Artificial bee colony algorithm–optimized error minimized extreme learning machine and its application in short-term wind speed prediction

Abstract

In order to improve the prediction accuracy of short-term wind speed, a short-term wind speed prediction model based on artificial bee colony algorithm optimized error minimized extreme learning machine model is proposed. The extreme learning machine has the advantages of fast learning speed and strong generalization ability. But many useless neurons of incremental extreme learning machine have little influences on the final output and, at the same time, reduce the efficiency of the algorithm. The optimal parameters of the hidden layer nodes will make network output error of incremental extreme learning machine decrease with fast speed. Based on the error minimized extreme learning machine, artificial bee colony algorithm is introduced to optimize the parameters of the hidden layer nodes, decrease the number of useless neurons, reduce training and prediction error, achieve the goal of reducing the network complexity, and improve the efficiency of the algorithm. The error minimized extreme learning machine prediction model is constructed with the obtained optimal parameters. The stability and convergence property of artificial bee colony algorithm optimized error minimized extreme learning machine model are proved. The practical short-term wind speed time series is used as the research object and to verify the validity of the prediction model. Multi-step prediction simulation of short-term wind speed is carried out. Compared with other prediction models, simulation results show that the prediction model proposed in this article reduces the training time of the prediction model and decreases the number of hidden layer nodes. The prediction model has higher prediction accuracy and reliability performance, meanwhile improves the performance indicators.

Keywords

Extreme learning machine artificial bee colony algorithm short-term wind speed prediction optimization

Introduction

Increasing natural energy resources, the reduction of emissions, climate change, and rising energy costs lead to a change from classic carbon or nuclear-based power supply to a concentration on renewable energy resources (Kramer et al., 2014). Wind energy is a typical renewable energy (Heinermann and Kramer, 2016). Accurate short-term wind speed prediction has important theoretical significance and practical application value for wind power industry (Tascikaraoglu and Uzunoglu, 2014; Troncoso et al., 2015). Short-term wind speed prediction means predict the future wind speed in advance (Ambach and Vetter, 2016). The persistence method is the simplest prediction method, that is, the measured wind speed value at the latest sampling time is taken as the prediction value of the next sampling time (Santamaria-Bonfil et al., 2016). Many prediction methods generally use the persistence method as a benchmark to evaluate the prediction accuracy. Some scholars have set up the mathematical model for wind speed and have achieved good prediction results (Bozkurt et al., 2014; Gottschall et al., 2006; Henriksen et al., 2013; Iversen et al., 2016; Luhur et al., 2015). These mathematical models need to consider a lot of input conditions and states, but in practical applications, these states or conditions are difficult to obtain, which greatly limits the actual implementation.

However, many prediction methods of short-term wind speed are mainly based on historical data and time series model. The author compares the use of wind speed time series as well as differences of subsequent measurements with random forests, support vector regression, and k-nearest neighbors. The results show that time series is suitable for wind speed prediction (Heinermann and Kramer, 2015). These prediction methods using historical data, through some linear models include multivariate adaptive regression model (Staid et al., 2015), autoregressive moving average model (Erdem and Shi, 2011), autoregressive integrated moving average model (Cadenas et al., 2016) or nonlinear model include support vector machine (SVM) (Ortiz-García et al., 2011; Shrivastava et al., 2016), least squares support vector machines (Mathaba et al., 2012), artificial neural network (Doucoure et al., 2016; Mishra and Dash, 2018; Ramasamy et al., 2015; Saleck and von Bremen, 2007; Sivachitra and Vijayachitra, 2014), and so on to predict short-term wind speed time series in the future. The research results show that the short-term wind speed has a strong nonlinear (Baloch et al., 2016; Soman et al., 2010), so the nonlinear model is more suitable for short-term wind speed prediction. But there is no uniform method to determine the parameters of SVM and least squares support vector machine (LSSVM) prediction model. The artificial neural network is difficult to determine the network structure and easy to fall into the local optimal value.

