Introduction

Background

Fossil fuel exhaustion is causing severe climate change due to the fast expansion of many sectors throughout the world. Climate change is affecting many people around the world. In Awosusi et al. (2022), the authors looked at how carbon emissions are affected by globalisation of trade, rents on natural resources, economic expansion, and financial sector development. The impact of the climatology parameters on the COVID-19 is explained in Ahmadi et al. (2020). The authors of Habeşoğlu et al. (2022) discussed the oil price’s impact on the amount of carbon emission levels in Turkey through financial regulation, energy use, and economic expansion. This gains the attention of the world to clean and endless resources such as solar energy, hydro energy, and wind energy, tidal energy (Council 2020). Especially, wind energy is the most popular renewable energy source with rapid development all over the globe. With 93.6 GW of new global wind installations in 2021, brings the total installed wind capacity to 837 GW. Wind turbine regulations control and wind power system dispatch are based on the dynamic wind speed. According to the cubic relation among wind power and speed, even a little change in wind speed causes a noticeable rise in wind power. Therefore, wind speed is essential for producing wind energy. Wind speed forecasting is challenging, nevertheless, due to the intrinsically nonlinear characteristics of wind speed fluctuations, such as intermittency. In order to increase the usage of wind energy sources, it is crucial and essential to improve the precision of wind speed forecasts. The point forecasting in which the difference between predicted and actual are calculate is focused more in the current literature. Most of the researchers emphases on point forecasting methods. But these point forecasting models possess demerits such as inadequate accuracy problems due to uncertainty in the forecasts and low reliability. However, as the data becomes more complicated, the performance declines. Also, point forecasting approaches, on the other hand, fail to account for uncertainties and do not produce the needed accuracy. To overcome the above demerits of point forecasting, interval prediction is employed, which gives intervals instead of point values.

Literature review

The models utilised for interval prediction in the literature are broadly classified as statistical models and machine learning (ML) models. The statistical models can predict the parameters of error distribution to calculate the upper and lower bounds of a certain confidence interval. The mean and variance of the response variable are predicted using the interval forecasting model in Nix and Weigend (1994) but the coverage of prediction intervals (PI) is quite low for this implemented model. In Khosravi et al. (2011a), the traditional Bayesian approach is developed for interval prediction, but this approach has the demerit of huge computational complexity. In Pullanagari et al. (2018), the interval forecasting approach is implemented using quantile regression. For linear sequences, conventional statistical models perform excellently, but they come up short when applied to non-linear data as they cannot estimate distribution function with a hypothesis, and also have computational complexity problems. Both linear and non-linear data can be handled by artificial intelligence-based machine learning and deep learning algorithms (Ahmadi et al. 2022). The support vector machine is one of the machine learning models that are most frequently employed in this sector. These models are more widely used as a result of the development of AI into time series applications. The performance of AI-based techniques to prediction is nevertheless constrained by the quasi nature of the wind speed. The lower upper bound evaluation (LUBE)-based ML model was developed for the interval prediction. By using optimisation techniques, different variants such as single-objective optimisation (SOO)-based LUBE model (Hu et al. 2017) and multiple objective optimisation (MOO)-based LUBE models (Shrivastava et al. 2016) are implemented. Shallow ML networks in combination with the LUBE framework are also developed for interval prediction. The computational time is unacceptable with the tuning of more hyperparameters for shallow ML network-based LUBE models. They also fail to provide the necessary information for the proper and efficient decision-making in the power system applications such as load management, spot pricing, and trading. Prediction interval forecasting techniques are used to address these issues since they reduce uncertainty and offer an indicator of accuracy. In comparison with shallow ML models, deep learning methods were given better performance for interval prediction (Khodayar et al. 2018). In Naik et al. (2019), multi-kernel robust ridge regression is used for interval forecasting of wind speed and wind power. The authors, in (Khosravi et al. 2011b; Quan et al. 2014), proposed a technique for constructing prediction intervals in neural network (NN) predictions that is both fast and trustworthy. The authors proposed a lower upper bound estimation (LUBE) method in which a NN with two outputs is built to estimate the prediction interval bounds. For nonparametric prediction intervals of wind power generation, the authors built a novel adaptive bilevel programming (ABP) model using extreme learning machine-based quantile regression in Zhao et al. (2020). The proposed ABP approach tries to reduce the mean interval width when good calibration is used. In He and Zhang (2020), authors used parallel quantile regression neural network wind power probability density forecasting model. This algorithm can improve the efficiency of quantile regression neural network. Results are evaluated by metrics of speed up and parallel efficiency. The authors in Pinson and Kariniotakis (2010) used a fuzzy inference model which allows integrating expertise on the properties of prediction errors for providing conditional interval forecasts. To improve the probabilistic forecast of wind farm levels and regional wind farms, a novel method based on Gaussian processes is developed (Xue et al. 2020). For wind speed interval prediction, a novel hybrid model depending on gated recurrent unit (GRU) with variational mode decomposition (VMD) was developed in Tang et al. (2019). For the forecast interval of wind power, a beta distribution-dependent long short-term memory (LSTM) neural network model has been proposed in Yuan et al. (2019). In the LSTM neural network model, a variation activation function is used, and the Beta distribution parameters are optimised using the PSO method. For wind power forecasting, Niu et al. (2022) use a data-driven strategy based on numerous factors and interval forecasting is done using kernel density estimation using a Gaussian functions. A BiLSTM model that is optimised via an attention mechanism is employed in this paper to increase point predicting accuracy. The authors of Zhang et al. (2020) offer a new interval model which depends on the fast correlation-based filter (FCBF) method, the optimised radial basis function (RBF) model, and the Fourier distribution for wind speed, which blends artificial intelligence techniques with statistical information. An interval prediction model was constructed in Zhang et al. (2022) using an improved whale optimisation algorithm (IWOA) and a fast learning network (FLN). Adjusting the nonlinear convergence factor, as well as incorporating adaptive inertia weights and a chaos search technique, improved the IWOA’s convergence speed and accuracy. The authors of Zhang et al. (2019) employed a multi-objective interval methods that rely on the conditional copula function, in which they completely exploited the correlations between variables to increase prediction accuracy without relying on an assumed probability distribution function. In Heydari et al. (2021), a wind power producer (WPP) in a competitive power market is given an interval prediction algorithm based on a new bidding technique based on optimal scenario making. Based on 39 years of data, the authors of Narayanan Natarajan (2021) check the effectiveness of nine prominent probability distribution methods for an evaluation of wind speed distribution (WSD) at ten sites in Tamil Nadu, India.

