Interval model of a wind turbine power curve

The wind turbine power curve model is critical to a wind turbine’s power prediction and performance analysis. However, abnormal data in the training set decrease the prediction accuracy of trained models. This paper proposes a sample average approach-based method to construct an interval model of a wind turbine, which increases robustness against abnormal data and further improves the model accuracy. We compare our proposed methods with the traditional neural network-based and Bayesian neural network-based models in experimental data-based validations. Our model shows better performance in both accuracy and computational time.

1 Introduction

Wind power has become a significant renewable power source of global energy systems (Gilbert et al., 2020). To ensure operation safety and high efficiency, it is critical to monitor the operation conditions of wind turbines to predict wind power output and detect potential faults (Wang and Liu, 2021). The Supervisory Control and Data Acquisition (SCADA) system provides wind turbine data, for example, wind speed and power, for establishing models for wind turbine power prediction and fault detection (Y. Wang et al., 2019). A significant proportion of the SCADA data is abnormal due to communication failures, maintenance, and other reasons (Morrison et al., 2022). A model trained by a dataset with abnormal data is biased from a real model and suffers from reduced accuracy (Ye et al., 2021). It is necessary to consider the data cleaning-based method to obtain a model with improved robustness against abnormal data.

Prior results of the data cleaning method were based on clustering algorithms (Zheng et al., 2015; 2010; Yesilbudak, 2016). In clustering algorithm-based methods, k-means, manifold spectral clustering, and other algorithms are applied to separate the wind power curve into partitions and then identify the outliers based on distances to the cluster centers. An alternative method is to determine the upper and lower boundaries of the wind power curve by boundary models. For example, Shen et al. (2019) used change point and quantile to estimate a contour for normal data. However, the aforementioned methods fail to identify the outliers when there are many cluster centers. The setting of the algorithm parameters, for example, cluster number in the clustering algorithm-based method, is also unexplainable. These reasons make the clustering algorithm-based and existing boundary methods suffer from issues of misidentification.

Normal distribution model-based methods, proposed to overcome issues of misidentification, use the normal distribution to fit the power data’s distribution and then calculate the probability contours. Data with low probability are regarded as abnormal data. Ouyang et al. (2017) calculated mean and standard deviation values based on exponential smoothing. Stephen et al. (2011) used bivariate joint distribution to fit the wind power data’s distribution.

However, the aforementioned existing methods for abnormal detection toward improving the model accuracy have the following disadvantages:

• Many real outliers, especially those near the normal region, cannot be recognized.

• There are too many hyperparameters to be set. The performance highly depends on the hyperparameters, while the function of each hyperparameter is unexplainable.

• To ensure performance, the prior information on the normal points should be available, which is not practical in general cases.

Another practical way to quantify uncertainty is to directly use a confidence interval-aware model, known as the interval predictor model (Campi et al., 2009; Garattia et al., 2019). The scenario approach presented by Calafiore and Campi (2006); Campi and Garatti (2019); and (2011); Campi et al. (2015) can be used to establish the interval predictor model. However, the scenario approach cannot give an exact confidence bound since a small number of samples will give a bound with high risk, and a large number of samples will give a conservative bound. Luedtke and Ahmed (2008) proposed a sample average approach to obtain an approximate solution that exactly converges to the original as the sample number increases. In this paper, we propose the sample average approach-based interval models and extend them into extreme learning machines to provide a fast algorithm for training neural networks that can exactly give the desired confidence interval. The proposed interval model solves the abnormal detection and wind power curve regression problems together in a direct way. We implement experimental data-based validation to compare the proposed methods with several existing methods.

The rest of this paper is organized in the following way: Section 2 briefly introduces the wind power curve and then gives a formal problem statement. In Section 3, extreme learning machine and the theory of sample average approach for chance-constrained optimization are briefly reviewed; Section 4 presents the proposed interval models combining the extreme learning machine and sample average approach; and Section 5 presents the results and discussions of experimental data-based validation. Finally, Section 6 concludes the paper.

2 Problem description

As shown in Figure 1, a wind turbine power curve has the following three critical points, namely, A, B, and C, which divide the wind turbine power curve into four segments (Marvuglia and Messineo, 2012; Shokrzadeh et al., 2014). Point A is the cut-in wind speed from where the wind turbine starts to output the power. Point B represents the rated power before which the output power increases as the wind speed increases. Point C is the cut-out speed from where the output wind power decreases even with the increased wind speed.

