In light of their potential environmental and economic benefits, renewable energy sources have garnered considerable attention for their projected influence on the future of power grids (Qian et al., 2020). These sources offer the potential to enhance voltage profiles and power supply quality for both energy suppliers and consumers (Li et al., 2023a), thereby contributing to reduced power losses, decreased pollution, and cost savings in specific scenarios. Within the array of energy sources, the assimilation of photovoltaic (PV) power generation into conventional power grids poses inherent stability challenges. This is mainly attributed to the intermittent characteristics of renewable energy sources and the variability introduced by meteorological factors (Hu et al., 2020). Even though meteorological and environmental elements impact photovoltaic (PV) output, resulting in noteworthy fluctuations, the extensive integration of PV brings forth considerable security challenges to the power grid system (Li et al., 2023b). In such contexts, precise PV power forecasting becomes indispensable not only for ensuring the reliable and stable utilization of renewable energy sources but also for minimizing the demand for additional balancing energy and reserve power to accommodate other forms of power generation.

The accuracy of photovoltaic (PV) power forecasting is influenced by a multitude of factors, adding intricacy to the prediction process. These factors encompass the forecasting horizon (Khuntia et al., 2016), the selection of inputs in the forecasting model (Ghofrani et al., 2015), and performance assessment (Choi et al., 2021). To enhance accuracy, it is crucial to undertake correlation analysis optimization (Li et al., 2015) and employ uncertainty estimation and approximations (Zhou et al., 2020). These processes map relationships between the input and output through historical parameter estimation in the development of PV power forecasting approaches. The accuracy of these models is intricately tied to both the quantity and quality of data, necessitating extensive datasets and meticulous data preprocessing (Rahman et al., 2023). By integrating deep learning, heuristic algorithms, and innovative frameworks (Hafeez et al., 2020a), the refined approaches not only improve accuracy but also tackle challenges related to stability and convergence rate, contributing to the dynamic evolution of smart grid technologies (Hafeez et al., 2020b). In recent years, deep neural networks (DNNs) have emerged as a viable option for PV power forecasting despite their limited nonlinear mapping capabilities. An advantage of using DNNs lies in their capacity to simplify high-dimensional and nonlinear problems in contrast to physical methods (Yildiz et al., 2021) that are well-suited for stable conditions. Hossain and Mahmood (2020) proposed a LSTM (long short-term memory) neural network with the synthetic irradiance forecast and algorithm, whichachieves over 30% accuracy improvement in PV power forecasting. Based on LSTM, with a focus on accuracy, Aslam et al. (2021) employed the Bayesian optimization algorithm in conjunction with LSTM for power forecasting, thereby demonstrating its efficiency. Shaqour et al. (2022) proposed a DNN combined with the bi-directional gated recurrent unit with fully connected layers (Bi-GRU-FCL), which achieves faster training time and fewer learnable parameters based on five aggregation levels based on a dataset in Japan. Lin et al. (2020) introduced an enhanced moth–flame optimization algorithm and applied it to optimize the support vector machine (SVM). The results demonstrated the strong performance of the model on both sunny and rainy days. Fekri et al. (2021) proposed an adaptive recurrent neural network (ARNN) prediction model for online learning. The model dynamically adjusts its hyperparameters in response to new data, allowing the prediction process to adapt to changing conditions. A comparative analysis of the integrating feature engineering (FE) and a modified fire-fly optimization (mFFO) algorithm with support vector regression (SVR) against benchmark frameworks highlights the superior performance in terms of accuracy, stability, and convergence rate (Hafeez et al., 2021). In summary, the above studies contribute to the evolving landscape of PV power and load forecasting, showcasing diverse approaches and improvements in accuracy through the integration of deep learning techniques and optimization algorithms.

