1. Introduction
Energy efficiency means the amount of energy saved determined by measuring or estimating consumption before and after the implementation of an energy efficiency improvement measure, while ensuring the normalization of external conditions affecting energy consumption [
1]. This definition applies to all branches of the European economy, but has a particular impact on the building sector, as energy consumption in the European building sector accounts for about 40% of the total energy demand of the European Union [
2]. According to [
3], the building sector consumes 35.3% of the final energy demand. This activity is responsible for the emission of about 36% of CO
2 pollution in the European Union [
2]. The European Union’s (EU) climate-energy policy, including its long-term vision to strive for EU climate neutrality by 2050 and the regulatory mechanisms to stimulate the achievement of effects in the coming decades, has a significant impact on shaping the energy strategy. Achieving the 2020 and 2030 climate and energy targets in the EU is key to a low-carbon energy transition. In line with the EU’s decarbonization ambition, in December 2020. The European Council approved in December 2020 a binding EU target to reduce net greenhouse gas emissions by at least 55% by 2030 compared to 1990 levels, thereby increasing the existing 40% reduction target. Additionally, there is a provision on increasing the share of renewable sources in the final energy consumption to at least 32% and increasing the energy efficiency by 32.5% [
4]. Reducing energy consumption and using renewable energy sources to a greater extent is intended to increase the security of energy supply and support technical development in individual countries. Reducing energy consumption in the residential sector is a priority action in the EU member states. These actions allow efficient and sustainable use of fossil fuel potential and reduce gas and dust emissions. In order to improve the efficiency of energy use for heating and air conditioning of buildings and reduce greenhouse gas emissions, in 2018 the European Parliament and the Council amended the Directive 2018/844 on the energy performance of buildings [
5]. The provisions set out in the Directive should contribute to the improvement of energy performance of buildings also in Poland, where the covering of thermal needs is done at the local level, therefore it is extremely important to ensure energy planning at the level of municipalities and regions—it is crucial for rational energy management, improvement of air quality and exploitation of local potential. A useful tool will also be the launch of a nationwide heat map (developed in connection with Article 14 of the revised Energy Efficiency Directive 2012/27/EU [
6]), which will facilitate planning for covering heat needs. National analyses show that current technical conditions [
7] do not guarantee energy consumption at the optimal, economically justified level. The analysis presented in the studies [
8,
9,
10] shows that the average value of the final energy demand indicator in Poland varies from 85 to 343 kWh/(m
2 year), depending on the type of building, with single and multi-family residential buildings obtaining values much higher than European averages. In residential buildings, the most important area of final energy consumption in a building is heating, which represents 76% of its total consumption [
7]. By minimizing heat losses and modernizing the heating system, energy efficiency in residential buildings can be improved. In order to achieve the objectives of Directive 2018/844 [
5], the construction sector in Europe is waiting for the transition to energy efficient and passive buildings. No less important is to improve the energy standard of existing buildings through their thermomodernisation. In the case of buildings existing prior to energy saving measures, an assessment of the energy performance of the buildings is necessary to obtain adequate knowledge of the existing energy consumption profile of the building or group of buildings, to determine how and in what quantity cost-effective energy savings can be achieved, and to inform about the results. Therefore, it is necessary to search for tools that will allow estimating the potential for savings in energy consumption for heating buildings. These tools will help policy makers to implement the intentions of the European Union’s energy policy, which aims to assess the energy saving potential and thus reduce pollutant emissions from the building sector.
2. Literature Review of Estimation Methods for Building’s Energy Demand
The assessment of energy efficiency in buildings can be done in different ways, which can be classified as: engineering calculations, statistical and data-based models (machine learning), and hybrid models [
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24,
25,
26,
27,
28,
29,
30,
31,
32,
33,
34,
35,
36].
Figure 1 shows the techniques of modelling and forecasting energy consumption in buildings with a division into individual calculation methods. Additionally, the diagram contains quotations of works in which the authors used a given method to forecast energy consumption for heating or to determine energy efficiency (energy performance) in various types of buildings.
Literature analysis has shown that statistical and artificial neural network-based models are most commonly used, while fuzzy logic-based models are used much less frequently.
