Prediction of Hourly Air-Conditioning Energy Consumption in Office Buildings Based on Gaussian Process Regression

Abstract

Accurate prediction of air-conditioning energy consumption in buildings is of great help in reducing building energy consumption. Nowadays, most research efforts on predictive models are based on large samples, while short-term prediction with one-month or less-than-one-month training sets receives less attention due to data uncertainty and unavailability for application in practice. This paper takes a government office building in Ningbo as a case study. The hourly HVAC system energy consumption is obtained through the Ningbo Building Energy Consumption Monitoring Platform, and the meteorological data are obtained from the meteorological station of Ningbo city. This study utilizes a Gaussian process regression with the help of a 12 × 12 grid search and prediction processing to predict short-term hourly building HVAC system energy consumption by using meteorological variables and short-term building HVAC energy consumption data. The accuracy R² of the optimal Gaussian process regression model obtained is 0.9917 and 0.9863, and the CV-RMSE is 0.1035 and 0.1278, respectively, for model testing and short-term HVAC system energy consumption prediction. For short-term HVAC system energy consumption, the NMBE is 0.0575, which is more accurate than the standard of ASHRAE, indicating that it can be applied in practical energy predictions.

1. Introduction

In recent years, with the improvement in people’s living standards, the increase in population, and the development of society, the world has generated a vast demand for energy supply and consumption, which further leads to severe environmental issues, including air pollution, water pollution, and the greenhouse effect [1,2]. When it comes to energy consumption reduction, the construction sector has received a great deal of attention. According to relevant statistics, building energy consumption accounts for nearly 40% of the total energy consumption in the United States [3] and about 20% in China [4]. Here, the building sector has become an area of concern in terms of minimizing the influence of massive energy consumption on environmental aspects [5]. The energy consumption of heating, ventilation and air-conditioning (HVAC) systems represents a considerable fraction of building energy consumption, accounting for nearly 50% of the total [6], and it still presents a continuous growth trend. IEA predicts that by 2050, global air-conditioning ownership will soar to two-thirds of all households [7], also resulting in an increase in global residential cooling demand of between 320% and 2270%, with the coupled effects of increased usage and longer cooling seasons [8]. Based on this high energy consumption, it is necessary to conduct in-depth research on reducing air-conditioning energy consumption. Recent studies have proposed two main means. On the one hand, more efficient control systems and techniques should be developed [9,10,11]. In recent years, scholars have researched the thermostatic control load to reduce energy consumption. The flexibility offered by the constant temperature control load could help the future energy system [12]; on the other hand, the use of air-conditioning can be adjusted by predicting the energy consumption of air-conditioning in order to reduce unnecessary waste [13].

Regression trees (RT), support vector machines (SVM), and artificial neural networks (ANN) are widely used to predict the energy consumption of buildings [14,15,16,17,18,19]. Zhun Yu et al. [20] developed a prediction model for building energy consumption based on the decision tree method, which was able to precisely classify and predict categorical variables and generate predictive models with a flowchart-like tree structure. This helps users quickly extract key information. Meanwhile, an accurate predictive model was developed by Zhitong Ma et al. using the SVM method. They used k-fold cross-validation with a grid searching method based on radial basis function kernel to explore the effect of the support vector machine on the two parameters c and g [21]. Additionally, Subodh Paudel et al. [22] used an artificial neural network to establish a short-term prediction model for building heating demand, considering the characteristics of occupancy profiles and operating heating power levels. Simultaneously, they proposed a new pseudo-dynamic transition model, which considered the time-dependent properties of operating power level characteristics and their impact on the model’s overall performance.

