sqrt{\frac{1}{N} {(w (i) - \bar{w} (i))}^{2}}}{\sqrt{\frac{1}{N} \sum_{i = 1}^{N} w {(i)}^{2}} + \sqrt{\frac{1}{N} \sum_{i = 1}^{N} \bar{w} {(i)}^{2}}}

(27)

h. The index of agreement (IA)

IA = 1 - \frac{\sum_{i = 1}^{N} {(w (i) - \bar{w} (i))}^{2}}{\sum_{i = 1}^{N} {(| \bar{w} (i) - w_{m} | + | w (i) + w_{m} |)}^{2}}

(28)

where, $N$ is number of sample, $w (i)$ is actual value of ultra-short-term wind speed, $\bar{w} (i)$ is predicted values of ultra-short-term wind speed, $w_{m}$ is the mean value of ultra-short-term wind speed.

At the same time, the Pearson’s test is introduced to test the accuracy of prediction from the statistical point of view. Pearson’s test can reflect the correlation strength between the actual value and the predicted value. If the Pearson’s correlation coefficient is closer to 1, the stronger the correlation between the actual value and the predicted value is, the better the prediction effect is. Otherwise, the closer the Pearson’s correlation coefficient is to 0, the worse the prediction effect of the prediction model is.

Comparison models

All the simulation verification is carried out on the Matlab Version 2018. The CPU is Intel i5-1050 with 3.10 GHz frequency, Windows 10 operating system (64 bit system), and 16 GB memory. In addition, in order to ensure the fairness comparison between different prediction models, the optimal parameters are allocated for different prediction models in all simulations.

This paper selects the following five models for comparison, including BSOA-LSSVM (Tian et al., 2020b), hybrid kernel LSSVM (Tian et al., 2018), ESN (Chitsazan et al., 2017), wavelet neural network (Shi et al., 2018) and ELM (Shamshirband et al., 2016). The first 400 groups sample data in last section are chosen as training set. After training, the parameters of these prediction models are as shown in Table 1.

Table 1.

The parameters of the comparison models.

Model	Parameters
BSOA-LSSVM	$γ$ : 25.3672; $σ^{2}$ : 10.5521
Hybrid kernel LSSVM	$a$ : 0.3220; $d$ : 1, $δ$ : 0.2033, $ε$ : 0.1836; $C$ : 26.3307
ESN	IS: 0.3051; SD: 0.6087; SR: 0.4115; N: 87
Wavelet neural network	Input neurons: 20; hidden layer neurons: 30; output layer neurons: 15
ELM	$λ$ : 1; L: 48

Results

The training set is processed by VMD algorithm. The decomposition level of VMD algorithm has a great influence on the decomposition effect of data. The algorithm does not clearly explain how to determine the optimal decomposition level. In this paper, through repeated experiments, the decomposition level of VMD is determined to be 8. The training set data is decomposed by VMD, as shown in Figure 4. It can be seen from Figure 4, the change of each component after VMD decomposition is relatively stable. Decomposition by VMD reduces the interference and coupling between original ultra-short-term wind speed time series. Therefore, VMD reduces the difficulty of prediction modeling and improves the prediction accuracy.

Figure 4.

The IMF components of ultra-short-term wind speed after VMD decomposition.

After getting each component, this paper uses the ELM model optimized by WOA to predict each component. The parameters of WOA are as follows: the maximum number of iterations is 100, the population number is 50, and logarithmic shape l is set as 1. The hidden layer node of ELM is 25, and the optimization range of input weights and bias is [0, 1]. After training the optimal parameters of eight ELM models, the optimized ELM model is used to predict each component of 100 groups of test data. Figure 5 below shows the comparison between the actual value and the predicted value of each component. It can be seen from Figure 5 that the ELM model optimized by WOA can better predict each IMF component. The predicted value is in good agreement with the actual value. The reason for better prediction accuracy is that the random parameters in ELM are optimized by WOA, and the optimized ELM model has higher prediction accuracy.

Figure 5.

The prediction results of each IMF component.

