Enhanced Signal Processing-Based Load Forecasting in Smart Grids Using Artificial Neural Networks and Heuristic Optimization

Zhang et al. [6] introduced an innovative real-time prediction model for smart grids, integrating a convolutional neural network (CNN) attention mechanism with bi-directional long- and short-term memory (BiLSTM). This model excels in spatiotemporal feature extraction, yielding higher prediction accuracy and enhanced adaptability compared to conventional methods such as ARMA and decision trees. To refine feature selection, Bayesian optimization is incorporated, allowing the model to efficiently learn from real-time load data. By utilizing real-time power system data—including power consumption, load variations, and meteorological factors—the model ensures accurate future load predictions. Additionally, Bayesian optimization fine-tunes hyperparameters, further strengthening predictive accuracy. Despite its efficiency in estimating real-time power loads for operational and dispatch applications, the model's computational complexity may affect execution speed.

Rabie et al. [7] proposed an Optimal Load Forecasting Strategy (OLFS), which leverages artificial intelligence (AI) for smart grid demand prediction. This methodology involves an initial preprocessing phase where feature selection and outlier detection are performed before forecasting. Advanced Leopard Seal Optimization (ALSO), a bio-inspired optimization technique, is employed for feature selection, while Interquartile Range (IQR) is utilized for detecting and eliminating statistical anomalies. For forecasting, the approach integrates the Weighted K-Nearest Neighbor (WKNN) algorithm, optimizing predictive performance while reducing root mean square error (RMSE). However, deep learning (DL) techniques were not explored, potentially limiting further improvements in forecasting accuracy.

To address the challenges in short-term power load forecasting, Wen et al. [8] designed a computational framework that integrates multiple deep learning (DL) models. This hybrid approach employs Gated Recurrent Unit (GRU) networks, which effectively capture long-term dependencies in time-series energy data. By leveraging pattern recognition and feature extraction, the model minimizes forecast errors and enhances accuracy, proving beneficial for grid planning and energy management. Furthermore, an attention mechanism is incorporated to prioritize key input components that significantly influence load prediction outcomes, improving model performance.

A hybrid short-term electric load forecasting model was introduced by Hafeez et al. [9], incorporating an improved Modified Mutual Information (MMI) technique. This method refines data preprocessing and feature selection by extracting abstract features from historical load records. The model is built upon a Factored Conditional Restricted Boltzmann Machine (FCRBM), a deep learning-based forecasting module. Additionally, Genetic Wind-Driven Optimization (GWDO) is employed to fine-tune model parameters, enhancing forecasting accuracy and convergence rates. Despite its effectiveness, the model’s high computational scalability presents a challenge in large-scale implementations.

To further enhance forecasting precision and system performance, Aly [10] developed a hybrid forecasting method by integrating multiple machine learning techniques with clustering algorithms. The model incorporates six forecasting schemes, each utilizing unique combinations of Kalman Filtering (KF), Wavelet Transformation, ANN, and Wavelet Neural Networks (WNN). While the approach achieves high accuracy levels, its training process is computationally intensive, requiring significant resources.

Mansoor Ali et al. [11] proposed a hybrid forecasting framework named WLANFIS, which merges Adaptive Neuro-Fuzzy Inference System (ANFIS), Neural Networks (NN), and Weighted Least Squares (WLS). The methodology integrates fuzzy logic and neural networks, where an optimized dataset—derived from NN and WLS—determines the optimal membership functions and fuzzy set boundaries. This approach effectively mitigates overfitting issues, yet its inability to integrate Deep Neural Networks (DNN) with Kalman filter-based state estimation remains a limitation.