Building upon the aforementioned studies, it has been demonstrated that employing appropriate optimization methods for DNNs can result in enhanced prediction accuracy, leading to improved forecasting outcomes. Addressing the inherent non-linear characteristics of data (Kleissl, 2013), research efforts have explored intelligent algorithms that do not rely on predefined mathematical models. Oudjana et al. (2013) and Konjić et al. (2015) conducted a comparative study involving an artificial neural network (ANN) model for PV output forecasting against three conventional mathematical approaches and regression models. The findings underscored the significantly superior forecast accuracy achieved by the ANN. Although each approach has its merits and limitations, genetic algorithm (GA)-based optimization emerges as the most popular and effective technique for optimizing weights and inputs in forecasting models (Ding et al., 2011), particularly when used in conjunction with an ANN (Deniz et al., 2016). Pedro and Coimbra (2012) reported enhancements in their ANN-based PV power forecasting model through GA optimization, resulting in improved forecast accuracy. In comparison to GA, GA-based optimization demonstrates easier convergence and requires fewer parameter adjustments (Viet et al., 2020). At present, particle swarm optimization (PSO) is widely used in function optimization, according to Ahmed et al. (2020), neural network training and parameter selection, fuzzy system control, and other applications as a substitute for GA. Nevertheless, mainstream optimization algorithms such as GA and PSO often display an increased vulnerability to getting ensnared in local extrema (Bamdad et al., 2017). The optimization direction is contingent upon parameter settings, thereby impacting the time and cost of optimization. Although the convolution operation can meet the requirements and yield a satisfactory performance, it introduces potential issues. The local influence of the convolution kernel poses a risk of introducing false information, and data transformation contributes to an escalation in the complexity of the prediction model.

Furthermore, traditional neural network prediction methods typically employ multiple hidden layers to construct forecasting models. Although these models efficiently learn and extract valuable features from input data, they are characterized by shallow architectures and limited capabilities in representing complex nonlinear functions. As a result, these models have a tendency to underfit complex prediction tasks, resulting in a decline in accuracy, particularly when dealing with large-scale data-driven problems. In contrast to DNNs, the broad learning system (BLS), as an alternative method, enhances the generalization performance by expanding the width of a single hidden layer rather than improving its approximation capabilities through the extension of deep architectures (Feng et al., 2022). Some researchers utilize BLS as a method to enhance the accuracy of forecasting when dealing with large-scale datasets (Chen and Liu, 2017; Cheng et al., 2022; Zhu et al., 2022). The expanded broad structure of BLS ensures its strong approximation capability of nonlinear mapping (Wang et al., 2020) to ensure the accuracy of prediction results while significantly reducing computational costs, and the accuracy can be maintained using proper optimization techniques (Gong et al., 2022).

It is crucial to note that the effectiveness of certain methodology comparisons may be compromised because various prediction models are included in the comparison sections, each employing different hyperparameter tuning approaches and distinct hyperparameter configurations. To uphold the effectiveness of the comparison results, the research concentrates on tuning hyperparameters within the same forecasting model while allowing for adjustability in the algorithms. In the present research, the BLS model is employed as a framework for short-term PV power forecasting. The self-organizing map (SOM) is introduced to evaluate the spatiotemporal correlations of five PV stations. The backtracking search optimization algorithm (BSOA) examined in this study demonstrated effectiveness in addressing engineering optimization problems (Civicioglu, 2013), given that the initialization step does not demand specific tuning. However, BSOA is not without drawbacks as it requires substantial information in the evolution process to regulate a correct optimization search direction. Moreover, ensuring the population diversity poses challenges, potentially resulting in convergence on the local optima. Consequently, the subsequent sections detail improvements aimed at addressing the limitations of BSOA. An improved backtracking search optimization algorithm (IBSOA) is employed to tune the hyperparameters of BLS. The improved backtracking search algorithm-based broad learning system (IBSOA-BLS) method is compared to other meta-heuristic algorithm variants, including genetic algorithm-based learning system (GA-BLS), backtracking search algorithm-based learning system (BSOA-BLS), and bird swarm algorithm-based learning system (BSA-BLS), to demonstrate its superior prediction accuracy.

Thus, the contribution of this paper includes the following:

• Employment of the BLS to predict PV power generation, which is a computationally efficient method that consistently delivers high forecasting accuracy compared to other neural network-based prediction approaches.

The remaining sections of the paper are organized as follows: Section 2 initiates with the definition of the broad learning system. Section 3 introduces the enhancements to the backtracking search optimization algorithm and evaluates the performance of IBSOA. Section 4 outlines the IBSOA-BLS prediction model, covering data preprocessing and clustering. Section 5 delves into the power forecasting results using the proposed method and conducts comparisons. Lastly, Section 6 concludes the paper.

