Applying an artificial neural network to assess thermal transmittance in walls by means of the thermometric method

1 2 3 4 5 6 7 8 9 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 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Applying an artificial neural network to assess thermal transmittance in walls by means of the thermometric method Abstract Most of the existing building stock has a deficient energy behaviour. The thermal transmittance of façades is among those aspects which most affect this situation. In this paper, the calculation procedure with correction for storage effects from ISO 9869-1 was applied to the thermometric method to determine the U-value. Due to the need for determining the number and type of layers that compose the wall to apply the calculation, a multilayer perceptron has been developed to estimate the U-value. From the different model configurations suggested, the most adequate architecture was the one with 14 nodes in the hidden layer without making transformations in the input variables. Valid results have been obtained by the multilayer perceptron for the case studies analysed from different building periods, with deviations lower than 20% between the measured value and the expected one, varying the test duration according to the thermal resistance of the wall and the temperature variations. Furthermore, it is not necessary to carry out a data post-processing for the model, so this fact simplifies and hastens the calculation procedure. Keywords Thermal transmittance; thermometric method; thermal mass factors; artificial neural network; multilayer perceptron Nomenclature Symbols : thermal capacity of the layer [J/(m2·K)] : total internal thermal mass factor [J/(m2·K)] : total internal thermal mass factor using thermal resistance values corrected by CCF [J/(m2·K)] : internal thermal mass factor for each layer of the wall [J/(m2·K)] : moisture correction factor [dimensionless] : total external thermal mass factor [J/(m2·K)] : total external thermal mass factor using thermal resistance values corrected by CCF [J/(m2·K)] : external thermal mass factor for each layer of the wall [J/(m2·K)] : temperature correction factor [dimensionless] : total internal heat transfer coefficient [W/(m2·K)] : specific heat flux through the element at the instant j [W/m2] R: thermal resistance of the wall [(m2·K)/W] : thermal resistance of the material by applying the CCF correction [(m2·K)/W] : sum of the internal thermal resistances from the k-1th layer to the internal air [(m2·K)/W] : thermal resistance of the layer [(m2·K)/W] : sum of the external thermal resistances from the k+1th layer to the external air [(m2·K)/W] s: thickness [m] : internal air temperature at the instant j [K] : external air temperature at the instant j [K] internal surface temperature of the wall at the instant j [K] U: thermal transmittance [W/(m2·K)] : thermal transmittance obtained by the thermometric method [W/(m2 ·K)] : thermal transmittance obtained by the thermometric method with correction for storage effects [W/(m2·K)] : thermal transmittance obtained by the multilayer perceptron [W/(m2·K)] : weight of the bias neuron of the hidden layer of the multilayer perceptron : weights of the hidden layer of the multilayer perceptron : weights of the output layer of the multilayer perceptron : values of the input layer of the multilayer perceptron : input value of the bias neuron of the hidden layer of the multilayer perceptron : output of the final layer of the multilayer perceptron Greek symbols : time interval between the measurements [s] : difference between the average internal air temperature from the 24 h before measuring the observation and the 1 2 3 4 5 6 7 8 9 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 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 average internal air temperate from the first 24 h of the test [K] : difference between the average external air temperature from the 24 h before measuring the observation and the average external air temperature from the first 24 h of the test [K] : thermal conductivity [W/(m·K)] : activation function of the multilayer perceptron Abbreviations ANN: artificial neural networks AVRA: Agency for Housing and Rehabilitation in Andalusia BFGS: Broyden-Fletcher-Goldfarb-Shanno algorithm BP: back propagation CCF: conductivity correction factors CTE: Spanish Technical Building Code ECM: energy conservation measure MAE: mean absolute error MLP: multilayer perceptron MW: mineral wool P: building period P1: building period anterior to NBE-CT-79 P2: building period between NBE-CT-79 and CTE P3: building period posterior to CTE PUR: polyurethane R2: linear correlation coefficient RMSE: root mean square error THM: thermometric method XPS: extruded polystyrene 1. Introduction Society is more and more aware of the effect that everyday processes have on nature, causing the acceleration of global