In order to improve accuracy and better account for the non-stationary characteristics of wind variables, hybrid models (Ahmadi et al. 2022; Ghoushchi et al. 2017) for such WSF have been developed. Intelligent models and signal decomposition techniques are some of the ways utilised to generate hybrid models (Cui et al. 2020; Sun and Xiaoxuan Wang 2022; Zhang and Pan 2020; Gupta et al. 2021). The ensemble empirical mode decomposition was used to pre-process the data (Artin et al. 2021; Cui et al. 2020). The bat algorithm was then used to optimise the back propagation neural network’s connection weights and thresholds for forecasting. In Sun and Xiaoxuan Wang (2022), the wind speed sequence is decomposed using the wavelet transform, and the resulting detailed coefficients are further decomposed by using symplectic geometry mode decomposition. BPNN is optimised by using the marine predator’s algorithm is then applied for WSF. To produce a decomposition result, variational mode decomposition (VMD) is applied to the original wind speed data in Zhang and Pan (2020), and the combined prediction approach using elman radial basis function is employed for prediction. In Zhang and Wang (2022), instead of initialising the parameters, an optimisation algorithm is used. In this paper, an improved PSO algorithm is used. And a rolling training prediction method is employed for wind speed prediction (WSP). In Barenya et al. (2022), authors utilised an hourly wind speed data for WSP and WSP is carried out by using wavelet kernel-based least square twin support vector regression. In Chen (2022), CEEMDAN is used in combination of singular value decomposition (SVD) to deconstruct and denoise the actual data, after which optimised Elman and ARIMA models are employed to forecast the wind speed components.