Random vector link neural networks (RVFLNNs) serve as the foundation of BLS. BLS operates by extracting features from the input data through feature mapping, subsequently transforming these feature nodes into enhancement nodes using nonlinear transformations, which further employs an incremental learning method to update the output weights of these enhanced nodes. The approach differs from traditional RVFLNN, where input data are directly accepted and enhancement nodes are established. In deep learning networks, improved fitting capabilities result from the addition of extra network layers rather than simply increasing the number of nodes. Based on RVFLNN, BLS training facilitates swift updates and refinements to the prediction system, contrasting with deep neural networks, where learning time progresses monotonically. By extending the feature and enhancement nodes, BLS improves the network performance and model fitting capabilities. The system consists of three fundamental components: 1) mapping feature nodes, 2) enhancement nodes, and 3) an output matrix as shown in Figure 1.

BLS training comprises two primary phases: first, the weights for the mapping and enhancement nodes are randomly generated, and second, the weights between the hidden layer and the output layer are calculated. In simpler terms, during training, the sole weight that requires acquisition is the one connecting the final output layer to the hidden layer as the weights of the BLS random layer are generated randomly. In the initial stage, BLS forwards the original input data as mapping features, following which the structure can be expanded using the enhancement nodes. The random feature mapping stage distinguishes BLS from other established learning techniques, such as the single-layer feedforward network (SLFN), which updates its parameters through gradient descent, and the support vector machine (SVM), which utilizes kernel functions for feature mapping. The mapping feature nodes can be represented as follows: M n = ϕ X W f n + β f n , n = 1,2 , … , N f , (1) where X is the input data and W _f and β _f are the weighting matrix and bias of feature nodes, respectively. Via the sparse autoencoder (Yang et al., 2021), BLS uses the mapping function ϕ to extend the enhancement nodes. The m^th group of the nodes can be expressed as follows: H m = ζ M n W e m + β e m , m = 1,2 , … , N e , (2) where ζ is the nonlinear transformation function and W _e and β _e are the weight matrix and bias of enhancement nodes, respectively. Then, the output of BLS can be obtained from the following equation: W n m = M n ∣ H m † Y , (3) where W n m is the output weights, † represents the Moore–Penrose pseudoinverse of matrix, and Y is the output data.

Numerous researchers have enhanced the generation of random mapping feature nodes to align with diverse industrial requirements. Empirical experiments have shown that diverse activation functions enable the model to acquire a range of nonlinear expression capabilities. The commonly used transformation functions are listed in Table 1 and the Gaussian function is used for the enhancement node calculation in this research.

In this research, only sunny days are considered. The results are shown in the manner of four seasons. Each season begins at 5:35 in spring, 5:50 in summer, 6:30 in autumn, and 7:10 in winter. The daytime ends at 18:45 in spring, 19:50 in summer, 17:50 in autumn, and 18:40 in winter. The algorithms are executed 10 times, and the average is taken to ensure validity; the mean values are used for evaluation. The measured values and prediction of PV power in different seasons are shown in Figures 5 and 6. The computational efficiency of the algorithm-based BLS is shown in Table 5. The simulation results are implemented and completed via MATLAB R2023a on a PC with Intel Core i7-7700 CPU @ 3.4 GHz and 16 GB RAM. The configurations of the four algorithms are shown in Table 6. The hyperparameter optimization processes of BLS in spring are illustrated in Figure 7. The optimization performance of IBSOA yields better outcomes when searching for the best fitness within 100 iterations. This is attributed to the enhancement of BSOA, which incorporates historical data (mean and the best fitness in each iteration) during the selection and mutation processes to regulate the correct optimization search direction and maintain dynamic population diversity. The enhancement also improves the accuracy of the forecasting model by obtaining optimized hyperparameters for the given iteration through the enhanced search ability of BSOA. The forecasting results illustrate that the prediction deviations of each model are more significant during the period from 20 to 40 min compared to the deviations from 12:00 to 18:00 during sunny periods. The phenomenon can be attributed to the movement of clouds in the early morning, which impacts the electricity generation at the PV power station, leading to increased power fluctuations and, thus, making prediction more challenging. Notably, the proposed IBSOA-BLS exhibits superior predictive performance compared to the other three models, while GA, BSA, and IBSOA also demonstrate relatively better predictive capabilities. To provide an accurate assessment of the prediction performance of the four algorithms, the mean square error (MSE), the root-mean-square error (RMSE), the mean absolute error (MAE), and the mean absolute percentage error (MAPE) are employed for a thorough analysis of each model’s prediction effectiveness.  MSE  = 1 N ∑ i = 1 N x i − x i ′ 2 , (18)  RMSE  = 1 N ∑ i = 1 N x i − x i ′ 2 , (19)  MAE  = 1 N ∑ i = 1 N | x i − x i ′ | , (20)  MAPE  = 1 N ∑ i = 1 N x i − x i ′ x i × 100 % , (21) where N is the sample number, x is the measured value of power, and x′ is the prediction result. The results of the Diebold–Mariano (DM) test are presented in Appendix B, providing a comprehensive evaluation and indicating potential enhancements to the prediction model.