In the domain of load and price forecasting, Naz et al. [12] developed two predictive models: Enhanced Logistic Regression (ELR) and Enhanced Recurrent Extreme Learning Machine (ERELM). ELR represents an optimized variant of logistic regression, whereas ERELM employs the Grey Wolf Optimizer (GWO) for fine-tuning model weights and biases. The study also utilized Recursive Feature Elimination (RFE), Relief-F, and Classification and Regression Trees (CART) for feature selection and extraction. While the models achieved superior forecasting accuracy with larger datasets, their high computational complexity posed a challenge for real-time applications.

Dai and Zhao [13] developed a hybrid model incorporating intelligent techniques to optimize parameters and select features for power load forecasting. Given the increasing impact of real-time pricing on electricity consumption patterns, they considered real-time pricing and other influential variables as candidate features. Minimal redundancy maximal relevance was applied to extract informative features, while the weighted gray relation projection approach was used to select historical load sequences, making the selection more general. Finally, repulsion particle swarm optimization (PSO) and second-order oscillation were employed to optimize support vector machine parameters. Although, the approach requires more training iterations, it achieves noticeably higher accuracy.

The key challenges faced by existing methods:

Data quality and availability issues, complexity of Smart Grids, and non-linear relationships between load demand and variables, like weather and time hinder accurate forecasting. Uncertainty and variability of load demand, scalability, and interpretability of models also pose challenges. Real-time processing requirements, integration with other systems, and cybersecurity concerns add to the complexity. Compliance with regulatory and policy frameworks, balancing accuracy and computational efficiency, and handling special events like weather anomalies or grid failures are also crucial. Moreover, long-term forecasting needs and accounting for distributed energy resources and electric vehicle charging must be addressed. To overcome these challenges, advanced forecasting techniques, data preprocessing, and robust modeling approaches are necessary. By addressing these challenges, Intelligent Load Forecasting can support efficient and reliable grid operations, optimize energy distribution, and enable a sustainable energy future. Effective solutions will require collaboration among researchers, industry experts, and policymakers to develop and implement innovative forecasting methods and technologies.

This study introduces a innovative approach for load forecasting in smart grids by integrating an Artificial Neural Network (ANN) with the SFWWO algorithm. The SFWWO algorithm synergistically combines the strengths of Smart Flower Optimization Algorithm (SFOA) and WWO, enabling efficient search for optimal solutions and robust handling of complex data patterns.

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Figure 2. Preview of load forecasting in smart grids using ANN_SFWWO

The methodology follows a structured approach starting with data collection from smart grids. This is followed by missing value imputation and preprocessing to enhance data integrity. Feature extraction techniques identify key technical indicators such as Simple Moving Average (SMA), Rate of Change (ROC), and Adaptive Moving Average (AMA). Next, feature selection is performed using Ruzicka analysis to determine relevance, while Motyka analysis assesses feature importance. The Dice similarity coefficient is then applied for feature fusion, ensuring redundancy reduction and improved classification accuracy. The final forecasting is executed using ANN, which is optimized by SFWWO to improve model training, accelerate convergence, and enhance predictive performance. Figure 2 illustrates the overall workflow of the ANN_SFWWO-based load forecasting approach which can be summarized as follows:

Preprocessing and Feature Extraction: The preprocessing stage enhances reliability by handling missing values, filtering noise, and detrending data. Imputation ensures completeness, reducing bias and improving accuracy. Key technical indicators like SMA, ROC, and MACD capture load variations and trends in smart grid data [14].

Feature Selection and Fusion: To enhance forecasting accuracy, extracted features are selected and fused using Ruzicka for relevance, Motyka for significance, and Dice similarity to refine fusion, remove redundancy, and improve classification.

ANN Training with SFWWO Optimization: The ANN model, optimized with SFWWO, merges SFOA’s adaptability and WWO’s search efficiency for precise weight tuning, ensuring faster convergence and higher accuracy in load forecasting. Designed for dynamic grid conditions, ANN_SFWWO offers a reliable solution for smart grid energy management.

