Nonlinear Ensemble Deep Learning Model for Energy Consumption Prediction with Bayesian Optimization

Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 155 https://internationalpubls.com Nonlinear Ensemble Deep Learning Model for Energy Consumption Prediction with Bayesian Optimization Ejigu Tefera1, Kula Kekeba1, Ravindra Babu.B2, and M. Mart´ınez-Ballesteros3 1Big Data and HPC Center of Excellence, Department of Software Engineering, Addis Ababa Science & Technology University, Addis Ababa P.O. Box 16417, Ethiopia, [email protected], [email protected] 2Distributed Systems Research Group (SIG), Adama Science and Technology University, P.O. Box 1888, Adama, Ethiopia, [email protected] 3Department of Computer Science, University of Seville, ES-41012 Seville, Spain, [email protected] Article History: Received: 10-07-2024 Revised: 23-08-2024 Accepted: 06-09-2024 Abstract: Accurate prediction of electric energy consumption is crucial for efficient load dispatching, energy utilization, and grid operation. Traditional statistical and classical machine learning methods struggle with the nonlinear nature of energy consumption data, often leading to higher prediction errors. Additionally, deep learning models using a single approach face challenges such as convergence to local minima and poor generalization. This paper proposes a nonlinear ensemble deep learning model for residential energy consumption prediction, incorporating Bayesian optimization for hyperparameter tuning. The model combines Long Short-Term Memory (LSTM), Bidirectional LSTM (BiLSTM), and 1D Convolutional Neural Networks (1D-CNN), leveraging their powerful nonlinear feature learning capabilities. A k-means clustering approach is used to preprocess and reduce variability in the data, enhancing the ensemble model's performance. The ensemble model was tested on real energy consumption data from two districts in Addis Ababa, showing significant improvements in prediction accuracy with lower MAE, RMSE, and MAPE values compared to single models and unclustered data. The integration of clustering and Bayesian optimization further enhanced model generalizability and minimized overfitting, demonstrating the effectiveness of a nonlinear approach in capturing complex energy consumption patterns. Keywords: Bayesian Optimization, Deep Learning, Ensemble Learning, Hyperparameter tuning, k-means Clustering. 1. Introduction Nowadays, people are highly dependent on the supply of sufficient and stable eclectic energy to live comfortably [31]. Consequently, electricity consumption demand has been rising due to the growth of urbanization along with the rapid growth of the human population throughout the world [32]. More importantly, energy demand in Africa keeps growing annually at an average rate of 4%, the highest in the world [13]. Similar to Africa, energy consumption demand steadily growing in Ethiopia. From the Ethiopian context, the residential consumption demand accounts for 39% which is the largest followed by the industrial (34%) and commercial (27%) sectors. Specifically, household energy demand is expected to exceed population growth because economic improvements have driven households to have appliances and become owners of energy dependent technological devices that use energy continuously. However, energy supply and distribution are characterized by frequent power interruption, inefficient utilization, and substantial waste [6]. Apart from the high demand frequent power outages problems, and the high level of energy demand especially in Ethiopia, once it is Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 156 https://internationalpubls.com generated, storing and preserving the produced electricity energy sufficiently using the current energy storage technology [1] is difficult. In other words, energy waste will occur if the electricity is adequately distributed and consumed as soon as it is produced following the consumption demand in each district. To this end, given the large contribution of the residential sector to total energy demand, it is feasible to study consumption trends and develop accurate models using state-of-the-art data-driven algorithms for effective planning and demand-supply management. Moreover, for reliable and efficient grid systems, effective load dispatching, and efficient energy utilization, accurate electricity consumption forecasting has become indispensable for energy companies. In this regard, machine learning and deep learning models have been widely used for electric load forecasting, power system monitoring, and anomalous energy usage detection [21]. However, the existing machine learning methods are incapable of capturing nonlinear energy consumption data and cannot yield accurate prediction results [5, 15]. Moreover, deep learning methods with a single model have been plagued by a poor capacity for generalization and a tendency to converge to local minima [11, 29]. In existing methods, little attention is given to the fine graining of the input data, which accounts for model