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

Abstract

Analyzing the energy consumption behavior in buildings is essential for implementing energy-saving and efficient energy use measures without losing attention to the comfort inside the buildings. In this study, a statistical analysis and time series forecast of the energy situation of a group of buildings in a university academic unit in Mexico City was conducted. Seasonal Autoregressive Integrated Moving Average (SARIMA) models were used for the forecast with electrical energy consumption data from 55 months. Training and test partitions were created with these data to generate two SARIMA models. The results showed a strong dependence on the school cycle of electricity consumption, in addition to a shift in the cycle in the first year of the study. The mean absolute percentage error (MAPE) for the training partitions created shows that the best fit is provided by the SARIMA (3,1,1) (1,0,0)12 model for the 48-month separation. In comparison, the SARIMA (2,1,2) (1,0,0)12 model does so for the 43-month test partition. The confidence intervals for the 7- and 12-month forecast are less wide for the SARIMA (3,1,1) (1,0,0)12 model than for the SARIMA (2,1,2) (1,0,0)12 model. Statistical analysis and time series modeling allows a better understanding of the building stock's energy performance and strengthens the energy audit to design or implement energy saving or efficient energy use measures.

Full text

Científica, vol. 26, núm. 2, pp. 01-22, julio-diciembre 2022, ISSN 2594-2921, Instituto Politécnico Nacional MÉXICO
DOI: https://doi.org/10.46842/ipn.cien.v26n2a03
1
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
José Luis Reyes Reyes1, Guillermo Urriolagoitia Sosa2, Francisco Javier Gallegos Funes3,
Beatriz Romero Ángeles4, Israel Flores Baez5, Misael Flores Baez6
Instituto Politécnico Nacional
1https://orcid.org/0000-0002-7645-7280 | jreyesre@ipn.mx
2https://orcid.org/0000-0001-7867-7386 | guiurri@hotmail.com
3https://orcid.org/0000-0002-4854-6438 | fgallegosf@ipn.mx
4https://orcid.org/0000-0001-6345-3726 | bromero@ipn.mx
Universidad Politécnica de Tecámac | Instituto Politécnico Nacional, MÉXICO
5https://orcid.org/0000-0002-3339-1912 | israelfb364@yahoo.com.mx
6https://orcid.org/0000-0001-7657-5298 | misaelfloresbaez@yahoo.com.mx
Recibido 24-05-2022, aceptado 30-08-2022
Abstract
Analyzing the energy consumption behavior in buildings is essential for implementing energy-saving and efficient energy
use measures without losing attention to the comfort inside the buildings. In this study, a statistical analysis and time series
forecast of the energy situation of a group of buildings in a university academic unit in Mexico City was conducted. Seasonal
Autoregressive Integrated Moving Average (SARIMA) models were used for the forecast with electrical energy consumption
data from 55 months. Training and test partitions were created with these data to generate two SARIMA models. The results
showed a strong dependence on the school cycle of electricity consumption, in addition to a shift in the cycle in the first
year of the study. The mean absolute percentage error (MAPE) for the training partitions created shows that the best fit is
provided by the SARIMA (3,1,1) (1,0,0)12 model for the 48-month separation. In comparison, the SARIMA (2,1,2) (1,0,0)12
model does so for the 43-month test partition. The confidence intervals for the 7- and 12-month forecast are less wide for
the SARIMA (3,1,1) (1,0,0)12 model than for the SARIMA (2,1,2) (1,0,0)12 model. Statistical analysis and time series
modeling allows a better understanding of the building stock's energy performance and strengthens the energy audit to
design or implement energy saving or efficient energy use measures.
Index terms: energy consumption, scholar buildings, time series forecasting, SARIMA models.
Resumen
Analizar el comportamiento del consumo energético en edificios es fundamental para la implementación de medidas de
ahorro y uso eficiente de la energía, sin perder atención al confort al interior de estos. En este estudio se realizó un análisis
estadístico y de pronóstico con series de tiempo de la situación energética de un conjunto de edificios de una unidad
académica universitaria de la Ciudad de México. Para el pronóstico se utilizaron modelos Estacionales Autorregresivos
Integrados y de Medias Móviles (SARIMA) con datos del consumo de energía eléctrica de 55 meses y con estos se crearon
particiones de entrenamiento y prueba que generaron dos modelos SARIMA. Los resultados mostraron una gran dependencia
en el ciclo escolar del consumo de electricidad, además de un corrimiento en el ciclo en el primer año de estudio. El
porcentaje de error absoluto medio (MAPE) para las particiones de entrenamiento creadas muestra que el mejor ajuste lo
tiene el modelo SARIMA (3,1,1) (1,0,0)12 para la partición de 48 meses, mientras que el modelo SARIMA (2,1,2) (1,0,0)12 lo
hace para la partición de prueba de 43 meses. Los intervalos de confianza para el pronóstico a 7 y 12 meses son menos
amplios para el modelo SARIMA (3,1,1) (1,0,0)12 que para el modelo SARIMA (2,1,2) (1,0,0)12. Finalmente, el análisis
estadístico y el modelado de series de tiempo permiten un mejor entendimiento del comportamiento energético del conjunto
de edificios y fortalece la auditoría energética con miras a diseñar o aplicar medidas de ahorro o uso eficiente de la energía.
Palabras clave: consumo de energía, edificios educativos, pronóstico de series de tiempo, modelos SARIMA.