4.1 Acquisition of time series data

The time series data is regarded as an input for additional processing in this context. The dataset, including multiple series data is provided by,

$A=\left\{ A{{}_{h1}},{{A}_{h2}},...,{{A}_{hp}},...,{{A}_{hv}} \right\}$    (1)

where, A_hv is the v^th time series data, A_h1 is the initial time series data at time period h, and A_hp is the p^th time series data for further processing.

4.2 Pre-processing

Pre-processing of time series data is crucial for load forecasting in smart grids as it ensures accurate and reliable predictions by handling missing values, removing noise and outliers, normalizing and scaling data, detrending and adjusting for seasonality, extracting relevant features, and improving model performance. This enables smart grids to optimize energy distribution, manage peak demand, and integrate renewable energy sources effectively. By pre-processing time series data, smart grids can improve forecast accuracy, reduce energy waste, and enhance overall grid efficiency, ultimately leading to cost savings, improved customer satisfaction, and a more sustainable energy future.

In order to reduce the amount of redundant data, the time series data A_hp is pre-processed using missing value imputation. In addition, it reduces bias and prediction errors by substituting estimated values for missing data points to produce a complete dataset. Additionally, during the pre-processing stage, the raw data input is converted into a readable data format, greatly improving the prediction accuracy. In this case, feature average in non-missing data is used to identify missing values. At last, the absent values are filled in. Here, the term Q represents the pre-processed data output.

4.3 Extraction of technical indicators

Extraction of technical indicators is important for load forecasting in smart grids because it transforms raw time series data into meaningful features that capture patterns, trends, and seasonality, enabling accurate predictions. Technical indicators, such as Simple Moving Average (SMA), Rate of Change (RoC), Adaptive Moving Average, Keltner Channels, Moving Average Convergence Divergence (MACD), Exponential Moving Average (EMA), and Double Exponential Moving Average (DEMA), which are extracted from A_hp for forecasting. By extracting these indicators, smart grids can identify key factors influencing load demand, improve forecast accuracy, and optimize energy management, ultimately leading to efficient grid operations, reduced energy waste, and enhanced customer satisfaction. Table 1 lists the technical indicators parameter setting.

Table 1. Configuration of technical indicators

SMA: The SMA [14] is calculated by summing up the load values over a specified period, and then dividing by the total count of time periods and the SMA indicator output is denoted as ${{b}_{1}}$.

EMA: EMA [14] is expressed as the weighted average of recent load values, giving more weight to recent observations, to better forecast load in smart grids.

$EMA=\left( a-EM{{A}_{a}} \right)*S+EM{{A}_{a}}$    (2)

The weighted average for the current period is denoted by a, the EMA for the previous period is shown by EMA_a, the smoothing constant is denoted by S, and the EMA indicator output is indicated by b₂.

DEMA: DEMA [14] is an indication that offers a smoothed mean with less lag than EMA. It is symbolized as,

$DEMA=\left( 2*EMA(t) \right)-\left( EMA(t)\,of\,EMA(t) \right)$     (3)

The DEMA indicator notation is b₃.

ROC: ROC [14] calculates the percentage change in load values over a specified period. The result is a measure of the rate of change in load, which can be used to identify trends, patterns, and anomalies in load data.

$ROC=\left( \left( {d}/{e}\; \right)-1.0 \right)*100$    (4)

where, d is the current load value, and $e$signifies the previous load value from a specified time ago, and the ROC indicator output is expressed as b₄.

MACD: The differences between the two MAs of different periods, such as fast MA and slow MA, are taken into account while computing the MACD indicator [15]. The MACD indicator can be written as:

${{b}_{5}}=FastMA-SlowMA$    (5)

where, FastMA stands for the shorter MA, SlowMA for the longer MA, and the MACD indicator is shown as b₅.

AMA: Another technical indicator that uses scalable constants rather than set constants to smooth data is the AMA. The symbol for this indicator is b₆.