complexity and larger prediction errors [35]. In general, despite several studies have been conducted for electric load forecasting based on deep learning and ensemble methods, enhancement is required to get optimal prediction performance by ensembling multiple deep learning algorithms with clustering and fine graining of the input data to learn nonlinear and complex energy data effectively [29, 25]. In this regard, the electricity consumption prediction method is imperative to ensure efficient load dispatching, scheduling, and efficient energy utilization [19]. This paper aims to investigate the effectiveness of an ensemble deep learning model for energy consumption prediction with fin-graining of input data including identifying optimal clusters of residential energy consumption profiles. The contributions of this paper can be summarized as follows: 1. K-means clustering was applied for energy consumption profile characterization to acquire a more thorough understanding of how power consumption patterns of users behave. Moreover, optimal clusters were identified that will lead the subsequent ensemble model to learn the detail features and intrinsic behaviors of energy consumption data. 2. Optimal hyperparameter combination is searched using a Bayesian optimization algorithm to get an improved prediction model. 3. The robust ensemble model has been developed based on optimal cluster-generated energy consumption data. 4. The ensemble deep learning model’s effectiveness in predicting the monthly aggregate residential energy consumption is evaluated and verified against the base models using MAE, RMSE, and MAPE. 2 Related Works Accurate electric energy consumption forecasting at both long-term and short-term horizons is necessary to establish a more stable supply-and-demand equilibrium [30]. To this end, several studies have been conducted on energy consumption forecasting problems. Wen et al. [35] proposed a deep- Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 157 https://internationalpubls.com learning model to forecast the load demand for residential buildings with a one-hour resolution. Hyperopt hyperparameter tuning was employed to find the optimal hyperparameter combination. Moreover, accurate forecasting of electricity consumption is a very challenging task due to the high volatility of energy consumption. A. Salam and A. El Hibaoui in [24] introduced an improved intelligent energy prediction model based on deep feedforward neural networks and Long Short-Term Memory. M. Cai et al. [2] proposed deep neural network models, namely recurrent neural networks (RNN) and convolutional neural networks (CNNs). The proposed model is compared with the Seasonal ARIMAX model’s accuracy, computational efficiency, generalizability, and robustness. Among all the investigated deep learning techniques, the gated 24h CNN model achieved the best performance, improving the forecasting accuracy by 22.6% compared to the seasonal ARIMAX. N. Somu, et al. in [27] proposed a hybrid model for building energy consumption forecasting using long short-term memory networks. In this work, a novel Haar wavelet-based mutation operator was introduced to improve the divergence nature of the sine cosine optimization algorithm while dealing with hyperparameter tuning using the sine cosine optimization algorithm. On the other hand, a hybrid of wavelet transform and machine learning model is proposed in [26] to estimate electrical load consumption using the historical time-series information of energy usage. To investigate the effectiveness of combining different deep learning algorithms for estimating residential household energy consumption, Authors in [14] employed a hybrid ensemble model consisting of CNN, multilayer LSTM, and BiLSTM algorithms, by which the CNN framework can extract spatial and nonlinear patterns of the energy data and multilayer LSTM used to learn temporal dependencies. An ensemble method [36] is developed to forecast the residential short-term energy consumption. Vector auto-regression, Gaussian process regression, and the long short-term memory neural network model were trained as base learners. Another ensemble method is proposed in [23] by combining the deep LSTM and Auto-regressive Integrated Moving Average (ARIMA) models. In this work, the ARIMA was used to capture the stationary pattern of load data, and the nonlinearity of the complex energy consumption data was tackled using LSTM architecture. The performance of the proposed model surpasses existing short-term load forecasting models with less computation complexity. Moreover, ensemble learning methods provide a powerful tool for improving accuracy and stability in power load forecasting by leveraging the strengths of multiple predictor techniques [30, 11, 29]. Hadjout et al. [10] introduced an ensemble model for monthly industrial energy consumption forecasting. The proposed model combines LSTM, GRU, and TCN based on weighted averages. Similarly, W. Khan et al. [16] developed an effective ensemble model but at this time