These three machine learning methods show advantages in prediction; however, the disadvantages cannot be ignored. Regression trees are suitable for predicting classification outcomes, but cannot provide a satisfactory outcome for non-linear datasets and are also susceptible to other interventions [23]. They are generally unsuitable for application to time series data unless there are clear trends and sequential patterns [15]. The neural network has strong parallelism and adaptability, and can obtain accurate prediction results; however, the algorithm’s convergence has not been proved. In most studies, many parameters are only set by means of a rule of the thumb. Using ANN algorithms requires additional processes to avoid overfitting, which may result in unsatisfactory results and errors. Moreover, tuning a neural network for load forecasting can be a lengthy process. The SVM method requires a great deal of computational time when solving large-scale data [24], which is not ideal for HVAC energy consumption prediction. The selection of the kernel function of SVM dramatically influences the computational results, but there has not been a solid reference for scholars to rely on [25].

This study adopted the Gaussian Process Regression (GPR) to avoid the above disadvantages. It is relatively simple to implement, and only requires a few modeling parameters, while having good adaptability for dealing with complex problems such as small samples, nonlinearity, and high dimensions; it has been verified to be reliable and is commonly used as a prediction method by scholars in many fields. Minghui Cheng et al. proposed a method for predicting residual stress on machined surfaces based on GPR, which provides a robust framework for surface residual stress prediction [26]. Weihan Li et al. found in the online prediction of lithium-ion battery power that the calculation speed of the iterative algorithm could be increased by 50% through the integration of Gaussian process regression [27]; Aaron Zeng et al. used this method to predict office energy consumption and found that the method had high prediction accuracy and required less calculation time [28]. Based on this, short-term forecasting could be carried out to adjust the energy system in real time and match the end of energy supply and demand according to the predicted value, thereby helping the system to operate effectively and generate operational and regulatory strategies for building energy supply systems for better distribution [29]. Hongfang Lu et al. also stated that short-term prediction of building energy consumption is a future research trend when conducting short-term building energy consumption prediction [30].

In this study, we focused on the effect of the Gaussian process regression method in the short-term prediction of building HVAC energy consumption. Then, regression trees (RT), support vector machines (SVM), and artificial neural networks (ANN) proceeded to verify the model accuracy under the same conditions. Due to the unfeasibility of studying all of the algorithms, only the main algorithms mentioned above are considered. This study aims to effectively and rapidly predict building air-conditioning energy consumption despite the small sample of building air-conditioning energy consumption data. Recent studies have focused on long-term prediction with large samples, and few scholars have paid attention to short-term prediction using training data of one month or less, as such predictions have many uncertainties and are challenging to apply in practice.

Therefore, it is of great significance to develop a precise prediction of short-term building air-conditioning energy consumption based on a small sample size.

This paper is structured as follows: In addition to the introduction shown in Section 1, the second chapter mainly discusses the research methodology. The first subsection introduces the machine learning method used in this paper, namely Gauss process regression, and explains its principle and main derivation formula in detail. The second subsection is about the model accuracy verification method, which introduces the principle and application process of ten-fold cross verification. In addition, three accuracy indexes—R², CV-RMSE, and NMBE—are proposed, and their calculation formula and accuracy requirements are described. The third subsection introduces the variables used in the case study, and the fourth subsection describes the basic process of modeling. In Section 3, the formation of the Gaussian process regression model and the method of improving accuracy used in the modeling process, as well as the results, are introduced and discussed. Then, the results obtained from the regression tree, support vector machine, and artificial neural network are compared with those from Gaussian process regression. The rest of this section introduces the advantages of this research method by comparing it with similar existing research methods. Section 4 mainly summarizes the purpose, achievements, and contributions of this research, its limitations, and future research guidelines.

2. Methodology

2.1. Gaussian Regression Analysis

Gaussian process regression, a nonparametric model, was applied to analyze data [31]. It is a random process in probability theory and statistics that allows any limited subset of random variables to have a multivariate Gaussian distribution, and is an effective means for dealing with complex regression problems and classifying initial distributions. One of the most critical characteristics of Gaussian processes is the diversity of covariance functions, which makes for the creation of functions of different degrees or types of continuous structure and provides the possibility for researchers to choose correctly [32]. The regression process provides a possible nonparametric modeling approach that can be used to solve various engineering problems [33]. The following derivation formula refers to Gaussian processes in machine learning [31].