warming, climate change and the extinction of different species [1]. One of the fundamental aspects which would change this situation is the reduction of greenhouse gas emissions. Recently, the European Union has published a roadmap for moving to a low carbon economy with the aim of reducing pollutant gas emissions by 80% by 2050 [2]. To achieve this, it is estimated that the building sector should cut its emissions by 90% due to its high energy consumption [2]. In this sense, from the different activities and procedures carried out nowadays, residential buildings were responsible for 24.79% of the total primary energy consumption in Europe in 2014 [3]. The main consumption source in these buildings is the heating [4]. Therefore, it is necessary to cut the level of the greenhouse gas emissions by reducing the energy demand from the existing building stock. Within this context, the adoption of the energy conservation measures (ECMs) in old buildings is one of the most significant performances [2]. In order to configurate correctly the ECMs, it is essential to determine the thermal transmittance (U-value) of the envelope [5,6], since it usually exceeds the maximum admissible levels from the technical standard in most cases [7] and has a major impact on façades, in which the maximum admissible value of energy loss can be exceeded by more than 100% [8]. The determination of thermal transmittance can be carried out by both theoretical and experimental methods. The theoretical method is included in ISO standard 6946 [9], and it calculates thermal transmittance of a conventional wall through the thickness and the thermal conductivity of each of its layers, as well as the internal and external surface thermal resistances. It is a method with a high uncertainty level, since it is difficult to know accurately the layers of the wall unless an endoscopic analysis can be carried out [10] or technical data are available [11]. Regarding experimental methods, in scientific literature there is the heat flow meter method included in ISO 9869-1 [12] as well as quantitative methods of infrared thermography, with different approaches from both exterior and interior [13]. Recently, Andújar Márquez et al. [14], Bienvenido-Huertas et al. [15], and Kim et al. [16] validated the use of the thermometric method (also known as air–surface temperature ratio method) to determine the U-value by means of in situ tests. This method differs from the heat flow meter method in that the heat flux is not measured, but the internal surface temperature of the wall. However, no using the heat flux plate does not affect the representation of results, as differences lower than 5% between the thermometric method and the heat flow meter method can be obtained in steady state conditions [16]. 1 2 3 4 5 6 7 8 9 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 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Obtaining thermal transmittance value by means of the measurement using temperature sensors can reduce the error associated with the test because of putting the heat flux plate. In this sense, Meng et al. [17] determined that the maximum error due to the use of surface temperature probes in the U-value is of 6%, whereas using the heat flux plate increases the maximum error to 26%. Regarding the thermometric method (THM), another aspect to highlight is the fact that no using the plate avoids the disturbance that probe can cause to thermal behaviour of the wall [10,18]. With respect to infrared thermography methods, THM has the advantage of not being significantly influenced by the environmental factors. In this sense, the influence of input variables on the results of infrared thermography methods is very significant, leading to obtain atypical results [19,20]. Moreover, variations of 50% in the measurement of internal, external, and surface air temperature can lead to errors of 5%, 50%, and 50%, respectively [19], whereas a deviation of 1 °C in the measurement of the reflected temperature can cause errors up to 100% [21]. For walls with a high thermal resistance, the thermal storage effect should be considered [12]. The thermal mass is the capacity to store thermal energy, that is, the effect that the components with a large heat capacity can produce in order to alleviate the temperature oscillations and reduce the rate of change [22]. The thermal storage characteristics and the material transmission of a component should be studied as a complete system, since any material influences the other [23]. The high thermal mass of walls limits the use of experimental methods [24]. The fundamental limitation of experimental methods is that the change of amount of heat stored in the wall is required to be low in comparison with the amount of heat going through it [25]. However, this requirement is difficult to achieve, especially in less cold climate regions [26]. In scientific literature, there are studies