Stress analysis of elastic bi-materials by using the localized method of fundamental solutions

1.   Introduction

The tendency toward clean power production is in an ongoing surge due to its beneficial impact on the environment. Thus, countries are embracing renewable energy by establishing supportive regulations to foster their development ^[1]. Solar energy is an essential renewable energy source, and its use has increased rapidly in recent years. However, one of the drawbacks of using solar electricity is its intermittent nature, which makes accurate forecasting of solar power output challenging ^[2,3]. Hence, reliable solar forecasting is essential for efficient solar energy integration into power networks. Solar forecasting improves power grid management, energy trading decisions, and power system planning and operation ^[4]. Accurate solar forecasting also encourages optimal solar energy consumption, which is critical for the global transition to a sustainable energy system. As a result, much research has been carried out in order to produce dependable and accurate solar forecasting models. Hence, this study focuses on forecasting Global Horizontal Irradiance (GHI).

GHI values are greatly influenced by the weather parameters, such as humidity, pressure, air temperature, wind speed, and cloud cover. The primary determinants of these variables are the site's geographic location and climate. In addition, four primary categories are taken into account while determining the GHI forecasting horizon ^[5]: Ultra-short-term forecasting (1 second to < 1 hour), short-term forecasting (1−24 hours), medium-term forecasting (1 week−1 month), and long-term forecasting (1 month−1 year). The goals and applications of GHI forecasting may vary depending on the stakeholders involved and the time horizon of interest. The ultra-short-term prediction has gained constant attention in energy-based real-time applications. For instance, the goal of a 5min GHI forecast might be to enable real-time control of power generation and consumption in a microgrid or a building ^[6]. A 15min GHI forecast, on the other hand, might be helpful in energy trading and market participation ^[7]. Power generators and retailers could use the forecast to optimize their bidding strategies in a day-ahead or intraday electricity market. A 30min solar forecast could be helpful in scheduling energy resources and optimizing energy management in a building, a microgrid, or a community. In contrast, a 60min GHI forecast could be helpful for long-term energy planning and grid integration ^[8]. Hence, this research paper builds GHI models considering different forecasting horizons.

1.1.   Related work

In addition, considering the GHI forecasting methods, statistical techniques, machine learning algorithms, and physical models are just a few examples of forecasting algorithms that can be utilized ^[8]. The statistical techniques are divided into (i) machine learning (ML) algorithms, such as support vector regression (SVR) and artificial neural networks (ANN), and (ii) time series models, such as autoregressive, moving average, exponential smoothing, and autoregressive moving average (ARMA), are frequently used in the energy sector. Physical models use mathematical equations to model the physical processes influencing GHI output ^[9]. An example of a physical model is the numerical weather prediction (NWP). Each of these algorithms has advantages and disadvantages, and the choice of method is determined by aspects such as data availability, forecasting horizon, and the desired level of accuracy. K. Omer ^[10], for instance, examines the performance of the particle swarm optimization (PSO) algorithm, ANNs, and bagged tree (BT) methods in forecasting seasonal solar irradiance. Data from 2007 to 2020, encompassing variables like air temperature, precipitation, snow mass, air density, and cloud cover fraction, are used to predict solar irradiance. The findings indicate that the BT method exhibited the most favorable statistical accuracy. Specifically, the BT model showcased superior performance, revealing a coefficient of determination (R²) of 0.992, root mean square error (RMSE) of 0.00339, and mean absolute error (MAE) of 0.0199. Solano et al. ^[11] explored the use of ML models, namely SVR, extreme gradient boosting (XGBT), categorical boosting (CatBoost), and voting-average (VOA), for solar radiation forecasting in Brazil using input parameters such as dry bulb temperature, relative humidity, wind speed, atmospheric pressure, and time of the day. Results revealed that VOA outperformed other models in terms of accuracy with RMSE in the winter and summer of 0.2417 and 0.2877, respectively. Lee et al. ^[12] also presented ensemble learning-based solar irradiance forecasting models using weather data. They used boosted trees, BT, random forest (RF), and generalized RF, and compared their performance in short-term prediction of solar irradiance with Gaussian process regression and SVR. Results indicated that ensemble approaches led to reliable forecasting outcomes for all the considered locations.

The paper in ^[13] presents a method for predicting hourly GHI using extraterrestrial radiation alongside limited weather forecast data. The study compared the performance of various prediction models—BP network, SVM, and the light gradient boosting machine (LightGBM). The LightGBM model demonstrated superior performance with the lowest RMSE in the testing set of 126.1 W/m². Moreover, the study explored the influence of weather types on the prediction outcomes. The analysis revealed that weather patterns were not the primary influencers on the LightGBM model's prediction outcomes. Interestingly, the model's accuracy remained unchanged even after excluding weather predictors, where the RMSE was found to be 135.2 W/m². Furthermore, the difficulties of projecting the power generation of distributed, small-scale solar PV systems at various horizons and resolutions were examined in ^[14]. The authors presented and assessed many forecasting methodologies, such as particle swarm optimization (PSO)-based prediction combinations and base forecasters. The assessment procedure compared how well the forecasting techniques work when trained on varying data sets and tested in different environments and periods. The findings demonstrated that forecast combinations, especially at high resolutions and short horizons, can enhance the performance of forecasting models for solar PV power output. The forecasting models are assessed using the median absolute scaled error (MASE). The results demonstrated that the proposed PSO-based forecast combination approach performed better than the base forecasters and other benchmark models at all resolutions and horizons, with a 3.81% reduction in MASE.