the authors employed a stacking-based ensemble approach using simple neural networks (ANN) and LSTM as base learners for solar energy forecasting. XGBoost algorithm was used as a meta-learner to combine the base models and the proposed model exhibited better consistency and stability in different cases. In general, most of the related works [16, 23, 24] have employed grid search for optimization tasks to improve deep learning and ensemble model performance. However, this optimization approach is highly criticized for high computation time requirements and is ineffective when the number of hyperparameter spaces and the type of hyperparameters have been increased. In addition, the presence of outliers in a dataset degrades model prediction performance and reduces model generalization abilities. This problem has been observed in the above-mentioned related works that little attention is Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 158 https://internationalpubls.com given to the fine-graining of the input data, which accounts for model complexity and larger prediction errors [35]. 3 Methodology The proposed method comprises three main phases: (1) Data preprocessing and clustering for energy consumption pattern identification; (2) Deep learning model hyperparameter tuning using a Bayesian optimization algorithm and (3) Individual base model training and model fusion to develop ensemble model and performance evaluation. In general, figure 11 shows the details of the proposed model. 3.1 Description of Data and Data Preprocessing The data for this study was collected from the Ethiopian electric utility. The collected data is about two districts of Addis Ababa city, South and West Addis Ababa (hereafter South AA and West AA) district’s monthly residential energy consumption data ranging from May 2019 to January 2021. Since each month’s consumption data was obtained from a different monthly bill report in separate Excel files for each month, it is necessary to combine the individual monthly file into a single, sequentially arranged dataset for each district as a Figure 1 shows. Figure 1: Data Aggregation There were many missing values in each month’s data because some customers may not have paid their consumption charge within the specified billing period. Consequently, users who have zero electric consumption for at least one month out of the 20 months or users who are absent for at least one month out of 20 months have been removed using filtering techniques because they are not good representatives of the samples. After filtering out the missing values, Table 1 shows the size of the input observation. Therefore, given the number of customers, C in each month for each selected district, and the number of months, M, the input observations or dataset D for each case study data is: Dataset,D = C * M (1) where C is the number of customers in each month and M is the number of months considered in each district. Furthermore, Table 1 summarizes the descriptive statistics of each district dataset. Table 1: Descriptive statistics of the load consumption dataset Dataset Count Mean Max. Min. Std. Skewness Kurtosis South AA 309303 304.00 919 0.100 182.74 0.849 0.29 West AA 313491 279.72 9120 0.020 215.87 11.24 7.93 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 159 https://internationalpubls.com Moreover, Figure 2 shows the average monthly electricity consumption over 20 months for the South AA district on the left and West AA district on the right. Figure 2: Aggregate Monthly Energy Consumption of west and West AA The data indicates that there is significant variation in electricity consumption habits from month to month in both districts. Furthermore, the pattern of electricity usage in each district is non-linear and irregular making difficult accurate predictions of this data using classical machine learning models and traditional statistical techniques [7]. In other words, this kind of data requires an effective preprocessing method such as a k-means clustering algorithm to discover the optimal clusters comprising more stable and similar consumption profiles. Moreover, integrating advanced data preprocessing techniques such as k-means clustering will enable the subsequent ensemble deep learning model [5, 17, 22] to learn the nonlinear complex association between energy consumption features and make accurate predictions. 3.2 Experiment Setup This section tried to discuss experimentation phases of our study which include data processing and ensemble deep learning model development based on the cluster-generated data. In this phase, the Keras framework on top of TensorFlow was selected to utilize a deep learning framework, hyperparameters, and a Bayesian optimization algorithm based on the BayesSearchCV interface. 