analysing the application of the heat flow meter method in high thermal mass buildings with significant deviations in the results obtained. Asdrubali et al. [27] assessed the thermal transmittance of six walls with a high thermal resistance using the ISO standard 9869-1. Results obtained deviations up to 75% for the case studies analysed. Nardi et al. [28] evaluated two walls with low thermal transmittance values, obtaining deviations between 47 and 83% in the results. Samardzioska and Apostolska [29] analysed the thermal transmittance of a constructive system (Fragmat NZ-1) in three different walls. Results obtained deviations lower than 25%, although authors excluded certain monitored days, since the required conditions of data analysis were not achieved. Gaspar et al. [30] determined that steady state conditions with temperature differences higher than 19 °C can lead to obtain results with deviations of 1.9%, although it is difficult to guarantee this thermal gradient in actual conditions. Traditionally, the different steady state methods try to limit the thermal storage effect by extending the test duration to at least 72 h, and in some cases more than two weeks [31]. Duration is influenced by the wall typology. In this sense, Ficco et al. [32] determined that the sample period should be extended for those elements with a high thermal inertia, particularly in unfavourable conditions, such as a low thermal gradient or heat flow inversions. These test durations cause an excessive consumption of time resources as well as disturbances for dwellings´ tenants [33], and do not guarantee the limitation of thermal storage effect [34]. In this sense, Bienvenido-Huertas et al. [15] showed the difficulties to obtain representative results using THM when steady state conditions were not achieved during the tests. Kim et al. [35] indicated that thermal resistance values of walls and the difficulty in guaranteeing steady state conditions could make the correct estimation of test durations difficult in order to obtain representative results for THM, so the deviation of the result obtained can be high. The data analysis methodology used plays an important role. According to Cesaratto and Carli [36], the accuracy of thermal transmittance results could cause variations higher than 20%, depending on the method employed. A theoretical methodology to determine the U-value considering this effect is described in section 7.2. from ISO 9869-1. In this methodology, it is necessary to know the thermal characteristics of the layer components. Deconinck et al. [25] determined that using this methodology reduced test durations for the heat flow meter method. In this line, Choi and Ko [37] stated that test duration was reduced to two days by using this method, obtaining more accurate results than the average method. Thus, implementing this methodology for THM can be an opportunity to analyse walls with high thermal mass. Nevertheless, the composition of the wall is unknow in most cases [32], since a reliable technical documentation is not available or an endoscopy cannot be made. So, it is essential to have data analysis techniques that allow to determine the U-value of a wall without knowing exactly its layers to apply the corrections for storage effects. In this sense, the regression techniques have become very important to estimate thermal transmittance. From the different existing regression techniques, artificial neural networks (ANNs) are one of the typologies ensuring greater reliability and viability [38]. ANNs imitate the hardware structure of the nervous system with the aim of building parallel and adaptive processing information systems which can estimate an efficient response. Along with the Expert Systems, ANNs are one of the study gaps on artificial intelligence in the last years [39]. In the field of building energy problems, they have been widely used with correlation coefficients and error values adequate for each case [40]. Moreover, they have been extensively used as a prediction tool of energy demand and consumption [41], although other researches focused on the characterisation of the thermal properties of components can be highlighted, such as: (i) 1 2 3 4 5 6 7 8 9 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 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Chudzik [42] applied an ANN to determine the heat transfer coefficient on the specimen surface to establish the thermal properties of thermal insulating materials; (ii) an ANN was designed by Buratti et al. [43] to predict the U-value of wooden windows; (iii) Sablani et al. [44] designed two ANN models to estimate the heat transfer coefficient for a liquid-solid set through its internal temperature; (iv) Singh et al. [45] determined the effective thermal conductivity of porous materials filled with different liquids using ANNs; and (v) Mitra et al. [46] predicted the thermal resistance of