However, these meteorological variables may not be available due to the high cost of weather monitoring devices. This presents a significant barrier to generating an accurate forecasting model, particularly for regions with limited financial resources. Therefore, one of the significant research gaps in GHI forecasting is the possibility of relying solely on lag observations of historical GHI data without integrating any weather or environmental variables. This method is also known as persistent forecasting or naive forecasting. This approach assumes that the GHI's future reading will be identical to its recent historical output without accounting for any external influences that may affect the output. In comparison to more complex algorithms that combine weather and environmental data, the use of lag observations alone for GHI forecasting has gotten very little attention in the literature despite its simplicity and ease of implementation. This approach, however, may offer potential advantages in terms of computational speed and ease of implementation, particularly for short-term solar forecasting applications when the influence of external factors may be negligible. As a result, this research aims to evaluate the possibility of forecasting GHI future observations using only lag observations.

Furthermore, machine learning models continue to confront difficulties when processing large amounts of input data, frequently facing complications such as vanishing or expanding gradients ^[8,15]. The rapid expansion of artificial intelligence approaches has resulted in a continued emphasis on deep learning (DL), which is known for its excellent performance in tasks such as image recognition ^[16,17] and machine translation ^[18,19]. To solve these inherent issues, deep learning has been implemented into solar prediction. Deep learning models surpass conventional machine learning models in terms of accuracy due to their greater feature learning capacity and ability to handle large datasets. S. Tajjour et al. ^[20] conducted a study focused on short-term solar irradiation forecasting utilizing DL models. Employing eleven years of NASA satellite data, they evaluated the effectiveness of three specific deep learning models: multilayer perceptron (MLP), LSTM, and gated recurrent unit (GRU). The results indicated that all three models exhibited comparable accuracy levels, with a mean square error (MSE) near to 0.017 kWh/m²/day. Despite containing more layers, the GRU model demonstrated higher training speed compared to LSTM. The MLP model emerged as the most efficient, attributed to its fewer parameters (49,281) when contrasted with GRU (1,025,793). In addition, M. Elizabeth et al. ^[21] presented a novel multistep CNN-stacked LSTM model designed for short-term solar irradiance prediction. Through comparisons with CNN and LSTM models, their proposed approach demonstratesd superior performance among contemporary DL models. Moreover, they benchmarked the proposed method against traditional ML techniques like linear regression (LR), SVR, and ANN using the same dataset. In forecasting solar irradiance, their framework yielded the lowest RMSE and R² values, achieving 0.36 and 0.98 W/m², respectively.

Moreover, the study in ^[22] presented a study on predicting solar radiation using a hybrid CNN-categorical boosting (CNN-CatBoost) model. They used extra-atmospheric solar radiation and three weather variables (temperature, humidity, and total cloud volume) to predict solar radiation. The study compared the performance of boosting models (XGBoost and CatBoost) and recurrent neural network (RNN) models (LSTM and GRU). The results indicated that the hybrid CNN-CatBoost model provided accurate predictions of solar radiation by a reduction in MAE values from 0.1104 to 0.1027. V. Sansine et al. ^[23] utilized also a hybrid deep learning model that combined CNN and LSTM algorithms (CNN-LSTM) for predicting solar irradiance. Additionally, the study compared the performance of the hybrid model with other stand-alone models, including ANN, CNN, and LSTM. The results showed that the CNN-LSTM hybrid model outperformed other models, with the best statistical error results for probabilistic forecasting. For test data, the CNN-LSTM model achieved an RMSE of 91.73 W/m² and MAE of 60.46 W/m², with an R² of 87. The authors in ^[24] also provided a short-term PV forecasting model using the variational autoencoder (VAE) model. They used data from two different locations (a parking lot in the US with a size of 243 kW and a PV system in Algeria with total capacity of 9 MW). For comparison purposes, they compared VAE with seven DL methods, namely the recurrent neural network (RNN), LSTM, bidirectional LSTM, the convolutional LSTM network, gated recurrent units, stacked autoencoder, and the restricted Boltzmann machine, and two well-known ML methods, namely LR and SVR. The findings showed that DL techniques outperformed other ML techniques, while VAE consistently beat the other techniques.

For time series forecasting, particularly GHI forecasting, CNN has grown in popularity. Unfortunately, few research efforts have currently concentrated on the improved CNN in GHI forecasting. One of the primary disadvantages of using CNN is their vulnerability to certain parameters. The CNN architecture is made up of a set of memory cells that can learn and store information over long periods of time, making it ideal for capturing temporal dependencies in sequential data. The accuracy and dependability of GHI forecasts can be considerably impacted by the best choice of CNN architecture, which entails selecting the number of convolution layers, filters, and the learning rate. The number of convolution layers assists the model in learning complicated connections in the data but also raises the possibility of overfitting ^[25]. Similarly, optimizing learning rates is critical for effective model convergence since it regulates the step size in weight updates during training ^[26]. The combination of these parameters is crucial; insufficient convolution layers or incorrectly set learning rates can impair the model's capacity to comprehend the temporal complexities associated with GHI data, resulting in suboptimal predictions. Therefore, since finding the optimal design of CNN is fundamental for achieving accurate and reliable GHI forecasts and there is little attention in the literature on this aspect, the current research discovers the best CNN model architectures that yield the best forecasting results.

