Data-Driven Building Energy Consumption Prediction Model Based on VMD-SA-DBN

Abstract

Prediction of building energy consumption using mathematical modeling is crucial for improving the efficiency of building energy utilization, assisting in building energy consumption planning and scheduling, and further achieving the goal of energy conservation and emission reduction. In consideration of the non-linear and non-smooth characteristics of building energy consumption time series data, a short-term, hybrid building energy consumption prediction model combining variational mode decomposition (VMD), a simulated annealing (SA) algorithm, and a deep belief network (DBN) is proposed in this study. In the proposed VMD-SA-DBN model, the VMD algorithm decomposes the time series into different modes to reduce the fluctuation of the data. The SA-DBN prediction model is built for each mode separately, and the DBN network structure parameters are optimized by the SA algorithm. The prediction results of each model are aggregated and reconstructed to obtain the final prediction output. The validity and prediction performance of the proposed model is evaluated on a publicly available dataset, and the results show that the proposed new model significantly improves the accuracy and stability of building energy consumption prediction compared with several typical machine learning methods. The mean absolute percent error (MAPE) of the VMD-SA-DBN model is 63.7%, 65.5%, 46.83%, 64.82%, 44.1%, 36.3%, and 28.3% lower than that of the long short-term memory (LSTM), gated recurrent unit (GRU), VMD-LSTM, VMD-GRU, DBN, SA-DBN, and VMD-DBN models, respectively. The results will help managers formulate more-favorable low-energy emission reduction plans and improve building energy efficiency.

1. Introduction

With increasing global energy demand and environmental challenges such as global warming, emphasis on energy-saving emission reduction is inevitable [1]. Urban buildings, as the main body of energy consumption, consume 40% of the world’s energy annually [2]. Moreover, as the quality of life improves, building energy consumption will continue to increase [3]. Therefore, an accurate prediction of energy consumption is essential [4], which can provide basic information for decision-making on scheduling, operation, and detection issues and effectively reduce energy waste caused by lack of efficient operation and management in buildings, thereby improving building energy utilization efficiency [5,6].

The prediction of building energy consumption by mathematical modeling has become a popular research topic with the emergence of machine learning and data mining techniques [7,8]. Due to its usability and applicability for fast optimization, prediction models driven by historical data can learn autonomously to predict energy consumption by judging energy use patterns, which solves the non-linear problem [9,10]. For example, Anurag Verma et al. [11] developed an energy consumption data-driven prediction model using artificial neural network (ANN) and TRNSYS software. L. Wang et al. [12] compared adaptive linear filtering algorithms (least mean square (LMS), normalized least mean square (nLMS), and recursive least mean square (RLS)) with Gaussian mixed model regression (GMMR) models for predicting the hourly energy consumption of buildings with higher accuracy and concluded that GMMR has a higher prediction accuracy. M. W. Ahmad et al. [13] compared deep highway networks (DHNs), extremely randomized trees (ETs), and support vector regression (SVR) to predict the hourly heating, ventilation, and air conditioning (HVAC) energy consumption of a hotel building. Jihoon Moon et al. [14] proposed a short-term load prediction model based on a deep neural network (DNN) using the stacked ensemble method.

On the basis of individual prediction methods, many optimization algorithms have been introduced for hyperparameter optimization, thereby improving prediction accuracy, shortening computation time, or avoiding overfitting [15,16,17,18]. J. F. Torres et al. [19] optimized least square support vector machine (LSTM) hyperparameters for electricity consumption prediction using the coronavirus optimization algorithm (CVOA), which is a metaheuristic algorithm developed by the SARS-CoV-2 virus propagation method. Hamed Tabrizchi et al. [20] forecast residential energy consumption using a multi-verse optimizer (MVO)-optimized support vector machine (SVM). S. Chou et al. [21] compared two decomposition methods, the ensemble empirical modal decomposition (EEMD) and wavelet transform (WT), combined with LSTM for predicting building energy consumption, and the former was found to be superior. M. Meng et al. [22] decomposed the original sequence by discrete wavelet transform (DWT) and combined ANN and the grayscale model to predict building energy consumption. S. Wei et al. [23] proposed a hybrid building energy consumption prediction model that integrates singular spectrum analysis (SSA) for data preprocessing, convolutional neural network (CNN) for deep feature extraction, and a bidirectional gated recurrent unit (BiGRU) neural network for time series forecasting.