A hybrid framework is proposed in this paper for wind speed interval prediction to address the following challenges. The present literature focuses on point forecasting, in which the difference between anticipated and actual values is computed. However, as the data becomes more complicated, the performance declines. Also, point forecasting approaches, on the other hand, fail to account for uncertainties and do not produce the needed accuracy. They also fail to provide the necessary information for the proper and efficient decision-making in the power system applications such as load management, spot pricing, and trading. Prediction interval forecasting techniques are used to address these issues since they reduce uncertainty and offer an indicator of accuracy. A hybrid framework is proposed in this paper for wind speed interval prediction (WSIP) to address these challenges. The primary contributions of this study are as follows: (1) A novel hybrid ICEEMDAN-ATCN-BiLSTM approach is proposed by integrating the neural network architecture in the LUBE framework. (2) An efficient data preprocessing ICEEMDAN algorithm reduces noise in the input data and enhances the signal-to-noise ratio. ATCN extracts the important and dominating spatial and temporal features from the denoised wind speed. Bi-LSTM model interprets the important features bidirectionally to forecast the high-quality prediction intervals (PIs). (3) The integration of attention mechanism to the TCN layers enhanced feature extraction.

The rest of the manuscript is organised as follows: The proposed approach methodology and working are presented in Section “Proposed hybrid approach for wind speed interval prediction”. Section “Experimental results” discusses the experimental results as well as a comparison to other methodologies. Section “Conclusions” outlines the conclusions.

Proposed hybrid approach for wind speed interval prediction

A novel hybrid framework using ICEEMDAN, TCN with attention (ATCN), and BiLSTM approach is proposed to enhance the quality of the WSIP. This section demonstrates the architecture of the proposed hybrid approach. The proposed hybrid approach is mainly divided into three sections: presupposition and noise elimination using ICEEMDAN, feature extraction using ATCN, and BiLSTM for WSIP. The proposed approach is illustrated in Fig. 1.

Fig. 1
figure 1

Complete framework of the proposed approach for WSIP

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Presupposition and data decomposition

Prediction interval forecasting differs from deterministic forecasting. As a result, there is no way to train forecasting networks directly. To address this problem, the boundaries of the input wind speed data must be presupposed for the training to develop a framework for the WSIP using a construction interval strategy. Ui and Li are the upper and lower bound of the wind speed data, formulated using Eqs. 1 and 2.

$$ U_{i}=x+R_{c} $$
(1)
$$ L_{i}=x-R_{c} $$
(2)

where x is the wind speed point, and Rc is the bound coefficient which is calculated by Eq. 3. α is the width coefficient pertaining to [0,1].

$$ R_{c}=\alpha*(\max(x)-\min(x)) $$
(3)

However, the upper and lower bounds formed are highly complex and non-linear making the WSIP more difficult. Thus, the ICEEMDAN decomposition method is used to decompose the signals to produce the denoised upper and lower bounds of the wind speed data for WSIP.

The ensemble empirical mode decomposition (EEMD) has been developed to solve the mode mixing problem occurred in the empirical mode decomposition (EMD). However, the presence of the residual noise in the EEMD affects the performance. Hence, complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) is developed. But, intrinsic mode functions (IMFs) produced consist of the residual noise and spurious modes. Thus, improved CEEMDAN is implemented to address the disadvantages of the CEEMDAN approach. Figure 2 shows the flowchart for ICEEMDAN approach (Bouhalais and Nouioua 2021). By lowering the number of trials, this ICEEMDAN overcomes the following issues:

  • a) mode mixing problem,

  • b) frequency aliasing problem,

  • c) residual noise problem.

Fig. 2
figure 2

Flowchart of improved CEEMDAN (Bouhalais and Nouioua 2021)

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In this approach, the wind speed data bound is decomposed into eight IMFs. As IMF1 is a very nonlinear signal, it is discarded, and the other IMFs are combined to generate the wind speed data without any external noise.

Attention TCN-based Bi-LSTM approach for enhanced prediction interval forecasting

The proposed forecasting network consists of two steps: feature extraction using ATCN and PI forecasting using BiLSTM. The two steps are discussed in the following subsections.

