Statistical Analysis and SARIMA Forecasting Model Applied to Electrical Energy Consumption in University Facilities

Análisis estadístico y modelo de pronóstico SARIMA aplicado al consumo de energía eléctrica en instalaciones universitarias

Global warming and climate change result from fossil fuel consumption as a source of energy. The energy demand is increasing day by day; the development of countries requires the consumption of energy from different carriers. However, the predominant sources of primary energy are still oil (33.1%), coal (27%), and natural gas (24.2%) [1]. Among the different sectors, the buildings sector contributes almost one-third of final energy consumption and continues to grow, driven by the economic development of the countries [2]. However, the E.U. has reduced energy consumption in buildings through energy efficiency policies. This is not the case in the U.S. [3] and Canada [4], where there are increases in demand for commercial and residential buildings. In the case of Mexico, final consumption in commercial, public, and residential buildings has remained relatively stable [5]. For 2018, CO2 emissions from energy consumption in buildings were 29% of a total of 33.9 Gt of CO2 [6]. To reduce energy consumption in buildings, it is necessary to implement measures conducive to this end, thereby reducing the negative impacts on the planet. The IEA establishes five measures applicable to buildings, including energy efficiency [7]. Energy efficiency in buildings allows better management of economic and material resources; it leads to maintenance improvements, achieving both environmental and economic benefits. However, to establish which energy efficiency actions should be taken, an energy diagnosis of the current situation of the building or group of buildings is necessary. Knowing what type of energy and how it is consumed is essential to implement energy-saving and efficient energy use measures, always ensuring that the comfort conditions inside the buildings are adequate for the performance of human activities. In the case of public buildings, such as schools, it is necessary to ensure that actions to achieve energy efficiency and economic savings do not somehow decimate the conditions suitable for the realization of the activities of each type or educational level [8]. Before opting for any measure according to the circumstances or existing ones [9] in the refurbishment of buildings to improve their energy efficiency, it is important to analyze the consumption pattern and forecast it.

For this purpose, energy forecasting models are used to establish energy-saving and efficient energy-use measures without altering the proper operation or service of the facilities. It is essential that the ability of the forecasting model can learn from past energy consumption patterns and accurately predict the future. This would allow the management and maintenance of the building to find corrective measures to possible variations in demand. Various methods and models are used to model energy behavior [10], [11], [12]. There are statistical methods to analyze the energy performance of buildings [13], [14], [15],[16], regression models [17], [18] and others that use specialized software for energy analysis [19], [20], [21], [22]. But for some years now, methods based on time series have been used [23], [24], and among these, the autoregressive integrated moving average (ARIMA) and SARIMA models that by themselves require fewer parameters and resources in their application [25], [26]. These models have been used in combination with physical models [27] and others, such as artificial neural networks (ANN) and supported vector machine (SVM) [28], [29] and machine learning (ML) [30].

Among the range of data prediction techniques, some are the most recurrent in the energy analysis of buildings, all of which have advantages and disadvantages in their application. For example, ANN models can be applied to nonlinear processes without knowing the relationship between input and output variables. However, evaluating the estimated parameters' relevance is impossible since there are no p-values. ARMA and ARIMA models are characterized by their ease of application and interpretation of parameters; they are more accurate than regression models, provide more reliable confidence intervals in predictions, require few computational resources, and use historical data. However, many models may need to be tested to fit, and although it is possible to determine the relationship between variables, their causal mechanism is not. Finally, these models are affected by outliers, and the forecast horizon may be short. In Decision Tree models, DT, rules are obtained that can be interpreted together with logical statements. However, they do not work well for nonlinear processes; they are susceptible to noise and unsuitable for time series. In the case of SVM models, they easily adapt to various problems, and optimal solutions are obtained; they can transform a nonlinear problem into a linear one. However, it is sometimes difficult to determine the kernel function and they can be computationally inefficient. Another type of model is the Fuzzy model, which, among its advantages, is its ability to be conducted without a training phase. This means that it can be used on data not contained in the training set. Fuzzy logic is derived from Boolean logic, and its rules for a model are usually not difficult to structure.