handloom cotton fabrics using an ANN. However, all these researches used fixed values for the input variables of ANNs, so ANNs were not used as a data analysis method for time series obtained by monitoring an element. Therefore, this paper is aimed at filling the existing gap in the state of the art. A new data analysis method is suggested for the thermometric method by means of an ANN allowing to characterise thermal transmittance of walls, considering the effect of thermal storage and without having the need of knowing the exact composition of the wall (fundamental requirement to carry out the calculation procedure). This article gives a new approach to deal with data analysis limitations to determine the thermal transmittance existing in scientific literature, suggesting an efficient methodology to be applied in order to characterize the U-value of walls. In addition, aspects never discussed in scientific literature are suggested, such as using ANNs as data analysis methodology to determine thermal transmittance and implementing a calculation methodology with correction for storage effects for THM. The key contributions of this paper can be briefly summarized as follows:  Applying the calculation procedure with correction for storage effects for THM.  Developing and analysing ANN models to determine thermal transmittance with correction for storage effects without the need of previously knowing the thermophysical properties of the materials of the wall.  Determining limitations to assess thermal transmittance by means of the model developed, analysing the time required and the need of performing a post-processing.  Proposing a new data analysis methodology obtained for THM, as well as assessing the potential of using ANNs as data analysis methodology to quantify thermal transmittance. This article starts by explaining the basis of the calculation methodology of THM in section 2. Section 3 details the design of both ANN models developed in this research. Then, Section 4 describes training and testing datasets used in this study. Section 5 analyses and discusses the calculation procedure, the ANN architecture and model with the best behaviour as well as the applicability of this model to new case studies. Finally, section 6 sums up the main conclusions of this study. 2. Theory and calculation methods 2.1. Average method The theoretical framework of THM is explained in detail in the publications by Andújar Márquez et al. [14], BienvenidoHuertas et al. [15], and Kim et al. [16]. Therefore, a brief description of the equations is given in this section. THM is based on the average method from ISO 9869-1 [12]. In this standard, the thermal transmittance of an element is defined as the heat flux that goes through the element divided by the difference between the internal and external air temperatures under the conditions of a stationary regime [12], and it is obtained as follows: (1) Where [W/m2] is the specific heat flux through the element at the instant j, and and [K] are the internal and external air temperature at the instant j, respectively. THM applies Newton’s Law of Cooling provided that the wall is the area of heat transfer (Eq. (2)). Considering that the heat transfer by conduction is the same as the heat transfer by convection and radiation in steady state (Eq. (3)), Eq. 4 is obtained following the progressive average procedure to determine the U-value. (2) (3) (4) 1 2 3 4 5 6 7 8 9 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 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Where [W/(m2·K)] is the total internal heat transfer coefficient, and [K] is the internal surface temperature of the wall at the instant j. Thus, the way to obtain the thermal transmittance is by measuring, on the one hand, the external and internal air temperatures and, on the other hand, the internal surface temperature. Hence, the measurement errors associated with the use of the heat flux plate are avoided [15]. For , the reciprocal of the tabulated value of internal surface resistance established in ISO 6946 [9] is used, being 7.69 W/(m2·K) the value associated for façades. This value is obtained by the mechanisms of heat transfer by convection and radiation, with a value of 2.5 W/(m2K) for the convective heat transfer coefficient and a value of 5.19 W/(m2K) for the radiative heat transfer coefficient [47], being applicable to surfaces with a medium-high emissivity [9]. Thus, it is applied to most of the buildings, since values of emissivity of most materials used in the façades are between 0.90 and 0.96 [48]. It is important to highlight that, from the two heat transfers which define the total internal heat transfer coefficient, transfer by convection is the one that has a less rigorous development, existing a wide variety of expressions for correlation coefficients due to temperature differences, wind speeds or dimensionless numbers [49]. In this sense, the