1.2.   Motivation and contributions of the study

Based on the discussion above, the primary objective of this research work is to offer a CNN-based framework aimed at estimating the GHI. The framework consisted of several steps: Data collection and preprocessing, data partitioning, CNN model architecture, model training, model testing, and model deployment. This framework creates a model that accurately and reliably predict a GHI output. Therefore, the following states are the main differences of this study compared to other published works in the literature:

● Optimal selection of CNN architecture: This study considers the best CNN architecture for GHI forecasting using solely historical GHI data. This is significant because the choice of CNN's architecture significantly impacts performance. To identify a suitable design, the CNN is tested under various combinations of layers, filters per layer, and learning rates.

● Use of past data of GHI only: In this study, we only used past data of GHI as input to the CNN model for forecasting. This differs from many other works that use weather data and GHI data for forecasting. This approach is functional when weather data is not available or is unreliable.

● Comparison with other forecasting algorithms: In this study, the effectiveness of the proposed CNN model is compared to that of several well-known forecasting methods, including the RNN, ANN, RF, and SVR. This comparison sheds light on how different algorithms compare in terms of forecasting GHI.

● Forecasting horizon: In this study, we concentrated on forecasting GHI over various time horizons, including 5, 15, and 30min. This is important because the accuracy of the forecasting algorithms may vary depending on the forecasting horizon.

The structure of this study is as follows: Section 2 provides a comprehensive discussion of the problem statement, framework, CNN algorithm, and data preparation techniques utilized. Section 3 focuses on the sensitivity analysis employed. Sections 4 and 5 present the key findings and provide a thorough discussion of the study results. In Section 6, a potential real-world application of the proposed CNN-based forecasting model is explored, while Section 7 highlights the study's conclusions.

2.   Materials and methods

This section includes the problem statement, a thorough explanation of the research framework, and an overview of the CNN algorithm.

2.1.   Problem statement

The growing significance of solar energy as a renewable energy source has increased, necessitating accurate projections of GHI for effective energy management. Precise estimation of the GHI can help utilities and grid operators balance the supply and demand of energy, optimize energy storage, and reduce costs associated with energy imbalance. However, forecasting GHI becomes challenging due to the lack of meteorological data either by their unavailability or reliability. This leaves an open opportunity for further research into the idea of relying purely on lag observations of past GHI data without incorporating any weather or environmental variables. In addition, traditional forecasting models, such as statistical models, have limitations in capturing the non-linear relationships between the input variables and the GHI observation. The CNN algorithm has recently shown promise in forecasting GHI. However, there is still a need for research to investigate the effectiveness of CNN-based models in GHI forecasting and to compare their performance with other forecasting models. Additionally, research is required to determine how various data sources, model architectures, and hyperparameters affect the precision and dependability of GHI forecasts. By filling in these knowledge gaps, forecasting of GHI may be made more accurate and reliable, and more effective energy management tactics can be supported.

2.2.   Study framework

The methodology for forecasting GHI using CNN with different forecasting horizons using only lag observations of CNN is shown in Figure 1 and described below:

● Step 1: Data Collection: The first step is to collect the historical data of GHI. The data should be collected at a high temporal resolution, such as every 5min. The data should cover a sufficiently long period to include seasonal patterns.

● Step 2: Data Preprocessing: The collected data should be preprocessed before feeding it to the CNN and other forecasting algorithms. The preprocessing steps include data cleaning and normalization and splitting the data into training, validation, and testing sets. In this study, we only use the lag observations of GHI, meaning that the model only uses past GHI values as inputs.

● Step 3: Forecasting Horizon Analysis: In this study, we evaluate the performance of the CNN model with different forecasting horizons. We generate forecasts for 5, 15, and 30min ahead. The performance metrics are calculated for each forecasting horizon, and the results are compared to identify the best forecasting horizon.

● Step 4: CNN Model Design: The CNN model is intended to capture temporal dependencies in GHI data. The model is made up of numerous CNN layers that are followed by a fully connected layer. The number of CNN layers, neurons in each layer, and the activation functions are all hyperparameters that should be tuned.

● Step 5: Model Training: The designed CNN model is trained on the training data set. During training, the model's weights are modified using an optimization technique such as Adam. When the validation loss stops improving, the training process ends.

● Step 6: Model Evaluation: The trained model is evaluated on the testing data set, to compare the CNN model's performance with other forecasting algorithms. The evaluation metrics used include the coefficient of determination (R²), root mean square (RMSE), normalized root mean square (nRMSE), mean absolute error (MAE), normalized mean absolute error (nMAE), and mean absolute percentage error (MAPE).

● Step 7: Implementation: The CNN model is implemented using a programming language. In this study, we used the MATLAB environment to build the CNN model.