Note that the training set is $D=\left\{\left(x_1,y_1\right),\dots ,\left(x_i,y_i\right),\dots ,\left(x_n,y_n\right)\right\}$. $x$ is an independent variable, and $y$ is a dependent variable. The input matrix is $x_i\in R^d$ and the output matrix is $y_i\in R$. The general regression model is shown in Equation (1).

(1)$y=f\left(x\right)+ϵ$

where $ϵ$ is residual or noise. If the function is not fixed, it could be referred to as the latent function. Each value of the latent function is a measure of the function-space. GPR takes the prior of this function space as a Gaussian process, which is expressed as a Gaussian process here with 0-mean, as shown in Equation (2).

(2)$f\left(x\right)~GP\left[0,k\left(x,x^{\prime }\right)\right]$

where $x$ is the training sample, and its measures in the Gaussian process are finite-dimensionally distributed. According to the definition, the finite-dimensional distribution is referred to as the multivariate normal distribution: $\forall t\in n, P\left[f\left(x_1\right),\dots ,f\left(x_t\right)\right]~N\left[0,k\left(x,x^{\prime }\right)\right]$. $k\left(x,x^{\prime }\right)$ is the kernel function, and the kernel function determines the Gaussian process of 0-mean.

If $ϵ$ is under the Gaussian distribution, and the mean and variance are 0 and ${\sigma }_n{}^2$ respectively, it can be denoted as $ϵ~N\left(0, {\sigma }_n{}^2\right)$. For (1), $ϵ$ is independent of $f\left(x\right),$ if both $f\left(x\right)$ and $y$ follow the Gaussian distribution, the set of the joint distribution of finite observations could form a Gaussian process, as shown in Equation (3):

(3)$y~GP\left[0,k\left(x,x^{\prime }\right)+{\sigma }_n{}^2{\delta }_{ij}\right]$

where ${\delta }_{ij}$ is the Kronecker delta function. If the covariance function is expressed in matrix form, it can be written as Equation (4).

(4)$C(X,X)=K(X,X)+{\sigma }_n{}^{2}I$

where $I$ presents the identity matrix of $N\times N$, and $C(X,X)$ presents the covariance matrix of N × N. K (X, X) represents the kernel matrix of N × N, known as the Gram matrix. Its element $K_{ij}=k\left(x^i,x^j\right)$.

According to the Bayesian principle, G.P. establishes the prior function in the set of given data $D_m$ and is converted to a posterior distribution in the set of m given test data $D_m=\left\{\left(x_{n+1},y_{n+1}\right),\dots ,\left(x_i,y_i\right),\dots ,\left(x_{n+m},y_{n+m}\right)\right\}$. The multivariate Gaussian distribution between the output vector $f$ of the training data vector $X_m$ and the output vector ${\overline{f}}_m$ of the test data is shown in Equation (5).

(5)$\left(\begin{array}{c}y\\ f_m\end{array}\right)~N[0,\left[\begin{array}{cc}K\left(X,X\right)+{\sigma }_n{}^{2}I& K\left(X,X_m\right)\\ K\left(X_m,X\right)& K\left(X_m,X_m\right)\end{array}\right]]$

Therefore, G.P. regression equation is obtained as shown in Equations (6)–(8).

(6)$f_m|X,y,X_m,{\sigma }_n{}^2~N[f_m|{\overline{f}}_m,cov\left(f_m\right)]$

(7)${\overline{f}}_m=K\left(X_m,X\right){\left(K\left(X,X\right)+{\sigma }_n{}^{2}I\right)}^{-1}y$

(8)$cov\left(f_m\right)=K\left(X_m,X_m\right)-K\left(X_m,X\right){\left(K\left(X,X\right)+{\sigma }_n{}^{2}I\right)}^{-1}K\left(X,X_m\right)$

where ${\overline{f}}_m$ is the output of the G.P. regression model or can be referred to as the predictive value of output vector $f_m$.