value from ISO 6946 has a behaviour similar to the convective heat transfer coefficient proposed by Khalifa and Marshall [50], with deviations lower than 6% [47], whereas deviations can reach 20% for other correlations. In a recent study, Kim et al. [35] analysed the use of the value of from ISO 6946 for THM. Tests obtained deviations lower than 5% with respect to the heat flow meter method, so this value is the most adequate with respect to other correlation values or coefficients existing in scientific literature [35]. Furthermore, the use of this value allows to establish linear correlations between the difference of internal and external air temperatures as well as the difference of internal surface temperature and internal air temperature for each U-value (see Fig. 1), making the behaviour of the input variables for each value of the thermal transmittance more understandable in the same way that for the heat flow meter method (Eq. (1)). Fig. 1. Relation between the difference of internal and external air temperatures (x-axis), and the difference of internal air temperature and the internal surface temperature of the wall (y-axis) for an internal convective coefficient of 7.69 [W/(m2·K)]. The use of the average method requires the change of the heat stored in the wall to be insignificant in comparison with the heat that goes through the wall, so the test duration should last between 3 and 14 days. However, there are difficulties to fulfil those conditions, such as [35,51]: (i) oscillations in temperature values and difficulties to achieve steady state conditions; (ii) the time needed to carry out the tests; (iii) significant variations in the convective heat flux due to operation cycles of heating systems; and (iv) putting the tenants of the analysed dwellings to great inconvenience. Thus, an improvement in the calculation procedure, where the thermal mass of the wall is taken into account, allow to reduce the error of the obtained result as well as to shorten the period of test performance [37]. 2.2. Average method with correction for storage effects The standard of the heat flow meter method [12] establishes a calculation procedure of thermal transmittance by correcting the heat storage effects for walls having both high resistance values and thermal mass. This calculation procedure should be applied when the U-value obtained at the end of the test varies more than 5% with respect to the value obtained 24 h before or to the data from the first two days [12]. To apply this procedure, it is fundamental to know the thermal characteristics of the layers composing the wall with the aim of also knowing its thermal resistance (Eq. (5)) and thermal capacity. (5) Where [m] and [W/(m·K)] are the thickness and the thermal conductivity of the material, respectively. For each layer of the wall, the thermal resistance is calculated, and the internal and external thermal mass factor are obtained (Eqs. (6) and (7)). (6) (7) Where [(m2·K)/W] is the sum of the external thermal resistances from the k+1th layer to the external air, [(m2·K)/W] is the sum of the internal thermal resistances from the k-1th layer to the internal air, [J/(m2·K)] is the external thermal mass factor of the layer , [J/(m2·K)] is the internal thermal mass factor of the layer , [J/(m2·K)] 1 2 3 4 5 6 7 8 9 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 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 is the thermal capacity of the layer , [(m2·K)/W] is the thermal resistance of the layer , and [(m2·K)/W] is the thermal resistance of the wall. Thermal mass factors of the wall are obtained by means of the sum of thermal mass factors of the N layers of the wall: (8) (9) Where [J/(m2·K)] is the total internal thermal mass factor, and [J/(m2·K)] is the total external thermal mass factor. To apply the correction, the numerator from Eq. 1 should be replaced by the following expression: (10) Where [K] is the difference between the average internal air temperature from the 24 h before measuring the observation and the average internal air temperate from the first 24 h of the test, [K] is the difference between the average external air temperature from the 24 h before measuring the observation and the average external air temperature from the first 24 h of the test, and [s] is the time interval between the measurements. Considering that the heat flux density of an element is equivalent to the density transferred from the internal air to the surface of the element analysed (Eq. (2)), Eq. (10) can be modified: (11) This expression (Eq. (11)) is applied to Eq. (4), obtaining the following equation: (12) Therefore, the method applies a correction factor which increases or reduces the numerator from Eq. (4) according to the variations of the average internal and external temperatures. Determining the composition of the wall can be carried out by several techniques, such as the analogy of similar