Statistical Analysis and SARIMA Forecasting Model Applied to Electrical
Energy Consumption in University Facilities
José Luis Reyes Reyes, Guillermo Urriolagoitia Sosa,
Francisco Javier Gallegos Funes, Beatriz Romero Ángeles,
Israel Flores Baez, Misael Flores Baez
Científica, vol. 26, núm. 2, pp. 01-22, julio-diciembre 2022, ISSN 2594-2921, Instituto Politécnico Nacional MÉXICO
DOI: https://doi.org/10.46842/ipn.cien.v26n2a03
2
I. INTRODUCTION
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

Statistical Analysis and SARIMA Forecasting Model Applied to Electrical
Energy Consumption in University Facilities
José Luis Reyes Reyes, Guillermo Urriolagoitia Sosa,
Francisco Javier Gallegos Funes, Beatriz Romero Ángeles,
Israel Flores Baez, Misael Flores Baez
Científica, vol. 26, núm. 2, pp. 01-22, julio-diciembre 2022, ISSN 2594-2921, Instituto Politécnico Nacional MÉXICO
DOI: https://doi.org/10.46842/ipn.cien.v26n2a03
3
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.
TABLE 1. MODEL, TIME SCALE, TYPE OF ENERGY ANALYZED, LENGTH OF MEASUREMENT, AND TYPE OF MODEL
INPUT DATA USED IN ENERGY ANALYSIS.
Model
Energy type
Time scale
Data source
Measure
length
ANN
• Electric Consumption [31], [32]
• Cooling energy consumption
[33]
• Heating and cooling
consumption [34]
• Cooling demand [35]
• Hourly [32]
• Daily [31], [33]
• Day type/Temp/Hum/Wind/Rad
• Historical data
1-2 years
2 months
1 year
45 weekdays
SVM
• Electric consumption [36], [37]
• Electric consumption [38], [39]
• Cooling load [40]
• Hourly [37]
• Monthly [36]
• Daily and Half-
hourly [38]
• Daily [40]
• Monthly [39]
• Calendar/historical data/Temp
• Sensor/Temp/holidays indicator
• Meters
• Measure data
• Historical data
2 years
1 year
6 months
3 years
ARIMA
model
• Thermal load [41]
• HVAC [42], [43]
• Electric consumption [44]
• Electricity demand [45]
• Hourly [41], [43]
• Daily [43]
• Yearly [42]
• Monthly [44]
• Monthly [45]
• Tem/Hum/Wind/Rad/Occu
• Building performance
• Temperature
6 months
1 year
7 years
10 years, 10
months
Fuzzy
model
• Electric consumption [46]
• Heating demand, electricity
demand [47]
• Energy consumption [48]
• Daily [46]
• Hourly [47]
• Day type/Temp/Rad
• Energy bills
• Survey
4-6 months
3 years
45 years
k-NN
• Electric consumption [49]
• Power demand [50]
• Electric load [51]
• Minutes-daily
[49]
• Daily [50]
• Minutes [51]
• Meter and meteorological
variables [1]
• Smart Meter [2], [5]
1 year
19 months
Temp: Temperature; Hum: Humidity; Rad: Radiation; Occu: Occupancy

Statistical Analysis and SARIMA Forecasting Model Applied to Electrical
Energy Consumption in University Facilities
José Luis Reyes Reyes, Guillermo Urriolagoitia Sosa,
Francisco Javier Gallegos Funes, Beatriz Romero Ángeles,
Israel Flores Baez, Misael Flores Baez
Científica, vol. 26, núm. 2, pp. 01-22, julio-diciembre 2022, ISSN 2594-2921, Instituto Politécnico Nacional MÉXICO
DOI: https://doi.org/10.46842/ipn.cien.v26n2a03
4
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].
II. METHODOLOGY
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, , can be explained as a function of a linear
combination of p past values , 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)
   