All the above studies achieved excellent results. However, considering the fluctuation of the original data and the stability of the prediction algorithm, there is room for further optimization of the building energy consumption prediction model [24]. This study constructs a novel prediction model with advanced signal processing, intelligent optimization, and deep learning algorithms to improve building energy consumption prediction accuracy. The main contributions are as follows:

• A short-term prediction model combining variational mode decomposition (VMD), SA, and DBN is proposed, with the VMD algorithm reducing the data fluctuation, the SA algorithm alleviating the lag in the data prediction, and the DBN network deep mining the hidden features.
• For the time series data of building energy consumption with complexity, non-linearity, and non-stationarity, the experiments show that the proposed model is effective and accurate for prediction.
• Accurate energy consumption prediction provides evidence for short-term power regulation and supports rational planning of building energy distribution, dispatch, and maintenance.

This paper is organized as follows. In Section 2, the VMD-SA-DBN prediction model is proposed based on the basic models, and the data are analyzed and processed. In Section 3, the results of the case study are analyzed and discussed, and Section 4 presents the conclusions.

2. Methodology

In this section, firstly, the structure of the building energy consumption data used is described, and the appropriate preprocessing methods to reduce data fluctuation and improve prediction accuracy are analyzed. Secondly, the prediction model framework is proposed based on the data characteristics, and the composition and functions of each unit are described.

2.1. Data Description

The experimental dataset was acquired from the Open Power System Data (OPSD) public dataset [25], which recorded the energy consumption data of several small businesses and residential households in southern Germany from 11 December 2014 to 1 May 2019, with a sampling frequency of 15 min. This study selected energy consumption data from industrial buildings consisting of several smaller loads in October 2015 for model training and validation, as shown in Figure 1. The region has a significant temperature difference between day and night during this season, with significant energy consumption ups and downs and a large drop in energy consumption distribution between weekdays and rest days due to the absence of holiday effects. The data structure characteristics are more challenging for short-term energy consumption prediction models. The statistical description of the data is shown in

Table 1. Descriptive statistics of energy consumption data.

Statistical Parameter	Min. (kWh)	Max. (kWh)	Mean (kWh)	Std.	Skewness	Kurtosis
	3.073	10.808	5.366	1.489	1.101	0.187

, including mean, maximum, minimum, standard deviation (Std.), skewness, and kurtosis, which express the terms of central tendency, discrete degree, and distribution shape of data. In addition, considering that building energy consumption is susceptible to geography and weather, these time series data usually exhibit complexity, non-linearity, and non-stationarity. Therefore, adopting the VMD method to preprocess the signal and improve the prediction accuracy by reducing the data fluctuation is a practical design.

2.2. Preprocessing Method

VMD is an entirely non-recursive method of modal variation and signal processing proposed based on empirical mode decomposition (EMD) [26], which can adaptively determine the number of modal decompositions and effectively reduce the non-smoothness of time series data with high complexity and strong non-linearity. The specific steps include two parts.

The first part is to construct the variational problem: assume that the original signal S is decomposed into K intrinsic mode function (IMF) components u. The decomposition sequence is guaranteed to be a finite-bandwidth modal component with central frequency, while the sum of estimated bandwidths of each mode is minimized, and the constraint is that the sum of all modes is equal to the original signal. The corresponding constrained variational expressions are as follows:

$$s.t. {{\displaystyle \sum }}_{k=1}^K{u}_k=S$$

(1)

$$\underset{\left\{u_k\right\},\left\{{\omega }_k\right\}}{\min }\left\{{{\displaystyle \sum }}_{k=1}^K\Vert {\partial }_t\left[\left(\delta \left(t\right)+\frac{j}{\pi t}\right)\ast u_k\left(t\right)\right]e^{-j{\omega }_{k}t}{\Vert }_2^2\right\}$$

(2)

where $\delta \left(t\right)$ is the Dirac function, $u_k$ is the decomposition to obtain the mode components, ${\omega }_k$ is the central frequency corresponding to the different decomposed mode components, and $e^{-j{\omega }_{k}t}$ is the exponential term of the central frequency.