The traditional neural network learning algorithm uses the gradient descent method to train the network. The learning speed of algorithm is slow, and all the parameters of the network need to be adjusted iteratively. This problem seriously restricts the development of forward neural network. Huang et al. (2006b) proposed a new neural network—extreme learning machine (ELM) algorithm. The ELM algorithm uses the random mechanism to reduce the parameter setting and choice. It is one kind of simple feasible fast learning algorithm. Compared with other traditional neural network learning algorithms or SVM, ELM algorithm has the advantages of fast learning speed and strong generalization ability (Huang et al., 2012; Lan et al., 2010). The authors point out that computing time of ELM is usually several thousand times faster than back propagation (BP) neural network or SVM (Naji et al., 2016). Therefore, the ELM algorithm is also applied to the short-term wind speed prediction (Bouzgou, 2014; Salcedo-Sanz et al., 2014; Shamshirband et al., 2015). However, the optimal number of hidden nodes in the standard ELM algorithm is determined by many experiments, so it is not only time-consuming but also random.

In recent years, many scholars have studied the network structure of ELM deeply. From the current research results, the main research work is focused on the node incremental ELM (I-ELM). I-ELM algorithm starts from a relatively small-scale neural network, gradually adds the hidden node to the network. When the output error of the network meets the accuracy of the actual problem, the node increase process will stop. Huang proposed extreme learning incremental machine (I-ELM) algorithm (Huang et al., 2006a) and enhanced incremental extreme learning machine (EI-ELM) algorithm (Huang and Chen, 2007, 2008), Wang proposed enhanced convex incremental extreme learning machine (ECI-ELM) algorithm (Wang and Zhang, 2012), and Feng proposed error minimized extreme learning machine (EM-ELM) algorithm (Feng et al., 2009). All of the above algorithms added hidden layer units in the network. The number of hidden nodes of I-ELM, CI-ELM, EI-ELM, and ECI-ELM algorithms increased with the increase in the number of iterations. The hidden nodes of EM-ELM algorithm can be added one by one or quantity with increase in the number of iterations. Although the incremental ELM algorithm have not the optimal number of hidden layer node and over fitting problems, many neurons in the incremental ELM algorithm have a very small effect on the final output. The useless neurons increase the number of iterations, which reduces the efficiency of the algorithm.

Based on the above discussions and EM-ELM algorithm, this article uses the artificial bee colony (ABC) algorithm to optimize the parameters of the hidden layer nodes in EM-ELM. The proposed algorithm can reduce the number of useless neurons in EM-ELM, decrease the complexity of the network, and improve the efficiency of the algorithm. The simulation results show that the method has better prediction effect through the actual short-term wind speed data.

The preliminaries

EM-ELM algorithm

The structure of ELM neural network algorithm is similar with the single-hidden layer feed forward neural networks (SLFNs). The difference is that ELM can randomly select its training parameters and get a complete network training model only through the output weights obtained by the least squares method.

ELM algorithm contains the following three steps (Wang and Zhang, 2012). 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 $g (x)$ , and the number of hidden nodes $L$ :

Step 1: the output weights $a_{i}, i = 1, \dots, L$ and offset $b_{i}, i = 1, \dots, L$ of the hidden units are randomly generated $a_{i}, b_{i}, i = 1, \dots, L$ .

Step 2: calculating output matrix of the hidden unit $H$ .

Step 3: calculating output weight $β$ . $β = H T$ . Where $H^{†}$ is Moore–Penrose pseudoinverse of the hidden layer output matrix H . Moore–Penrose pseudoinverse in ELM can be calculated by factorization method based on singular value

β = {(H^{T} H)}^{- 1} H^{T} T

(1)

where $H^{†} = {(H^{T} H)}^{- 1} H^{T}$ . Because $H^{†} H = I$ , $H$ is left pseudoinverse matrix of $H$ . Therefore, the general minimum mean square estimation of a linear system is expressed as follows

β = {(H^{T} H)}^{- 1} H^{T} T

(2)

The initial network structure of EM-ELM neural network has zero hidden layer nodes. The network structure can also be determined by the user. Under the premise of the initial structure determination, before increasing the number of nodes in the hidden layer, the training data should be used to train the network. The number of hidden layer nodes is increased after the initial output weights and the training errors are obtained. There are two methods on how to increase the nodes number of the hidden layer, single or quantity increasing. In the case of a single increase, the algorithm randomly generated input weights and thresholds of node. The node is added to the existing network structure. In the case of quantity increasing, there are two methods to increase the number of hidden layer nodes. The first method is to increase the fixed number of hidden layer nodes each time. The other is to increase the number of hidden layer nodes, which has a linear relationship with the number of existing hidden layer nodes. After obtaining the number of nodes that need to be increased, it will be added to the network and the corresponding input weights and thresholds can be added to the existing input weights and thresholds matrix. After iteration, the network needs to update the output weights and training errors of the whole network. Repeat to increase the number of hidden nodes until achieving the maximum hidden layer nodes number or the desired training error.