(1)
where  is an error term, which is assumed to be approximately a white-noise process, and  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 , 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)
  
(2)

Statistical Analysis and SARIMA Forecasting Model Applied to Electrical
Energy Consumption in University Facilities
José Luis Reyes Reyes, Guillermo Urriolagoitia Sosa,
Francisco Javier Gallegos Funes, Beatriz Romero Ángeles,
Israel Flores Baez, Misael Flores Baez
Científica, vol. 26, núm. 2, pp. 01-22, julio-diciembre 2022, ISSN 2594-2921, Instituto Politécnico Nacional MÉXICO
DOI: https://doi.org/10.46842/ipn.cien.v26n2a03
5
And where  is a white-noise process of the series and  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
     
(3)
where  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 equation and  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 ,
for the second differencing 󰇛󰇜󰇛 󰇜 which removes linear and quadratic trends from the
series. In general terms, this differentiation process can be written as equation (4)
󰇛󰇜󰇛 󰇜
(4)
where  is 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

(5)
And that by multiplying equation (5) with itself, the second differentiation is obtained, resulting in equation (6).
 
(6)
And that for a number d of differentiations, one has

(7)
From equation (7), the first differentiation can be rewritten as equation (8)
󰇛󰇜
(8)
And that, in general, for differentiation of order d with the operator B, we have the expression of equation (9)
󰇛󰇜
(9)
If the operator B is applied to the process AR(p) represented by equation (1), one has

(10)
Or equation (10) can be simplified in the form of equation (11)
󰇛󰇜
(11)

Statistical Analysis and SARIMA Forecasting Model Applied to Electrical
Energy Consumption in University Facilities
José Luis Reyes Reyes, Guillermo Urriolagoitia Sosa,
Francisco Javier Gallegos Funes, Beatriz Romero Ángeles,
Israel Flores Baez, Misael Flores Baez
Científica, vol. 26, núm. 2, pp. 01-22, julio-diciembre 2022, ISSN 2594-2921, Instituto Politécnico Nacional MÉXICO
DOI: https://doi.org/10.46842/ipn.cien.v26n2a03
6
where
󰇛󰇜
(12)
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
󰇛󰇜
(13)
Defining the moving average operator by means of equation (13) as
󰇛󰇜
(14)
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.
󰇛󰇜󰇛󰇜󰇛󰇜
(15)
Nevertheless, if the time series contains seasonal variations between periods, then the series  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)
󰇛󰇜󰇛󰇜󰇛󰇜
(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  is a white noise process with mean zero.
󰇛󰇜 
(17)
󰇛󰇜 
(18)
Equations (17) and (18) derived from equation (16) represent the seasonal regressor and moving average
operator, respectively, with  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
󰇛󰇜󰇛󰇜󰇛󰇜󰇛󰇜󰇛󰇜󰇛󰇜
(19)

Statistical Analysis and SARIMA Forecasting Model Applied to Electrical
Energy Consumption in University Facilities
José Luis Reyes Reyes, Guillermo Urriolagoitia Sosa,
Francisco Javier Gallegos Funes, Beatriz Romero Ángeles,
Israel Flores Baez, Misael Flores Baez
Científica, vol. 26, núm. 2, pp. 01-22, julio-diciembre 2022, ISSN 2594-2921, Instituto Politécnico Nacional MÉXICO
DOI: https://doi.org/10.46842/ipn.cien.v26n2a03
7
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.
II. 1 DESCRIPTION OF THE ACADEMIC BUILDINGS
The facilities considered in the study are part of the professional unit of the National Polytechnic Institute
located in Mexico City, in the center area of the Valley of Mexico, whose geographical coordinates are 19.5°
N and 99.14° W. The unit was built more than 60 years ago, although not all the buildings were constructed
simultaneously. The facilities studied have ten buildings of linear geometry with four stories and an annex
building with a different purpose, where academic, research and administrative activities are carried out. The
structure of the buildings is made of steel, and the walls are made of prefabricated material. Fig. 1 shows the
distribution and location of all the buildings of the unit that make up the study. All of them were built at distinct
stages according to academic and professional needs. In the red box, you can see nine buildings in parallel and
one transverse building, the longest one. They all have the same building structure, as shown in Fig. 2. The
smaller red boxes show classroom buildings, offices, and teachers' cubicles on three-stories and a four-stories
foreign language teaching center. The laboratories and workshops for maintenance and miscellaneous services
in the green box are single-story buildings with high walls and a laminated roof. The administrative buildings
in the blue box are one-story, and only the building shown in the lower part of the same box has two stories. In
the yellow box, the national library was built on three levels, a basement, and a two-levels auditorium, with
glass envelopes. Finally, there are five coffee shops, one on each side of the parallel buildings. Although the
unit has more buildings and various facilities, the set under study was considered because it is connected to the
same electrical system. In contrast, the rest has three independent electrical supply networks.
Fig. 1. Satellite image of the Unidad Professional Adolfo López Mateos, Zacatenco.
Administrative buildings.
National Library and main
auditorium.
Buildings for service
workshops.
Classroom, laboratory, library,
auditorium and office
buildings.