The second part is to solve the constrained variational expression: introduce the penalty parameter α, the Lagrange multiplicative operator λ, and transform the constrained variational problem into an unconstrained variational problem to obtain the augmented Lagrange expression as follows:

$$\begin{gathered} L\left[\left\{u_k\right\},\left\{{\omega }_k\right\},\lambda \left(t\right)\right]=\alpha {{\displaystyle \sum }}_{k=1}\Vert {\partial }_t\left[\left(\delta \left(t\right)+\frac{j}{\pi t}\right)\ast u_k\left(t\right)\right]e^{-j{\omega }_{k}t}{\Vert }_2^2+ \\ \Vert S\left(t\right)-{{\displaystyle \sum }}_{k=1}^K{u}_k\left(t\right){\Vert }_2^2+\langle \lambda \left(t\right),S\left(t\right)-{{\displaystyle \sum }}_{k=1}^K{u}_k\left(t\right)\rangle \end{gathered}$$

(3)

By solving the saddle point of the incremental Lagrange function in Equation (3), which is the optimal solution for the constrained variation, the optimal decomposition of IMF is implemented.

In this study, through the fully non-recursive adaptive decomposition of VMD, the time series of the data is mode decomposed, and each mode’s central frequency and fixed bandwidth are determined to obtain 5 IMFs, which can reflect the local characteristics of the data. The main parameters are configured as follows: Nstd is 0.01, NR is 1; MaxIter is 100; K = 5, and Moderate bandwidth constraint α is 2000. The decomposition results are shown in Figure 2. IMF1 significantly demonstrates the peak-to-valley fluctuation pattern of the original data, and the effective separation of IMF2–IMF5 by different finite bandwidths realizes the frequency domain division of the signal and reflects the characteristics of the IMF components from high to low frequencies. The decomposed IMFs have a slight fluctuation range and good smoothness compared with the original building energy consumption data, which is helpful to the effectiveness of the prediction model.

2.3. VMD-SA-DBN Model

Combined with the analysis in Section 2.3, a building energy consumption prediction method based on the new VMD-SA-DBN model is proposed. The model framework is shown in Figure 3; firstly, VMD is used to decompose the non-stationary building energy consumption time series into different modes to extract the features effectively; secondly, the SA algorithm is used to optimize the DBN parameters for the high-fluctuation characteristics of data on a short-term scale to improve the prediction accuracy of individual modal components; finally, the prediction results of each mode component are summed and reconstructed to obtain an optimized prediction result.

The DBN used is a deep learning classifier composed of a multilayer restricted Boltzmann machine (RBM) and a back propagation neural network (BPNN) [27], which achieves high-accuracy classification and recognition by fitting complex functions through a non-linear neural network structure. First, the DBN model performs bottom-up unsupervised pretraining for each RBM separately to obtain the weight w between each layer and the bias b for each layer. Second, fine-tune the parameters in the whole DBN from top to bottom to improve the model performance. As an essential part of the DBN, an RBM is a two-layer random neural network in which the visible and hidden layers are fully linked and can propagate in both directions. In an RBM, the energy function of the RBM can be derived based on the interaction of energy deviations between the layers and the interaction between the layers. The specific energy function can be defined as follows:

$$E\left(v,h\right)=-({{\displaystyle \sum }}_i{a}_i{v}_i+{{\displaystyle \sum }}_j{b}_j{h}_j+{{\displaystyle \sum }}_{i,j}{w}_{ij}{v}_i{h}_j)$$

(4)

where $E\left(v,h\right)$ is the energy function, $v_i$ is the state of the ith neuron in the visible layer and $h_j$ is the state of the jth neuron in the hidden layer, $a_i$ and $b_j$ are the offset vectors in the hidden and display layers, respectively, and $w_{ij}$ is the bidirectional weight. In addition, the SA algorithm for optimizing DBN parameters is a heuristic algorithm inspired by the principle of annealing of solids [28]. Starting from a specific higher initial temperature, the solution gradually stabilizes as the temperature parameter decreases. Optimal local solutions are eliminated in the SA algorithm, with a certain probability of finding the global optimal solution of the objective function. The algorithm has a robust local optimization capability and prevents the search process from falling into local optimal solutions [29], which is better in dealing with complex network problems with a large number of configurations and various local minima [30]. The prediction process based on the VMD-SA-DBN model is as follows:

• VMD: The original data are decomposed by the VMD method to obtain K sub-modes, and the value of K is determined according to the proximity of the mode center frequencies.
• Data normalization: To reduce the data dispersion in the components and decrease the training time, the data of each component is normalized, as shown in Equation (5):

  $$y=\left(y_{max}-y_{min}\right)\frac{x-x_{min}}{x_{max}-x_{min}}+y_{min}$$

  (5)

  where x is the original data; y is the normalized data; and $y_{max}$, $y_{min}$ are the normalized maximum and minimum values, respectively.
• SA-DBN: The SA algorithm is performed on each component in turn to optimize the number of hidden-layer nodes. The DBN network structure is determined, and the network parameters are initialized. The DBN error is selected as the objective function, and then the relevant SA algorithm parameters are set. The network parameters are iteratively updated according to the inner and outer loop conditions in the SA algorithm, and the optimal network parameters can be obtained. The SA-DBN model is trained and parameter tuned using the training dataset to obtain the optimal component prediction model. The test dataset is input into the trained SA-DBN model to obtain the component prediction results.
• Data output: The subsequence components at different scales are superimposed to obtain the predicted value of total building energy consumption.

3. Experiment and Analysis

The hidden features of the building energy consumption history data are deep mined by SA-DBN through hierarchical pretraining and feedback fine-tuning, which can have better results for predicting IMFs after VMD decomposition. In this section, the data shown in Figure 1 are used for experimental verification and comparative analysis of the prediction effect. The ratio of the training set to the testing set is 7:3.

3.1. Experimental Setup

For the experimental hardware environment, the CPU is Intel Core i5-8350, the running memory is 8 GB, and MATLAB 2020b is used for simulation programming. Parameters in the VMD-SA-DBN model are configured as shown in

Table 2. Hyperparameter configuration in the VMD-SA-DBN model.

Model Layer	Hyperparameter Configuration
VMD	Same as Section 2.3
SA	Markov chain length = 3; decay coefficient = 0.5; step factor = 0.1; initial temperature = 5
DBN	The number of hidden layers = 2; the value interval for the number of nodes in each hidden layer: [hmin, hmax] = [1, 20]
	Pretraining epoch = 10; pretraining batch size = 16; pretraining learning rate = 0.1; pretraining momentum = 0;
	Fine-tuning epoch = 100; fine-tuning batch size = 16; fine-tuning learning rate = 0.1

. Considering the complexity and data size, epochs are set to 100 and batch size is 16.

To analyze the prediction effect, several error-evaluation indicators are selected for comparison, including mean absolute error (MAE), root mean square error (RMSE), mean absolute percent error (MAPE), and coefficient of determination (R²). MSE is the mean variance between the predicted value and the actual value, RMSE is the square root of the MSE, and MAPE represents the relative relationship between the predicted error and the actual value. R² represents the ratio of the MSE of the prediction error to the variance of the actual dataset.

$$MAE=\frac{1}{n}{{\displaystyle \sum }}_{i=1}^n\left|y_{forecasting,i}-y_{observed,i}\right|$$

(6)

$$RMSE=\sqrt{\frac{1}{n}{{\displaystyle \sum }}_{i=1}^n{\left(y_{forecasting,i}-y_{observed,i}\right)}^2}$$

(7)

$$MAPE=\frac{1}{n}{{\displaystyle \sum }}_{i=1}^n\left|\frac{y_{forecasting,i}-y_{observed,i}}{y_{observed,i}}\right|\ast 100$$

(8)

$$R^2=1-\frac{{{\displaystyle \sum }}_{i=1}^n{\left(y_{forecasting,i}-y_{observed,i}\right)}^2}{{{\displaystyle \sum }}_{i=1}^n{\left(y_{observed,i}-{\overline{y}}_{observed}\right)}^2}$$

(9)

Among them, n is the sample size, $y_{forecasting,i}$ is the predicted value, $y_{observed,i}$ is the actual value, and ${\overline{y}}_{observed}$ is the average value of the actual value. The model achieves better prediction performance when MSE and RMSE tend to 0 and the value of R² tends to 1.

3.2. Result and Analysis

For verification of the prediction effectiveness of the VMD-SA-DBN model, LSTM, gated recurrent unit (GRU), VMD-LSTM, VMD-GRU, DBN, SA-DBN, and VMD-DBN are introduced for comparison with the proposed model, and the error evaluation indexes of different model predictions are compared as shown in

Table 3. Basic statistics of energy consumption data.