constructions [11], the endoscopy [52], or the use of reliable technical documentation [11]. As there are difficulties to access to the project details as well as the impossibility of making an endoscopy due to the havoc caused in the wall, determining the configuration of the wall by means of either the analogy of similar constructions or based on previous experiences is the technique most used in most professional cases [11], although the associated error is high. 2.3. Average method with correction for storage effects using CCF The thermal conductivity does not always coincide with the tabulated values included in technical catalogues and projects, since it can vary depending on the environmental temperature and moisture [53]. Most database establish average temperature and moisture conditions to carry out the calculation procedure of thermal conductivity from ISO 10456 [54]. Due to this, Pérez-Bella et al. [55] achieved conductivity correction factors (CCF) for each of the province capitals in Spain according to the external ambient conditions. CCF can simplify the application of ISO 10456 by combining a temperature conversion factor and a moisture conversion factor (Eq. (13)), both applied to the thermal conductivity of the material with the result of obtaining a more representative thermal resistance (Eq. (14)). (13) (14) Where [dimensionless] is the conductivity correction factor for a certain province capital in Spain published by PérezBella et al. [55], [dimensionless] is the temperature correction factor, [dimensionless] is the moisture correction factor, and [(m2·K)/W] is the thermal resistance of the material by applying CCF correction. So, Eq. (12) can be modified by using thermal mass factors, which are modified by applying CCF to the province capital where tests are performed. Thus, the following expression is obtained: 1 2 3 4 5 6 7 8 9 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 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 (15) Where [J/(m2·K)] and [J/(m2·K)] are the total internal and external thermal mass factor using thermal resistance values corrected by CCF, respectively. 3. Estimation of U-value using ANN To apply the proposals of the average method with correction for storage effects (Eqs. (12) and (15)), a previous knowledge or a good estimation of thickness, thermal conductivity, and thermal capacity of the layers of the wall are required to calculate thermal mass factors. As mentioned above, consulting project details or making endoscopies are the most adequate methods to know the thermal properties of the wall, but these options are not always possible, and it is necessary to resort to the estimation by means of either the analogy of similar constructions or based on previous experiences. For this reason, it is essential to have analysis techniques predicting thermal transmittance taking into account the thermal storage effect of the wall and without having the need of knowing its constructive composition. From the different existing typologies to perform an advanced data analysis, ANNs are the ones which best features have [38], and the multilayer perceptron (MLP) is the most used structure. The MLP structure is formed by three or more layers: an input layer, one or several hidden layers, and an output layer. In each layer, there are neurons or nodes connected to those of the following layer by means of weighted connection with different weights. The neurons of the input layer do not make any calculation but send information. Each neuron of the hidden layer adds all the inputs together, so if that value is higher than the activation value, an output is produced. The outputs go over to the output layer where the process is repeated, and the system response is obtained. The MLP operation is mathematically expressed as follows: (16) Where is the output of the final layer, are the weights of the output layer, is the activation function, are the weights of the hidden layer, are the values of the input layer, and and are the weight and the input value of the bias neuron of the hidden layer, respectively. To ease the understanding of the steps followed in the research, Fig. 2 shows the flow-chart representing the main steps of the research. Fig. 2. Flow-chart of the procedure followed in the research. 