Statistical Analysis and SARIMA Forecasting Model Applied to Electrical
Energy Consumption in University Facilities
José Luis Reyes Reyes, Guillermo Urriolagoitia Sosa,
Francisco Javier Gallegos Funes, Beatriz Romero Ángeles,
Israel Flores Baez, Misael Flores Baez
Científica, vol. 26, núm. 2, pp. 01-22, julio-diciembre 2022, ISSN 2594-2921, Instituto Politécnico Nacional MÉXICO
DOI: https://doi.org/10.46842/ipn.cien.v26n2a03
8
Fig. 2. Classroom, laboratory, library, auditorium and office buildings, and offices with linear steel structure.
On the other hand, since it is an educational institution, its operation is determined by the seasonality of the
academic periods, in this case, two-semester periods; one takes place from September to February, while the
other from March to August. The buildings with classrooms are used Monday through Friday from 7:00 am to
10:00 pm, while the library operates Monday through Friday from 8:30 am to 8:00 pm; Saturday and Sunday
from 9:00 am to 4:30 pm; the foreign language center Monday through Friday from 7:00 am to 9:00 pm;
Saturday and Sunday from 7:00 am to 12:00 pm. In the case of laboratories and services, only Monday through
Friday from 7:00 am to 8:00 pm. The spaces dedicated to research do not have a limited and established schedule
of activities.
II.2 DESCRIPTION OF THE DATA
The professional unit is supplied with electrical energy. Other energy sources, such as gas or fuels, are of
specific consumption and are not relevant to the study, so only information from the electrical energy source
was collected. Electricity consumption data were obtained from the General Services Department of the
Institute and came from the billing provided by the electricity supply company. These are monthly for a period
of five years from 2015 to 2019, meeting the requirements for a building energy analysis; monthly billing data
and covering a period of two or more years, sufficient for a level 0 or 1 audit, as established by the American
Society of Heating, Refrigerating and Air-Conditioning Engineers (ASHRAE) [52]. On the other hand,
information before 2015 was not considered due to events inside the Institute that forced the closure of the
facilities for more than two months in 2014 and distorted the behavioral pattern of electricity consumption,
producing harmful outliers for the prediction models. As shown in Fig. 3, the annual electricity consumption in
the academic unit has a strong dependence on school periods and a tendency to decrease from year to year.
On the other hand, there was an irregularity in the annual consumption cycle in 2015, caused by the events
mentioned in 2014 that forced changes in the school calendar. As a result, in March 2015, activities concluded,
and a one-week vacation began in April, leading to a drop in consumption. When activities resumed, consumption
increased until July, and in August, there was a drop in consumption due to the summer vacation period. With
the start of activities in September, consumption increased until it dropped significantly in December because
only the first two weeks of the year were worked.