FIGURE 1

FIGURE 1. Brief illustration of the wind turbine power curve and estimation biases using abnormal data.

The segment between points B and C gives the rated level of the wind power output. The segment between points A and B shows a non-linear correlation between wind speed and power. Let v be the wind speed and P be the wind power. The non-linear correlation can be described approximately by the following equation:

$P=\frac{1}{2}{C}_{\mathrm{p}}\rho \pi R^2{v}^3,(1)$

where $C_{\mathrm{p}}$ is the coefficient of wind turbine power, ρ is the air density, and R is wind rotor’s radius.

A dataset obtained by the SCADA system can be defined by the following equation:

${\mathcal{D}}_N^{\mathsf{S}}={\left\{\left(v_i,P_i\right)\right\}}_{i=1}^N,(2)$

where N is the number of the data samples. As shown in Figure 1, the SCADA system’s data have normal and abnormal data. In other words, ${\mathcal{D}}_N^{\mathsf{S}}$ is a corrupted dataset. Using the normal data, we can get an estimated curve (blue dashed line) close to the real curve (purple solid line). However, with the corrupted data, the estimated curve (red dashed line) is biased from the real curve.

Therefore, it is necessary to investigate a robust power curve estimation method for the abnormal data. In this paper, we address the following problem.

$\begin{array}{cc}\hfill & \underset{\mathbf{f}}{\min }{\displaystyle \sum _{i=1}^{N_{\mathsf{c}}}}|P_i-\mathbf{f}\left(v_i\right){|}^2\hfill \\ \hfill & \mathsf{s}.\mathsf{t}.\left(v_i,P_i\right)\in {\tilde{\mathcal{D}}}_{N_{\mathsf{r}}}.\hfill \end{array}(3)$

Here, the set ${\tilde{\mathcal{D}}}_{N_{\mathsf{r}}}$ is a subset of ${\mathcal{D}}_N^{\mathsf{S}}$, which is obtained by cleaning the abnormal data. Thus, we have to address the following two sub-problems:

• data cleaning problem to obtain a cleaned data ${\tilde{\mathcal{D}}}_{N_{\mathsf{r}}}$;

• regression problem described by (3).

3 Preliminaries

This section briefly reviews the extreme learning machine and sample average approach as a preparation for introducing our proposed interval models.

3.1 Extreme learning machine

Extreme learning machine is a fast algorithm to train a single-layer neural network (Huang et al., 2006). A single-layer neural network has an input layer, a hidden layer, and an output layer. Let a positive integer L be the number of neurons. The hidden layer can be defined as a vector function by the following expression:

$\mathbf{h}\left(\mathbf{x}\right)≔{\left[h_1\left(\mathbf{x}\right),\dots ,h_L\left(\mathbf{x}\right)\right]}^{\top }.$

Each h_i(x), i = 1, …, L is a neuron. Often, we choose the neuron as follows:

$h_i\left(\mathbf{x}\right)≔G\left({\mathbf{a}}_i,{\mathrm{b}}_i,\mathbf{x}\right),$

where a_i, b_i are the hyperparameters in the i − th neuron. The neuron can be a sigmoid function or a Gaussian function, etc. Let

$\mathit{\beta }≔{\left[{\beta }_1,\dots ,{\beta }_L\right]}^{\top }$

be the coefficient of the output layer. Then, we can write the single-layer network as follows:

$\mathrm{y}\left(\mathbf{x}\right)={\mathit{\beta }}^{\top }\mathbf{h}\left(\mathbf{x}\right)={\displaystyle \sum _{i=1}^L}{\beta }_i{h}_i\left(\mathbf{x}\right).(4)$

As a summary, the parameters that need to be trained are coefficient vector β and hyperparameters ${\left({\mathbf{a}}_i,{\mathrm{b}}_i\right)}_{i=1}^L$. The extreme learning machine is used to train β, ${\left({\mathbf{a}}_i,{\mathrm{b}}_i\right)}_{i=1}^L$ with a giving set ${\mathcal{D}}_N$.