But the network output value $β$ of EM-ELM is obtained by least square. The algorithm can guarantee the minimum of $| | β | |$ , which means that $| | β | |$ approaches to 0 after several iterations. This leads to network output difference of EM-ELM and is very small between using $M$ neurons and $M + 1$ neurons. That means a lot of neurons have a very small effect on the performance of the network. At the same time, the authors proved that the optimal hidden layer parameters $a^{*}, b^{*}$ can make network output error decrease with the fastest speed (Alade et al., 2017). If we can obtain these two optimal parameters, we can greatly improve the learning speed of EM-ELM. Based on the above discussions, this article will use the ABC algorithm to optimize the parameters of the hidden layer.

ABC algorithm

The ABC algorithm is an intelligent optimization algorithm which comes from the behavior of honey bees (Karaboga and Basturk, 2007). Compared with the genetic algorithm, the differential evolution algorithm, and the particle swarm optimization algorithm, the ABC algorithm is very competitive (Karaboga et al., 2012). This article will use ABC algorithm to optimize optimal parameters of hidden layers of EM-ELM. The EM-ELM prediction model is constructed with the obtained optimal parameters.

The nectar source of ABC algorithm is abstracted as a point in the solution space. The quality of nectar source $(i = 1, 2, \dots, S N)$ is chosen as the fitness of the solution. Assuming the dimension of the problem to be solved is $D$ . For the $t th$ iteration, the location of nectar source can be expressed as $X_{i}^{t} = [x_{i 1}^{t} x_{i 2}^{t} x_{i D}^{t}]$ , $x_{i D} \in (L_{d}, U_{d})$ . Where $L_{d}$ is the lower bound of the search space, and $U_{d}$ is the upper bound of the search space. The location of nectar source is randomly generated in the search space according to equation (3)

x_{i d} = L_{d} + r a n d (0, 1) (U_{d} - L_{d})

(3)

At the start of the search phase, a new nectar source around nectar source is generated by employed foragers according to equation (4)

v_{i d} = x_{i d} + φ (x_{i d} - x_{j d})

(4)

where $j \in {1, 2, \dots, S N}$ , $j i$ . It represents a random selection of nectar source which is different with $i$ . $φ \in [- 1, 1]$ . When fitness of new nectar source is better than $X_{i}$ , greedy algorithm is used to carry out $V_{i}$ replace $X_{i}$ . Otherwise, $X_{i}$ is remained. Then, onlooker bees share information according to nectar source of scout bees. The probability is as follows

p_{i} = \frac{f i t_{i}}{\sum_{i = 1}^{S N} f i t_{i}}

(5)

That means onlooker bees generate a random number belong to [0,1] and compare it with $p_{i}$ . If this random number is smaller than $p_{i}$ , then generate a new nectar source according to equation (4). In the searching process, if the nectar source $X_{i}$ cannot find a better new nectar source after several iterations, then $X_{i}$ is abandoned, the corresponding employed bees change to scout bees. A new nectar source will be random generated according to equation (6)

x_{i}^{t + 1} = {\begin{cases} L_{d} + r a n d (0, 1) (U_{d} - L_{d}), trail \geq limit \\ x_{i}^{t}, trail < limit \end{cases}

(6)

The implementation steps of ABC algorithm are as follows:

Step 1: training data sample set generation and parameter initialization. Those include the number of nectar source, SN, the maximum number of iterations, MCN, and the maximum number of nectar source mining, $l i m i t$ . The values of the parameters to be optimized are given. Let $t = 1$ .

Step 2: an employed forager is assigned for nectar source. The searching process is begun according to equation (4). A new nectar source $V_{i}$ will be generated.

Step 3: the fitness value is calculated by samples data. The nectar source will be retained according to greedy algorithm.

Step 4: the probability of nectar source be followed is calculated by equation (5). The onlooker bees are searching and retaining the nectar source according to greedy algorithm.

Step 5: the algorithm determines whether the nectar source should be discarded. If true, the onlooker bees are changed into scout bees. Otherwise, go to step 7.

Step 6: the scout bees will generate new nectar source according to equation (6).