The kernel function of Gaussian process regression is added: the initial values for the kernel parameters are designated as a specific vector, the parameter estimation method for the GPR model for the approximation of a subset of regressors, the method used for prediction was fully independent conditional approximation along with indicators to standardize data [34].

2.2. Model Accuracy Validation

2.2.1. Ten-Fold Cross-Validation

The accuracy validation method of the model involved here is 10-fold cross-validation, in which the data set is randomly divided into ten parts, nine of which are used as training sets, and one of which is used as test data. Each test will receive a corresponding accuracy index, and the average value of the 10 results is used to estimate the model accuracy. The general process is shown in Figure 1.

2.2.2. Accuracy Index of Model

Two statistical indicators, namely determination coefficient (R²) and CV-RMSE, were used to assess the performance of the Gaussian process regression model, as shown in Equations (9) and (10).

(9)$R^2=1-\frac{{\displaystyle \sum }{\left(y_{fit}-y\right)}^2}{{\displaystyle \sum }{\left(y-\overline{y}\right)}^2}$

(10)$CV\text{-}RMSE=\frac{\sqrt{\frac{{\displaystyle \sum }{\left(y_{fit}-y\right)}^2}{n}}}{\overline{y}}$

where $y_{fit}$ is the predicted energy consumption of air-conditioning; y is the actual energy consumption of air-conditioning; $\overline{y}$ is the average value of air-conditioner energy consumption samples. n is the total number of samples. R² is often used to determine accuracy. The closer it is to 1, the higher the prediction accuracy will be [35]. ASHRAE Guideline 14 [36] defined the acceptance range of CV-RMSE for monthly and hourly model prediction accuracy; that is, CV-RMSE per hour model prediction accuracy should be less than 30%.

To make the short-term prediction using the optimal Gaussian process regression model more convincing, NMBE was introduced as a validation model accuracy index. Its calculation formula is as follows:

(11)$NMBE=\frac{\frac{1}{n}{\displaystyle \sum }(y_{fit}-y)}{\overline{y}}$

where $y_{fit}$ is the predicted energy consumption of air-conditioning; y is the actual energy consumption of air-conditioning; $\overline{y}$ is the average value of air-conditioner energy consumption samples. n is the total number of samples. ASHRAE Guideline 14 defines the acceptance range of NMBE for monthly and hourly model prediction accuracy; that is, NMBE per hour model prediction accuracy should be less than 10%.

2.3. Case Study

A government office building (see Figure 2) is located in the East New Town of Ningbo, with 22 stories and a floor area of 72,600 m², containing 500 occupants. It is equipped with a central air-conditioning system, including water chiller cooling tower, cooling water circulation systems, and air-conditioning units. The insulation of the walls is in the form of external wall insulation. The maximum allowed energy consumption intensity of government office buildings located in the hot summer and cold winter climatic zone is 70 kWh/(m²∙a). The building in this case study only consumed approximately 26.15 kWh/(m²∙a), which meets the standard. According to GB/T51161-2016 Energy Consumption Standard for Civil Buildings [37], the building could be defined as an energy-saving office building, satisfying the national building energy efficiency standard.

In this study, hourly air-conditioning power consumption and outdoor air temperature, solar radiation intensity, relative humidity, wind speed, wind direction, PM2.5, PM10, and other meteorological data for 18 days (4 July 2021–21 July 2021) of hourly air-conditioning energy consumption in a government office building in Ningbo were used as research objects. The air-conditioning energy consumption of the building was obtained from the building energy consumption monitoring platform of the Ningbo construction data and archives management center, and the meteorological data were obtained from the Ningbo meteorological station. In such government office buildings, the occupants’ behaviors and energy consumption variation in the building are relatively regular, and less affected by other factors, especially economic development. The variation in the energy consumption of the HVAC system with time is shown in Figure 3.

Figure 3 shows that the energy consumption of the HVAC system changes periodically with time. Hourly and weekly profiles were considered supplements for improving the model accuracy.