3.1. Phase 1. Design and data collection The design of MLPs with two or more hidden layers does not usually generate a better behaviour of the model, increasing the risk of converging to a local minima [56]. For this study, the authors proposed three layers as architecture for the MLP (see Fig. 3). The MLP estimated the U-value of the wall by means of the time series of internal ( and external air temperatures ( , and the internal surface temperature ( ). Other variables required to characterise thermal transmittance were the average difference temperatures from the first 24 h, both internal and external , as well as the thickness of the wall (s), the time interval between the measurements ( ), and the building period (P). Regarding this last variable, the most predominant construction typologies in Spain (from the middle of the 20 century to nowadays) can be divided into three building periods [57]: (i) anterior to the normative NBE-CT-79 [58]; (ii) posterior to NBE-CT-79, and anterior to the Spanish Technical Building Code (CTE) [59]; and (iii) posterior to CTE. Buildings constructed anterior to NBE-CT-79 are characterised by being formed by one or two brick layers, without insulation and with the possibility of having an air gap [7], while with NBE-CT-79 coming into force, insulating material was added to the constructive solutions with thicknesses lower than 3 cm [60]. The implementation of CTE increased the limit values of thermal resistance, so the insulation thickness of the buildings from this period is higher [60]. With respect to data input, three discrete values were used for this variable: P1 (anterior to NBE-CT-79), P2 (between NBE-CT-79 and CTE), and P3 (posterior to CTE). This can be quickly determined by consulting the cadastral data of the building. 1 2 3 4 5 6 7 8 9 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 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Considering the existing linear correlations between the temperature differences for each thermal transmittance value (Fig. 1), another MLP was developed by transforming the variables , and into the difference of internal and external air temperature ( ), and the difference of internal air temperature and internal surface temperature ( ). Therefore, two MLPs were designed: MLP1 for temperature variables without being modified and MLP2 for transformed variables (see Fig. 3). Fig. 3. Schematic diagram of the MLPs designed. 3.2. Phase 2. Training and validating the MLP, and selecting the best architecture The MLP was trained by back propagation (BP) [61]. The Quasi-Newton Method was used as algorithm of second order, using Broyden-Fletcher-Goldfarb-Shanno (BFGS) [62] updates due to the high accuracy achieved by both of them in results [63]. Test data indicated in section 4 of this paper were used as training and testing dataset. For both hidden and output layer, a sigmoidal activation function was used. During the training, input data had the correct U-value as classification value. The result of the MLP depends on the number of neurons of the hidden layer. Therefore, to obtain an optimal result, the numbers of neurons generating a better behaviour should be analysed [40]. For this reason, architectures of the MLP in which the number of neurons in the hidden layer oscillated between 4 and 15 were analysed. Furthermore, a momentum of 0.2 and a learning rate of 0.3 were used as fixed parameters. The number of training time was between 500 and 120,000 until determining the optimal configuration, using a 10-fold cross validation. The assessment of the correct number of nodes from the hidden layer as well as the performance of the MLP were determined by the linear correlation coefficient (R2) (see Eq. (17)), the mean absolute error (MAE) (see Eq. (18)), and the root mean square error (RMSE) (see Eq. (19)). The use of these parameters was useful to determine efficiently the performance of the models [64]. Both MAE and RMSE should be as small as possible, whereas R2 should have values close to 1, since its value scale goes from 0 (null correlation) to 1 (total correlation) [38]. The determination of the nodes composing the hidden layer to achieve an optimal performance is discussed in the result section of this paper. (17) (18) (19) Where n is the number of observations in the training dataset, is the value predicted by the ANN, and is the measured value. 3.3. Phase 3. Comparing models and testing the model selected in new case studies After determining the most optimal architectures of MLP1 and MLP2, the behaviour of both models was compared using the statistical parameters of R2, MAE and RMSE. Then, the MLP with the most adequate behaviour was determined, and three new case studies no included in the training process were used for testing its behaviour in new case studies belonging to the three building periods considered. 