A fuzzy approach for determining the energy performance of buildings was applied by Nebot and Mugica [
34], where two methods were used for prediction: fuzzy inductive inference (FIR) and adaptive neuro-fuzzy inference system (ANFIS). The study was performed on a set of simulated residential buildings developed by Tsanas and Xifara [
30]. The computational results obtained by the authors [
34] have been compared with the results obtained for the same input data (for simulated buildings) using static and artificial neural network-based methods [
25,
28,
30,
31]. The results showed that the use of methods based on fuzzy logic gives very good results in predicting energy consumption [
34]. The presented forecasting models are mainly focused on estimating energy consumption, energy characteristics in simulated objects, for which reliable data on building envelope insulation, ventilation air flows and number of inhabitants can be obtained [
28,
30,
31,
32,
33,
34]. The presented research results are promising and indicate high accuracy in estimating energy consumption in buildings, however, there is a lack of studies on real buildings [
14,
25,
26,
27,
29], for which it is difficult to obtain unambiguous data. Individual energy demand in residential buildings is more difficult to estimate due to the lack of data on building occupancy and complexity of occupant behavior. In particular, there is a lack of studies on prefabricated buildings. Buildings erected in the 1960s and 1970s of the last century in the large panel technology, by definition, are not able to meet the thermal and energy requirements of today [
29]. Conducted research on the analysis of thermal insulation of external walls of buildings built in large panel technology, confirmed the deterioration of their thermal insulation, mainly due to the use of concrete with increased density. Defects and errors resulting from the execution or damage of thermal insulation layers and the location of thermal bridges also had an impact. The occurrence of these places was caused by insufficient thickness of thermal insulation or its absence [
26]. An additional factor making it difficult to estimate energy consumption for heating is the way the apartments are used. It results from inappropriate approach to ventilation of rooms, e.g., covering ventilation grilles, which directly influences the size of ventilation air stream, which also leads to dampness of partitions. A frequent problem in thermal calculations is the lack of complete architectural and building documentation, which hinders correct estimation of energy consumption for heating [
26,
29]. Therefore, in our research we want to use fuzzy set theory supported by neural network to predict the energy efficiency of real multifamily residential buildings made with prefabricated technology that have undergone thermal improvement process. We believe that the fuzzy logic-based method provides a convenient tool to describe the uncertainty and imprecision of the input data inherent in the energy demand prediction process. In this research, we propose to apply models with Takagi-Sugeno type inference for residential energy efficiency forecasting. It should be emphasized that, as shown in the literature review, it has not been applied in energy assessment of existing buildings before, and thus is new to this type of research. Due to the different availability and accuracy of data describing buildings, we want to investigate and test different configurations of input variables in order to obtain a trade-off between the auditor’s effort to obtain them and the accuracy of the forecast. The results obtained will be compared with other forecasting methods (Neural Methods and Rough Set Theory) [
26,
29], which were based on the same input data characterizing a set of real residential buildings undergoing thermal improvements.
3. Materials and Methods
The research was carried out on a group of multi-family residential buildings built in the 1960s to 1980s, which were constructed in the technology of large plates (prefabricated elements). The energy characteristics of the analyzed group of buildings have been discussed in the paper [
29]. Before proceeding to the analyses, the collected database was subjected to preliminary processing. It consisted among others in elimination of those buildings, which did not have complete information. Thanks to that, it was possible to create a group including 109 buildings. These buildings were characterized by high variability of parameters influencing energy consumption, which is presented in
Table 1.
In the next step, qualitative variables such as, information about which partition was undergoing thermal improvement were converted into a quantitative variable. Each qualitative variable could be in two response options “Yes” or “No”. These words were replaced by numbers: “1” and “0”. The value “1” was put when the event occurred (the partition was thermally improved). Otherwise, when the event did not occur the variable took the value “0”. Since zero-one variables sum to unity the resulting matrix is singular. In order to eliminate the linear relationship between the variables in further studies, the variable that most often took the value of “1” was omitted. Next, the independent variables that were not statistically significantly correlated with the explanatory quantity were removed from the collected database. In the last step of pre-processing the data was randomly divided into a learning set and a test set. The learning set was formed from 75% of observations and the rest was the test set. In order to ensure comparability of results obtained for particular methods during the research no objects were moved between particular sets. From the data base prepared in this way, independent variables were selected for four purposefully chosen sets describing the process of energy demand change in buildings after thermomodernization. The individual sets of variables were created based on the statistical analyses performed, in which the requirement was that the independent variables be statistically significantly correlated with the dependent variable, while the degree of correlation between the independent variables could not be greater than 0.3. On this basis, four sets of variables that met these requirements were identified. The required effort to collect all the necessary information was also taken into account when creating each group of variables. The buildings subjected to analysis were characterized by different availability of data, some of the buildings in use had full documentation describing their technical condition and equipment enabling monitoring of energy needs. For older buildings, there has often been a problem in obtaining reliable and up-to-date data on their energy demand.
