The unique feature is the information process from the past to the future (forward flow) and from the future to the past (backward flow) (Graves and Schmidhuber, 2005). It consists of two LSTM layers, one for the forward information flow and the other for the backward information flow. BiLSTM can extract more complex relationships from the input sequence by considering historical and longitudinal data. By combining the two LSTM layers, the final output is obtained. The prediction may be required to be decided jointly by the previous and subsequent inputs. Improvement in prediction performance uses forward and backward transitions. Thus, making the network learn better and more feasible for real-time applications. Building the deep architecture model involves stacking more than one BiLSTM. Figure 1 shows the simplified BiLSTM model.

BiLSTM simplified mathematical model representation in terms of matrix form. G F G I G C G O = f s i g f s i g f tan f s i g V R F V R I V R C V R O R n + W S F W S I W S C W S O S n − 1 + B F B I B C B O (1)

Mathematical model of BiLSTM and gate updating Forget Gate , G F = f s i g V R F R n + W S F S n − 1 + B F (2) Input Gate , G I = f s i g V R I R n + W S I S n − 1 + B I (3) Cell agent  G C = f tan V R C R n + W S C S n − 1 + B C (4) Output Gate , G O = f s i g V R O R n + W S O S n − 1 + B O (5) Hidden State  Q n = G O ∘ f tanh G n (6) Cell State  Memory cell output   G n = G F ∘ G n − 1 + G I ∘ G C (7)

The BiLSTM output vector P n = V S → P Q → n + V S ← P Q ⃖ n + B P (8) Q → n = f s i g V R S → R n + V S → S → Q → n − 1 + B Q → (9) Q ⃖ n = f s i g V R S ← R n + V S ← S ← Q ← n − 1 + B Q ← (10)

Where, f s i g - Sigmoidal activation function for gates, f tanh - Tangent activation function, ∘ - Hadamard Product (element-wise product), V -weighted connection of the gates, respectively forget gate, input gate, cell gate and output gate, W -weighted connection between the output state to the input state cell, S n − 1 -time stamp n-1 past hidden state output, Q n - time stamp n hidden state, R n - current time stamp input, B - bias of the respective gates, G n - Cell state, Q → n -forward flow information, Q ⃖ n -backward flow information, P n - Output Vector.

Selecting an activation function f s i g and f tanh is essential for managing the non-linearity of the output. The forward LSTM Q → n has access to historical data, but the backward LSTM Q ⃖ n represents the model’s memory at time step n. The model may produce predictions based on both temporal contexts by combining these hidden states. The BiLSTM may determine dependencies from both directions in the sequence due to this bidirectional structure. The degree to which the input affects the current hidden state is determined by this V matrix. The impact of the hidden state from the previous time step on the present one is determined by this W matrix. The output layer’s bias term B enables the final prediction to change by the patterns the model has learned. It considers the context of the future. The model may generate predictions based on both temporal contexts by combining these hidden states. Equations 1–10 state the mathematical model of BiLSTM.

The Bayesian optimisation algorithm solves difficulties by finding the optimal parameters that minimise the objective function in a finite area, with lower and upper bounds on each variable, as given by Equation 11 (Pelikan et al., 1999). r * = argmin r ∈ R f r (11)

Where, f r -a score that should be minimised, R-domain of the hyperparameter values, and r -the combination of hyperparameters that yields the lowermost value of the score f r .

Two fundamental elements that constitute the Bayesian Optimisation Algorithm (BOA):.

• Probabilistic (surrogate) framework: BAYES’ THEOREM serves as the basis for BOA, which approximates the objective function via a surrogate framework in each iteration efficiently sampled. A Gaussian process serves as the best stand-in framework for choosing a desirable group of hyperparameters to assess the actual objective function. The objective function is determined using a surrogate framework and utilised to direct upcoming sampling.

This section is devoted to describing the proposed framework. Deep BiLSTM hyperparameter tweaking is significant in achieving the best model performance. The disadvantages of automatic optimisation strategies like grid and random search are that they take a long time for more giant parameter sets and do not always provide the best finding, respectively. In contrast, the Bayesian optimisation algorithm (BOA) is a well-informed method that evaluates simply the most promising models using a surrogate framework. With fewer sampling points and a quicker computing time, the Bayes theorem is applied to construct the posterior distribution of the objective function.

To enhance the long-term prediction of GHI, the best hyperparameters for deep BiLSTM, such as its number of hidden layers, hidden nodes, learning rate, regularisation, and other significant hyperparameters, were found using the Bayesian optimisation method (BOA). The Bayesian optimisation algorithm was used to optimise Deep BiLSTM concerning the optimal and promising hyperparameter and the performance evaluated on the two datasets for the GHI prediction in the long-term horizon. Figure 2 illustrates the proposed BOA-D-BiLSTM prediction model workflow and model built according to the set values and parameters presented in Table 2. The author uses 10 times the number of hyperparameters optimised as the initial samples and then processed with complete samples. The convergence is determined by the minimisation of the MSE.