2.3.   Convolution Neural Network (CNN)

The CNN stands as a fundamental DL algorithm that has significantly advanced the field of computer vision and image processing ^[27]. CNNs are specifically designed to process and analyze visual data, which renders them ideal for applications such as image recognition, object detection, and image classification ^[28]. One of the advantages of CNNs lies in their capacity to automatically learn hierarchical representations of features from raw data ^[28]. This is achieved through the use of specialized layers, including convolutional layers, pooling layers, and fully connected layers (see Figure 2). The convolutional layers play a fundamental role in feature extraction by applying adaptable filters or kernels to the input data ^[29]. These filters are convolved with the input to detect patterns, edges, and textures, enabling the network to capture meaningful visual information. In contrast, pooling layers execute downsampling operations on the feature maps derived from convolutional layers, reducing spatial dimensions while retaining crucial features ^[30]. Popular pooling techniques like max pooling and average pooling assist in minimizing computational complexity and prevent overfitting. Finally, the fully connected layers process the extracted features to perform classification or regression tasks, allowing the network to learn complex relationships in the data ^[31].

The initial step involves feeding the input data into the input layer to initiate the process of feature transformation. Subsequently, the convolutional and pooling layers work in extracting relevant features from the input data. These extracted details are then amalgamated through the fully connected layers. Finally, the output layer communicates the result of the feature extraction process. The goal of each convolutional layer is specifically geared toward extracting spatial patterns from the input variables correlated with the target variable, GHI. This process is illustrated as follows ^[22]:

where $f$ is the specified activation function, ${W}^{k}$ represents the kernel weight, and $\times$ refers to the convolution process operator.

2.4.   Data cleaning and normalization

Data cleaning is an essential step in developing a successful forecasting model. Solar datasets should be cleaned and filtered before being fed into the forecasting models. In GHI forecasting, the night hours are removed from the database, and only the ones that occur between sunrise and sunset are saved. A solar elevation-based pre-processing operation is carried out to accomplish this because data near sunset and dawn are frequently incorrect. Hence, solar radiation data is excluded for solar elevations less than 10 ^[32]. Furthermore, normalizing input data is necessary before examining forecasting models' performance. The objective here is to mitigate the likelihood that characteristics with substantial numerical values outweigh those with comparatively lower numerical values. Equation (2) is used to normalize the input data between 0 and 1.

where ${x}_{i}$ is the measured GHI value; ${x}_{i}^{n}$ is the normalized GHI, while ${x}_{max}$ and ${x}_{min}$ are the highest and lowest values corresponding to the measured GHI that exists in the input dataset, respectively.

2.5.   Model evaluation metrics

The precision and effectiveness of the forecasting techniques are assessed using the following statistical indicators: R², RMSE, nRMSE, MAE, nMAE, and MAPE. These metrics reflect the degree to which the measured values agree with the GHI values generated by the forecasting models. The formulas in Eqs (3)−(8) define these metrics [33−35].

In the above equations, $n$ represents the volume of the testing datasets; ${y}_{i}$ denotes the measured value of the GHI; ${y}_{i, max}$ corresponds to the highest value within the testing dataset, while ${f}_{i}$represents the forecasted value produced by the forecasting models. The mean of the measured GHI values of ${y}_{i}$ is represented by $\tilde{y}$. In regression problems, a model's R² indicates how well it fits a set of observations ^[36]. The MAE, known as the mean absolute value of the residuals (forecasting errors), measures the average magnitude of errors ^[37]. On the other hand, the RMSE quantifies the divergence between actual GHI readings and forecasted values by considering their squared differences, while MAPE is frequently used to determine the forecasting model's performance accuracy using a percentage form ^[38].

2.6.   Study site and dataset source

Solcast is a corporation that offers solar irradiance data worldwide ^[39]. Researchers can obtain valuable data from it, and the public can freely access these data. The public can access many atmospheric parameters via their website (https://solcast.com/). It is possible to acquire various meteorological variables over a number of time intervals (5, 30, and 60 minutes), including GHI, diffuse horizontal irradiance (DIF), direct normal irradiance (DNI), air temperature, solar zenith angle, solar azimuth angle, cloud capacity (a percentage ranging from 0% to 100% completely cloudy), pressure, wind speed, and wind direction. The solar data are collected at Riyadh, Saudi Arabia, with the location with the following coordination: latitude: 24.90689 $°N$ and longitude: 46.39721$°E$ (see Figure 3). The GHI data are gathered in 5min intervals for the period between Jan 1^st, 2022, and Dec 31^st, 2022. The maximum GHI reading from the system was found to be on May 15^th, 2022, at 10:45 A.M. with a value of 1076 W/m², while the average of the GHI readings in 2022 was found to be 506.75 W/m².

3.   Sensitivity analysis

In this section, a sensitivity analysis is conducted to examine the influence of the different lengths of the dataset, the resolution of data, and the seasonal variation of solar radiation on the future forecasting output of the GHI readings.

3.1.   Analysis of different lengths of datasets

Most previous studies used at least one year of data for hour-ahead solar radiation. This amount of data is helpful in training the forecasting model, yet it requires a long time to generate the ultimate GHI forecasting model. This could hinder its applicability in real-word applications. Hence, this study investigates different lengths of datasets, including 1 day, 1 week, 1 month, 2 months, and 3 months, for the goal of generating high-accuracy models in a shorter time. A thorough grasp of the temporal dynamics and patterns present in solar irradiance data is made possible by investigating several temporal spans. For instance, shorter datasets—such as those covering one day or one week—offer information on short-term patterns and instantaneous fluctuations, which are essential for comprehending the quick changes in GHI brought on by variations in the weather. Longer datasets, on the other hand, covering 1, 2, or 3 months, reflect seasonal patterns, long-term climate impacts, and possible cyclic patterns that affect solar irradiance. Hence, more resilient and flexible forecasting models are made possible by the model's ability to learn from and adapt to a variety of temporal variables through the analysis of these different dataset lengths. In this study, different combinations of historical observations of GHI were selected as the input feature, as follows:

- 5-min: Previous 5min of GHI readings

- 15-min: Previous 15min of GHI readings at 5-minute intervals

- 30-min: Previous 30min of GHI readings at 5-minute intervals

- 45-min: Previous 45min of GHI readings at 5-minute intervals

- 60-min: Previous 60min of GHI readings at 5-minute intervals

In terms of training dataset volume, each of the above combinations of historical observations of GHI were trained using historical data of 1 day, 1 week, 1 month, 2 months, and 3 months. A comparison study was conducted in this research work to determine the optimum training dataset and feature set.