Statistical Analysis and SARIMA Forecasting Model Applied to Electrical
Energy Consumption in University Facilities
José Luis Reyes Reyes, Guillermo Urriolagoitia Sosa,
Francisco Javier Gallegos Funes, Beatriz Romero Ángeles,
Israel Flores Baez, Misael Flores Baez
Científica, vol. 26, núm. 2, pp. 01-22, julio-diciembre 2022, ISSN 2594-2921, Instituto Politécnico Nacional MÉXICO
DOI: https://doi.org/10.46842/ipn.cien.v26n2a03
9
Fig. 3. Electricity consumption of the academic unit from 2015 to 2019.
It is not until the following year, 2016, that the annual cycle, due to calendar adjustments, begins to regularize,
and the minimums in consumption occur in March 2016, April 2017, and April 2018 because of the one-week
vacation period. Not so the case of April 2019, where the decrease in energy consumption due to the holiday
period is marginal. This is because there were more academic and work activity days in April. From May to June,
consumption increased due to the activities, concluding at the end of June. July saw the summer vacation period,
reducing annual consumption from 2016 to 2019. After the break, by August, consumption increases until
reaching its maximums, which corresponds to the beginning of activities of the second school period of the year.
After that, consumption decreases until the December holiday as only the first two weeks are worked.
III. RESULTS
The results of this study were divided into three parts: seasonal analysis, autocorrelation analysis, and modeling.
III.1 Seasonal analysis
Fig. 4 shows the grouping of the frequency unit, i.e., the same month of each year. Thus, the average in each
frequency group was examined, showing that, on average, monthly consumption varies except in May and June,
where it is similar. On the other hand, the annual behavior during the school cycles; July and December were
the months where activities are reduced; the vacation periods of April and December decrease consumption,
and the peak occurs in October, the month of greatest academic, research, and administrative activity in the
academic unit. The variations in the monthly frequency averages have their origin in the trend of the series. The
decreasing trend presented by the series did not significantly modify the monthly difference since it decreased
the series in the same proportion. In addition, the variance (standard deviation) of the monthly averages of each
frequency decreased since it aligns each monthly observation closer to its frequency peers.
400
500
600
700
800
2015 2016 2017 2018 2019 Year
Electrical energy consumption (MWh)

Statistical Analysis and SARIMA Forecasting Model Applied to Electrical
Energy Consumption in University Facilities
José Luis Reyes Reyes, Guillermo Urriolagoitia Sosa,
Francisco Javier Gallegos Funes, Beatriz Romero Ángeles,
Israel Flores Baez, Misael Flores Baez
Científica, vol. 26, núm. 2, pp. 01-22, julio-diciembre 2022, ISSN 2594-2921, Instituto Politécnico Nacional MÉXICO
DOI: https://doi.org/10.46842/ipn.cien.v26n2a03
10
Fig. 4. Graph of the monthly average by frequency
Fig. 5. Graph of consumption behavior per cycle
However, the series of monthly averages did not show how consumption behaved by month and year. By month,
July and August presented significant volatility, produced by the events of 2014, Fig. 5, while the effect due to
the cycle shift in the first quarter of 2015 was presented from June to August 2015, as shown in Fig. 6. The
cyclical pattern is also distinguishable with the 2015 shift. This shift was the major contributor to the adjustment
error due to the large dispersion of observations for July and August, as shown in Fig. 7.
700
650
600
550
500
750
Media (MWh)
Jun Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Month
2015
2016
2017
2018
800
750
700
650
600
550
500
450
400
By frequency cycle
(MWh)
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Month

Statistical Analysis and SARIMA Forecasting Model Applied to Electrical
Energy Consumption in University Facilities
José Luis Reyes Reyes, Guillermo Urriolagoitia Sosa,
Francisco Javier Gallegos Funes, Beatriz Romero Ángeles,
Israel Flores Baez, Misael Flores Baez
Científica, vol. 26, núm. 2, pp. 01-22, julio-diciembre 2022, ISSN 2594-2921, Instituto Politécnico Nacional MÉXICO
DOI: https://doi.org/10.46842/ipn.cien.v26n2a03
11
Fig. 6. Consumption behavior graph by month and cycle
Fig.7. Box plot: Shows the dispersion of observations and median
Fig.7. Box plot: Shows the dispersion of observations and median
Jun
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
650
400
2015 2015,5 2016 2016,5 2017 2017,5 2018 Year
By frequency unit (MWh)
800
750
450
600
550
500
700
800
750
700
650
600
550
500
450
400
By frequency unit (MWh)
Jun Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Month

Statistical Analysis and SARIMA Forecasting Model Applied to Electrical
Energy Consumption in University Facilities
José Luis Reyes Reyes, Guillermo Urriolagoitia Sosa,
Francisco Javier Gallegos Funes, Beatriz Romero Ángeles,
Israel Flores Baez, Misael Flores Baez
Científica, vol. 26, núm. 2, pp. 01-22, julio-diciembre 2022, ISSN 2594-2921, Instituto Politécnico Nacional MÉXICO
DOI: https://doi.org/10.46842/ipn.cien.v26n2a03
12
III.2 Autocorrelation analysis
The autocorrelation function, ACF [56], [57], [58] is a tool for analyzing the linear dependence, stationarity,
and trend of variables in a time series. The autocorrelation function measures the correlation between two
variables separated by k periods, i.e., the lags of the series or distance between periods.
󰇛󰇜󰇛󰇜
󰇛󰇜󰇛󰇜
(20)
Equation (20) shows the form of the autocorrelation function in which 󰇛󰇜 is the covariance between
and , 󰇛󰇜 and 󰇛󰇜 the variances of and , respectively. Along with the ACF there is also
the partial autocorrelation function, PACF, which, unlike the ACF, the partial autocorrelation function measures
the correlation between two variables separated by k periods when the effect of other lags is removed, i.e., the
dependence created by the lags existing between the two variables is not considered. Partial autocorrelation is
defined in the form of equation (21)
󰇛
 