2.4. The Basic Process of Modeling

For the basic process of modeling, we took the Gauss process regression method as an example, and the initial data of 14 days (5–18 July) of hourly air-conditioning power consumption and outdoor air temperature, solar radiation intensity, relative humidity, wind speed, wind direction, PM2.5, PM10, and other meteorological data as training data for establishing the model. The training data were replaced with the Gaussian process regression method to establish the model, and then ten-fold cross-verification was used. If the effect was not satisfactory, the prediction results and data were processed to improve the accuracy until the required accuracy was achieved (e.g., zeroing the predicted negative energy consumption and 12×12 grid search). Finally, the hourly air-conditioning energy consumption was predicted using hourly meteorological data from 19 July to 21 July and using a 10-fold cross-validation method for verification. If the obtained accuracy did not meet the standard, the data were further processed to improve the accuracy. The specific flow chart is shown in Figure 4.

3. Results and Discussion

3.1. Modeling Using the Gaussian Progress Regression Method (GPR) and Results

The Gaussian process regression method was applied to the training datasets to predict the energy consumption of the HVAC. After ten-fold cross-validation, each accuracy value R² was obtained in Figure 5.

The average R² and CV-RMSE were 0.9261 and 30.2%, respectively. However, the CV-RMSE is not less than 30% required by ASHRAE Guideline 14, hence requiring further processing of the prediction data to improve the accuracy. The negatively predicted energy consumption should return to zero, since the energy consumption is positive. The original data for temperature (T) and solar radiation intensity (S.R.I.) can be used as variables to improve the model accuracy, since the dry bulb temperature and solar radiation intensity of the first few hours will affect the indoor environment in the following hours in a thermally isolated room with high thermal mass. In this case, when the floor is exposed to solar radiation and gradually heating up, the heated surface transfers the heat to the air contact the surface, which combines with the effect of the heat transfer due to temperature difference, further affecting the room temperature and influencing HVAC energy consumption. To reflect the influence of the previous hours on the data, the values of the environmental parameters of the previous hours are accumulated with the values of the current hour. In this case, the data are processed by using the grid search method. The sun and moon orbit a point on the earth for about 12 h in general, only the dataset from hours 0 to 12 will be considered. To deal with values from 1 to 12 h, a 12 × 12 grid is applied for accumulation. This approach is known as a 12 × 12 grid search. The specific process is shown in Figure 6.

SRIi presents the sum of the solar radiation intensity of the previous i-1 h and the current hour; Ti means the sum of temperature for the previous i-1 h and the current hour; the accuracy index here is the average accuracy of ten-fold cross-validation. A total of 144 training simulation data were obtained through the grid searching method. The regression model accuracy index is therefore shown in Figure 7 with the Gaussian process regression method.

Results show that the model with a total of four steps forward after the accumulation of temperature and a total of eight steps forward after the accumulation of solar radiation provided the most accurate prediction model. The average R² of the model was 0.9741, and the CV-RMSE was 0.2117. The average R² was close to 1, and the CV-RMSE accuracy also met the ASHRAE Guideline 14 requirement of being less than 30%. The accuracy value for each time in the ten-fold cross-validation is shown in Figure 8.

Figure 9 presents the fitted curve for training and testing. The two-fold Gaussian process regression model had the highest accuracy, with R² of 0.9917 and CV-RMSE of 0.1035.

It can be seen that the temperature in the first 3 h and the solar radiation in the first 7 h also played a considerable role in predicting the air-conditioning energy consumption of the building. The temperature and solar radiation intensity at a given moment as training data for the building made it possible for the Gaussian process regression model to achieve the highest accuracy. The training set with two-fold cross-validation showed the best performance in predicting energy consumption, and the model created could be considered the optimal Gaussian process regression model.

3.2. Comparison of Different Machine Learning Methods

To verify the outcome provided by Gaussian process regression, three machine learning methods—support vector machine, regression tree, and artificial neural network—were used. The grid search method was similarly used to obtain 144 training simulation data. Figure 10 shows the prediction accuracy index using the three methods.