Step 7: let $t = t + 1$ . If the termination conditions are satisfied, then optimal parameters are outputted. Otherwise, go to step 2 and continue to execution.

The short-tem wind speed prediction model based on ABC-optimized EM-ELM

The prediction model

The hidden layer parameters $a^{*}, b^{*}$ in EM-ELM algorithm are chosen as optimal parameters to be optimized. In order to maintain consistency with the ABC algorithm, equation (7) is chosen as fitness function of ABC-optimized EM-ELM prediction model

f i t_{i} = \frac{1}{1 + f_{i}}

(7)

where $f_{i}$ is network training error of EM-ELM algorithm. The network training error $E$ can be expressed as equation (8)

E = ‖ H β - T ‖

(8)

The implementation steps of hidden parameters of EM-ELM optimized by ABC algorithm can be described as following:

Step 1: for a given training set ${(x_{i}, t_{i})}_{i = 1}^{N}$ including N training samples, activation function of the hidden layer is $g (x),$ initial hidden layer node number is $L_{0}$ , maximum number of nodes in hidden layer is $L_{\max}$ , and expected training error is $E$ .

Step 2: in the numeral range of input weights and thresholds, the corresponding input weights and thresholds are generated randomly for $L_{0}$ hidden layer nodes.

Step 3: the output matrix $H_{0}$ of the initial network is calculated

\begin{matrix} H_{0} (a_{1}, \dots, a_{L}, b_{1}, \dots, b_{L}, x_{1}, \dots, x_{N}) \\ = {[\begin{matrix} G (a_{1}, b_{1}, x_{1}) & \dots & G (a_{L}, b_{L}, x_{1}) \\ ⋮ & \dots & ⋮ \\ G (a_{1}, b_{1}, x_{N}) & \dots & G (a_{L}, b_{L}, x_{N}) \end{matrix}]}_{N \times L_{0}} \end{matrix}

(9)

Step 4: training error $E_{0}$ of initial network is calculated

E_{0} = ‖ H_{0} H_{0}^{†} T - T ‖

(10)

Step 5: let iteration numbers $k = 0$ .

Step 6: the algorithm judge whether the residual error of the network is less than the user expected training error $U$ or the number of nodes in the hidden layer reaches the maximum value $L_{\max}$ . If the end condition is satisfied, stop iterating, save the network that has been trained. Otherwise, go to step 7.

Step 7: let $k = k + 1$ .

Step 8: the number of hidden layer nodes is updated according to equation (11)

L_{k} = L_{k - 1} + δ L_{k - 1}

(11)

Step 9: input weight vector $a_{i}$ and bias $b_{i}$ of hidden layer node parameter are optimized by ABC algorithm. The new output matrix $δ H_{k - 1}$ of hidden layer node will be obtained. Then, the output matrix $H_{k}$ of whole network hidden layer node can be expressed as $H_{k} = [H_{k - 1}, δ H_{k - 1}]$ . That is

δ H_{k - 1} = {[\begin{matrix} G (a_{L_{k - 1} + 1}, b_{L_{k - 1} + 1}, x_{1}) & \dots & G (a_{L_{k}}, b_{L_{k}}, x_{1}) \\ ⋮ & \dots & ⋮ \\ G (a_{L_{k - 1} + 1}, b_{_{L_{k - 1} + 1}}, x_{N}) & \dots & G (a_{L_{k}}, b_{L_{k}}, x_{N}) \end{matrix}]}_{N \times δ L_{k - 1}}

Step 10: the output weights of whole network are calculated

D_{k} = ((I - H_{k} H_{k}^{†}) δ H_{k})^{†}

(12)

U_{k} = H_{k}^{†}^{} (I - δ H_{k}^{T} D_{k})

(13)

β_{k} = H_{k}^{†} T = [\begin{array}{l} U_{k} \\ D_{k} \end{array}] T

(14)

Step 11: the training error $E_{k}$ after new nodes are added in network is calculated according to equation (15), go to step 6 and continue to execution

E_{k} = ‖ H_{k} β_{k} - T ‖

(15)

Convergence analysis of the algorithm

The following two lemmas are introduced.

Lemma 1

Given arbitrary $ε > 0$ and activation function. For $q$ discrete samples $(x_{i}, t_{i})$ , $x \in R^{n}$ , $t \in R^{m}$ , $q$ hidden layer neurons meet

| | H (a, b, x) β - t | | < ε

(16)

where $a = [a_{1}, a_{2}, \dots, a_{q}]$ , $b = [b_{1}, b_{2}, \dots, b_{q}]$ , and $β = t H^{T} / H H^{T}$ .