After the predicted values of each IMF component are obtained, the final predicted results can be obtained by adding these predicted values. The comparison between the actual and predicted values of the final test set is shown in Figure 6. In order to compare the prediction performance, the prediction results of the five comparison models are also shown in Figure 6. It can be seen from the comparison results in Figure 6 that the predicted values of the proposed prediction model can better fit the variation trend of ultra-short-term wind speed, and can accurately predict the wind speed at each sampling time.

Figure 6.

The actual value and prediction value comparison between the proposed model and comparison models.

Meanwhile, the prediction error of each prediction model is shown in Figure 7. The results in Figure 7 further show that the designed prediction model has smaller prediction error, which means that the prediction accuracy of the proposed prediction model is the highest. It can also be seen from Figure 7 that the prediction error range of the prediction model in this paper is [−0.28, 0.26], the prediction error range of BSOA-LSSVM is [−0.67, 0.49], the prediction error range of hybrid kernel SVM is [−0.72, 0.62], the prediction error range of ESN is [−0.55, 0.52], the prediction error range of Wavelet neural network is [−0.76, 0.78], and the prediction error range of ELM is [−0.82, 0.80]. After comparison, compared with other comparison models, the prediction error of this paper is reduced from 46.93% to 65.85%.

Figure 7.

The prediction error comparison between the proposed model and comparison models: (a) proposed model,(b) BSOA-LSSVM, (c) hybrid kernel SVM, (d) ESN, (e) Wavelet neural network, and (f) ELM.

Figure 8 shows the histogram of the prediction error distribution. It can be seen from this figure that the prediction error distribution of the proposed prediction model is more concentrated around the abscissa zero, which means that the number of smaller prediction errors is more than that of larger prediction errors. Therefore, the prediction error distribution of the proposed prediction model is centralized rather than decentralized, and the prediction performance is better.

Figure 8.

The histogram of the prediction error distribution: (a) proposed model, (b) BSOA-LSSVM, (c) hybrid kernel SVM,(d) ESN, (e) Wavelet neural network, and (f) ELM.

Table 2 show the statistical analysis results of prediction errors of mentioned prediction models for the ultra-short-term wind speed. The absolute value of the prediction error is compared, and the results of the maximum, minimum, average, and variance value of the prediction error are given. It can be seen from the results in Table 2 that the prediction model in this paper has better effect on the statistical indicators of prediction error.

Table 2.

The statistical analysis results of absolute value of prediction errors for ultra-short-term wind speed.

Forecasting model	Maximum (m/s)	Average (m/s)	Variance ((m/s)²)
Proposed model	0.2824	0.0625	0.0040
BSOA-LSSVM	0.6649	0.1783	0.0210
Hybrid kernel LSSVM	0.7263	0.1547	0.0244
ESN	0.5457	0.1465	0.0150
Wavelet neural network	0.8423	0.1803	0.0280
ELM	0.8225	0.2147	0.0371

The following Table 3 shows the comparison of the eight performance indicators between the designed model and other comparison models. It can be seen from the results in the table that the performance indicators of this paper are better than other comparison models. Specifically, as can be seen from the comparison results in Table 3, RMSE, MAE, MAPE, RRMSE, SSE, and TIC values of the proposed prediction model are smaller than the other comparison models. The smaller the values of these performance indicators, the better the prediction performance of the prediction model. Meanwhile, $R^{2}$ and IA value of the proposed prediction model is closer to one than the comparison models. The closer the value of $R^{2}$ and IA is to 1, the better the regression prediction performance of the model is. Therefore, the prediction accuracy of the proposed prediction model for ultra-shot-term wind speed is better than the other comparison models.

Table 3.

The comparison of the performance indicators between the designed model and other comparison models.