In the support vector machine model, the predicted accuracy index when the model with a total of one step forward after the accumulation of temperature and a total of four steps forward after the accumulation of solar radiation was the highest. The average R² of the model was 0.5421, and the CV-RMSE was 0.9340. In the regression tree model, the model with a total of 10 steps forward after the accumulation of temperature and a total of two steps forward after the accumulation of solar radiation resulted in the highest accuracy. The average R² of the model was 0.9139, and the CV-RMSE was 0.3867. In the artificial neural network model, the model with a total of two steps forward after the accumulation of temperature and a total of three steps forward after the accumulation of solar radiation achieved the most accurate performance. The average R² of the model was 0.6612, and the CV-RMSE was 0.7931. The optimal training simulation data obtained by the four machine learning methods with grid search and their average precision indicators are summarized in Table 1.

By comparing the predictive accuracy index shown in Figure 6 and Figure 9, it can be seen that the GPR model had the best overall value of R², close to 1, and the overall value of CV-RMSE was the lowest, followed by RT., SVM, and then ANN. The values of R² and CV-RMSE of the artificial neural network model were unstable and fluctuated significantly. Table 1 also verifies that the GPR can be used to develop an accurate model by implementing the grid search method, followed by the RT, ANN, and SVM.

To better show the superiority of the Gaussian process regression method in predicting the energy consumption of HVAC in a building, the fitting effects of the optimal models of the four machine learning methods were compared. After being trained, the accuracy values at each time during ten-fold cross-validation is shown in Figure 11.

The SVM model with three-fold cross-validation had the highest accuracy, with R² of 0.7106 and CV-RMSE of 0.7023. The regression tree model with three-fold cross-validation likewise had the highest accuracy, with R² of 0.9691 and CV-RMSE of 0.2296. Different from the previous two machine learning methods, the ANN with the one-fold and four-fold cross-validation were the most accurate. For the one-fold model, R² was 0.822 and CV-RMSE was 0.6832, while R² was 0.7822 and CV-RMSE was 0.5337 for the four-fold model. The fitting curve and relative values are shown in Figure 12 and Table 2.

It can be seen that the solar radiation in the first 3 h also plays a considerable role in predicting the building’s air-conditioning energy consumption at a given moment by using a support vector machine and using these data as training data. In this case, the support vector machine model has the highest accuracy. At the same time, the training data with three-fold cross validation were the most suitable for predicting building air-conditioning energy consumption, and could be employed to create the optimal support vector machine model; when using regression tree modeling, the temperature in the first 9 h and the solar radiation in the first 1 h could be used to predict energy performance, and when the data were used for training, the most optimal accuracy was ensured. Similar to using SVM, the training data with three-fold cross-validation could be used to create the optimal regression tree model. When using artificial neural networks in modeling, the temperature in the first 1 h and the solar radiation in the first 2 h can be used to predict the air-conditioning energy consumption in a building, and using this data set helps to ensure that the highest accuracy is achieved. The use of training data with one-fold or four-fold cross-validation was the best for predicting building air-conditioning energy consumption, and the model could be considered an optimal artificial neural network model. As shown in Figure 9 and Figure 13, as well as in Table 2, the Gaussian process regression model is closer to the measured values, followed by the RT, and finally by the ANN and the SVM.

The energy consumption of the HVAC system from 19 July to 21 July was further predicted to verify the prediction model generated by the GPR for short-term prediction, and the accuracy index R² was 0.9863, CV-RMSE was 0.1278, and NMBE was 0.0575. Since R² was close to 1, the CV-RMSE accuracy met the ASHRAE Guideline 14 requirement of being less than 30%, and the NMBE accuracy met the ASHRAE Guideline 14 requirement of being less than 10%, the feasibility of this predictive model for use in practical engineering projects was confirmed. The comparison between test values and predicted values within the period from 19 July to 21 July is shown in Figure 13.