On the other hand, its disadvantages are that the model cannot be better than the expert because it cannot be trained and is challenging to fit with noise, in addition to its high computational complexity and lack of stability. The k-nearest neighbor (k-NN) models are characterized by the fact that they do not require prior training, which results in faster processing and ease of implementation for various problems. However, how the nearest neighbors are calculated, i.e., the distance function, is difficult to determine. Therefore, they are unsuitable for large data sets and overly sensitive to outliers and noise. Table 1 shows some previously described methodologies used in building energy analysis, classifying them by energy scale, energy type, time scale and type of input data for building energy analysis.

For example, Chae [31] uses ANN to forecast electricity consumption in commercial buildings. The data collected for the study came from a management system; power and electricity consumption were measured at one-minute and 15-minute intervals, respectively. In addition, weather variables and operating conditions were incorporated, which requires a large data set and significant computational power [32], [33], [34], [35]. Zhang [38], using SVR develops an electrical load forecasting model for a university building using time series of electrical energy consumption with two types of intervals: daily and half-hourly. The information from the management system corresponds to one year of consumption. Dong [36] forecasts electricity consumption using an SVR algorithm for a set of commercial buildings. The input variables were the monthly electric service billing and weather data for four years [37], [38], [39], [40], [41], [42], [43]. Kaur and Ahuja [44] predict the electricity consumption of a healthcare institution with ARIMA models using monthly, bimonthly, and quarterly periods of historical consumption data for more than 10 years [45]. Li [46] uses two databases with electricity consumption and meteorological variables to predict electricity consumption using Fuzzy+ANN models. In this case, the data of the environmental variables do not correspond to the same period of the consumption data [47], [48], [49]. Finally, Valgaev [50] uses hourly meter load data to forecast the next day's load using k-NN models applicable to all buildings. This work aims to study the energy performance of the buildings that constitute the academic unit based on statistical analysis and energy consumption forecasting using univariate SARIMA models. The advantage of this technique over others is that the modeling can be built with few parameters, it does not require special personnel and equipment, nor significant computational capacity. Historical consumption data are used as predictive variables, which could facilitate the preliminary energy use analysis (PEA), or a level 1 audit [51], [52], [53].

The study was carried out using statistical research methods divided into two phases. The first is the seasonal and correlation analysis, to analyze the seasonal behavior of the data using descriptive statistics that allow characterizing the data and examining the existence of patterns in the structure of the data over time to determine the seasonal component and its frequency. In addition, the relationship between the data set was primarily related to the immediate past. The second phase consisted of modeling the data as a time series using SARIMA processes to obtain a univariate predictive model of the series. Training and test partitions were created with the data, and to establish the partitions, the criterion followed was that the length of the test partition should not exceed 30% of the data. Otherwise, there would be a risk of not having enough information for the model training process. The two phases of the study were conducted with the statistical programs R [54] and RStudio [55].

The SARIMA models are derived from autoregressive and moving average models. The autoregressive models are based on the idea that the current value of the time series, X_t, can be explained as a function of a linear combination of p past values X_t, X_t-1, X_t-2, ..., X_t-p, where p determines the number of lags needed to forecast a current value [56]. The autoregressive models of order p, AR(p), are expressed as equation (1)

where $\textcolor{red}{}{\epsilon }_t$ is an error term, which is assumed to be approximately a white-noise process, and$\textcolor{red}{}{\phi }_1$,$\textcolor{red}{}{\phi }_2$, ..., $\textcolor{red}{}{\phi }_p$ are the parameters of the model, being applicable in a time series if, and only if, the series in question is stationary, that is, a time series whose properties do not depend on the time in which it is observed. On many occasions, time series present patterns that the model used for their prediction cannot represent. In this case, a known moving average process with a number q of past error terms can capture the patterns in the series. Moving average models are defined by an external information source, where the actual value of the series X_t, is determined or influenced by values from a random white noise process [56]. These moving average models of order q, MA(q), are defined by the equation (2)