Feature extraction using attention-based TCN method

For forecasting the prediction intervals, optimal feature extraction is essential to decrease the uncertainty and enhance the PI’s quality. For this reason, ATCN layers are used for the feature extraction. The temporal convolutional network (TCN) is derived from the base convolutional neural network (CNN). The difference is the convolution operation. They are causal in the TCN whereas non-causal in the CNN model. TCNs use causal convolutions on the sequential data for the feature extraction. In this sequential method, the sequential inputs (X) are mapped to the sequential outputs (Y ) through a non-linear mapping function (f ). The main principle of the causal condition is that the output prediction (yt) is depended only on the past data (x0,x1,...,xt) but not on the future data (xt+ 1,xt+ 2,...,xT) at the instant t. In this study, the input of the ATCN is the denoised data and the output represents the features extracted from the denoised wind speed.

$$ \bar{Y_{0}}, \bar{Y_{1}}, \bar{Y_{2}}, ...., \bar{Y_{t}} = f(X_{0},X_{1},X_{2}, ..., X_{t}) $$
(4)

The TCN is trained in the supervised manner for decreasing the loss function L. There is no data leakage from the future since the output length is equivalent to the input length. Dilated convolutions are present in the TCN to exponentially extend the receptive field, allowing for more past data to be included when forecasting. There is no data leakage from the future data. Equation 5 represents the dilated convolution function. The representation of the dilated convolution with k= 2 and d= 1,2,4,8 is shown in Fig. 3.

$$ (X^{*}_{d}f)(s)= \sum\limits_{j-1}^{i=1}f(i).X_{s-d.i} $$
(5)

where x is the 1-dimensional sequence; k is the filter size; ∗ is the convolution operator; and d is the dilation factor.

Fig. 3
figure 3

Representation of dilated convolution with d= 1,2,4,8 and filter k= 2

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Normally high dilation factor and filter size are used to produce bigger receptive field to interpret more information from the past useful in long predictions. The fully connected 1D layer, and various residual blocks are present sequentially in the TCN for training the network. The dilated convolutions are performed using Eq. 5. For feature extraction, a generic residual block made up of layers of causal convolution is used for each layer. The output from the last residual block are used as the input to the model’s next layer. And the output from the last residual block are fed to the fully connected (FC) layers present in the TCN model. These FC layers convert the high-dimensional features to lower dimensional features representing the features. The ReLU activation function is used in the residual blocks. Batch normalisation and spatial dropouts are used for the regularisation. The output vector of the FC layer is the OVec, in which each value representing the feature from the denoised wind speed data. Enhancing the spatial and temporal feature extraction property of the TCN, this study used attention mechanism (AM) to amplify the important features and suppress the irrelevant features from the vector OVec. The primary principle of the AM is to imitate the human brain’s ability to comprehend things consciously. The implementation of the AM is represented in Eqs. 6 to 8.

$$ \begin{array}{@{}rcl@{}} \text{Va}_{i}&=& \mathrm{attten(OVec}_{i}) \end{array} $$
(6)
$$ \begin{array}{@{}rcl@{}} \mathcal{E}_{i}&=& \mathrm{softmax(Va}_{i}) = \frac{\mathrm{exp(Va}_{i})}{{\sum}_{i=q}^{Q} \mathrm{exp(Va}_{i})} \end{array} $$
(7)
$$ \begin{array}{@{}rcl@{}} \mathcal{X}&=&\mathcal{E}^{T}*\text{OVec} \end{array} $$
(8)

The function atten() is used for obtaining the importance of the individual element in the vector OVec. The importance of the each element is denoted by the new vector Vai, where i denotes the index value of the element in the vector OVec. Then, the Vai is normalised using the softmax function obtaining the attention weight vector \(\mathcal {E}\). Q indicates the vector Va length. The weight vector consists of the weights corresponding to the importance of the element in the vector OVec. More important feature will have more weight, and vice versa. Finally, the attention vector \(\mathcal {X}\) is produced by multiplication of \(\mathcal {E}\) and OVec. Thus, this final feature vector is fed to the Bi-LSTM that interprets the features for WSIP.

Wind speed interval prediction by Bi-LSTM

The features extracted with the ATCN model from the denoised wind speed data are given as the input to the bidirectional LSTM (Bi-LSTM) model for interpreting the features and forecasting. The LSTM is a recurrent neural network which is developed to work with the long-term sequences by integrating gate mechanism and memory unit. However, the LSTM transmits the information in only one way, i.e. it interprets only the past information. Hence, a derivative of LSTM, i.e. BiLSTM is developed to consider both the future sequence and past sequence information, i.e. it transmits the information bidirectionally. It is divided into forward LSTM and backward LSTM to extract the features and it concatenates the hidden features to achieve the extraction bidirectionally. In one LSTM, the input sequence is fed and to the other LSTM, the reverse sequence is fed. The implementation process of the Bi-LSTM is shown in the equations below, where Eq. 9 represents the implementation for the forward LSTM, and Eq. 10 for the backward LSTM. The structure of BiLSTM is represented in Fig. 4.