3.3. Comparison with Existing Research

In the results predicting the energy consumption of buildings, Yong Zhou et al. used school heating energy consumption over two months to predict short-term building heating load using 15 machine learning methods. The GPR was also considered to develop a model with high accuracy, with an R² of 0.939 during training and an R² of 0.956 during testing [38]. However, unlike in our research, where the grid search method was adopted, the training data here were not further processed, which led to a lack of credibility. Yuan Gao et al. studied the energy consumption of three buildings over three years. They also collected related information, including weather. Using two sequence-to-sequence modeling methods (seq2seq) and two-dimensional (2D) convolutional neural network (CNN), the model accuracy CV-RMSE was between 0.1394 and 0.5592 [39]. Although the model’s accuracy index was high, the quantity of the training data required was huge compared to the cases researched in this study. This paper only used meteorological variables and 18-day building HVAC energy consumption data. Thulasi Ram Khamma et al. collected the energy consumption data of a high-rise office building from January to June 2015 to perform predictions using three machine learning methods—support vector machine, classification, and regression tree—and their model accuracy CV-RMSE was above 0.2 [40]. M.A. Rafe Biswas et al. adopted a neural network approach for the prediction of energy consumption for residential buildings with an R² of 0.906 [41]. Compared with the prediction models developed by XJ Luo, Thulasi Ram Khamma, and MA Rafe Biswas, our developed predictive model has higher accuracy, with an R² of 0.9863, and lower CV-RMSE, at 0.1278. In conclusion, compared with the above-mentioned studies, this study has certain advantages in terms of the quantity of sample data, precision, and the comprehensive consideration of methods.

4. Conclusions

This paper aimed to utilize meteorological variables and short-term building historical HVAC energy consumption data to predict short-term building HVAC energy consumption. After predicting the energy consumption of air-conditioning, strategies for reducing its energy consumption can be studied to reduce the environmental problems caused by building energy consumption without affecting people’s quality of life. The case studied here is that of a typical office building in China, and Gaussian process regression was used with a 12 × 12 grid search method. The obtained predictive model using the method had an R² of 0.9917 and a CV-RMSE of 0.1035 during model testing. The R² of the short-term building HVAC energy consumption reached 0.9863, which is close to 1; meanwhile, CV-RMSE was below 0.3, as required by ASHRAE Guideline 14, reaching as low as 0.1278, and NMBE was below 0.1, as required by ASHRAE Guideline 14, namely 0.0575.

Using two variables, temperature and solar radiation intensity, with a 12 h scale, and implementing a 12 × 12 grid search improved the model’s accuracy. The model studied can be used with only the help of learning software such as MATLAB. The research results can be of technical help for the follow-up research on building energy-saving strategies, and the research methodology can provide a reference for researchers studying other building or energy types. However, the adopted method still has disadvantages. It takes longer than other machine learning methods, and the Gaussian process regression in this method is not applicable for predictions with large sample sizes. Only seven meteorological data indicators monitored by the meteorological station of Ningbo were considered, and the training and validation data pool was rather limited, which creates its own limitations. Furthermore, this study only examined short-term air-conditioning energy consumption predictions in office buildings, and while the outcome might apply to residential buildings, it would not apply to commercial buildings, including hotels and entertainment venues. The procedure described here can be used for short-term predictions for residential buildings due to the regularity of occupants’ behavior and the changes in their energy use. Additionally, the buildings in question are less likely to undergo changes in air-conditioning equipment or refurbishment, or to be affected by fluctuations in economic development; therefore, this research is valid for making short-term predictions. In contrast to residential and office buildings, predictions cannot be made for commercial buildings in the short term due to the irregularity of people’s mobility and the greater impact of the economy. Future research could therefore be conducted for other building types and would need to overcome the challenges of time duration, structural complexity, and insufficient factors to establish a fast and effective prediction method. Furthermore, more representative data should be used to make the research findings more convincing and address their scaleability.