Keltner Channels: The Keltner channel is a volatility-based technical indicator that typically displays three lines: an upper band, a middle line, and a lower band. The EMA is the middle line of the Keltner channel, with the other two bands situated above and below it. Keltner channel equations are therefore represented by,

$W={{E}_{1}}$     (6)

${{B}_{R}}={{E}_{1}}+2*D$     (7)

${{B}_{J}}={{E}_{1}}+2*D$     (8)

where, D stands for average true range, W for the center line of the keltner channel, B_R for the upper band, B_J for the lower band, and E₁ for the EMA. The indication of the Keltner channel is denoted by b₇.

Rate of Change: The ROC is used to estimate the rate of alternation or change in load quality for a previous time interval.

${{b}_{8}}=\frac{X\left( q \right)}{X\left( q-m \right)}*100$    (9)

where, load quality is represented by X, load quality at a given period q by X(q), and load quality variation over time by X(q-m). The ROC curve, represented by b₈.

Thus, joining the whole extracted technical indication yields the final feature vector of the technical indicator. Consequently, the expression is modeled as,

$G=\left\{ {{b}_{1}},...,{{b}_{8}} \right\}$     (10)

As a result, the next step in the feature selection procedure uses the input as the retrieved technical indicator G.

4.4 Feature selection

Feature selection is crucial after extracting technical indicators for load forecasting in smart grids as it reduces dimensionality, removes irrelevant features, improves model interpretability, enhances model performance, and reduces the risk of multicollinearity. By selecting the most relevant features, understand the relationships between technical indicators and load demand, improve model accuracy, reduce noise, and increase training speed. This step helps build a more robust, accurate, and interpretable load forecasting model, enabling smart grids to optimize energy management, reduce energy waste, and enhance customer satisfaction. Here, the retrieved technical indicator output $G$ is considered as input. In this case, the dice similarity metric is fused with the Ruzicka and Motyka to choose the essential features.

Ruzicka similarity: The Ruzicka similarity (also known as weighted Jaccard similarity) measures the degree of overlap between feature distributions while accounting for their magnitude differences. It has the potential to analyze uneven data distribution, which makes it suitable for smart grid applications where the load and energy patterns vary continuously. This is then expressed as:

${{P}^{RUZ}}=\frac{\sum\limits_{g=1}^{c}{\min ({{N}_{g}}{{K}_{g}})}}{\sum\limits_{g=1}^{c}{\max ({{N}_{g}}{{K}_{g}})}}$   (11)

where, N_g stands for candidate features and K_g for class label. It determines the highest-value features by calculating each feature's Ruzicka similarity. Thus, C_n×q where p>q is the output.

Motyka similarity: The Motyka similarity evaluates the similarity between two data sequences. It defines ratio of summation of minimum values of corresponding components in the two data sequences to the sum of all elements, as formulated in Eq. (12). In the presented work, it was deployed for selecting relevant attributes for ANN by estimating features that are similar across various data sequences. Here, $G$ is regarded as input in this instance as well. Each pair of two input collections has its similarity calculated using the Motyka similarity, which is provided by:

${{P}^{MOT}}=\frac{\sum\limits_{g=1}^{c}{\min ({{N}_{g}}{{K}_{g}})}}{\sum\limits_{g=1}^{c}{\left( {{N}_{g}}+{{K}_{g}} \right)}}$     (12)

where, K_g denotes the target class and N_g denotes candidate features. It chooses the best features with the highest value after calculating each feature's Motyka similarity. F_n×r where p>r is the output's symbol.

Fusion by dice coefficient: Features obtained by means of the Motyka and Ruzicka poses several types of representation. Thus, efficiency can be completely destroyed by a simple concatenation. Therefore, the majority of rich detail information is encoded in the characteristics that the Ruzicka obtains, while the Motyka gathers context information. In order to create features that are more effective, the output:

${{P}^{DICE}}=\frac{2\sum\limits_{g=1}^{c}{{{N}_{g}}{{K}_{g}}}}{\sum\limits_{g=1}^{c}{N_{g}^{2}}+\sum\limits_{g=1}^{c}{K_{g}^{2}}}$    (13)

where, K_g stands for Target class and K_g stands for candidate features, which are the output features produced by combining Ruzicka and Motyka. C_n×k represents the output in the case where (q+r)>l.