Lemma 2

Given any bounded continuous or piecewise continuous activation function, for any continuous objective function, if

β_{L} = \frac{t H_{L}^{T}}{H_{L}^{} H_{L}^{T}}

(17)

then there is arbitrary output matrix $H$ and SLFNs makes the following equation (18) to be satisfied. At the same time, network error $| | E_{L} | |$ decreases monotonically with the decrease in the number of neurons, $L$

\lim_{L \to + \infty} | | E_{L} | | = \lim_{L \to + \infty} | | H_{L} β_{L} - t | | = 0

(18)

Theorem 1

Given discrete sample $(x_{i}, t_{i})$ , $x \in R^{n}$ , $t \in R^{m}$ , for arbitrary $ε > 0$ . If the parameters of EM-ELM neural network are optimized by ABC algorithm, there must be an SLFN that has $q$ neurons which make $| | H_{q} β_{q} - t | | < ε$ be satisfied.

Proof

1.If $δ L_{k - 1}$ does not exceed the maximum allowable number of neurons, this means ABC-optimized EM-ELM is I-ELM algorithm.

Assuming the maximum number of iterations of the ABC algorithm is $M C N$ , the number of hidden layer nodes in the network is $L$ . If $L = 0$ and $k = 0$ , the corresponding output error $| | E_{L} (H_{k}) | | > ε$ . It is assumed that the ABC algorithm is iterated $K$ times, then the corresponding output error $| | E_{L} (H_{k}) | | | | E_{L} (H_{0}) | |$ . At this time, the hidden layer node is increased and $L = 1$ , and $| | E_{1} (H_{0}) | | | | E_{0} (H_{K}) | |$ according to the literature (Huang et al., 2006a). Therefore, when $L = N$ and $k = K$

| | E_{N} (H_{K}) | | \leq | | E_{N} (H_{0}) | | \leq \dots \leq | | E_{1} (H_{0}) | | \leq | | E_{0} (H_{K}) | | \leq | | E_{0} (H_{0}) | |

(19)

According to the literature (Huang et al., 2006b), if $\sum_{L = 1}^{q} L > N$ exists and $H_{q}$ is invertible

| | E_{0} (H_{0}) | | \geq \dots \geq | | E_{q} (H_{K}) | | = 0

(20)

Therefore, there exists $q < N$ , which makes $| | H_{q} β_{q} - f | | < ε$ to be satisfied.

2.If $δ L_{k - 1}$ is greater than or equal to the maximum allowable number of neurons, ABC-optimized EM-ELM in this article becomes standard ELM algorithm. This means that all the parameters of the hidden layer and the output weights can be obtained at a time.

Assuming the maximum number of neurons is $L_{\max}$ . The number of iterations is $k = 1$ , the corresponding network output error is $| | E_{L_{\max}} (H_{1}) | | > ε$ . If ABC algorithm is iterated to $K$

| | E_{L_{\max}} (H_{K}) | | \leq | | E_{L_{\max}} (H_{K - 1}) | | \leq \dots \leq | | E_{L_{\max}} (H_{1}) | |

(21)

According to Lemma 1, constant $q$ , $L_{\max} < q < N$ , makes the corresponding network error $| | E_{q} (H_{1}) | | \leq ε$ . At the same time, the number of iterations is increased to $k = K$

| | E_{q} (H_{1}) | | \geq | | E_{q} (H_{2}) | | \geq \dots \geq | | E_{q} (H_{K}) | |

(22)

Therefore, there exists $q < N$ , which makes $| | H_{q} β_{q} - f | | < ε$ to be satisfied. From the above proof process, ABC-optimized EM-ELM proposed in this article is stable and convergent.