3.2.1 Energy Consumption Clustering and Analysis In this study, K-means++ clustering was employed to discover the optimal clusters because kmeans++ is developed as an enhanced version of k-means clustering in initial cluster center identification and can give faster computation advantages [33]. In this study, K-means++ clustering was employed to discover the optimal clusters because k - means++ is developed as an enhanced version of k -means clustering in initial cluster center identification and can give faster computation advantages [5]. The silhouette coefficient is used to evaluate how effective a clustering method is. It has a value between -1 and 1 . From this 𝑎(𝑖) represents the average distance from an item 𝑖 in the cluster 𝐴 to all other objects in 𝐴, and 𝑑(𝑖,𝐶) represents the average distance from an object 𝑖 to all objects in the cluster 𝐶≠𝐴. After computing 𝑑(𝑖,𝐶) and 𝐶≠𝐴 for each cluster, the smallest cluster is chosen as described below. 𝑏(𝑖)=min 𝐶≠𝐴 𝑑(𝑖,𝐶) with 𝑖∈𝐴 (2) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 160 https://internationalpubls.com The value 𝑏(𝑖) denotes to what extent a data point 𝑖 is dissimilar to its nearest neighbor cluster. Thus, the silhouette values, silh(i) are given in Equation (3): silh⁡(𝑖)= 𝑎(𝑖)−𝑏(𝑖) max{𝑎(𝑖),𝑏(𝑖)} (3) Another cluster validation metric is the DBI which is used to determine the goodness of clusters. The DBI for 𝐾 clusters 𝐶𝑖 with 𝑖=1,…,𝐾 is defined according to Equation (4): 𝐷𝐵𝐾=1𝐾∑ 𝐾 𝑖=1 max 𝑗≠𝑖 𝑓𝑖,𝑗 (4) where: 𝑓𝑖,𝑗=diam⁡(𝐶𝑖)+diam⁡(𝐶𝑗) 𝑑(𝐶𝑖,𝐶𝑗)(5) and, in this case, the diameter of a cluster is defined as: diam⁡(𝐶𝑖)=(1 𝑛𝑖∑ 𝑥∈𝐶𝑖 ∥ ∥ 𝑥−𝑧𝑖 ∥ ∥ 2)12(6) with 𝑛𝑖 the number of data points and 𝑧𝑖 the centroid of cluster 𝐶𝑖. The DBI will achieve very small values, which guarantees the presence of high-quality clusters. Therefore, the ideal number of clusters is discovered when this index is minimized depending on the input dataset. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 161 https://internationalpubls.com As indicated in Table 2, each district data is grouped into the best similar cluster, and each cluster has a different observation size. For example, in the south AA district (Cluter 1 = 111503, Cluster 2 = 152620, and Cluster 3 = 45183 observations). In the case of West AA data ( Cluster 1 = 169800, Cluster 2 = 109227, and Cluster 3 = 34464 observations). Moreover, from Table 2, it has been indicated that the silhouette score for each cluster is > 0.5, which is higher, and the data points are correctly grouped in their proper cluster. Furthermore, the similarity (cohesion) of data points in a cluster is also very high as the larger silhouette score reveals the closeness of data points in a cluster. In general, the cluster validation results in Table 2 show that k-means clustering is a viable solution to characterize the energy consumption profiles and generate optimal clusters that will improve the prediction accuracy of the subsequent ensemble models. Table 2: K-means clustering validation results District Dataset #Cluster Silhouette_Score DBI_Score South AA 309303 3 0.5478 0.5785 West AA 313492 3 0.550 0.593 A dataset with normal distribution has skewness and kurtosis values of 0 and 3, respectively. However, as Table 1 and Figure 3, Figure 4, and Figure 5 show, our dataset is positively skewed. This kind of asymmetrical data distribution and complicated energy consumption patterns [7] requires efficient data clustering and an ensemble deep learning model that can handle much better than the classical machine learning models and statistical techniques. Moreover, k-means clustering results in Figure 7 illustrate that South AA data is grouped into 3 clusters of energy consumption profiles. Accordingly, Cluster 1 contains the medium size energy consumption profiles and the user’s monthly energy usage is between 275kW and 575kW. Next to Cluster 1, Cluster 2 is indicated in the brown box and contains lower energy users; their monthly energy consumption is between 0.1kW and 275KW, but the largest observation or energy consumption profiles are grouped in this category. The last cluster consists of the group of consumption profiles that comprises the highest monthly energy consumption profiles whose monthly consumption revolves between 575 kW and 925 kW, but this group accommodates the smallest number of observations. Figure 3: Skewness test for South AA after clustering. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 162 https://internationalpubls.com Figure 4: Skewness of West AA Data before Clustering and After Clustering Figure 5: Skewness test for West AA data after clustering. Figure 6: K-means clustering Results. 3.3 Deep Learning Model Deep learning methods have gained greater attention because of their remarkable performance in image classification, natural language processing, and nonlinear electric load consumption prediction [9, 36]. Convolutional neural networks (CNN) [18], Long Short-Term Memory networks (LSTM) [8], and Gated Recurrent networks (GRU) [17] are the most widely used deep learning algorithms. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 163 https://internationalpubls.com 3.3.1 Long-Short Term Memory Neural Networks (LSTM) Long short-term memory network (LSTM) is an improved version of a recurrent neural network frequently used in time forecasting and natural language processing. With its special memory cell, LSTM is capable of storing information involving long-range temporal dependencies [31]. LSTM network can establish long-term temporal correlation information and overcome vanishing gradient problems of RNNs networks as Figure 8 shows the sequential learning capabilities of LSTM. The selfconnection of the LSTM memory block, referred to as the cell state, preserves (remembers) longerrange temporal dependencies of the data. Moreover, LSTM architecture is equipped with multiplicative gate modules which include an input gate, forget gate, and output gate [25]. The gate units are responsible for regulating the flow of information while sequential data processing is dealt with LSTM model. Figure 7: LSTM Sequential Learning Process 𝑦ˆ𝑡+1=𝑓∑⁡(𝑊𝑖𝑥𝑖∗𝑏𝑖)(7) where the 𝑊𝑖 and 𝑥𝑖 are the updated weight vector and the input data respectively. Furthermore, f is the activation function. 3.3.2 Bidirectional LSTM (BiLSTM) Bidirectional Long Short-Term Memory (BiLSTM) is the defamation of the LSTM algorithm which is capable of learning sequential data in both forward and backward directions as Figure 9 shows. It is more effective in learning the past and the future context information in two ways forward and backward directions allowing it to capture the context from both past and future information. This makes Bidirectional LSTM well-suited for tasks involving sequential data such as natural language processing and time series forecasting. The forward and backward operation of the BiLSTM can be expressed using Equations 8-10. ℎ 󰇍 𝑡=LSTM(𝑥𝑡,ℎ 󰇍 𝑡)⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(8) ℎ ←𝑡=LSTM(𝑥𝑡,ℎ ←𝑡)⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(9) 𝑦ˆ𝑡+1=𝑤→𝑦ℎℎ 󰇍 𝑡+𝑊←ℎ𝑦,ℎ𝑡 ← +𝑏𝑦⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(10) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 170 https://internationalpubls.com Figure 11: MAE Values for Clustered and Un-clustered, South AA Data Figure 12: MAE Values for Clustered Vs Un-clustered, West AA Data Figure 11 and Figure 12, shows that the performance of the ensemble model significantly improved in the case of cluster generated data with MAE(69.696%) error decrease south AA case study data and MAE(55.595%) error decrease on the west AA data. From these results, we can conclude that the potential of k-means clustering to find the optimal clusters of the energy data significantly contributes to the ensemble deep model to learn complex energy data effectively and exhibits lower prediction errors in the new dataset compared to the model performance on un-cluster data concerning MAE performance metric. To generalize the training of deep learning and their ensemble models with postclustering data affirms the better performance. This is because in addition to outlier treatment, clustering of highly variable energy consumption data [17, 34] into more similar consumption patterns enables the proposed model to learn the detailed features of the input data. 4 Conclusion In this study, the effectiveness of deep learning models (1D-CNN, LSTM, BiLSTM and GRU) and ensemble model is investigated for aggregate energy consumption prediction focusing on the residential users category. The model’s performance was assessed on both the un-cluster and postclustered energy datasets. The integration of k-means clustering with an ensemble model to find the optimal cluster that minimizes the high variability and complexity of energy consumption data has been investigated. The viability of the clustering technique to group energy consumption data into a more similar consumption profile was validated and promising results were found which has enabled the ensemble deep model to learn the complete and intrinsic nature of the energy consumption data. Hence, the integration of clustering approach with deep learning and ensemble techniques significantly improves the prediction performance of the proposed model with very low prediction errors when compared to the performance obtained without clustering. Furthermore, while properly combining the capabilities of multiple deep learning algorithms, results indicate that the proposed ensemble model has outperformed the optimal base model performance in all case study data sets used in this study. In addition, enhanced by the metaheuristic Bayesian based hyperparameter tuning method, the proposed ensemble deep learning model has demonstrated the best performance and better generalization abilities without facing the problem of model overfitting while the trained model is exposed to test data. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 171 https://internationalpubls.com Overall, the ensemble model proposed in this study has demonstrated better capabilities for learning the complex energy consumption data and provides a significant MAE, RMSE and MAPE error decrease in both case study data as compared to base algorithms, i.e., LSTM, BiLSTM and 1D-CNN performance. 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