Fig. 4
figure 4

Representation of Bi-LSTM

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$$ \begin{array}{@{}rcl@{}} \overrightarrow{i_t} &=& \sigma (\overrightarrow{W_{ih}}\overrightarrow{h_{t-1}}+\overrightarrow{W_{ix}}\overrightarrow{x_t}+ \overrightarrow{W_{ic}}\overrightarrow{c_{t-1}}+ \overrightarrow{b_i}) \\ \overrightarrow{o_t} &=& \sigma (\overrightarrow{W_{oh}}\overrightarrow{h_{t-1}}+\overrightarrow{W_{ox}}\overrightarrow{x_t}+ \overrightarrow{W_{oc}}\overrightarrow{c_{t-1}}+ \overrightarrow{b_o}) \\ \overrightarrow{f_t} &=& \sigma (\overrightarrow{W_{fh}}\overrightarrow{h_{t-1}}+\overrightarrow{W_{fx}}\overrightarrow{x_t}+ \overrightarrow{W_{fc}}\overrightarrow{c_{t-1}}+ \overrightarrow{b_f}) \\ \overrightarrow{\tilde{c_t}}&=&\tanh(\overrightarrow{W_{ch}}\overrightarrow{h_{t-1}}+\overrightarrow{W_{cx}}\overrightarrow{x_t}+ \overrightarrow{b_c})\\ \overrightarrow{c_t}&=& \overrightarrow{f_t} * \overrightarrow{c_{t-1}}+ \overrightarrow{i_t} * \overrightarrow{\tilde{c_t}}\\ \overrightarrow{h_t}&=& \overrightarrow{o_t} * \tanh(\overrightarrow{c_t}) \end{array} $$
(9)
$$ \begin{array}{@{}rcl@{}} \overleftarrow{i_t} &=& \sigma (\overleftarrow{W_{ih}}\overleftarrow{h_{t+1}}+\overleftarrow{W_{ix}}\overleftarrow{x_t}+ \overleftarrow{W_{ic}}\overleftarrow{c_{t+1}}+ \overleftarrow{b_i}) \\ \overleftarrow{o_t} &=& \sigma (\overleftarrow{W_{oh}}\overleftarrow{h_{t+1}}+\overleftarrow{W_{ox}}\overleftarrow{x_t}+ \overleftarrow{W_{oc}}\overleftarrow{c_{t+1}}+ \overleftarrow{b_o}) \\ \overleftarrow{f_t} &=& \sigma (\overleftarrow{W_{fh}}\overleftarrow{h_{t+1}}+\overleftarrow{W_{fx}}\overleftarrow{x_t}+ \overleftarrow{W_{fc}}\overleftarrow{c_{t+1}}+ \overleftarrow{b_f}) \\ \overleftarrow{\tilde{c_t}}&=&\tanh(\overleftarrow{W_{ch}}\overleftarrow{h_{t+1}}+\overleftarrow{W_{cx}}\overleftarrow{x_t}+ \overleftarrow{b_c})\\ \overleftarrow{c_t}&=& \overleftarrow{f_t} * \overleftarrow{c_{t+1}}+ \overleftarrow{i_t} * \overleftarrow{\tilde{c_t}}\\ \overleftarrow{h_t}&=& \overleftarrow{o_t} * \tanh(\overleftarrow{c_t}) \end{array} $$
(10)

where it and ht indicate the input and hidden layer vector at the time t. Similarly, ht− 1, ct− 1 represents the hidden layer and memory cell value at the time t − 1. bi, bo, bf, and bc represent the bias of the input, output, forget gate, and memory cell respectively. W indicates the weight matrices for the different gates such as input, cell state, output, and forget gates. tanh and σ represent the activation functions. The → and \(\leftarrow \) represent the forward and backward propagations in the LSTM network. The output of the ATCN is a feature set of optimal characteristics from the input denoised data. Thus, to interpret the features for WSIP, Bi-LSTM is adopted, because of its high performance in time series forecasting applications. In the Bi-LSTM, the hidden layer is formed by concatenating the forward and backward propagation of LSTM as shown in Eq. 11.