Simulation

In order to verify the predictive performance of the ABC-optimized EM-ELM, the short-term wind speed data are used to validate the predictive ability. In order to illustrate the effectiveness of proposed prediction method, four groups’ short-term wind speed data are collected from a power plant located in Liaoning Province, China. The sampling period is 1 h, 30 min, 20 min, and 10 min, respectively. The length of the four datasets is 300. 300 groups of short-term wind speed data with a sampling period of 1 h were collected from 6 o’clock on 1 March 2013 to 18 o’clock on 13 March 2013. The first 276 groups of data are used to train the model (from 6 o’clock on 1 March 2013 to 18 o’clock on 12 March 2013), and the latter 24 groups of data (from 19 o’clock on 11 March 2013 to 18 o’clock on 12 March 2013) are used as the test set to verify the accuracy of the prediction model. 300 groups of short-term wind speed data with a sampling period of 30 min were collected from 6 o’clock on 15 March 2013 to 12 o’clock on 21 March 2013. The first 276 groups of data are used to train the model (from 6 o’clock on 15 March 2013 to 24 o’clock on 20 March 2013), and the latter 24 groups of data (from 0 o’clock on 21 March 2013 to 12 o’clock on 21 March 2013) are used as the test set. 300 groups of short-term wind speed data with a sampling period of 20 min were collected from 6 o’clock on 15 March 2013 to 10 o’clock on 19 March 2013. The first 276 groups of data are used to train the model (from 6 o’clock on 15 March 2013 to 2 o’clock on 19 March 2013), and the latter 24 groups of data (from 2 o’clock on 19 March 2013 to 10 o’clock on 19 March 2013) are used as the test set. 300 groups of short-term wind speed data with a sampling period of 10 min were collected from 6 o’clock on 15 March 2013 to 8 o’clock on 17 March 2013. The first 276 groups of data are used to train the model (from 6 o’clock on 15 March 2013 to 4 o’clock on 17 March 2013), and the latter 24 groups of data (from 4 o’clock on 17 March 2013 to 8 o’clock on 17 March 2013) are used as the test set. 300 groups’ short-term wind speed data with four sampling periods are shown in Figure 1.

Figure 1.

Short-term wind speed time series.

The proposed ABC algorithm optimized EM-ELM prediction model is used to predict four groups’ short-term wind speed data. The prediction steps of wind speed data are 24. The parameters of ABC algorithm–optimized EM-ELM prediction model are $δ = 1.2$ , $L_{0} = 1$ , $U = 0.001$ , and $L_{\max} = 100$ . The number of nectar source SN is 20, maximum number of iterations $M C N$ is 100, and the maximum number of nectar source mining $l i m i t$ is 50. Compared with standard ELM algorithm (the number of hidden nodes is 45 by trial and error method); in EM-ELM algorithm, the initial hidden layer node is 1, each adding four hidden nodes, the node limit is set to 100; and in EI-ELM, the hidden layer node limit is set to 100. All the activation functions of the algorithm are Sigmoid functions $(λ = 1)$ . The range of input weights and bias values are $[- 1, 1]$ . All inputs and outputs are normalized to $[- 1, 1]$ . Program operating environment are CPU (i7-4770, 3.4 GHz), 8GB memory, Windows 7 operating system, and simulation software is MATLAB 2010b. Ten simulation experiments are carried out under the same conditions. CPU training time, training error, test error, the average number of hidden layer nodes is chosen as evaluation index. The training error and test error are judged by the root mean square error. Table 1 shows the operation results of several prediction models of wind speed data with a sampling period of 1 h. Table 2 shows the operation results of several prediction models of wind speed data with a sampling period of 30 min. Table 3 shows the operation results of several prediction models of wind speed data with a sampling period of 20 min. Table 4 shows the operation results of several prediction models of wind speed data with a sampling period of 10 min.

Table 1.

The operation results of several algorithms of wind speed data with a sampling period of 1 h.

Prediction models	Training time(s)	Training error	Test error	The average number of hidden layer nodes
ABC-optimized EM-ELM	0.6135	1.3354	0.1154	19
ELM	1.7234	2.3621	0.2045	45
EM-ELM	1.0785	1.9144	0.1569	28
EI-ELM	1.4126	2.1763	0.1910	34

ABC: artificial bee colony; EM-ELM: error minimized extreme learning machine; ELM: extreme learning machine; EI-ELM: enhanced incremental extreme learning machine.

Table 2.

The operation results of several algorithms of wind speed data with a sampling period of 30 min.

Prediction models	Training time(s)	Training error	Test error	The average number of hidden layer nodes
ABC-optimized EM-ELM	0.6432	1.2647	0.1028	21
ELM	1.6254	2.5217	0.2237	48
EM-ELM	1.1125	1.9258	0.1768	29
EI-ELM	1.4128	2.3157	0.2017	38

ABC: artificial bee colony; EM-ELM: error minimized extreme learning machine; ELM: extreme learning machine; EI-ELM: enhanced incremental extreme learning machine.