󰇜
󰇛
󰇜󰇛 
󰇜
(21)
Where the expressions 
 and 
 represent the regressions for  and .
By applying this correlation analysis, it was found that the time series of the consumption data has a trend (Fig.
3), the series is not stationary. Fig. 8 shows the ACF correlogram for the series, where the lags 1, 2, 3, and 4
represent seasonal lags and correspond to the periods of 12, 24, 36, and 48 months. As can be seen, the first
seven lags presented significant correlation, where the blue line in the graph shows the 5% critical values at ±
1.96n1/2 under the null hypothesis of white noise, being n the sample size. The way in which the peaks are
decreasing is due to the trend of the series, while the change in the direction of the peaks demonstrates their
seasonality. The PACF in Fig. 9 confirms the seasonal behavior of the series and its non-stationarity.
Fig. 8. ACF correlogram of electricity consumption time series
Fig. 8. ACF correlogram of electricity consumption time series.
1.0
0.6
0.2
-0.2
0 1 2 3 4 Lag
Autocorrelation Function, ACF

Statistical Analysis and SARIMA Forecasting Model Applied to Electrical
Energy Consumption in University Facilities
José Luis Reyes Reyes, Guillermo Urriolagoitia Sosa,
Francisco Javier Gallegos Funes, Beatriz Romero Ángeles,
Israel Flores Baez, Misael Flores Baez
Científica, vol. 26, núm. 2, pp. 01-22, julio-diciembre 2022, ISSN 2594-2921, Instituto Politécnico Nacional MÉXICO
DOI: https://doi.org/10.46842/ipn.cien.v26n2a03
13
Fig. 9. PACF correlogram of the electricity consumption time series
Fig. 9. PACF correlogram of the electricity consumption time series
III.3 Modeling
The modeling strategy consisted of building a forecast model from training and test partitions. First, the series
of consumed electric power data underwent a differencing process, as described in the methodology, to stabilize
the mean and variance. Then, the data were exponentially smoothed to minimize the impact of the irregular
behavior of the first months of 2015. Subsequently, training and test partitions were created, following the
criterion that the length of the test partition should not exceed 30% of the time series data. Due to the number
of observations in the time series (55 months), two models were built: one with a training partition of 48 months
(from January 2015 to December 2018) and a test partition of 7 months (from January to July 2019). Another
was with a training partition of 43 months (from January 2015 to July 2018) and a test partition of 12 months
(from August 2018 to July 2019). The procedure described above was performed using R statistical software
and the RStudio platform. Once the order of the SARIMA model (i.e., the values of p, d, q, P, D and Q) was
found, the estimation of the parameters followed, using the maximum likelihood estimation (MLE). This
technique finds those parameter values that have the maximum likelihood of obtaining the observed data,
employing the Bayesian Information Criterion, BIC = -2ln (maximum likelihood estimate) + zln (n), where z
represents the number of model parameters and n the number of observations used in the model. The best model
is obtained by minimizing the value of BIC.
So, among the fit models tested in the 48-month training partition, the best fitting model was SARIMA (3,1,1)
(1,0,0)12, which showed the BIC value = -63.77039. Given the model, the residuals were analyzed, and it was
found that they have a normal distribution and the ACF (Fig. 10) that the lags showed no correlation. On the
other hand, to test whether a series of observations in a specific period are independent, the Ljung-Box test was
used. The Ljung-Box test showed a p-value of 0.1128 above 0.01 as the significance test value and thus
confirmed that there is no correlation in the residuals. The results show that, even though the model worked
with the training partition data, the forecast accuracy of the test data is better, as demonstrated by all the metrics.
The same procedure was performed to model the 43-month training partition, and the SARIMA (2,1,2) (1,0,0)12
model was obtained, with a BIC = -59.10216. Fig. 11 corresponds to the residual analysis of the model. Its
distribution is normal, and the Ljung-Box test is p-value = 0.03941, which confirms that there is no correlation
between lags.
0.1
-0.1
0.3
-0.3
0 1 2 3 4 Lag
Partial autocorrelation function, PACF