And where $\textcolor{red}{}{\epsilon }_t$ is a white-noise process of the series and$\textcolor{red}{}{\theta }_1$,$\textcolor{red}{}{\theta }_2$, ..., $\textcolor{red}{}{\theta }_q$ are the model parameters, and like the AR models, the MA are applicable to seasonal series. On the other hand, although both AR and MA processes can be used in time series separately, the combination of both allows working with more complex time series. The combination of AR(p) and MA(q) processes is known as ARMA (p, q) processes and can be written as

where X_t in equation (3) represents the time series, p defines the number of lags for the regression, q the number of past error terms used in the equationand$\textcolor{red}{}{\phi }_1$,$\textcolor{red}{}{\phi }_2$, ...,$\textcolor{red}{}{\phi }_p$,$\textcolor{red}{}{\theta }_1$,$\textcolor{red}{}{\theta }_2$, ..., $\textcolor{red}{}{\theta }_q$ the parameters to be determined from the model. However, ARMA (p, q) models, like AR(p) and MA(q) are limited in their application to seasonal time series. To deal with the problem of the non-stationarity of a series, techniques such as the logarithmic transformation and differencing, which consists of differentiating the time series using its lags, and where parameter estimation is not required. The first differencing, for example, is represented as $\textcolor{red}{}\Delta {\text{X}}_t={\text{X}}_t-{\text{X}}_{t-1}$, for the second differencing (X_t - X_t-1) - (X_t-1 - X_t-2) which removes linear and quadratic trends from the series. In general terms, this differentiation process can be written as equation (4)

where X_dis the d differentiation of the series. These differences can be worked out through an operator known as the backward shift operator and defined in the form

And that by multiplying equation (5) with itself, the second differentiation is obtained, resulting in equation (6).

And that for a number d of differentiations, one has

From equation (7), the first differentiation can be rewritten as equation (8)

And that, in general, for differentiation of order d with the operator B, we have the expression of equation (9)

If the operator B is applied to the process AR(p) represented by equation (1), one has

Or equation (10) can be simplified in the form of equation (11)

where

Equation (12) is known as the autoregressive operator. A similar result can be obtained for the processes MA(q), which can be written from equation (2) in the form

Defining the moving average operator by means of equation (13) as

Therefore, if the time series with which we are working is not stationary and we want to model it with ARMA (p, q) processes, we add a differentiation process called the integration process. The model that arises from this integration is known as ARIMA (p, d, q) [57], [58] and using the autoregressive and moving average operators of equations (12) and (14) in equation (3), equation (15) is obtained.

Nevertheless, if the time series contains seasonal variations between periods, then the series $\textcolor{red}{}{\epsilon }_t$ will not be white noise since it contains correlations between periods. However, a time series with a seasonal component strongly related to its seasonal lags can be modeled with an ARIMA model using these lags, being represented in the form of equation (16)

The coefficient D, represents the past seasonal degree lag of the seasonal differencing of the series, while S denotes the seasonality of the model and $\textcolor{red}{}{\omega }_t$ is a white noise process with mean zero.

Equations (17) and (18) derived from equation (16) represent the seasonal regressor and moving average operator, being the coefficients of the seasonal autoregressive and moving average processes, SAR(P) and SMA(Q), where P represents the past seasonal lags and Q are the past error terms. We denote the parameterization of these models as SARIMA (p, d, q) (P, D, Q) [57] and where p and q are the parameters of the nonstationary AR and MA processes, respectively. In contrast, d and D define the degree of differencing for nonstationary and seasonal lags, respectively. Similarly, P and Q are the order of the SAR(P) and SMA(Q) processes for seasonal lags. By combining both models to model the time series, the general expression for a SARIMA model is obtained in the form

Equation (19) represents the combination of seasonal and non-seasonal autoregressive and moving average models used to model the electricity consumption series of this study.

The results of this study were divided into three parts: seasonal analysis, autocorrelation analysis, and modeling.