Model	LSTM	VMD-LSTM	GRU	VMD-GRU	DBN	SA-DBN	VMD-DBN	VMD-SA-DBN
MAE (kWh)	0.444	0.279	0.433	0.416	0.305	0.238	0.264	0.169
RMSE (kWh)	0.616	0.346	0.522	0.555	0.411	0.328	0.349	0.224
MAPE (%)	7.651	5.217	8.050	7.886	4.966	3.870	4.352	2.774
R2	0.897	0.967	0.911	0.889	0.932	0.957	0.965	0.982

. Comparison between DBN and SA-DBN shows that the latter significantly improves the model prediction accuracy, which is evidence that optimizing the DBN network structure parameters for prediction by the SA algorithm is suitable for improving the prediction accuracy. In contrast to DBN and VMD-DBN, it is observed that the prediction strategy of decomposing the original time series, predicting the subcomponents in turn, overlaying, and reconstructing is feasible. The VMD algorithm not only decreases the time complexity of the original building energy consumption data but also effectually avoids waveform mixing. After establishing the prediction model for each mode, the prediction accuracy increases significantly by summation and reconstruction. To compare the VMD optimization effect, VMD associated with other models was added, namely VDM-LSTM and VMD-GRU. For LSTM, the prediction accuracy improved considerably after reconstruction by VMD decomposition, with MAPE improvement percentages of 31.8%. On the contrary, for GRU, the improvement of the prediction effect after reconstruction by VMD decomposition was not apparent. Overall, the prediction error evaluation metric of the VMD-SA-DBN model showed a greater degree of reduction than the other seven prediction models, and the MAPE of the VMD-SA-DBN model was 63.7%, 65.5%, 46.83%, 64.82%, 44.1%, 36.3%, and 28.3% lower than the one of the LSTM, GRU, VMD-LSTM, VMD-GRU, DBN, SA-DBN, and VMD-DBN models, respectively.

The prediction effects of the VMD-SA-DBN model and the other seven comparison models are further represented by visualization, and the fit plots of the predicted data of different models to the actual data are shown in Figure 4a. It is highly apparent that the proposed prediction model has the best fit effectiveness but slightly deviates in the prediction of some peak data. Considering the clarity of observation, the better prediction methods VMD-LSTM, SA-DBN, VMD-DBN, and VMD-SA-DBN were selected to plot the line graphs with the actual energy consumption, as shown in Figure 4b. The prediction curves of the VMD-SA-DBN model are more consistent with the original time series of energy consumption, and the model can achieve better prediction results for smoothed data and better ability for peak-to-valley variation tracking than other models. It can be noticed that, in the prediction results of the testing set, all methods had errors at the second peak and failed to follow it completely. The performance at the other peaks was more accurate for the proposed model. Meanwhile, the impact was not significant in practical applications. Moreover, the proposed VMD-SA-DBN prediction model had excellent robustness for the data. Simultaneously, the model had an average computation time within 60 s, which is sufficient for prediction data with a step size of 15 min.

4. Conclusions

In this study, a novel VMD-SA-DBN model is proposed for short-term building energy consumption prediction. The VMD method is used to decompose and denoise the original building energy consumption data, and the sub-modes are forecasted by the SA-DBN model. The experimental results show that the VMD algorithm can separate the trend, period, and noise components from the original building energy consumption series and can increase the prediction accuracy. The DBN network structure optimized by the SA algorithm significantly predicts the time series.

The proposed model can accurately capture the non-linear variation of the time series and perform better for the peaks and valleys of the building energy consumption data, which alleviates the lag in the prediction of the polar data. The results can provide more accurate data support for short-term power regulation and contribute to the rational planning of building energy allocation, scheduling, and maintenance. Meanwhile, the accurate prediction for demand measurement enables feedbacks and adjustments toward the supply side. It is conducive to improving building energy efficiency and helping managers formulate more-favorable low-energy emission reduction plans. In the future, further improvements will be applied to the mathematical model to track peaks and valleys better and enhance the model prediction accuracy. Model construction and prediction for other variable parameters and software development will be actively experimented with in the subsequent work.