Statistical Analysis and SARIMA Forecasting Model Applied to Electrical
Energy Consumption in University Facilities
José Luis Reyes Reyes, Guillermo Urriolagoitia Sosa,
Francisco Javier Gallegos Funes, Beatriz Romero Ángeles,
Israel Flores Baez, Misael Flores Baez
Científica, vol. 26, núm. 2, pp. 01-22, julio-diciembre 2022, ISSN 2594-2921, Instituto Politécnico Nacional MÉXICO
DOI: https://doi.org/10.46842/ipn.cien.v26n2a03
14
Fig. 10. Graph of the residual analysis of the 48-month partition from SARIMA (3, 1, 1) (1, 0, 0)12
Fig. 11. Residual analysis of the 43-month training model from ARIMA (2, 1, 2) (1, 0, 0)12 with shift
0.1
0.0
-0.1
-0.2
-0.3
ACF
0.2
0.3
0.1
0.0
-0.1
-0.2
-0.3
2015 2016 2017 2018 2019 Year
5 10 15
15
10
5
Count
-0.2 0.0 0.2
0
Lag
Residual
0.1
0.0
-0.1
-0.2
-0.3
2015 2016 2017 2018 Year
0.2
12.5
10.0
0.0
Count
5.0
-0.2
7.5
ACF
2.5
0.0
Residual
Lag
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
2 4 6 8 10 12 14

Statistical Analysis and SARIMA Forecasting Model Applied to Electrical
Energy Consumption in University Facilities
José Luis Reyes Reyes, Guillermo Urriolagoitia Sosa,
Francisco Javier Gallegos Funes, Beatriz Romero Ángeles,
Israel Flores Baez, Misael Flores Baez
Científica, vol. 26, núm. 2, pp. 01-22, julio-diciembre 2022, ISSN 2594-2921, Instituto Politécnico Nacional MÉXICO
DOI: https://doi.org/10.46842/ipn.cien.v26n2a03
15
III.4 Prediction precision criteria
Once the models were built, their predictive capacity was analyzed, and for this purpose, the mean absolute
percentage error, MAPE, was used, which is calculated by a term-to-term comparison of the relative error of
the prediction value with respect to the real observed value, as shown in equation (22).
  
󰈅

󰈅


(22)
Where  are the actual values and 
 are the predicted values, and n is the number data observations considered.
Table 2 shows the error metric of both the training and test partitions for the two models. When analyzing the
values in the error metric, it is observed that the fit to the training partition was better for the SARIMA (2,1,2)
(1,0,0)12 model. In comparison, the test partition was better fitted by the SARIMA (3,1,1) (1,0,0)12 model. This
result is confirmed by analyzing the plots in Fig. 12 and Fig. 13. The test values are closer to the actual values
of the series for the SARIMA (3,1,1) (1,0,0)12 model. While the fitted values for the training partition are better
represented by the SARIMA (2,1,2) (1,0,0)12.
TABLE 2. ERROR METRIC FOR THE TRAINING AND TEST PARTITIONS
Model
SARIMA (3,1,1) (1,0,0) 12
SARIMA (2,1,2) (1,0,0) 12
Partition
Training
Test
Training
Test
Parameter
MAPE
1.09405146
0.9320904
0.93786799
1.62500404
Fig. 12. Current vs. forecast values and adjustment; SARIMA (3,1,1) (1,0,0)12
Actual
Fitted
Forecasted
6.7
6.6
6.5
6.4
6.3
6.2
6.1
6.0
2015 2016 2017 2018 2019 Year
Logarithm of energy consumption

Statistical Analysis and SARIMA Forecasting Model Applied to Electrical
Energy Consumption in University Facilities
José Luis Reyes Reyes, Guillermo Urriolagoitia Sosa,
Francisco Javier Gallegos Funes, Beatriz Romero Ángeles,
Israel Flores Baez, Misael Flores Baez
Científica, vol. 26, núm. 2, pp. 01-22, julio-diciembre 2022, ISSN 2594-2921, Instituto Politécnico Nacional MÉXICO
DOI: https://doi.org/10.46842/ipn.cien.v26n2a03
16
Fig. 13. Current vs. forecast values and adjustment; SARIMA (2, 1, 2) (1, 0, 0)12
III.5 Confidence intervals
To show how accurate the forecast model is, confidence intervals were used. This is a statistical approximation
method to express a range of possible values in which the observed value of the series lies with a certain degree
of certainty, i.e., with a given probability. However, any percentage of probability can be used in the confidence
interval. For this study, the usual gaps of 80% and 95% were considered. As can be seen in Fig. 14, the confidence
intervals for the 7-month forecast of the SARIMA (3,1,1) (1,0,0)12 model is extensive, both for the 80% and 95%
levels and the same situation occurs for the SARIMA (2,1,2) (1,0,0)12 model (Fig. 15).
Fig. 14. SARIMA (3,1,1) (1,0,0)12 7-month forward consumption forecast.
2015 2016 2017 2018 2019 Year
6.0
Observed
80% confidence
95% confidence
Forecasted
6.7
6.6
6.5
6.4
Logarithm of energy consumption
6.3
6.2
6.1
5.9
6.7
6.6
6.5
6.4
6.3
6.2
6.1
6.0
2015 2016 2017 2018 2019 Year
Logarithm of energy consumption
Actual
Fitted
Forecasted