4.5 Load forecasting using ANN_SFWWO

Load forecasting plays a vital role in enhancing energy management and minimizing energy waste within smart grids. By predicting energy demand with high precision, smart grids can dynamically adjust supply to meet demand in real-time, ensuring efficient and reliable power distribution while optimizing overall grid performance. In the proposed work, an optimized framework was developed for accurately predicting loads. The designed framework leverages the optimization capacity of hybrid optimizer named “Sun Flower Water Wave Optimization (SFWWO)” with ANN. The proposed SFWWO combines the efficiency and advantages of SFOA and WWO. SFOA is a meta-heuristic optimization technique inspired from the characteristics of sunflowers for searching the best orientation towards the sun. In this algorithm, the pollination process is simulated based on random generation of seeds, which ensures adaptability to real-world constraints. Also, this algorithm outperformed the conventional optimization techniques such PSO, genetic algorithm (GA), etc., by easily solving the complex problems in different fields. On the other hand, WWO is a nature-inspired meta-heuristic algorithm modeled based on the shallow water wave theory for resolving the world optimization problems. This approach includes water wave phenomena like refraction, propagation and refraction. This approach has the efficiency of searching in a high-dimensional solution space, which makes it an optimal solution for problems in large networks like smart grids. Also, this approach regulates the searching process with minimal control parameters, reducing the computational power and computational time. These advantages make the WWO approach more efficient than conventional techniques like PSO, GA, ACO, etc. [15, 16]. Thus, the integration of SFOA and WWO into single optimization algorithm ensures adaptability, complex problem solving, less computational demands and high-dimensional searching, making it a suitable solution for smart grid-based problems. This synergy enables smart grids to better anticipate and respond to changing energy demands, reducing the risk of power outages and minimizing energy waste. By improving load forecasting, smart grids can optimize energy management, reduce costs, and enhance customer satisfaction, making this research essential for the development of high-performance renewable energy models.

Structure of ANN: An Artificial Neural Network (ANN) [17] is an adaptable model capable of identifying patterns by repeatedly analyzing data, allowing it to generalize and make predictions on previously unseen information. In this context, C_n×k serves as the input for the ANN. In supervised learning, the network is guided by human intervention, where specific data relationships are defined to enable accurate learning and decision-making. The network attempts to learn the input-output connection by adjusting its free parameters after being provided a set of inputs and corresponding desired outputs. Here, the activation function u(z) is the sigmoid function, which can be found by:

$u\left( z \right)=\frac{1}{1+\exp \left( -z \right)}$    (14)

$z=\sum\limits_{f=1}^{j}{{{x}_{if}}{{l}_{f}}}+{{\alpha }_{i}};i=1\,to\,p$    (15)

$z=\sum\limits_{i=1}^{j}{{{x}_{ib}}{{q}_{i}}}+{{\alpha }_{b}};b=1\,to\, \text{K}$    (16)

where, α is threshold, x is synaptic weight, l is input node value, p is hidden note value, p is number of hidden nodes, f is number of output nodes, and j is the number of input nodes. Figure 3 shows the structure of ANN.

The learning algorithm of a back-propagation neural network consists of two phases.

Phase I: The network input layer is given with a constraining input pattern.

Phase II: After that, the input pattern is spread throughout the network's layers until the output layer generates the output pattern. When the produced pattern does not match the expected output, an error is calculated and then propagates backward from the output layer back to the input layer across the network. This backpropagation process adjusts the network’s weights, refining them based on error magnitude and optimization criteria, ultimately enhancing the model’s accuracy and performance. In contrast to other neural networks, a back-propagation neural network is based on the activation function, the connections between the neurons, and the adjustment of weights by the intended output level. A back-propagation network typically consists of three or more tiers in a multi-layer architecture. Every neuron in each layer is connected to every other neuron in the forward levels that surround it, and the layers are coupled.