Model	RMSE (m/s)	MAE (m/s)	MAPE (%)	RRMSE	SSE ((m/s)²)	$R^{2}$	TIC	IA
Proposed model	0.0889	0.0625	1.8328	0.0214	0.7911	0.9979	0.0118	0.9998
BSOA-LSSVM	0.2294	0.1783	5.5491	0.0644	5.2612	0.9864	0.0304	0.9989
Hybrid kernel LSSVM	0.2193	0.1547	4.7873	0.0581	4.8110	0.9875	0.0291	0.9990
ESN	0.1906	0.1465	4.5534	0.0524	3.6310	0.9904	0.0253	0.9992
Wavelet neural network	0.2453	0.1803	5.2821	0.0640	6.0172	0.9838	0.0325	0.9987
ELM	0.2877	0.2147	6.3671	0.0725	8.2780	0.9779	0.0381	0.9982

Pearson’s test is introduced to measure the association strength between the actual value and the prediction value of ultra-short-term wind speed. If Pearson’s correlation coefficient is equal to 1, it indicates that the actual value and the prediction value of the prediction model have a linear relationship. On the other hand, if Pearson’s correlation coefficient is equal to 0, there is no relationship between the actual value and the prediction value of the prediction model. Table 4 gives the results of the Pearson’s test between the proposed model and comparison models. The result clearly shows that the result of Pearson’s test of the proposed prediction model is higher than those of the other prediction models.

Table 4.

The Pearson’s test results between the proposed model and comparison model.

Model	Pearson’s test coefficient
Proposed model	0.9989
BSOA-LSSVM	0.9933
Hybrid kernel LSSVM	0.9938
ESN	0.9952
Wavelet neural network	0.9920
ELM	0.9889

In a word, the comparison between the predicted value and the actual value, the comparison of the prediction error and its histogram distribution, the statistical analysis results of prediction errors, the comparison of the performance indicators and the Pearson’s test results all show that the prediction model proposed in this paper has better prediction accuracy and prediction effect than other comparison models.

Discussions

It can be seen from the results of the previous section that the prediction performance of the proposed model is better than other comparison models. The results are analyzed and discussed as follows.

The comparison between the actual value of ultra-short-term wind speed and the predicted value of the proposed model shows that the predicted value can better fit the change of the actual value. Furthermore, the prediction error of the prediction model in this study is smaller than other models, and the distribution of the prediction error is more reasonable. Therefore, the prediction model in this paper is superior to other comparison models only from the fitting curve.

All the prediction performance indicators not only show that the established prediction model is better than the comparison models, but also clearly explain why the proposed prediction model shows certain prediction performance. The prediction model proposed in this paper can better reveal the internal variation law of ultra-short-term wind speed, so as to improve its performance indicators.

The fundamental reason for the improvement of prediction performance lies in the decomposition of ultra-short-term wind speed by VMD algorithm and the selection of ELM model. Each component decomposed by VMD is more stationary than the original ultra-short-term wind speed series, so the difficulty of prediction is reduced. Meanwhile, the optimization of ELM parameters by WOA improves the adaptability of the model, so the prediction performance is improved.

Conclusions

In this paper, aiming at the problem of ultra-short-term wind speed prediction, a decomposition and combination prediction model based on VMD and WOA optimized ELM is proposed. The original ultra-short-term wind speed is non-linear and non-stationary, which contains many different frequency components. By decomposing the ultra-short-term wind speed with VMD algorithm, the stationary of the components with different frequencies is improved. ELM is chosen as the prediction model for each component. WOA is introduced to optimize the input weights and bias of the ELM model. The combination of VMD, ELM and WOA gives full play to their respective advantages and improves the prediction accuracy of the final model. The real collected ultra-short-term wind speed data is taken as the research object, and the prediction effect of the prediction model is verified. The results show that, compared with other prediction models, the prediction model proposed in this paper is the best in both curve fitting degree and performance indicators. The prediction model in this paper can provide a certain degree of technical support for the prediction of wind speed and wind power of the actual wind farm.

Although, VMD algorithm is used to reduce the non-stationary of ultra-short-term wind speed, and then combination model is used to improve the prediction accuracy, but there are still some details to be further adjusted. The future work includes: (a) The number of hidden layers of ELM neural network is determined by repeated experiments, which is a waste of time. How to quickly and effectively determine the number of hidden layers of ELM is the research goal of the future. (b) How to choose the decomposition level of VMD algorithm to achieve the best decomposition state effect is also a problem worthy of study.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is partially supported by the Doctoral Scientific Research Foundation of Liaoning Province (Grant No. 20180540050).

ORCID iD

Lian Lian

Data Availability Statement

The data used to support the findings of this study are available from the corresponding author upon request.

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