3.2.   Impact of higher-forecasting horizons

Most of the previous studies that focus on short-term forecasts of GHI are in 1-hour intervals ^[41]. The available data from Solcast are in 5 minutes, enabling the exploration of shorter resolutions of data on the accuracy of predicting hour-ahead GHI forecasting. Therefore, this study investigates the accuracy of forecasting GHI values at the 5min, 15min, and 30min horizons. To accomplish this, the lag observations of GHI mentioned in Subsection 3.1 are used to create the multistep forecasting models. In 15min and 30min forecasting horizons, the 1, 2, and 3 months are used only as they led to the best forecasting models in case of 5min (see Section 4).

With all forecasting horizons, the training and testing datasets are divided using the sliding window approach. In the sliding window technique, for example, the lag of 30min at 5min intervals (window size) are employed as an input and the future 15min at 5min intervals (forecast horizon) are used as an output variable (see Figure 4). Through using the sliding window technique, the CNN is enabled to use supervised learning. In addition, different ML algorithms are compared with CNN using the same set of input features (see Algorithm 1).

3.3.   Impact of seasonal change

In the literature, most of the studies divide the yearly data into 80% for training and 20% for testing to develop the forecasting model. This testing data is unnecessary to reflect all the seasonal variations during the year, and the generated model could not be generalized. Hence, the impact of seasonal change must be investigated to examine the performance of a forecasting algorithm.

Analyzing seasonal variations in GHI values across a range of meteorological scenarios is crucial for determining the reliability of a forecasting model. These various weather scenarios illustrate the changing pattern of solar irradiance throughout the year and depict a range of meteorological circumstances that are common across seasons. It is essential to comprehend how the model reacts to and predicts GHI in various weather conditions and seasons in order to verify the model's generalizability and dependability. This study, therefore, explored the performance of the CNN algorithm with different seasonal changes in GHI observations across varied weather conditions, including rainy, cloudy, partially cloudy, partially sunny, and sunny days. In this study, therefore, a total of 25 independent models were generated for each type of day. Each day was examined with 5 different volumes of dataset (1 day, 1 week, 1 month, 2 months, and 3 months) in which there were 5 different combinations of historical observations of GHI.

3.3.1.   Classification of day type

Many studies have used weather or day-type categorization to forecast GHI, aiming to organize vast datasets characterized by significant fluctuations ^[42,43]. Most of these studies divided the type of day according to the general metrological conditions. In this paper, nevertheless, the seasonal variation was captured by classifying days into five groups based on the incident solar radiation (W/m²). Equation (9) determines the type of day using the ratio (R_day) that compares the daily measured GHI to the daily clear sky GHI data, which are collected from CAMS ^[44]. After obtaining the value of R_day, the type of day was classified based on the R_day range shown in Table 1 ^[45].

4.   Results and discussion

This section compares the CNN forecasting models based on a number of error metrics to assess how well they performed in estimating the GHI output. The results of the CNN models with different time ahead horizons (5min, 15min, and 30min) forecasting are listed in Tables 3 and 7, respectively. Figures 5, 11, and 12 display graphical representations of the five selected days, with each forecasting model based on 5min ahead and multi-step forecasting within 15min and 30min forecasting horizons, respectively.

4.1.   Case 1: Hour-ahead forecasting based on 5min

This section discusses the results of the hour-ahead forecasting of the GHI based on 5min. This section covers the following topics: Choosing the optimal feature set, optimizing hyperparameters, comparing the proposed CNN with other widely used forecasting algorithms, predicting outcomes, and examining the execution time of the proposed CNN model.

4.1.1.   Input features selection

Variations in the number of lag observations could significantly affect the accuracy of the GHI forecasting in the future. Furthermore, the amount of trained data may result in accurate prediction and faster generation of the subsequent GHI reading, which is essential for real-time applications. Therefore, for every type of day, 25 independent models were created. Every day was analyzed using five distinct dataset volumes—1 day, 1 week, 1 month, 2 months, and 3 months—each containing five possible combinations of GHI's historical observations—lag 5min, 15min, 30min, 45min, and 60min, each at 5min intervals. The evaluation herein is accomplished with the initial hyperparameters displayed in Table 2.