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Figure 3. ANN architecture

Training ANN by SFWWO: Although ANN offers promising results in pattern recognition, its prediction performances highly rely on the appropriate selection of hyperparameters in its training phase. Hence, hyperparameter optimization was introduced for finding the best combination of hyperparameters for a machine learning model to maximize its performance. In the presented work, the ANN training is performed using SFWWO, a novel hybrid algorithm combining SFOA [18] and WWO [19]. Sunflowers serve as the inspiration for the SFOA, which clarifies and idealizes the growth of young sunflowers. The outcomes validated the ability and effectiveness in identifying the best. It reveals its usefulness in resolving practical issues with ambiguous search spaces. It helps with a range of engineering design problems. Flowers bloom and grow towards sunlight, maximizing their exposure to light and nutrients. This process is optimized through natural selection, ensuring the fittest flowers survive and reproduce. Meanwhile, the WWO propagates and interacts with their environment, adapting to obstacles and boundaries. This algorithm captures this dynamic behavior to search for optimal solutions. Thus, the mixture of SFOA and WWO aids to elevate overall efficiency and leads to obtain global optimal solution. The steps of SFWWO are described below.

Initialization: The search agent update in SFOA is performed using an immature sunflower growth model and is expressed as,

$M=\left[ \begin{matrix}{{s}_{1,1}} & {{s}_{1,2}} & \cdots  & {{s}_{1,\dim}}  \\{{s}_{2,1}} & {{s}_{2,2}} & \ldots  & {{s}_{2,\dim}}  \\ \vdots  & \vdots  & \vdots  & \vdots   \\{{s}_{H,1}} & {{s}_{H,2}} & \ldots  & {{s}_{H,\dim}}  \\\end{matrix} \right]$     (17)

where, H denotes the quantity of young sunflowers and $\dim$ the quantity of variables.

Identify error: To get the optimal answer, the error in each solution is found and is provided as:

$MSE=\frac{1}{k}\,\left[ \sum\limits_{p=1}^{k}{{{\beta }_{p}}-L} \right]$    (18)

where, k is the total amount of data, β_p is the estimated output, and L is the output generated by the ANN.

Immature sunflower growth simulation: According to SFOA, two modes, like sunny and cloudy or rainy, are employed to simulate the growth of immature sunflowers. The initial mode can be written as:

$I_{new,M}^{y+1}=\left\{ \begin{align}& I_{old,M}^{y}+o\times \operatorname{Sin}(\eta )\times \left[ Dzp\times I_{best,M}^{y}-I_{old,M}^{y} \right] \\& I_{old,M}^{y}+o\times \operatorname{Sin}(\eta )\times \left[ I_{best,M}^{y}-I_{old,M}^{y} \right] \\\end{align} \right.$    (19)

where, o denotes the damping parameter, $I_{old,M}^{y}$ discloses the old element in iterations, and $I_{best,M}^{y}$ is the best element in iterations. The stopping of the stem elongation of the sunflower is reflected by the damping attribute, which is expressed as:

$o=dam{{p}_{\max }}-y\times \frac{(dam{{p}_{\max }}-dam{{p}_{\min }})}{{{y}_{\max }}}$    (20)

Here, y is the current iteration, damp_max and damp_min represent the maximum and lowest values of the damping parameter, and y_max is the maximum number of iterations. Rainy or overcast days cause the young sunflowers to experience heliotropic effects at reduced rates. When a day is cloudy or wet, it means that Auxin has not advanced. The second option is used, where the sun attribute is set to 0, to replicate these scenarios. It is stated as:

$\begin{align}& I_{new,M}^{y+1}=I_{old,M}^{y}+o\times \operatorname{Sin}(\eta )\times \left[ I_{best,M}^{y}-I_{old,M}^{y} \right]; \\& at\,\,any\,\,hours\,day \\\end{align}$    (21)

$I_{new,M}^{y+1}=I_{old,M}^{y}\left[ 1-o\operatorname{Sin}(\eta ) \right]+o\times \operatorname{Sin}(\eta )\times I_{best,M}^{y}$    (22)

However, the SFOA has a slow convergence speed and a high memory consumption. To solve this, the WWO incorporates the SFOA to find the best solution. The revised equation derived from WWO is provided by:

${{A}_{c,\,s}}(a+1)={{A}_{c,\,s}}(a)+Gaussian\left( 0,\,1 \right).\,\delta .\,{{Z}_{s}}$    (23)

Assume, ${{A}_{c,s}}\left(a+1\right)=I_{new,M}^{y+1},and~{{A}_{c,s}}\left( a+1 \right)=I_{old,M}^{y}$ substitute in Eq. (23):

$I_{new,M}^{y+1}=I_{old,M}^{y}+Gaussian\left( 0,\,1 \right).\,\delta .\,{{Z}_{s}}$    (24)

$I_{old,M}^{y}=I_{new,M}^{y+1}-Gaussian\left( 0,\,1 \right).\,\delta .\,{{Z}_{s}}$     (25)

Substitute Eq. (25) in Eq. (22),

$\begin{align}& I_{new,M}^{y+1}=I_{new,M}^{y+1}-Gaussian\left( 0,\,1 \right).\,\delta .\,{{Z}_{s}}\left[ 1-o\operatorname{Sin}(\eta ) \right] \\& +o\times \operatorname{Sin}(\eta )\times I_{best,M}^{y} \\\end{align}$      (26)

$\begin{align}& I_{new,M}^{y+1}-I_{new,M}^{y+1}\left[ 1-o\operatorname{Sin}(\eta ) \right]= \\& -Gaussian\left( 0,\,1 \right).\,\delta .\,{{Z}_{s}}\left[ 1-o\operatorname{Sin}(\eta ) \right]+o\times \operatorname{Sin}(\eta )\times I_{best,M}^{y} \\\end{align}$     (27)

$\begin{align}& I_{new,M}^{y+1}\left[ 1-1+o\operatorname{Sin}(\eta ) \right]= \\& -Gaussian\left( 0,\,1 \right).\,\delta .\,{{Z}_{s}}\left[ 1-o\operatorname{Sin}(\eta ) \right]+o\times \operatorname{Sin}(\eta )\times I_{best,M}^{y} \\\end{align}$     (28)

$\begin{align}& I_{new,M}^{y+1}\left[ o\operatorname{Sin}(\eta ) \right]= \\& -Gaussian\left( 0,\,1 \right).\,\delta .\,{{Z}_{s}}\left[ 1-o\operatorname{Sin}(\eta ) \right]+o\times \operatorname{Sin}(\eta )\times I_{best,M}^{y} \\\end{align}$     (29)

ANN_SFWWO final update is provided by,

$\begin{align}& I_{new,M}^{y+1}=-Gaussian\left( 0,\,1 \right).\,\delta .\,{{Z}_{s}}\left[ 1-o\operatorname{Sin}(\eta ) \right] \\& +o\times \operatorname{Sin}(\eta )\times I_{best,M}^{y}\left[ \frac{1}{o\operatorname{Sin}(\eta )} \right] \\\end{align}$     (30)

Re-compute error: Eq. (18) is used to reevaluate the error and produce the best possible solution by adjusting the ANN optimal weights.

Termination: When an iteration reaches a higher iteration count, the process ends with termination. Algorithm 1 represents the ANN_SFWWO pseudocode.

Here, the best prediction result is achieved by adjusting the ANN weight and bias using the SFWWO.