Environmental variables such as temperature, wind speed, pressure, relative humidity, cloud cover, precipitation of water content, wind direction, dew point, and sun irradiance (GHI and DNI (Direct Normal Irradiance (DNI)) are closely related to solar energy fluctuations. Table 3 shows the statistics of the dataset. This research experiment used data from India’s two regions, namely, Chennai and Jammu. We used two data sets of Chennai and Jammu locations in India that contain 10 years of datasets consisting of GHI and other environmental variables for our investigation and simulation evaluation. The daily resolution for each meteorological parameter (Dataset 1 and Dataset 2) is included in the data, encompassing 10 years from 01 January 2013 to 31 December 2022 hourly collected data, which is obtained from the NOAA (National Oceanic and Atmospheric Administration).

In this investigation, the collected real-time data was processed in the range (0, 1) using the Min-Max normalisation approach, as represented by the Equation 12. R n = R 0 − R min R max − R min (12)

Where, R n are the values of R after normalisation, R 0 is the current value for variable R, R min and R min are the minimum and maximum data points in the variable R of the input dataset.

The proposed models possess the ten input nodes: GHI, DNI, temperature, wind speed, pressure, relative humidity, cloud cover, precipitation of water content, wind direction, and dew point. The predicted GHI is the output node. The collected two region datasets (1 and 2) consist of 87,000 data points of all considered inputs, which are separated into training and testing sets based on the ratio 70:30, respectively.

This paper used RMSE (Root Mean Squared Error), R² (R-squared), MSE (Mean Squared Error) and MAE (Mean Absolute Error) as evaluation error metrics to evaluate the proposed Bayesian Optimisation algorithm-based Optimised Deep Bidirectional Long Short-Term Memory (BOA-D-BiLSTM) and other baseline prediction model performance. Equations 13–16 state the evaluation metric formulations. RMSE = 1 N ∑ n = 1 N R n − P n 2 (13) MAE = 1 N ∑ n = 1 N R n − P n (14) MSE = 1 N ∑ n = 1 N R n − P n 2 (15) R 2 = 1 − ∑ n = 1 N R n − P n 2 ∑ n = 1 N R n − R ¯ 2 (16)

Let, N - the total number of the samples, R n - the real target GHI, R ¯ - the mean target GHI, and P n - the predicted GHI.

Grid and random search are the traditional methods used to identify the optimal parameter sets and comparative analysis. In the context of the two data sets from the India region, this research constitutes a deep bidirectional long short term memory network that is resilient and optimised for long-term global horizontal radiation (GHI) prediction. The results achieved are tabulated in Table 4. From experimentation observation, it was noticed that grid searches are notoriously time-consuming. The trial-and-error method results in uncertain efficiency, and the grid search is exorbitant in computational complexity. In order to optimally use the network’s capability, hyperparameters must be appropriately configured. Grid Search and Random Search (RS) are common approaches for hyperparameter optimisation. Still, they are computationally costly, may take a long time to analyse, may result in inappropriate hyperparameters, and produce an extensive variance during computation. The advantage of Bayesian optimisation algorithms is that they prevent being trapped in the optimal solution.

The Bayesian optimisation algorithm optimises the Deep BiLSTM hyperparameters to overcome the traditional optimisation limitations. The findings suggest that the models’ GHI prediction capabilities are considerably enhanced by adjusting the hyperparameters using random search, grid search, and Bayesian optimisation algorithms. The experimental results show that compared to the grid search, random search-based identified hyperparameters-based Deep BiLSTM results are better but do not surpass the Bayesian optimisation algorithm-based identified hyperparameters-based BiLSTM prediction model for long-term solar GHI prediction application. Random search-based BiLSTM achieves reduced RMSE, MAE, MSE, and R2 compared to grid search-based BiLSTM, but it is not supreme compared to Bayesian optimisation algorithm-based BiLSTM. As the number of hyperparameters rises, grid search algorithms become less effective, and computation time complexity becomes problematic. The approach employs the Bayesian optimisation algorithm to discover the optimal combination in an acceptable time. Moreover, the prediction performance of BiLSTM is greatly enhanced by tweaking hyperparameters during training and testing. With this, it was concluded that the optimised hyperparameter based on the proposed Bayesian optimisation algorithm incurred BiLSTM outperforms in terms of better performance and faster convergence. The Bayesian optimisation algorithm outperforms grid search and random search because it can intelligently navigate complex, high-dimensional parameter spaces with few iterations. Therefore, the authors considered the Bayesian optimisation algorithm accurate and simple compared to the traditional hyperparameter fine-tuning method, which was clearly inferred from Figure 3. The identified significant hyperparameters based on the Bayesian optimisation algorithm based identified significant hyperparameters are tabulated in Table 5. Finding the best set of parameters in an acceptable amount of time may be accomplished through hyperparameter tweaking using Bayesian optimisation. Additionally, it benefits in reducing model overfitting problems.