The algorithm of the extreme learning machine is summarized as follows:

• randomly generate hyperparameters ${\left({\mathbf{a}}_i,{\mathrm{b}}_i\right)}_{i=1}^L$;

• estimate β by solving

$\underset{\mathit{\beta }}{\min }{\displaystyle \sum _{t=1}^N}\Vert {\mathit{\beta }}^{\top }\mathbf{h}\left({\mathbf{x}}_t\right)-{\mathrm{y}}_t{\Vert }^2,(5)$

which gives the solution as

${\mathit{\beta }}_{*}≔{\left({\mathbf{H}}^{\top }\mathbf{H}\right)}^{-1}{\mathbf{H}}^{\top }{\mathbf{Y}}_N,(6)$

where $\mathbf{H}\in {\mathbb{R}}^{N\times L}$ is defined as

$\left[\begin{array}{ccc}\hfill G\left({\mathbf{a}}_1,{\mathrm{b}}_1,{\mathbf{x}}_1\right)\hfill & \hfill \dots \hfill & \hfill G\left({\mathbf{a}}_L,{\mathrm{b}}_L,{\mathbf{x}}_1\right)\hfill \\ \hfill \dots \hfill & \hfill \dots \hfill & \hfill \dots \hfill \\ \hfill G\left({\mathbf{a}}_1,{\mathrm{b}}_1,{\mathbf{x}}_N\right)\hfill & \hfill \dots \hfill & \hfill G\left({\mathbf{a}}_L,{\mathrm{b}}_L,{\mathbf{x}}_N\right)\hfill \end{array}\right](7)$

and Y_N is defined as

${\mathbf{Y}}_N≔\left[\begin{array}{c}\hfill {\mathrm{y}}_1\hfill \\ \hfill \dots \hfill \\ \hfill {\mathrm{y}}_N\hfill \end{array}\right].(8)$

Theorem 2.2 of Huang et al. (2006) provides the universal approximation property of an extreme learning machine-based single neural network regarding a dataset ${\mathcal{D}}_N$. We summarize it in Lemma 1.

Lemma 1. For any given small ɛ and activation function G(⋅) which is infinitely differentiable in any interval, there exists $\bar{N}$ such that, for $N>\bar{N}$ arbitrary distinct samples (x_t, y_t), for any ${({\mathbf{a}}_i,{\mathrm{b}}_i)}_{i=1}^L$ randomly extracted according to any continuous probability distribution, with probability one,

${\displaystyle \sum _{t=1}^N}\Vert {\mathit{\beta }}_{*}^{\top }\mathbf{h}\left({\mathbf{x}}_t\right)-{\mathrm{y}}_t{\Vert }^2<\epsilon .(9)$

Lemma 1 shows that we could use a single-layer neural network to approximate the wind power curve.

3.2 Sample average approach

Chance-constrained optimization seeks to optimize an objective under a stochastic constraint (Campi et al., 2015; Shen et al., 2020; 2021), which is written as follows:

$\begin{array}{cc}\hfill & \hfill \underset{z\in \mathcal{Z}}{\min }J\left(z\right)\\ \hfill & \mathsf{s}.\mathsf{t}.{\mathrm{Pr}}_{\xi }\left\{G\left(z,\xi \right)\le 0\right\}\ge 1-\alpha ,\xi \in \mathrm{\Xi },\hfill \end{array}(10)$

where $z\in \mathcal{Z}\subset {\mathbb{R}}^{n_u}$ denotes the input variable with the compact feasible set $\mathcal{Z}$, uncertainty is represented by $\xi \in \mathrm{\Xi }\subset {\mathbb{R}}^{n_{\xi }}$ defined on probability space $\left(\mathrm{\Xi },\mathcal{B}(\mathrm{\Xi }),{\mathrm{Pr}}_{\xi }\right)$, $J(z):\mathcal{Z}\to \mathbb{R}$ and $\forall \xi \in \mathrm{\Xi },G:\mathcal{Z}\times \mathrm{\Xi }\to {\mathbb{R}}^l$ are continuous and differentiable objective function and constraint function in z, respectively. Problem (10) is a relaxation of robust optimization in which α is zero. The optimal objective function of (10) is defined by $J_{\alpha }^{*}$. Let $Z_{\alpha }^{*}$ be the optimal solution set of (10).