Author Contributions

Conceptualization, Y.F. and Y.H.; methodology, Y.F.; formal analysis, Y.F.; investigation, H.S. and J.L.; writing—original draft, Y.F.; writing—review and editing, Y.F., Y.H., A.d.K., J.Y. and R.Z.; supervision, J.Y. and R.Z.; project administration, Y.F. and J.Y.; funding acquisition, J.Y. and R.Z. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

Nomenclature

ANN	Artificial neural network
ASHRAE	American Society of Heating Refrigerating and Airconditioning Engineers
CV-RMSE	Coefficient of variation of the root mean squared error
CNN	Convolutional neural network
GPR	Gaussian process regression
HVAC	Heating, ventilation, and air-conditioning
IEA	International Energy Agency
NMBE	Normalized mean bias error
R2	Determination coefficient
RT	Regression trees
SVM	Support vector machines
S.R.I.	Solar radiation intensity
seq2seq	Two sequence-to-sequence model methods
T	Temperature
2D	Two-dimensional

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Figures and Tables

Figure 1. Flow chart of 10-fold cross-validation.

Figure 2. A government office building in Ningbo.

Figure 3. Variation of air-conditioning energy consumption of a government office building in Ningbo during 4 July 2021 to 21 July 2021.

Figure 4. Modeling flow chart.

Figure 5. Accuracy index of the Gaussian process regression model with ten-fold cross-validation (using initial data). (a) R2. (b) CV-RMSE.

Figure 6. Flow chart of the 12 × 12 grid search method.

Figure 7. Accuracy index of the Gaussian process regression model with grid searching method. (a) R2. (b) CV-RMSE.

Figure 8. Accuracy index of the Gaussian process regression model with ten-fold cross-validation. (a) R2. (b) CV-RMSE.

Figure 9. Optimized fitting model of the Gaussian progress regression. (a) Training. (b) Test.

Figure 10. Accuracy index of Gaussian process regression model with the grid searching method. (a) R2 (SVM). (b) CV-RMSE (SVM). (c) R2 (RT). (d) CV-RMSE (RT). (e) R2 (ANN). (f) CV-RMSE (ANN).

Figure 10. Accuracy index of Gaussian process regression model with the grid searching method. (a) R2 (SVM). (b) CV-RMSE (SVM). (c) R2 (RT). (d) CV-RMSE (RT). (e) R2 (ANN). (f) CV-RMSE (ANN).

Figure 11. Accuracy index of Gaussian process regression model with ten-fold cross-validation (data processed by grid searching method). (a) R2 (SVM). (b) CV-RMSE (SVM). (c) R2 (RT). (d) CV-RMSE (RT). (e) R2 (ANN). (f) CV-RMSE (ANN).

Figure 12. Fitting curve with Gaussian progress regression. (a) Training (SVM). (b) Test (SVM). (c) Training (RT). (d) Test (RT). (e) Training (ANN with 1-fold cross-validation). (f) Test (ANN with 1-fold cross-validation). (g) Training (ANN with 4-fold cross-validation). (h) Test (ANN with 4-fold cross-validation).

Figure 12. Fitting curve with Gaussian progress regression. (a) Training (SVM). (b) Test (SVM). (c) Training (RT). (d) Test (RT). (e) Training (ANN with 1-fold cross-validation). (f) Test (ANN with 1-fold cross-validation). (g) Training (ANN with 4-fold cross-validation). (h) Test (ANN with 4-fold cross-validation).

Figure 13. Fitting of building air-conditioning energy consumption in the period 19–21 July.

Table 1

Optimal step number and accuracy index with grid searching using 4 machine learning methods.

Machine Learning Method	Steps for T	Steps for S.R.I.	R2	CV-RMSE
GPR	4	8	0.9741	0.2117
SVM	1	4	0.5421	0.9340
RT	10	2	0.9139	0.3867
ANN	2	3	0.6612	0.7931

Table 2

Predicted air-conditioning energy consumption, folding number, and precision index.

Machine Learning Method	Number of Folds	R2	CV-RMSE
GPR	2	0.9917	0.1035
SVM	3	0.7106	0.7023
RT	3	0.9691	0.2296
ANN	1	0.8220	0.6832
	4	0.7822	0.5337

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