In this paper, an attempt was made to estimate the energy efficiency of buildings undergoing thermal improvement on the basis of different sets of diagnostic variables. These sets differ not only in the number of variables, but also in the specificity of the information provided. Some of the variables are of typically energetic character, such as heating power demand or seasonal energy demand for heating, while others describe constructional parameters such as: surface areas of partitions through which heat losses occur, heated volumes, heat transfer coefficients through partitions. Utility parameters were also taken into account, such as the number of people using the building. Detailed characteristics of the individual sets of variables are presented in
Table 2. The first set contained the fewest indices, but they were the most strongly correlated with the dependent variable. They were the calculated heat power of the heating system before modernization and the annual final energy demand indicator (before modernization). This set, just like the other three sets, also contained information on which partition was subjected to thermal upgrading. Practical use of this set will be possible only in the objects in which the calculations of design heat load for heating have been carried out according to EN-ISO 12,831 [
35]. From the next set of variables, information concerning the demand for heating power has been eliminated and replaced by information concerning characteristic dimensions of particular parts of the building such as: surface area of particular partitions, surface and volume of the building and also indicators characterizing the given building (compactness coefficient of the building, number of people using the building, number of flats). The use of the above-mentioned sets (I and II) will be possible only in those facilities, where the measurements of final energy consumption for heating are performed and archived. The third set of variables, from which information concerning energy consumption was eliminated, was supplemented with heat transfer coefficients for individual rooms. Gathering such an extensive range of information allows for an accurate characterization of the facility but requires a great deal of effort to prepare it reliably. Due to the fact that the third set of variables was extensive and therefore collecting such an extensive range of data is time-consuming, in the last set of input data only variables with direct influence on heat losses in the object were left, such as heat transfer coefficients through partitions and surface areas of partitions through which heat losses occur.
Table 2 lists the individual parameters included in the sets of input variables, with the parameter’s membership in a given set denoted by 1.
The developed sets of variables were used to build a forecasting model. The study tested the usefulness of Takagi-Sugeno type neuro-fuzzy sets for predicting the energy demand index in residential buildings undergoing thermal upgrading.
The prediction of the variable based on the neuro-fuzzy model followed an iterative procedure. Its structure is schematically illustrated in
Figure 2.
The waveforms of the variables obtained during the measurements were divided into learning and test matrix (green area in the diagram). The learning matrix, containing the input data and the output variable, was the basis for formulating a predictive model with a Takagi-Sugeno structure. This model consisted of three basic functional modules (blue area in the diagram): blurring, inference, and sharpening. The former module was responsible for the blurring of input variables. It consisted in dividing variables into subsets described by membership functions. The second module—inference—was responsible for processing input data into appropriate value of output variable in the space of fuzzy variables. Inference process was based on rule base describing relations between input variables and output variable. In the last module of sharpening, the resultant value of output variable—forecast—was determined. The processes outlined above, in the blue area of the diagram, occurred automatically. The procedure, however, required the prior definition of a number of parameters related to the optimization of the model, such as: the type of the blurring function, the division into subsets, the type of the output function (linear or constant) and the number of learning epochs. In the next iteration step, a computer simulation was performed, during which the prediction calculated using the model was compared with the actual variables of the test matrix (purple area in the diagram). The result of the analysis was the information about the quality of the model (its measure is the error values described by relations 1–3, which are discussed below). The acceptance of the final version of the model (acceptable error value) meant the end of the iteration procedure—the transition to the stop stage. In the case when the error value exceeded the accepted assumptions, the prediction model was subjected to further optimization—there was a transition to the ANFIS stage (see diagram). At the beginning of this stage, the set of parameters related to the process of fuzzifying the input variables and to the output function was redefined (in comparison to the previous iterations). ANFIS (Adaptive Neuro-Fuzzy Inference System) using an adaptive five-layer neural network automatically performed Sugeno-type predictive model optimization using two algorithms: back propagation and least squares estimation. The error back propagation algorithm was used to determine the parameters of the fuzzy function. Through the mean square output error minimization algorithm (with respect to the learning data), the weights of the neural network were gradually updated [
36].