Since the academic unit is in a temperate climate region, the usual use of HVAC systems is limited to certain spaces, and its power source is electricity, the main energy source of the buildings. There is a direct relationship between electrical energy consumption and the activities within the academic cycle, but not this trend. The monthly average values show that a seasonal cycle in consumption is maintained, while the monthly and annual trend is to decrease. In the case of 2015, there is a shift explained by the fact that the school period underwent adjustments in 2014. July and August are the months with the most significant disparity in the data because of the shift in the school period, which forced the continuity of activities where there are usually vacations. The trend presented by electricity consumption has a steeper decreasing slope from January 2015 to May 2018, indicating a steep decline in electricity consumption. Subsequently, the slope decreased; consumption declined much lower than before June 2018. Although the institution has increased its enrollment, electricity consumption has been reduced due to measures such as replacing lighting fixtures and office equipment with lower consumption. Classrooms have been fitted with sensors to control lighting, energy saving, and efficient energy use measures that the Institute implemented based on the Comprehensive Energy Diagnosis conducted by the Mexican Center for Cleaner Production. In conjunction with the National Commission for the Efficient Use of Energy, conducted during 2015 and 2016.

Despite the above, no measures respond to seasonal consumption behavior, especially when consumption is higher. One possible explanation for this behavior is that between May and June, consumption increases due to the need to conclude academic and administrative activities before the first vacation period. While for August, the increase in consumption is due to the extra administrative work that is added to the other activities since it is that month that students enter the institution for the first time. From the results obtained by the models, the SARIMA (3,1,1) (1,0,0)₁₂ model had the best fit for the test data, with the most significant minimum values being those that the model could not fit more accurately. In its 12-month forecast, it is observed that it maintains the trend of the series at a constant mean and that it represents the behavior that would be expected given that consumption could not continue to be reduced under the current operating conditions of the facilities. The SARIMA (2,1,2) (1,0,0)₁₂ model for the 43-month training partition fits the most significant minimum values better than the previous model. However, its accuracy in predicting the test values is not good. Furthermore, when examining the 7- and 12-month forecasts, the model does not present the natural trend of the series, indicating that electricity consumption would continue its downward trend, which would be unrealistic. These results suggest that larger test partitions would improve the forecasting model, optimizing the buildings' energy audit.

In this study, we have worked with two approaches to analyze and forecast electrical energy consumption in educational buildings; the statistical approach and the univariate modeling with SARIMA processes. The univariate modeling of the time series of electricity consumption shows that, of the two best-evaluated models, SARIMA (3,1,1) (1,0,0,0)₁₂ best fits the real values, maintaining the seasonal behavior and the trend, which demonstrates its predictive capacity. Furthermore, in the medium-term projection of the model, it establishes that electric energy consumption will be a stationary process where its mean will be constant, which means that the trend will decay in such a way that it will cancel out. This is what would be expected in electricity consumption if the conditions of use do not change. For this model, a training partition of 48 months was used, indicating that a larger number of input data would result in a better-fitting model. However, it should be considered that as input data increases, the number of parameters to be calculated also increases.

On the other hand, from the statistical analysis, it is concluded that although there are actual values for 2015 that are presented as unusual due to the school calendar adjustment, the data do not contain outliers that could significantly affect the capacity of the predictive model. Furthermore, the monthly consumption averages project a seasonal behavior that can be used to establish electric energy efficiency strategies. For example, implementing ASHRAE Standard 100-2018, the months with the highest electricity consumption, May and June, have the highest daylighting, which means that artificial lighting time could be reduced through a building energy management system. Also, it would be possible to reduce lighting through devices that can vary light levels or dim when appropriate, along with implementing task lightings where needed, such as in offices and libraries. If possible, use occupancy, presence, or motion sensors in corridors and stairwells whose operation allows manual activation or turning on lighting at no more than 50% of capacity. Finally, upgrade indoor and outdoor lighting systems to provide demand response capacity to reduce lighting loads during peak electricity demand periods such as October.

The advantage of modeling with SARIMA processes is the ease of building and adjusting the model, making it efficient and a viable option to be implemented in building energy control and management systems. Also, to form a part of the processes for conducting energy audits that require an energy performance model. The complexity of the model and scope will depend on the needs of the audit.