In existing global horizontal prediction applications research, many prediction models were suggested using baseline models such as the persistence model, ARIMA (Autoregressive integrated moving average), RNN (recurrent neural network), BPN (backpropagation neural network), SVR (Support Vector Regression), Boosted Tree, LSTM (Long Short Term Memory) and BiLSTM (Bidirectional Long Short Term Memory). All these models predict values based on the present and past information, and the selection of hyperparameters is not optimal. Still, in real-time applications, consideration of future information along with past and present and optimal hyperparameter-associated neural networks leads to improved prediction performance. To address this issue, the paper uses the deep Bi-LSTM with the Bayesian optimisation algorithm-based optimised hyperparameters to predict the GHI in the long-term horizon time scale. The results of the proposed BOA-D-BiLSTM investigations are examined in the configuration as per Table 5; listed hyperparameters and other parameters of the baseline model are considered as per the literature but evaluated on the India region. The results attained are tabulated in Table 6.

The suggested BOA-D-BiLSTM model performed better than the baseline models when assessed using the MAE, RMSE, MSE and R². In this research, using the Bayesian optimisation approach, the author determines the best Deep BiLSTM hyperparameter values that lead to greater model precision, as noted in Figures 4–9. The combined use of deep BiLSTM and Bayesian optimisation algorithm in the proposed model yields better-reduced error metrics than the benchmark models, as demonstrated by our comparative analysis and experiment results. The persistence model prediction performance does not produce effective results with high evaluation error metrics compared to other prediction models. ARIMA and BPN models compete for dataset 1, where ARIMA performs better than BPN, and for dataset 2, where BPN performs better than ARIMA. RNN performs better than SVR, BPN, ARIMA and persistence.

LSTM has a reduced amount of evaluation metrics than Boosted Tree, but BiLSTM performs better than LSTM, Boosted Tree, SVR, BPN, ARIMA and persistence regarding reduced error. For linear and stationary time series, ARIMA works well; however, it cannot effectively represent the complex, non-linear patterns encountered in GHI data. Although non-linear relationships can be captured by simple LSTMs, such models frequently rely on human expertise or assistance for hyperparameter adjustment, which is challenging and not an optimal option for large-scale applications. To overcome the drawbacks of simpler models like ARIMA and basic LSTMs, the BOA-D-BiLSTM model incorporates the Bayesian Optimisation Algorithm (BOA). The BOA-D-BiLSTM model’s incorporation of BOA ensures optimal configurations with low computing complexity by optimising hyperparameter tweaking. Effectively handling varied and high-dimensional datasets increases scalability in addition to improving forecast accuracy. Regarding predicting, the Deep BiLSTM structure outperforms simpler models by capturing temporal dependencies in both forward and backward directions. The validity and generalisation ability are proved on the two different regions’ data sets. The proposed model performed well and had improved accuracy for both datasets. BOA-D-BiLSTM for the India region dataset 1 achieves MAE: 0.0016, RMSE: 0.0023, MSE: 6.6852 × 10⁻⁰⁶ and R²: 0.9994 meanwhile for dataset 2-based evaluations, achieve MAE: 0.0018, RMSE: 0.0030, MSE: 8.8628 × 10⁻⁰⁶ and R²: 0.9988. As a result, Table 6 demonstrates the feasibility of using a Bayesian optimisation approach to tune the Deep BiLSTM framework by identifying the perfect combination of hyperparameters that greatly enhances the proposed BOA-D-BiLSTM efficacy that leads the predicted values to exactly match with the actual target of GHI for the two data sets, as noted in Figures 4A–C and Figures 7A–C the zoomed-in view visualises the effectiveness of the proposed model in accurate prediction of GHI in the long-term horizon. Hence, prediction errors are near zero for both datasets 1 and 2, and it is clearly visualised with the help of the prediction error plot and the zoomed-in section view shown in Figures 5A, B and Figures 8A, B. The linear regression plot shows the linear relationship with R = 1 for both datasets, as observed in Figures 6A, B and Figures 9A, B, respectively. Compared to the frequently used time series models, namely, the Persistence Model, ARIMA, BPN, RNN, SVR, Boosted Tree, LSTM, and BiLSTM, the proposed hybrid model BOA-D-BiLSTM can affirm the significant elements of the input information and also integrate forward and backward transitions to improve solar irradiance (GHI) prediction and produce the minor error than the eight baseline models, which is illustrated in Figure 10 for better visualisation of the efficacy of the proposed model.