Missing data can be handled in different ways. The entire observation can be removed or replaced with the average, median or mode. These ways may reduce dataset variability. On the contrary, KNN imputation keeps dataset value and variability. Figure 2 shows an example of 3-nearest neighbors.

As shown in Figure 3 , Euclidean distances can be calculated for the data. The nearest neighbors are those having the shortest distance. At k = 1, A is the nearest neighbor to B.

Figure 4 shows how K-means-Clustering works. K-means clustering is an unsupervised machine learning algorithm used to group data points into distinct clusters. It aims to minimize the sum of squared distances between data points and their respective clusters. The algorithm involves initialization, assignment, update, and iteration until convergence. The algorithm assigns each data point to a cluster based on the nearest centroid, with the centroids representing the center points. However, the algorithm is sensitive to the initial placement of centroids, so it is often run multiple times with different initializations.

Recurrent Neural Networks (RNNs) have feedback connections and can process input sequences of arbitrary length. However, training RNNs can be challenging due to gradient-based weight update algorithms [19] . Long Short-Term Memory (LSTM) is a type of RNN that can work with sequential data, making it effective for time series forecasting in various fields [20] [21] .

The LSTM architecture, consisting of three logistic sigmoid gates and one tanh layer, is commonly used with a forget gate to limit information flow, with output typically ranging from 0 - 1. An LSTM cell has 3 inputs $h_{t-1}$, $x_t$ and $C_{t-1}$ and 2 outputs $h_t$ and $C_t$. For a given time t, $h_t$ is the hidden state, $C_t$ is the cell state or memory, $x_t$ is the current data point or input. The first sigmoid layer has two inputs $h_{t-1}$ and $x_t$ where $h_{t-1}$ is the hidden state of the previous cell [22] .

In an LSTM network, the forget gate decides how much of the previous cell state to retain, while the input gate determines what new information to add. A candidate layer generates values to update the cell state, which combines these inputs with the retained information from the forget gate. For output, a sigmoid layer selects relevant parts of the cell state, which is scaled by a tanh layer to produce the final cell output. Together, these gates and layers enable the LSTM to effectively manage information flow and maintain long-term dependencies [23] .

An LSTM network computes the output by mapping an input sequence $x=\left(x_1,\cdots ,x_T\right)$ to an output sequence $y=\left(y_1,\cdots ,y_T\right)$ and calculating the network unit activations using the following equations iteratively from t = 1 to T [24] :

$i_t=\sigma \left(W_{ix}{x}_t+W_{im}{m}_{t-1}+W_{ic}{c}_{t-1}+b_i\right)$(1)

$f_t=\sigma \left(W_{fx}{x}_t+W_{fm}{m}_{t-1}+W_{fc}{c}_{t-1}+b_f\right)$(2)

$c_t=f_t\odot c_{t-1}+i_t\odot g\left(W_{cx}{x}_t+W_{cm}{m}_{t-1}+b_c\right)$ (3)

$o_t=\sigma \left(W_{ox}{x}_t+W_{om}{m}_{t-1}+W_{oc}{c}_t+b_o\right)$(4)

$m_t=o_t\odot h\left(c_t\right)$(5)

$y_t=\varphi \left(W_{ym}{m}_t+b_y\right)$(6)

where W represents weights, $W_{ic}$, $W_{fc}$, $W_{oc}$ are diagonal weights for peephole connections, the b represents bias, $\sigma$ is the logistic sigmoid function, and i, f and o are the input, forget, output gates. Also, c is cell activation, while $\odot$ is element-wise multiplication of vectors. Finally, φ is the network output activation function.

Multi-variate time series forecasting is a challenging task due to complex factors in real-life sequential data, such as dependencies, randomness, and noise. Hyperparameters and learning properties also affect performance. Traditional manual fine-tuning can lead to sub-optimal model representations and ill-performed learning parameters. To address these challenges, a new approach incorporates automatic feature extraction and optimal network configuration identification into the vanilla LSTM structure. Convolutional Neural Networks (CNNs) are hybridized with LSTM to extract fundamental features and construct more accurate feature representations.

The GWO is an optimization algorithm inspired by nature, modelled after grey wolves’ social hierarchy and hunting mechanisms. It uses a four-class pack of wolves, alpha, beta, delta, and omega, with the best-fitting wolf being the alpha wolf. The algorithm updates the hierarchy each iteration, adjusting solutions based on the wolves’ fitness function. The algorithm further explains grey wolf hunting mechanisms mathematically [25] .

The search process uses random positions of α, β, and δ, with coefficient vectors A and C assigned random values to facilitate a global search for prey.

Once a prey is found, the wolves encircle it; as shown in Figure 1 . For i iteration, equations that represent the positions are:

$D=\left|C\cdot V_{p\left(i\right)}-V_{\left(i\right)}\right|$(7)

$V_{\left(i+1\right)}=V_{p\left(i\right)}-A\cdot D$(8)

where $V_p$ is vector of prey position, i is the current iteration, and $V$ denotes grey wolf position of a. $A$ and $C$ are computed as the following equations:

$A=2a\cdot r_1-a$(9)

$C=2r_2$(10)

Assuming that a grey wolf’s location is (X, Y) and that of its prey is (X’, Y’), the wolves can then adjust their positions in response to the prey’s location, as shown in Figure 5 .

The alpha wolf, typically the pack leader, detects prey locations. The Alpha, Beta, and Delta wolves provide mathematical solutions for optimum prey location [26] :

$\begin{array}{l}{D}_{\alpha }=\left|C_1\cdot V_{\alpha }-X\right|\\ D_{\beta }=\left|C_2\cdot V_{\beta }-X\right|\\ D_{\delta }=\left|C_3\cdot V_{\delta }-X\right|\end{array}$(11)

$\begin{array}{l}{V}_1=V_{\alpha }-A_1\cdot D_{\alpha }\\ V_2=V_{\beta }-A_2\cdot D_{\beta }\\ V_3=V_{\delta }-A_3\cdot D_{\delta }\end{array}$(12)

Next, as indicated by the following equation, the other wolves have to adjust their locations in accordance with the positions of the top three search agents (α, β, and δ):

$V_{\left(i+1\right)}=\frac{V_1+V_2+V_3}{3}$(13)

The grey wolves surround a prey, stop moving, and attack it. The value of A decreases, resulting in a decrease in the prey’s position. The next grey wolf position is between its current and prey location.

Figure 6 shows class diagram of the implementation. The class diagram, which shows properties like training_data, test_data, lstm_model, and gwo_optimizer for the main WPF class. Wind energy training and testing data are included in the dataset. The GWO technique is utilized for hyperparameter tuning, and the LSTM model is used for forecasting. Be aware that composition, also known as aggregation, is the relationship between classes. The composition relationship, shown by the diamond shape at the end of the association line, denotes a strong “whole-part” relationship between classes.

For downloading the Python source code and full-merged dataset, go to the repository https://github.com/dr-dos-ok/Wind-Power-Forecasting-onGithub.com .

Figure 7 shows components of the project. The file C1.csv is the merged data. WPF usually merges NWP with historical data for providing accurate weather prediction. GWO.py is the implementation of GWO algorithm, while model.py provides the implementation of LSTM.