Table 3 lists the statistical error results of each type of day with different combinations and volumes of historical datasets. It can be observed from Table 3 that 2 months of data with 5min lag observation has the best forecasting performance with each type of day. This indicates that seasonal trends and the long-term climatic effects of solar irradiance can be reflected in the 2 months of trained data. Furthermore, the preceding 5min data provides insights into short-term trends and immediate variations in sun irradiation. According to the statistical error measurement shown in Table 3, the average value of the R² of all the days is 0.9999, while the RMSE and MAE are found to be 2.714 W/m² and 2.249 W/m², respectively. In addition, 1 week and 1 month of data with 5min of previous GHI measurements could lead to satisfactory forecasting results, where R², RMSE, and MAE are 0.999, 2.997 W/m², and 2.372 W/m², respectively, for 1 week and 0.999, 4.903 W/m², and 4.617 W/m², respectively, for 1 month. On the other hand, 1 day of data performs poorly regardless of the type of day and the amount of trained data.

Comparing the lag observation of data, the previous 5min of GHI readings led to the best forecasting results for all days and volume of data—1 day, 1 week, 1 month, 2 months, and 3 months. The 15min and 30min at 5min intervals came in second and third place in generating models with high-accuracy outcomes, respectively. Hence, the 2 months of the trained dataset with the previous 5min of the GHI reading (2M-5min) is selected as the best feature set to predict the future 5min output of the GHI. In addition, and for further visualization, Figure 5 depicts the performance of different models when the measured GHI values are plotted against the predicted value of the GHI model.

4.1.2.   Hyperparameter tuning

Hyperparameter selection is an important step when using deep learning algorithms for prediction, such as CNN. This stage helps to improve overall precision and shorten the algorithm's execution time. A comprehensive evaluation of the previously mentioned validation criteria is combined with a heuristic technique to discover the optimal set of hyperparameters for the GHI forecasting using the CNN algorithm. The best-predicting results were obtained using the 2 months of the trained dataset with the previous 5min of the GHI reading, as mentioned in Subsection 4.1.1. Therefore, this set of features was selected to carry out the hyperparameter tuning for 5min forecasting horizon at the study site.

There are no set techniques when it comes to hyperparameter tuning. Nonetheless, the following order for fine-tuning the hyperparameters was chosen for the GHI forecasting based on the literature analysis and best practices: Number of convolution layers (ConvLayer), number of filters at each ConvLayer, and learning rate. For hyperparameter tuning, the falling leaf approach was used as it offers a more flexible and dynamic way to explore the hyperparameter space. In this approach, for instance, the process is continued with various combinations of several filters after determining the ideal number of ConvLayers. For example, it was found that the two ConvLayers with 32 filters at each layer produced the best forecasting outcomes out of the (1, 2, 3) ConvLayers. The two ConvLayers were fixed in the following phase, and various learning rates were examined.

Figures 6 and 7 show the performance comparison to obtain the optimal number of ConvLayers, filters, and learning rates, respectively. Figure 6 depicts the statistical error results of 1, 2, and 3 ConvLayers with the number of filters as 32, 64,100, and 128. It can be seen that a setup with 2 ConvLayers with 32 filters (2-ConvLayer (32)) had the best forecasting outcomes. Hence, 2 ConvLayers with 32 filters were selected to continue in the hyperparameter tuning process.

The performance comparison between two ConvLayers, each with 32 filters, is shown in Figure 7 in order to determine the ideal learning rate value. Compared to 0.1, 0.01, 0.001, and 0.0001, it can be seen that the learning rate of 0.001 produced better forecasting outcomes. As a result, Table 4 lists the ultimate, best CNN configurations chosen for 5min ahead of GHI forecasting of the study site.

4.1.3.   Comparing the proposed CNN with other forecasting algorithms

The forecasting performance of the developed forecasting model was evaluated against four popular forecasting algorithms, namely RNN, ANN, RF, and SVR. Table 5 contains the results of developed CNN, RNN, ANN, RF, and SVR. To ensure a fair comparison, the best input features (2M-5min) identified in the Subsection 4.1.1 were used as input to RNN, ANN, RF, and SVR. According to Table 5, the proposed forecasting models with optimal input features and configurations outperformed the other forecasting models in predicting the future values of GHI with low RMSE, MAE, and MAPE values for all the day types. Regarding models fitting accuracy with the CNN, the proposed model had the best prediction outcomes, where the average value RMSE for the five days was found to be 2.262 W/m², MAE was found to be 1.794 W/m², and MAPE was found to be 2.17%. The RNN algorithm showed promising performance with an average RMSE value of 3.062 W/m², MAE of 2.192 W/m², and MAPE of 2.169%. ANN, RF, and SVR came in third, fourth, and fifth, respectively.

In addition, Figure 8 illustrates the efficacy of the proposed CNN model in comparison to RNN, ANN, RF, and SVR for the five specified days. Figure 8(a) depicts that when input features and CNN hyperparameters were appropriately selected, the proposed CNN model exceled in accurately tracing the actual values of the GHI output, outperforming other models. Furthermore, the boxplots shown in Figure 8(b) were designed to offer a more comprehensive assessment of the forecasting models' predictive performance. A box and whisker plot (BWP) shows the distribution of the mean absolute error (MAE) when all of the predicted days are combined. While analyzing the BWP, an outlier is a data point that deviates quantitatively from the rest of the data (shown by the red cross). Consistent with earlier deductions, the proposed CNN models consistently outperformed the RNN, ANN, RF, and SVR. This superior performance is also highlighted in the scatter plots presented in Figure 9. This figure shows the measured versus predicted GHI output values acquired by the proposed CNN model compared to the RNN and SVR models for the five simulation days.