$$ \beta_t= \overleftarrow{h_t} +\overrightarrow{h_t} $$
(11)

The features extracted by the ATCN are the inputs and and the outputs are corresponding lower (Li) and upper (Ui) bound of the data point, i.e. yi=[Li, Ui]. However, in point or deterministic forecasting, the loss function of the network used is mean squared error (MSE). However, the MSE loss function is ineffective for predicting intervals. This paper proposes a novel loss function technique to solve this challenge, keeping the optimisation of PI as the criteria. Two evaluation criteria, prediction interval coverage probability (PICP) and coverage width (CW), are used to construct the custom loss function. The loss function is formulated in Eq. 12, where 𝜃 represents the parameters such as weights and bias of the network.

$$ \mathrm{Loss\ function= argmin} \begin{cases} \text{CW}(\theta)\\ \frac{1}{\text{PICP}(\theta)} \end{cases} $$
(12)

The loss function is developed in such a way that the proposed network can predict the optimal intervals while taking into account both PICP and CW. When the PICP is high and the CW is low, the PIs are regarded to be ideal. As a result, the loss function is developed using these two criteria. The weights are modified during the network’s training to achieve the best result. A case study is performed for evaluating the proposed approach.

Experimental results

A wind speed dataset of 2013 year from a wind farm located in Garden City, Manhattan, is used in this study (NREL 2022). A map of the investigated area is shown in Fig. 5. The wind speed is obtained in 5-min ahead samples. However, to evaluate the novelty of the proposed approach for WSIP, wind speed is re-sampled into 10-min and 30-min ahead samples. The characteristics of the wind farm are shown in Table 1. Literature demonstrated that the performance of the forecasting models decreases with the increase in the ahead. Hence, the proposed approach is tested with three different time interval data. The original input data are divided into testing, training data, where training data consists of the first 80% data, and the rest data is 20% data is used as test data.

Fig. 5
figure 5

Map of the Garden city, Manhattan, USA

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Table 1 Characteristics of Garden city, Manhattan wind farm
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Evaluation criteria for optimal PI

Evaluation criteria such as mean squared error (MSE), root mean squared error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE) are used in the point forecasting. However, in the prediction interval forecasting applications, the performance cannot be evaluated using conventional metrics. To deal with this, evaluation indices such as prediction interval coverage probability (PICP), PI normalised root-mean-square width (PINRW), and mean prediction interval width (MPIW) are used to evaluate the PI quality (Tang et al. 2019), (Khosravi et al. 2011a). PICP and PINRW represent the reliability and precision of PI. MPIW indicates the mean width of the PIs. The evaluation indices are formulated in Eqs. 13 to 15.

$$ \text{PICP}= \frac{1}{n} \sum\limits_{j=1}^{n}C_{j},\ C_{j}= \begin{cases} 1, y_{j} \in [L_{j},U_{j}] \\ 0, y_{j} \not\in [L_{j},U_{j}] \end{cases} $$
(13)
$$ \text{PINRW}= \frac{1}{R} \sqrt{\frac{1}{n}\sum\limits_{j=1}^{n} (U_{j}-L_{j})^{2}} $$
(14)
$$ \text{MPIW}= \frac{1}{n} \sum\limits_{j=1}^{n} (L_{j}-U_{j}) $$
(15)

where n is the number of samples and upper and lower bounds of the j th prediction interval are denoted by Ui and Li, respectively. R represents the range of the wind speed data. Ci indicates whether the actual wind speed data at j th point lies in the predicted interval [Lj,Uj]. An optimal PI should have higher PICP and lower PINRW (narrow PI). Because of their inverse relationship, PICP and PINRW cannot provide a perfect evaluation of PI quality. Thus, a new criteria, coverage width criterion (CWC), is developed by combining the PICP and PINRW as shown in Eq. 16.

$$ \text{CWC}=(1+\eta_{1}\text{PINRW})(1+\gamma(\text{PICP})e^{-\eta_{2}(\text{PICP}-\mu)})\\ $$
(16)
$$ \gamma(\text{PICP})= \begin{cases} 0, \text{PICP} \geq \mu \\ 1, \text{PICP} < \mu \end{cases} $$
(17)

where η and μ are the crucial hyperparameters controls the CWC index. η is used to magnify the variation of the PICP with respect to μ, which is the confidence level of the interval. Therefore, this hybrid index is used to evaluate the quality of the PIs. Smaller the CWC, better the quality.