The sample-based approximation is a practical way to solve chance-constrained optimization. This paper adopts the sample average approach presented by Luedtke and Ahmed (2008). In the sample average approach, samples are extracted from the sample space Ξ, and then, an approximate problem of the original chance-constrained optimization is established. Let (ξ⁽¹⁾, …, ξ^(N)) be an independent Monte Carlo sample set of the random variable ξ. After choosing $ϵ\in \left[\left.\mathrm{0,1}\right)\right.$ and η > 0, with sample set (ξ⁽¹⁾, …, ξ^(N)), the sample average approximation problem is defined as follows:

$\begin{array}{ll}\hfill & \underset{z\in \mathcal{Z}}{\min }J\left(z\right)\hfill \\ \hfill & \mathsf{s}.\mathsf{t}.z\in {\mathcal{Z}}_{ϵ,\gamma }^N,\hfill \end{array}(11)$

where ${\mathcal{Z}}_{ϵ,\gamma }^N$ is defined as follows:

${\mathcal{Z}}_{ϵ,\gamma }^N=\left\{z\in \mathcal{Z}|\frac{1}{N}{\displaystyle \sum _{i=1}^N}\mathbb{I}\left(G\left(z,{\xi }^{\left(i\right)}\right)+\gamma \right)\ge 1-ϵ\right\},(12)$

where Λ_η(⋅) is defined by the following equation:

$\mathbb{I}\left(t\right)=\left\{\begin{array}{cc}0,\hfill & t>0,\hfill \\ 1,\hfill & \text{if}t\le 0.\hfill \end{array}\right.(13)$

Let ${\tilde{J}}_{ϵ,\gamma }^N$ be the optimal objective function of (10) and ${\tilde{Z}}_{ϵ,\gamma }^N$ be the optimal solution. Note that both ${\tilde{J}}_{ϵ,\gamma }^N$ and ${\tilde{Z}}_{ϵ,\gamma }^N$ are decided by the sample set, and the sample set is randomly extracted. Thus, ${\tilde{J}}_{ϵ,\gamma }^N$ is a random variable, and ${\tilde{Z}}_{ϵ,\gamma }^N$ is a random set. The uniform convergence of ${\tilde{J}}_{ϵ,\gamma }^N$ and ${\tilde{Z}}_{ϵ,\gamma }^N$ should be addressed.

The following assumption on G(⋅) holds throughout this paper.

Assumption 1. There exists L > 0 such that

$|G\left(z,\xi \right)-G\left(z^{\prime },\xi \right)|\le L\Vert z-z^{\prime }{\Vert }_{\infty },\forall z,z^{\prime }\in \mathcal{Z}\mathrm{and}\forall \xi \in \mathrm{\Xi }.(14)$

Assumption 1 is reasonable since we could choose an activation function that makes the neural networks satisfy it and also preserve the universal approximation.

The uniform convergence of ${\tilde{J}}_{ϵ,\gamma }^N$ and ${\tilde{Z}}_{ϵ,\gamma }^N$ is summarized from Luedtke and Ahmed (2008).

Lemma 2. Suppose that Assumption 1 holds as N → ∞, γ → 0, and ϵ → α, ${\tilde{J}}_{ϵ,\gamma }^N\to J_{\alpha }^{*}$, and $\mathbb{D}({\tilde{Z}}_{ϵ,\gamma }^N,Z_{\alpha }^{*})\to 0$ with probability 1.

Lemma 2 shows that the approximate problem’s solution converges to one in the solution set of the original problem if the number of samples increases to infinite. In addition, for a certain bounded value, we could use a large enough sample number to ensure that the approximate problem’s solution is within that bound.

4 Proposed method

This section presents the proposed extreme learning machine with a confidence region. The convergence analysis is given. In addition, the proposed algorithm is presented.

4.1 Extreme learning machine with a confidence region

The previous extreme learning machine gives a single prediction value for a given input. In this paper, we investigate a computation method to give a confidence region for a given input with the center of the confidence region as the estimation of the curve and the normal data located in the confidence region with high probability. In this way, we can solve problem (3). The concept of the extreme learning machine with the confidence region is illustrated in Figure 2

FIGURE 2

FIGURE 2. Basic concept of the extreme learning machine with a confidence region.