4.1.4.   One week of forecasting

To further examine the performance of the proposed forecasting model, a randomly selected week (December 5–11, 2022) is forecasted using the optimal set of input features and CNN configurations. The forecasting accuracy results are shown in Table 6 and Figure 10. According to Table 6, the 5min ahead forecast led to an RMSE value of 2.2785 W/m², while the MAE and MAPE were found to be 1.59 W/m² and 2.913%, respectively. It can be inferred that the accuracy of forecasting models steadily declined, starting with its best forecasting result 5min ahead and ending at a 30min estimate. While the time horizon lengthens, the accuracy of various models gradually declined, and the uncertainty in observations of GHI forecasting grew.

4.1.5.   Analysis of execution time

The system used for the simulations consists of an Intel Core i7-7700@ 4.20GHz CPU, an NVIDIA GeForce GTX 1080 GPU, and 16 GB of RAM. The computer simulation environment is MATLAB, which permits the usage of GPUs that support the CUDA Toolkit. Utilizing a GPU greatly accelerated computing; therefore, to generate the forecasting model in fast execution time, especially with a large amount of training dataset, it is recommended to use high-performance GPUs. For 5min ahead of forecasting the GHI, the average running time of the optima prediction model was 57 seconds for the five selected days. Nevertheless, the model training consumed over 95% of the entire run duration. In a real-time scenario, loading a model that has already been trained can reduce the simulation time for the purpose of a 5min GHI forecast.

5.   Cases 2 and 3: Hour-ahead forecasting based on 15min and 30min

The outcomes of the hour-ahead GHI forecasting based on 15min and 30min horizons are covered in this section. The following topics are covered in this section: Selecting the best feature set, fine-tuning hyperparameters, evaluating the proposed CNN with other popular forecasting algorithms, predicting results, and investigating the recommended CNN model's execution time.

5.1.   Input features selection

Similar to what was conducted with a 5min prediction horizon, every day was analyzed to cover the seasonal variations and examine the performance of the CNN algorithm. Regarding the data volume, however, 1 month, 2 months, and 3 months were used as input features with 15min and 30min multistep forecasts. Each trained data contained five possible combinations of GHI's historical observations—lag 5min, 15min, 30min, 45min, and 60min, each at 5min intervals. Hence, in this case, and for each specific day, a total of 15 models were created. The analysis conducted for 15min and 30min multistep forecasts used the first set of hyperparameters shown in Table 2.

Table 7 presents the 15min multistep statistical error results for each type of day based on different combinations of historical dataset quantities and the lag readings of GHI readings. In comparison to other developed models, the results show that 2 months of data with a 15min lag in observation (2M-15min) performed the best for the 15min ahead forecast scenario. The average values of R², RMSE, MAE, and MAPE for all days are 0.9708, 35.776 W/m², 20.685 W/m², and 12.437%, respectively. On the other hand, the 3 months of training data with the previous 45min of input GHI values (3M-45min) outperformed other models for the 30min multistep forecasts of GHI. The error values of R², RMSE, MAE, and MAPE generated with this model were found to be 0.9276, 56.319 W/m², 36.891 W/m², and 19.711%. In addition, Table 7 indicates that regardless of the trained data, the input feature of lag 5min of GHI had the worst accuracy in predicting the 15min and 30min multistep of GHI forecasts. Furthermore, and for additional visualization, Figures 11 and 12 show how various models performed when the measured GHI values were plotted against the GHI model's predicted value for the 15min and 30min multistep, respectively.

5.2.   Hyperparameter tuning

The hyperparameter tuning process conducted with the 5min ahead forecast was also employed with the 15min and 30min multistep forecast of the GHI values. Figures 13, 14, and 7 show the performance comparison to obtain the optimal number of ConvLayers, filters, and learning rates for the 15min and 30min horizon forecasts, respectively. Figures 13 and 14 depict the statistical error results of 1, 2, and 3 ConvLayers with the number of filters as 32, 64,100, and 128 of the 15min and 30min horizons, respectively.

For the 15min and 30min forecasting horizons, it can be seen that the design with 3 ConvLayers, each with 100 filters (3-ConvLayer (100)), had the best forecasting outcomes. Hence, 3 ConvLayers with 100 filters were selected to continue in the hyperparameter tuning process for the cases of 15min and 30min. On the other hand, and for learning rate tuning, Figure 7 reveals that 0.001 and 0.0001 are optimal values for 15min and 30min multistep, respectively. Consequently, Table 4 presents the optimal CNN configurations selected for 15min and 30min of GHI forecasting of the study site.

5.3.   Comparing the proposed CNN with other forecasting algorithms

Here, a comparison is accomplished between the proposed CNN model and other forecasting algorithms. Since RNN and ANN performed the best compared to RF and SVR in the case of 5min ahead forecast, the RNN and ANN are selected to be compared with the optimal CNN model in the case of 15min and 30min.

According to Tables 8 and 9, the proposed CNN forecasting models with optimal input features and configurations outperformed RNN and ANN models in predicting the future values of GHI with noticeably low statistical error results for all the day types. For the 15min case, Table 8 shows that the average RMSE values of proposed CNN, RNN, and ANN were 30.569 W/m², 35.759 W/m², and 43.058 W/m², respectively. On the other hand, and for the 30min case, Table 9 indicates that the average R² values of proposed CNN, RNN, and ANN were generated to be 0.933, 0.919, and 0.914.