Results and discussions

To evaluate the performance of the proposed approach, in this paper, two categories of approaches are compared. The benchmark methods, such as ATCN-BiLSTM, MLP, LSTM, and CNN, fall into the first group of models. All these models from the first category are hybridised using the ICEEMDAN algorithm for denoising. As a result, ICEEMDAN-CNN, ICEEMDAN-LSTM, and ICEEMDAN-MLP fall into the second hybrid category of models. The proposed approach as well as the reference models are evaluated for WSIP, and the results are given in Tables 2, 3, and 4. Figure 6 illustrates the WSIP result by the proposed approach. Figure 7 illustrates the comparison of WSIP by all the hybrid approaches.

Table 2 Comparison of 5-min ahead WSIP statistical indices between benchmark approaches and proposed hybrid framework
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Table 3 Comparison of 10-min ahead WSIP statistical indices between benchmark approaches and proposed hybrid framework
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Table 4 Comparison of 30-min ahead WSIP statistical indices between benchmark approaches and proposed hybrid framework
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Fig. 6
figure 6

WSIP result of proposed approach using Garden city wind farm data

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Fig. 7
figure 7

Illustration of WSIP result of all models with ICEEMDAN algorithm for Garden city wind farm

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From Tables 2, 3, and 4 and Figs. 6 and 7, the hybrid model ATCN-BiLSTM achieved the best results compared to the first category models like CNN, MLP, and LSTM. In the 5-min ahead WSIP, ATCN-BiLSTM has an MPIW value of 1.9298, which is the least in the indices of first category models. Similarly, in PINRW, PICP, and CWC, this hybrid model achieved the best results compared to benchmark approaches. The CNN model occupies the second spot with a PICP of 0.8642. The MLP occupies the last spot with relatively higher CWC and PINRW indices. ATCN-BiLSTM approach, on the other hand, continues to dominate throughout the 10-min and 30-min ahead WSIP. LSTM and CNN approaches occupied the second spot in 10-min and 30-min WSIP, respectively. Therefore, the hybrid ATCN-BiLSTM’s dominant performance is confined to the first category models in the 5-min, 10-min, and 30-min WSIP. However, in the 10-min and 30-min WSIP, the hybrid ATCN-BiLSTM is having better performance than the comparative hybrid models using ICEEMDAN. Category 2 models with data denoising technique achieved better results than the category 1 models without denoising technique. In the second category of models, the proposed approach achieved the best results. For example, considering the 5-min ahead WSIP, the proposed approach observed a very high PICP, i.e. 0.9793, which is around 97%. The proposed approach achieved a lower CWC value of 0.0880, which is 36% lesser than the second-best value. Similarly, in MPIW and PINRW, the proposed approach observed best indices of 1.3885 and 0.1021, respectively. 25% and 24% improvement is achieved by the proposed approach in the MPIW and PINRW over the second best model. ICEEMDAN-based CNN observed the second-best results among the two categories of models. ATCN-BiLSTM model at the third position followed them. In 5-min interval, the improvement percentage of proposed approach over ATCN-BiLSTM model is 28%, 25%, and 37% in terms of MPIW, PINRW, and CWC. Comparing the 10-min ahead WSIP indices, the proposed approach achieved the best values of indices. For example, the CWC is 0.2306, which is 47% lesser compared to the second-best approach available. The hybrid ATCN-BiLSTM approach occupied the second spot, and MLP occupied the last rank in the 10-min ahead WSIP. The improvement of 19% and 23% in the MPIW, and PINRW is achieved by the proposed approach over the second best model in the 10-min ahead WSIP. However, as the time ahead increased from 5-min to 10-min ahead, the performance of all the models declined. On the other hand, the proposed approach observed betterment in the performance from 5-min to 10-min ahead WSIP. Coming to the 30-min ahead WSIP, the proposed approach maintained stable performance with a CWC of 0.4502. The proposed approach achieved the highest improvement percentage in terms of the evaluation indices in the 30-min ahead WSIP. For instance, the improvement percentages are 19%, 25%, and 17% in the MPIW, PINRW, and CWC respectively. MLP model ranked last in the indices of 30-min ahead WSIP among all the comparative approaches. In 30-min interval, the improvement percentage of proposed approach over ATCN-BiLSTM model is 20%, 28%, and 37% in terms of MPIW, PINRW, and CWC. The percentage improvement of proposed model over all other individual and hybrid models is shown in Table 5.