Table 3.

The operation results of several algorithms of wind speed data with a sampling period of 20 min.

Prediction models	Training time(s)	Training error	Test error	The average number of hidden layer nodes
ABC-optimized EM-ELM	0.7025	1.4018	0.1251	19
ELM	1.8147	2.4752	0.2127	46
EM-ELM	1.0571	1.9842	0.1638	27
EI-ELM	1.4325	2.2247	0.1816	39

ABC: artificial bee colony; EM-ELM: error minimized extreme learning machine; ELM: extreme learning machine; EI-ELM: enhanced incremental extreme learning machine.

Table 4.

The operation results of several algorithms of wind speed data with a sampling period of 10 min.

Prediction models	Training time(s)	Training error	Test error	The average number of hidden layer nodes
ABC-optimized EM-ELM	0.6785	1.4210	0.1268	22
ELM	1.8069	2.4758	0.2251	51
EM-ELM	1.1745	1.9365	0.1746	32
EI-ELM	1.5012	2.1952	0.1913	41

ABC: artificial bee colony; EM-ELM: error minimized extreme learning machine; ELM: extreme learning machine; EI-ELM: enhanced incremental extreme learning machine.

It can be seen from Tables 1 to 4, because the ABC algorithm reduces the number of hidden layer nodes in the iteration cycle, the time consuming and training time of the algorithm is better than the standard ELM, EM-ELM, and E-ELM. Also due to the reduction of the hidden layer node numbers, the complexity of the ELM network is reduced. The overall performance of the system is improved, which means that the method can obtain more simplified ELM network structure in the same or less training time.

Figure 2 shows comparison of the predicted values and the actual values of four groups’ short-term wind speed data in test set in one simulation by using four ELM models. From the graph, we can observe that the method in the article is superior to other three kinds of ELM in curve fitting. The prediction model in this article has a better regression prediction ability for short-term wind speed time series.

Figure 2.

Comparison results of four groups’ short-term wind speed data by four ELM prediction models.

In order to further compare the predictive effects of this method, method is compared to combined prediction model based on wavelet transform (Tian et al., 2015b), prediction model based on chaotic theory and online least square support vector machine (Tian et al., 2015a), and prediction model based on combination kernel function least squares support vector machine (Tian et al., 2014). At the same time, the proposed method is compared with persistence method (Santamaria-Bonfil et al., 2016), least squares support vector machines (Mathaba et al., 2012), autoregressive integrated moving average model (Cadenas et al., 2016), and neural network (Saleck and von Bremen, 2007). The parameters of least squares support vector machines can be obtained by LSSVM Toolbox. The parameters of autoregressive integrated moving average model can be obtained by Akaike information criterion (AIC) criterion. The simulation results of four groups’ wind speed data are shown in Figure 3. Figure 4 shows absolute error distribution of four groups’ wind speed data of the prediction models mentioned in this article. Figure 5 shows relative error distribution of four groups’ wind speed data of the prediction models mentioned in this article. From the results of Figures 2 to 5, we can see that the proposed prediction model is better than other models in prediction accuracy and prediction error.

Figure 3.

Comparison of the short-term wind speed prediction and actual value of four groups’ wind speed data.

Figure 4.

Absolute error distribution of four groups’ wind speed data.

Figure 5.

Relative error distribution of four groups’ wind speed data.

In this article, we introduce the following four kinds of performance indicators to measure the prediction accuracy of the prediction model:

RMSE (root mean square error)

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

(23)

MAE (mean absolute error)

MAE = \frac{1}{N} \sum_{k = 1}^{N} | w (k) - \hat{w} (k) |

(24)

MAPE (mean absolute percentage error)

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

(25)

Reliability

R^{(1 - a)} = [\frac{ξ^{(1 - a)}}{N} - (1 - a)] \times 100 %

(26)

where $N$ is the number of samples, $w (k)$ is actual value of wind speed, $\hat{w} (k)$ is predictive value of wind speed, and $ξ^{(1 - a)}$ is the number of confidence intervals in which the actual value falls under the confidence level $1 - α$ .

Table 5 shows the comparison of RMSE, MAE, and MAPE performance indicators of these prediction models. The results in Table 5 also show that the prediction model in this article is superior to other prediction models in the performance indicators.

Table 5.