After optimization using ANFIS, the quality of the model was checked again and on this basis a decision was made to either correct it further or terminate the iterative procedure. For a complete picture of the methodology, apart from its structure (
Figure 2), it is necessary to present also the data processing method during the forecast calculation. When formulating neuro-fuzzy models based on the learning matrix, calculations were initially performed, taking into account successively all types of membership functions available for the fuzzification stage (
Figure 3) [
37].
Initially, the minimum number of subsets sharing the input variable values and the minimum number of learning epochs (3 subsets and 3 epochs) were used during the fuzzification. Such an example of initial configuration of blurring parameters for one of the input variables is shown in
Figure 4.
As can be observed (
Figure 4), the set of values of the input variable
Ve (calculated from exterior measurements of the heated volume of building) is divided by the triangular membership function μ (
Ve) into 3 subsets: small S, medium M, and big B.
For the prepared models, the most accurate forecasts (with the smallest error) from particular data sets were selected on the basis of error values. For the selected forecasts in subsequent iteration steps, the model optimization parameters were increased by increasing the number of subsets and the number of epochs.
In the course of calculations, the number of learning epochs was maximally increased up to 50, because according to Nebot and Mugica [
34], the increase in the number of epochs above this value causes a significant extension (nearly six times) of the computer simulation time and, at the same time, does not affect the reduction of the forecast error value. The quality assessment of the developed models was based on the mean bias error (
MBE), coefficient of variance of the root mean square error (
CV RMSE) and coefficient of determination (
R2) which are accepted as statistical calibration standards by ASHRAE Guideline (American Society of Heating, Refrigerating and Air-Conditioning Engineers) 14-2014 [
38,
39,
40]. Other metrics frequently used in the literature such as
MAE (mean absolute error) and
MAPE (mean absolute percentage error) were also used for quality assessment [
38].
where:
yi is the actual value (quantity) in the facility
i, and
ypi is the forecast value (quantity) in the facility
i. The difference between
yi and
ypi is divided by the actual value
yi and
m is the index of number of test object,
ng is the number of objects (
m = 1, 2, 3, …,
ng).
Two linear ordering methods were used to assess the suitability of each set of variables to determine changes in the annual heat demand index for thermally improved buildings: Simple Additive Weighting (SAW) and a method based on a synthetic measure of development (SMR). MBE, CV RMSE, R2, MAE, and MAPE were used as diagnostic variables to determine the synthetic variable. Initially, the nature of the variables (stimulants, destimulants) and the weights of the variables were determined, taking only the R2 index as stimulant and equal weights for all diagnostic characteristics.
The conversion of variables of the nature of destimulants to stimulants was done during the normalization itself, using Weitendorf normalization [
41]:
Formula normalization of stimulants
Formula normalization of destimulants
—normalized value of the i-th variant according to the j-th criterion,
—value of the i-th variant according to the j-th criterion,
, —maximum or minimum value of the i-th variant according to the j-th criterion.
The SAW method was first presented in 1954 in a paper by Churchman and Ackoff [
42] and over time has become one of the most well-known algorithms for multicriteria analysis, mainly due to the simplicity of determining synthetic variables (
pi) and the intuitiveness of the method.
wi—weight for particular evaluation criteria, .
This paper also analyzed a linear ordering method based on the synthetic SMR measure, which is very popular in Poland. The author of this method is Hellwig, one of the precursors of Multiple Criteria Decision Making (MCDA) methods, and he described the algorithm of the method in 1968 in his paper [
43].
where:
—maximum value from the set of features,
n—number of objects.