Statistical Analysis and SARIMA Forecasting Model Applied to Electrical
Energy Consumption in University Facilities
José Luis Reyes Reyes, Guillermo Urriolagoitia Sosa,
Francisco Javier Gallegos Funes, Beatriz Romero Ángeles,
Israel Flores Baez, Misael Flores Baez
Científica, vol. 26, núm. 2, pp. 01-22, julio-diciembre 2022, ISSN 2594-2921, Instituto Politécnico Nacional MÉXICO
DOI: https://doi.org/10.46842/ipn.cien.v26n2a03
17
Fig. 15. SARIMA (2,1,2) (1,0,0)12 7-month forward consumption forecast
For the 12-month forecast horizon, as shown in Fig. 16, the SARIMA (3,1,1) (1,0,0,0)12 model predicted the
trend and seasonal behavior of the original series. This was not the case for the SARIMA (2,1,2) (1,0,0)12 model,
which showed a downward trend and did not reproduce the seasonal pattern of the original series. In Fig. 17,
its confidence intervals are more extensive than the SARIMA (3,1,1) (1,0,0)12 model, which means greater
uncertainty in the expected value.
Fig. 16. SARIMA (3,1,1) (1,0,0)12 12-month forward consumption forecast.
Observed
80% confidence
95% confidence
Forecasted
2015 2016 2017 2018 2019 Year
5.9
6.0
6.2
6.3
6.4
6.5
6.6
6.7
Logarithm of energy consumption
Observed
Forecasted
2016 2017 2018 2019 2020 Year
Logarithm of energy consumption
6.7
6.6
6.5
6.4
6.3
6.2
6.1
6.0
5.9
95% confidance
80% confidance

Statistical Analysis and SARIMA Forecasting Model Applied to Electrical
Energy Consumption in University Facilities
José Luis Reyes Reyes, Guillermo Urriolagoitia Sosa,
Francisco Javier Gallegos Funes, Beatriz Romero Ángeles,
Israel Flores Baez, Misael Flores Baez
Científica, vol. 26, núm. 2, pp. 01-22, julio-diciembre 2022, ISSN 2594-2921, Instituto Politécnico Nacional MÉXICO
DOI: https://doi.org/10.46842/ipn.cien.v26n2a03
18
Fig. 17. SARIMA (2,1,2) (1,0,0)12 12-month forward consumption forecast
IV. DISCUSSION
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,
Logarithm of energy consumption
6.7
6.6
6.5
6.4
6.3
6.2
6.1
6.0
5.9
2016 2017 2018 2019 Year
5.8
Observed
80% confidence
95% confidence
Forecasted

Statistical Analysis and SARIMA Forecasting Model Applied to Electrical
Energy Consumption in University Facilities
José Luis Reyes Reyes, Guillermo Urriolagoitia Sosa,
Francisco Javier Gallegos Funes, Beatriz Romero Ángeles,
Israel Flores Baez, Misael Flores Baez
Científica, vol. 26, núm. 2, pp. 01-22, julio-diciembre 2022, ISSN 2594-2921, Instituto Politécnico Nacional MÉXICO
DOI: https://doi.org/10.46842/ipn.cien.v26n2a03
19
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.
V. CONCLUSIONS
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.
ACKNOWLEDGEMENTS
This work has been possible thanks to the General Services Department of the Instituto Politécnico Nacional,
which provided the information, and to the Sección de Estudios de Posgrado e Investigación of the Escuela
Superior de Ingeniería Eléctrica y Mecánica of the Instituto Politécnico Nacional (SEPI-ESIME) for lending
the facilities for the study. This research did not receive any specific grant from funding agencies in the public,
commercial, or not-for-profit sectors.
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