Comparison of performance indicators of four groups’ wind speed data.

Prediction models	1-h dataset			30-min dataset			20-min dataset			10-min dataset
Prediction models	RMSE	MAE	MAPE	RMSE	MAE	MAPE	RMSE	MAE	MAPE	RMSE	MAE	MAPE
Proposed method	0.1154	0.1026	0.1587	0.1461	0.1188	0.1180	0.1169	0.1133	0.1283	0.1257	0.1106	0.0951
ELM	0.2045	0.1780	0.4199	0.2068	0.1785	0.2103	0.2246	0.1634	0.2453	0.2124	0.1570	0.1539
EM-ELM	0.1569	0.1330	0.3123	0.1594	0.1417	0.1942	0.1827	0.1374	0.2272	0.1866	0.1752	0.1414
EI-ELM	0.1910	0.1708	0.4403	0.1824	0.1588	0.2292	0.1669	0.1240	0.2353	0.2026	0.1554	0.1285
Combined prediction model	0.1541	0.1230	0.1890	0.1521	0.1195	0.1496	0.1454	0.1280	0.2754	0.1331	0.1298	0.1426
Chaotic theory and online LSSVM	0.1492	0.1222	0.1742	0.1664	0.1487	0.1547	0.1527	0.1345	0.1982	0.1407	0.1270	0.1328
Combination kernel function LSSVM	0.1509	0.1299	0.2064	0.1693	0.1492	0.1284	0.1563	1.2333	0.1876	0.1284	0.1347	0.1506
Persistence method	0.5740	1.5201	0.4796	1.5858	1.3400	1.5535	1.4449	0.7863	1.4160	1.5782	1.2320	1.2518
LSSVM	0.2484	0.3977	0.2282	0.1985	0.1770	0.1582	0.2149	0.2003	0.2299	0.1877	0.1533	0.1699
ARIMA	0.2459	0.2840	0.2166	0.1933	0.1551	0.2236	0.2341	0.1929	0.4494	0.2138	0.1894	0.1851
Neural network	0.2266	0.4653	0.1973	0.2131	0.1838	0.2483	0.2261	0.1933	0.3260	0.2403	0.2103	0.1512

RMSE: root mean square error; MAE: mean absolute error; MAPE: mean absolute percentage error; EM-ELM: error minimized extreme learning machine; ELM: extreme learning machine; EI-ELM: enhanced incremental extreme learning machine; LSSVM: least squares support vector machine.

Figure 6 shows reliability and confidence distribution of the prediction models mentioned in this article. It can be seen from this graph that the prediction model in this article has higher reliability under the same confidence level. It can be known that the reliability of ABC-optimized EM-ELM prediction model is better than other prediction models.

Figure 6.

Reliability and confidence distribution of four groups’ wind speed data.

In summary, from the point of view of the above modeling training results, curve fitting, error distribution, performance indicators, reliability, and so on, ABC algorithm–optimized EM-ELM of short-term wind speed prediction model has better prediction effect. The main reason for the improvement of prediction performance is that ABC algorithm optimizes the parameters of the hidden layer nodes of EM-ELM. It makes the network output error rapid decline, simplifies the structure of network, reduces the number of useless neurons, and improves the algorithm efficiency.

Conclusion

A prediction model based on ABC algorithm–optimized error minimized ELM is proposed in this article. This prediction model is applied to the prediction of short-term wind speed time series. The I-ELM is the main research direction to improve the performance of ELM, but there exist a large number of useless neurons in the I-ELM, which reduces the efficiency and performance of the algorithm. In order to simplify the network structure, improve the performance of the algorithm, this article uses the ABC algorithm with good performance to optimize the parameters of the hidden layer based on the EM-ELM. At the same time, the stability and convergence of the algorithm are proved. The simulation results are verified by the short-term wind speed time series. Compared with other prediction models, the simulation results show that the prediction model proposed in this article has better prediction effect and higher prediction accuracy.

In this article, the short-term wind speed time series is used as the simulation object. In fact, the proposed prediction model can also be used in the prediction of any time series such as sunspot number, river run-off, stock price, network traffic, and wind power. The future research of this article is to apply the prediction model to other time series and further improve the prediction accuracy of the prediction model.

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 was partially supported by the Science Research Project of Liaoning Education Department (No. LGD2016009) and the Natural Science Foundation of Liaoning Province of China (No. 20170540686).

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