5.4.   One week of forecasting

The 15min and 30min multistep forecasts were also conducted for the randomly selected week (December 5–11, 2022). This step is to further examine the performance of the developed CNN models. Table 6 lists the error values of the generated results based on one week, while Figure 10 shows the performance of the forecasting results when they are plotted against the observed GHI readings. For the 15min case, the RMSE and MAE values were 29.248 W/m² and 15.761 W/m², respectively, while RMSE and MAE values were found to be 60.040 W/m² and 33.486 W/m² for the case of 30min, respectively. As a comparison between the 5min, 15min, and 30min ahead forecasts, Table 6 reveals that as the forecasting horizon increased, more error was expected in the forecasting results. For example, the R² was found to be 0.999 for the 5min forecast horizon, while it was found to be 0.983 and 0.927 for 15min and 30min forecasting horizons, respectively. Hence, it is expected to obtain high error values if the forecasting horizons increase to be, for example, 60min at 5min intervals.

5.5.   Analysis of execution time

The system setup used for the simulations in the case of a 5min ahead forecast was used in the cases of 15min and 30min ahead forecasts of the GHI values. In the case of the 15min forecast, the optimal prediction model took an average of 42 seconds to run for the five days that were chosen, while it took an average of 51 in the case of the 30min forecast. However, most of the running time was consumed in the training phase.

6.   Hardware implementation and potential challenges

Our forecasting model using CNN with optimized architecture can be implemented into hardware to be used for energy management applications. This can be achieved by deploying the model into a microcontroller or a single-board computer, such as Raspberry Pi, and integrating it into an energy management system. The system can receive real-time data from a solar radiation device and use our proposed model to forecast the future output of the photovoltaic system. The forecasted output can then be used to optimize the energy management system, such as scheduling energy consumption and storage, or selling the excess energy to the grid. By implementing our model into hardware, we can provide a reliable and accurate forecasting tool for energy management, which can lead to cost savings and more efficient use of renewable energy sources.

There are a number of research constraints that must be considered while analyzing the study's possible limitations while creating a CNN prediction model for GHI. CNN initially requested a significant volume of GHI data. It can be challenging to gather trustworthy GHI data with shorter time periods because of measurement irregularities or sensor problems. Second, in this research work the CNN was trained using certain amounts of data (1 day, 1 week, 1−3 months) owing to the fact that CNNs learn features from raw data. However, determining the pertinent features or data volume can be challenging because it involves considerable thought and technical experience. Finally, when dealing with large datasets or intricate structures, training CNN models for GHI forecasting can be computationally demanding. For models to be trained effectively, sufficient computational resources—including strong GPUs—are required, which can be costly for certain users. Therefore, resolving these issues is essential to guaranteeing the model's dependability and practicality in real-world applications.

7.   Conclusions and future work

Forecasting solar irradiance has gotten a lot of interest owing to the growing demand for renewable energy. However, the expensive cost of climate observatories makes gathering meteorological data difficult, impeding the development of precision forecasting models. Therefore, this research intended to overcome this barrier by developing a framework to forecast GHI values even in the absence or inaccuracy of meteorological data. A forecasting model based on the convolution neural network (CNN) algorithm was developed using merely lag measurements of GHI as input, with no external variables. The CNN forecasting outputs with different network designs was investigated through a heuristic configurations paradigm. Furthermore, the performance of the developed model was then compared to that of other popular forecasting algorithms over predicting horizons of 5, 15, and 30min. By analyzing the outcomes derived from the most effective forecasting model and evaluating the performance of estimation algorithms, the conclusion can be summarized as follows:

- Based on the criteria for model accuracy, a duration of two months' worth of data proves sufficient for constructing high-accuracy forecasting models for 5min and 15min horizons. However, to achieve similarly good forecasting results for a 30min horizon, three months of data is recommended.

- Regarding the model fitting accuracy, the developed CNN forecasting models outperformed other forecasting models (RNN, ANN, RF, and SVR) in forecasting GHI output. The average RMSE prediction results under different forecasting horizons of 5min, 15min and 30min considering different types of days (rainy, cloudy, partially cloudy, partially sunny, and sunny) with CNN model were 2.262, 30.569, and 54.244 W/m², respectively.

- Forecasting model accuracy rapidly decreased, beginning with a high predicting result 5min ahead and ending with a 30min prediction. As the time horizon increased, the accuracy of various models steadily fell, and the uncertainty in GHI forecasting observations grew.

Finally, the framework developed in this study holds the potential for predicting GHI output in other countries, offering a valuable tool for enhancing energy management strategies. However, there exists opportunities for further exploration to enhance the accuracy of GHI prediction models. A hybrid deep learning model, such as CNN-LSTM and RNN-LSTM, can be investigated to acquired more spatial features in the GHI data. Furthermore, extending the duration of available data, such as spanning over 2 weeks or 6 months, warrants deeper examination, particularly in non-real-time applications. Another potential avenue involves leveraging metaheuristic optimization algorithms like particle swarm optimization and genetic algorithms. These could optimize CNN architectures, revealing the model's sensitivity to variations in CNN designs.

Use of AI tools declaration

The authors declare that they have not used artificial intelligence (AI) tools in the creation of this article.

Acknowledgments

The researchers would like to thank the Deanship of Scientific Research at Qassim University for funding the publication of this project.

Conflict of interest

The authors declare that they have no competing interests.