Table 5 Percentage improvement of proposed approach over all other models
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Based on the evaluation indices, it is evident that the hybridisation of category 1 models with ICEEMDAN resulted in the enhanced performance in WSIP. This is due to the denoising of highly non-linear signals. From Fig. 6, the actual wind speed lies precisely in the prediction interval obtained by the proposed approach, respectively. However, there is a slight decline in the quality of the prediction interval from 5-min to 30-min ahead. But, from Fig. 7, it is very apparent that the proposed model’s prediction interval quality is quite optimal compared to the other reference models, which are having an unstable performance for 5-, 10-, and 30-min ahead WSIP.

The proposed approach’s performance is also evaluated in terms of testing and training time efficiency with the comparative CNN and LSTM models. The proposed framework’s training time is 820.52 s, while the testing time is 0.041 s. When compared to the proposed approach, the CNN model requires 778.21 s for training and 0.033 s for testing. Similarly, the LSTM model took 801.23 s to train and 0.039 s to test. However, since the training is done offline and just once, the quantity of training time achieved is acceptable. The testing time is also significantly fast, with predictions taking 0.3 ms. Furthermore, when the evaluation indices are taken into account, the proposed approach achieves the best WSIP performance indices as already presented through Tables 2, 3, and 4. The proposed approach predicts high-quality intervals because it uses effective feature extraction, denoising, and feature interpretation to forecast intervals. As a result, the proposed approach meets the criterion for real-time WSIP.

The experimental findings from the WSIP results of all available methods reveal the following critical points.

  • (1) The approaches such as CNN, LSTM, and MLP failed to maintain the consistency for the 5-, 10-, and 30-min ahead WSIP.

  • (2) But, it is also observed that the proposed approach is consistently leading in all the three forecasting results with the best indices.

  • (3) The evaluation indices clearly show the efficiency of hybridisation using the ICEEMDAN method. Category 2 models are outperforming category 1 models in terms of prediction interval quality.

  • (4) The quality of the prediction intervals of the benchmark approaches hybridised with ICEEMDAN, on the other hand, is not adequate.

  • (5) The proposed approach achieved an increase in the improvement percentage from 5-min WSIP to 30-min ahead WSIP.

Conclusions

The primary prerequisite for wind energy grid management is accurate wind speed predictions. As noted in the literature, point forecasting fails to account for uncertainties and does not produce needed information for power system operations. As a result, in this paper, a novel approach consisting of ICEEMDAN and TCN with attention mechanism and Bi-LSTM is proposed for improved accuracy of WSIP. Effective elimination of auxiliary noise, feature extraction plays a crucial role in the forecasting performance. In the proposed approach, ICEEMDAN is used for decomposing the signal to eliminate the auxiliary noise, ATCN is used to extract important features from the decomposed wind speed, and Bi-LSTM forecasts the accurate prediction intervals. Addressing the decline of performance of the models with an increase in ahead values, the proposed approach is tested using 5-min, 10-min, and 30-min ahead WSIP. A comparative analysis is performed using two categories of models to evaluate the proposed approach performance. The evaluation indices from the experiment indicate that the proposed approach’s performance is consistent for 5-min, 10-min, and 30-min WSIP. The feasibility and performance of the proposed approach are investigated and confirmed during the experiments. The experimental results indicate the dominating performance of the proposed approach for 5-min, 10-min, and 30-min ahead WSIP. The proposed approach offers an improvement of 36%, 47%, and 17% for three time intervals WSIP.

The future work would demonstrate the application of the optimisation techniques for enhancing the performance of the proposed approach.