We want to establish an interval model to give a power curve’s interval for any given wind speed and require the correct wind power prediction to be within the interval at a given probability. This interval should be the smallest since a large interval has the issue of being too conservative. An interval model based on a single-layer neural network can be defined by the following equation:

${\mathcal{I}}_{\mathsf{power}}≔\left\{\tilde{P}={\mathit{\beta }}^{\top }\mathbf{h}\left(v\right)+e,\mathit{\beta }\in \mathcal{B}\subseteq {\mathbb{R}}^L,e\in \left[-\gamma ,\gamma \right],\gamma \in {\mathbb{R}}^{+}\right\}.(15)$

Let

$\mathit{\theta }≔\left(\mathit{\beta },e\right)(16)$

be the parameter vector that specifies the interval. We have a set of θ as $\mathrm{\Theta }=\mathcal{B}\times [-\gamma ,\gamma ]$. The wind power set for a given v and hyperparameters ${\left({\mathbf{a}}_i,{\mathrm{b}}_i\right)}_{i=1}^L$ can be obtained by Θ.

In this paper, we use a ball set for ${\mathcal{I}}_{\mathsf{power}}$. With a little manipulation of notation, ${\mathcal{I}}_{\mathsf{power}}$ is also used for the ball interval of wind power. Therefore, the set $\mathcal{B}$ is a ball. Let c and r be the center of $\mathcal{B}$. Then, the set $\mathcal{B}$ can be specified by the following equation:

$\mathcal{B}=\left\{\mathit{\beta }\in {\mathbb{R}}^L:\Vert \mathit{\beta }-\mathit{c}\Vert \le r\right\}.(17)$

With $\mathcal{B}$ defined by (17), we can rewrite ${\mathcal{I}}_{\mathrm{power}}$ in the following form:

$\begin{array}{ll}\hfill {\mathcal{I}}_{\mathrm{power}}\left(v,{\left({\mathbf{a}}_i,{\mathrm{b}}_i\right)}_{i=1}^L,\mathit{c},r,\gamma \right)& =\left[{\mathit{c}}^{\top }\mathbf{h}\left(v\right)-\left(r\Vert \mathbf{h}\left(v\right)\Vert +\gamma \right)\right.,\hfill \\ \hfill & \left.{\mathit{c}}^{\top }\mathbf{h}\left(v\right)+\left(r\Vert \mathbf{h}\left(v\right)\Vert +\gamma \right)\right].\hfill \end{array}(18)$

Then, the problem of solving the extreme learning machine with a confidence region is written as follows:

$\begin{array}{cc}\hfill & \underset{{\left({\mathbf{a}}_i,{\mathrm{b}}_i\right)}_{i=1}^L,\mathit{c},r,\gamma }{\min }\eta r+\gamma \hfill \\ \hfill & s.t.r,\gamma >0,\hfill \\ \hfill & \mathbb{P}\left\{P\left(u\right)\in {\mathcal{I}}_{\mathrm{power}}\left(v,{\left({\mathbf{a}}_i,{\mathrm{b}}_i\right)}_{i=1}^L,\mathit{c},r,\gamma \right),\forall v\right\}\ge 1-\epsilon .\hfill \end{array}(19)$

Here, η is a positive number. By using the extreme learning machine algorithm (Lemma 1), we can obtain ${\left({\mathbf{a}}_i,{\mathrm{b}}_i\right)}_{i=1}^L$ randomly. Then, after obtaining ${\left({\mathbf{a}}_i,{\mathrm{b}}_i\right)}_{i=1}^L$ randomly, we simplify the problem to the following one:

$\begin{array}{cc}\hfill & \underset{\mathit{c},r,\gamma }{\min }\eta r+\gamma \hfill \\ \hfill & s.t.r,\gamma >0,\hfill \\ \hfill & \mathbb{P}\left\{P\left(u\right)\in {\mathcal{I}}_{\mathrm{power}}\left(v,{\left({\mathbf{a}}_i,{\mathrm{b}}_i\right)}_{i=1}^L,\mathit{c},r,\gamma \right),\forall v\right\}\ge 1-\epsilon .\hfill \end{array}(20)$