Table 7 indicates that the predictions made by these models exhibit slight deviations from the actual values. The RMSE indicators of the proposed prediction method are consistently lower than the worst-case scenario in each season, decreasing by 3.2283 kW in spring, 3.9159 kW in summer, 1.3425 kW in autumn, and 1.4058 kW in winter. Similarly, the MAPE exhibits a reduction compared to the worst case, with decreases of 0.882% in spring, 1.2399% in summer, 1.803% in autumn, and 1.087% in winter. The results demonstrate that, in general, the predictive performance of the proposed IBSOA-BLS model surpasses that of the other three models. The MAPE for the IBSOA-BLS model ranges from approximately 0.769%–1.639% across four seasons, which is lower than the results obtained by the other three models (GA-BLS, BSA-BLS, and BSOA-BLS). The MAPE in spring reaches 1.639% due to power fluctuation caused by cloud motion. Despite the differences arising from the regular and relatively low volatility in PV power generation on sunny days, compared to the other three models, IBSOA-BLS exhibits superior robustness, leading to prediction outcomes that closely align with the actual values. The quantified results of prediction errors are associated with four seasons. It reveals that on sunny days with stable power fluctuations, the IBSOA-BLS-based prediction method outperforms the other three models. All models exhibit small RMSE and MAPE values for prediction errors, indicating relatively accurate predictions. Among the four season simulations, the proposed IBSOA-BLS method exhibits lower error quantization values and superior predictive performance compared to the other three methods. It also demonstrates good adaptability to environmental conditions, especially on sunny days. Additionally, it is observed from the experimental error index values that once the input variables and structural parameters of the prediction model are determined, RMSE and MAPE values for GA-, BSA-, BSOA-, and IBSOA-based BLS methods remain consistent, indicating more accurate predictive performance for IBSOA-BLS. In contrast, when other methods are employed under the same conditions for multiple predictions, the results of each prediction exhibit varying amplitudes and lack stability. This is because the proposed method achieves an improved accuracy by considering both the mean and the best hyperparameters during the optimization process of IBSOA.

This paper proposes a PV power forecasting model based on BLS and IBSOA. Compared to the three other traditional algorithms (GA, BSA, and BSOA), which do not consider the historical experience and the representation of the characteristics of extensive data, IBSOA provides sufficient information during the evolution process to regulate the correct optimization search direction and maintain dynamic population diversity, thereby mitigating the risk of converging on the local optima. These improvements are achieved through enhancements to the selection and mutation processes of BSOA. BLS, which reduces extra network layers, effectively addresses the challenges posed by deep architecture in traditional neural networks when dealing with large-scale data forecasting. The SOM is utilized not only to cluster the five PV plants based on their respective external conditions but also to capture the spatio-temporal dependence of PV power generation under varying conditions. The effectiveness of the forecasting model is validated on the actual data on PV units from the DKA solar center in Australia. Based on well-tuned hyperparameters of BLS by IBSOA, the results indicate that the proposed PV power forecasting model yields more reliable and accurate predictions when compared to those produced by GA-BLS, BSA-BLS, and BSOA-BLS. The future work will concentrate on adding rainy days in each season for training and testing the model on more complex case studies.