4. Experimental campaign and dataset used To do the training and the testing, a dataset composed by 69 subsets was used. Each of these subsets corresponded to tests carried out in different façades. The façades analysed were common wall typologies from the three building periods considered: P1, P2 and P3. The façades were multilayer walls with brick leaves, with or without insulation. It is a constructive typology widely used in several countries like Belgium [65], United Kingdom [66] or Italy [67], so the application of the data analysis method can be extended to other regions. Some of the tests were performed by the authors and the others by the Agency for Housing and Rehabilitation in Andalusia (AVRA) of the Regional Government of Andalusia under the Energy Improvement in the Public Housing Stock Program. Choosing the walls, selecting the equipment and mounting the probes were done according to the methodology described by Bienvenido-Huertas et al. [15]. Tests were 1 2 3 4 5 6 7 8 9 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 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 performed between 4 and 7 days, with a data acquisition interval of 15 minutes, and each measured instant of time was an observation of the subset. Observations from the first 24 h of each test were not included in the subsets generated. For each wall, there was a reliable technical documentation which allowed to determine the number, the material and the thickness of their layers exactly. The values of thermal conductivity and capacity were obtained from the constructive elements catalogue in Spain for the different materials [68]. Tests were performed in the locations of Cadiz, Cordoba, Granada, Huelva, Jaen, Malaga, and Seville, so the value of the used CCF varies (see Table 1). Table 1 CCF for the locations of the analysed walls. As indicated in section 3, from the 69 tests available, 66 were used for training and validating the MLPs, and therefore determining which method was the most adequate (see Fig. 4). Data included in subsets corresponded to input data from the MLP as well as to the classification value (see Table 2). With respect to the three new case studies included in the testing of the model selected, Table 3 and Fig. 4. show the sketch, technical characteristics and thermophysical properties of each one. Table 2 Input and output data compiled in datasets for both models of MLP. Table 3 Technical characteristics and thermophysical properties of the walls used for testing the MLP. Fig. 4. Scheme of the training and testing of the MLP. 5. Results and discussion First, the results obtained for the training dataset of the MLP were analysed with the aim of studying the limitations of Eqs. (12) and (15). It is important to highlight that all the tests included in the training dataset of the neural network had thermal transmittance values without correction (Eq. (4)) and with differences higher than 5% with respect to the observations from the previous 24 h, being in some cases higher than 30%. Table 4 shows the values obtained for Eqs. (4), (12), and (15) in two walls from each building period. These equations were applied to the data obtained in 2 and 3 days. Furthermore, the estimated value of thermal transmittance was obtained by applying ISO 6946 in order to determine the percentage error of the result obtained by the test, following the validation criteria from ISO 9869-1 [12]. As mentioned above, a precise technical documentation was acquired, so it was possible to determine exactly the dimensions and thermal properties of the materials composing the walls analysed. The uncertainty of the obtained results was determined by the combined standard uncertainty from ISO/IEC Guide 98-3 [69]. Table 4 Results obtained of some of the walls of the training dataset. By analysing the results, it was detected that the U-value corrected in the walls from P1 had results more approximate to the estimated value, with errors lower than 10%. Generally, the values obtained in THM without correction in the façades from this period were adequate because they presented a difference lower than 20% with respect to the estimated value. However, using correction for storage effects could reduce the deviation percentage. Regarding the walls from P2 and P3, the application of correction could reduce the deviation percentage in comparison with the method without correction in most cases. In the examples included in Table 4, the application of correction allowed to reduce the obtained error from 32% to 13% and from 5% to 4% for the data analysed in the same time period. Nevertheless, there were two tests in which the obtained results were no representative due to the difficulties they had to achieve a stability in the measurements with a high thermal gradient, although it could also be attributed to a possible degradation of the thermal performance of these walls. That degradation caused differences between the estimated value from ISO 6946 and the measured value. For these tests, the application of correction could reduce the error with respect to the estimated U-value, reducing the deviation percentage from 75% to 59% in W3 and from 90% to 33% in W5, although the result continued being non-representative. 1 Table 1 CCF for the locations of the analysed walls. Location CCF a Cadiz 1.0403 Cordoba 1.0367 Granada 1.0311 Huelva 1.0385 Jaen 1.0354 Malaga 1.0382 Seville 1.0385 a Conductivity correction factor given by Pérez-Bella et al. [33]. 1 Table 2 Input and output data compiled in datasets for both models of MLP. Model Input data Class MLP1 MLP2