Let z and ${\mathcal{Z}}_{\epsilon }$ be the decision variable and feasible region of (20), respectively. Let

${\ell }_{\epsilon }^{*}≔\underset{\mathbf{z}\in {\mathcal{Z}}_{\epsilon }}{\min }\eta r+\gamma (21)$

be the optimal objective value of problem (20) and

${\mathbf{Z}}_{\epsilon }^{*}≔\left\{\mathbf{z}\in {\mathcal{Z}}_{\epsilon }:\eta r+\gamma ={\ell }_{\epsilon }^{*}\right\}(22)$

be the optimal solution set. Let ${\mathbf{z}}_{\epsilon }^{*}\in {\mathbf{Z}}_{\epsilon }^{*}$ be an optimal solution. With ${\mathbf{z}}_{\epsilon }^{*}$, we can obtain ${\mathcal{I}}_{\mathrm{power}}^{*}$ for any wind speed v. We can reject all data outside of ${\mathcal{I}}_{\mathrm{power}}^{*}$ as abnormal data.

4.2 Sample-based approximation and proposed algorithm

Due to chance constraints, problem (20) is not intractable. With the sample set ${\mathcal{D}}_N^{\mathsf{S}}$, we can obtain an approximate problem of problem (20) by the following expression:

$\begin{array}{cc}\hfill & \underset{\mathit{c},r,\gamma }{\min }\eta r+\gamma \hfill \\ \hfill & s.t.r,\gamma >0,\hfill \\ \hfill & {\displaystyle \sum _{t=1}^N}\mathbb{I}\left\{P_t\in {\mathcal{I}}_{\mathsf{power}}\left(v_t,{\left({\mathbf{a}}_i,{\mathrm{b}}_i\right)}_{i=1}^L,\mathit{c},r,\gamma \right)\right\}\ge \left(1-{\epsilon }^{\prime }\right)N.\hfill \end{array}(23)$

Here, $\mathbb{I}\{\cdot \}$ is an indicator function defined by (13). Let ${\tilde{\ell }}_{{\epsilon }^{\prime }}^N$ and ${\tilde{\mathbf{Z}}}_{{\epsilon }^{\prime }}^N$ be the optimal objective value and optimal solution set of problem (23), respectively.

Theorem 1. As N → ∞ and ɛ′ → ɛ, ${\tilde{\ell }}_{{\epsilon }^{\prime }}^N\to {\ell }_{\epsilon }^{*}$, and $\mathbb{D}({\tilde{\mathbf{Z}}}_{{\epsilon }^{\prime }}^N,{\mathbf{Z}}_{\epsilon }^{*})\to 0$ with probability 1.

Proof. Theorem 1 can be proved by directly applying Lemma 2 since problem (20) satisfies Assumption 1.

The proposed algorithm of the extreme learning machine with a confidence region is presented as follows.

Algorithm 1.Proposed algorithm of the extreme learning machine with a confidence region.

 Inputs: Dataset ${\mathcal{D}}_N^{\mathsf{S}}$, ɛ′

  1: Solve Problem (23) to obtain ${\tilde{\mathbf{z}}}_{{\epsilon }^{\prime }}^N\in {\tilde{\mathbf{Z}}}_{{\epsilon }^{\prime }}^N$

  2: Abnormal data detection

${\tilde{\mathcal{D}}}_{N_{\mathsf{r}}}=\left\{\left(v_t,P_t\right)\in {\mathcal{D}}_N^{\mathsf{S}}:P_t\in {\mathcal{I}}_{\mathsf{power}}\left(v_t,{\left({\mathbf{a}}_i,{\mathrm{b}}_i\right)}_{i=1}^L,\tilde{\mathit{c}},\tilde{r},\tilde{\gamma }\right).\right\}(24)$

  3: Estimated curve

${\tilde{P}}_t={\tilde{\mathit{\beta }}}^{\top }\mathbf{h}\left(v_t\right),t=1,\dots ,N.(25)$

 Output: ${\tilde{\mathcal{D}}}_{N_{\mathsf{r}}}$, ${\tilde{P}}_t,t=1,\dots ,N$.

Notice that (25) gives the estimated curve. The confidence bound (upper and lower bounds) can be given using the interval obtained by solving Problem (23).

5 Experimental data-based validation

This section presents the results of experimental data-based validations. First, the experimental dataset is introduced. Then, the results given by the proposed method and several existing methods are compared.

5.1 Experimental data and settings

Figure 3 plots the data used in this validation. The dataset was collected from a wind farm in Hubei, China. Figure 3A shows all data, including abnormal data and normal data. Specialists were approached to give labels on the data set. The data with labels are plotted in Figure 3B.

FIGURE 3

FIGURE 3. Experimental dataset: (A) data without labels and (B) data with labels.

In this paper, we compare the performance of the following methods:

• TrueELM: extreme learning machine using normal data;

• SSELM: the proposed extreme learning machine combined with the sample average approach;

• ELM: extreme learning machine without data cleaning;

• BELM: Bayesian extreme learning machine proposed by Soria-Olivas et al. (2011);

• SNN: Neural network trained by the scenario approach presented by Sadeghi et al. (2019).

In BELM, the parameter β is assumed to obey some predefined distribution. First, a prior distribution is set. With new data, the prior distribution is adjusted to a posterior distribution so that the data have maximum likelihood. Then, the corresponding output will also have a conditional probability, from which the confidence interval can be calculated. In addition, SNN can be regarded as a special case of SSELM with ɛ′ = 0.

We evaluate the mean square error (MSE) and mean absolute error (MAE) regarding the normal data in the evaluations.

5.2 Results and discussions

Figure 4 gives the examples of boundaries estimated by SSELM, BELM, and SNN. For each method, 10,000 samples are used. For SSELM, the probability threshold is set as 0.09. Each method also gives a corresponding center point of the confidence region. The abnormal data detection performance of different methods is summarized in Table 1. SSELM shows a better performance than other methods. The reason that SNN shows a poor performance is that it includes more abnormal data since it essentially gives a completely robust interval.

FIGURE 4

FIGURE 4. Examples of boundaries: (A) SSELM, (B) BELM, and (C) SNN.

TABLE 1

TABLE 1. Abnormal data detection accuracy (%) of different methods.

The results of wind power predictions on a text set are plotted in Figure 5. Note that the proposed method, SSELM, gives a prediction that concentrates around the real value with a shorter distance. More comprehensive results of error statistics are summarized in Table 2, which shows that the proposed method, SSELM, performs very close to the results the method gave using normal data. This shows the effectiveness of the proposed method. The proposed method increases the robustness of the regression against the abnormal data since it can clean the abnormal data effectively and thus increases the accuracy of regression.

FIGURE 5

FIGURE 5. Results of wind power predictions on a test set by different methods: (A) ELM, (B) SSELM, (C) BELM, and (D) SNN.

TABLE 2

TABLE 2. Error statistics of different methods.

6 Conclusion

This paper proposes an interval model of wind turbine power curves to improve the accuracy of wind power prediction. The interval model combines an extreme learning machine and the sample average approach. Thus, the proposed interval model can give a confidence region and center point of the wind power prediction for a given wind speed. The confidence region can be used for abnormal data detection, and the center point can be used as the estimation of the wind turbine power curve point. Experimental data-based validations have been conducted to compare the proposed method with several existing methods. The results show that the proposed method improves the accuracy of both abnormal detection and wind power curve estimation.

Data availability statement

The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.

Author contributions

KZ: conceptualization, methodology, project administration, writing–original manuscript, and writing–review and editing. HH: data curation, formal analysis, software, supervision, validation, and writing–review and editing. JL: data curation, investigation, methodology, and writing–review and editing. YW: investigation, validation, visualization, and writing–review and editing. WT: data curation, formal analysis, resources, and writing–review and editing. FH: formal analysis, supervision, validation, and writing–review and editing. YL: methodology and writing–review and editing. RB: data curation, formal analysis, resources, and writing–review and editing. HZ: data curation, investigation, validation, and writing–review and editing. LJ: methodology, supervision, and writing–review and editing.

Funding

This work was supported by the Science and Technology Project of State Grid Hubei Electric Power Company (52152020003K).

Conflict of interest

Authors KZ, HH, JL, YW, WT, FH, YL, RB, HZ, and LJ were employed by State Grid Hubei Extra High Voltage Company.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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