Uncertainty and sensitivity analysis in building simulation:

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1 Uncertainty and sensitivity analysis in building simulation: A probabilistic approach to the real estate market of apartments in Santiago de Chile FELIPE ENCINAS PINO 1, FRANCISCO JOSÉ SANCHEZ DE LA FLOR 2, CARLOS AGUIRRE NUNEZ 3, SERVANDO ÁLVAREZ DOMINGUEZ 4 1 Architecture et Climat, Université catholique de Louvain, Louvain-la-Neuve, Belgium 2 Escuela Superior de Ingeniería, Universidad de Cádiz, Cádiz, Spain 3 Centro de Política de Suelo y Valoraciones, Universidad Politécnica de Cataluña, Barcelona, Spain 4 Escuela Superior de Ingenieros, DIE Grupo de Termotecnia, Universidad de Sevilla, Sevilla, Spain ABSTRACT: It is clear that the thermophysical properties of materials, occupancy patterns and internal gains represent some of the most important sources of uncertainty in the field of building simulation. Uncertainty and sensitivity analysis deals with this situation, since it can generate a great range of forecast values based on the distribution of the input variables. At the same time, these techniques allow to determine as each variable contribute to the total variance of output results. However, most of the building energy simulation programs are deterministic, rather than probabilistic and consequently their results frequently are not expressed in terms of probabilities. On the contrary, the probabilistic approach requires a more complex process, since parameters quantification requires not only an assessment of the point estimate, but also an assessment of the uncertainty. This research aims to define the uncertainty of the predicted energy performance by means of the comparison between factorial design and Monte Carlo Analysis for a sample of 7776 cases that belong to the real estate market of apartments in Santiago de Chile. A total of 9 input parameters constitute the basis for these analyses using the standard EN ISO as calculation algorithm for estimating the annual heating demand. Keywords: uncertainty and sensitivity analysis, building performance simulation 1. INTRODUCTION It is clear that the introduction of building performance simulation models represents an important contribution to decision-makers understanding and overview of the decision-problem in the field of energy performance. Indeed, many of the answers that can be formulated at the light of these models can improve the certainty level with respect to aspects such as the material definition of the envelope or the glazing ratio of the building. Several studies have shown that uncertainties in design evaluations may be quite substantial that makes it imperative to convey them to the decision makers [1] [2] [3] [4]. Consequently, if the design process is carried out based on informed decisions, the possibilities to reach the desirable level of energy performance and/or thermal comfort are considerable. This situation can be applied for both new and existing buildings. Moreover, the definition of assumptions for input parameters clearly can only be estimated with some degree of uncertainty. This uncertainty may arise, for example, from the lack of knowledge about the building components or on the distribution of these values. Indeed, there are several sources of uncertainty in the field of building performance simulation, which mainly depend on the model abstraction, databases and solution methods [5]. At the same time, most energy building simulation programs are deterministic rather than probabilistic, and as such, offer their results in the form of deterministic values [2]. Also, none of these programs offer methods to propagate uncertainty of the input parameters on the model output [6]. On the contrary, the probabilistic approach requires a more complex process, since parameters quantification requires not only an assessment of the point estimate, but also an assessment of the uncertainty. Probably this situation appears as particularly critical when the focus is place on a group of cases instead of a single building case. The probability theory allows conducting an uncertainty assessment of the simulation output (dependent variable Y) based on the uncertainty of one or more input parameters (independent variables, X = {X 1,...,X n }) and where Y = f(x). In this respect, literature identifies two closely related methods to assessing uncertainties, which can be defined as uncertainty analysis (UA) and sensitivity analysis (SA). UA assesses the uncertainty in model outputs that derives from uncertainty in inputs parameters. SA assesses the contribution of the inputs parameters to the total uncertainty in analysis outcomes. This research aims to define the uncertainty of the predicted energy performance as well as the input parameters that causes this uncertainty for a sample of 7776 cases that belong to the private real estate market of apartments in Santiago de Chile. A total of 9 input parameters constitute the basis for these analyses using the standard EN ISO for estimating the annual heating demand.

2 2. METHODOLOGY 2.1. Parameter sensitivity analysis utilizing two sampling methods In general, UA and SA are conducted by: (a) defining the model along with the independent and dependent variables, (b) assigning probability density functions to each input parameter, (c) generating an input matrix through an appropriate sampling method, (d) creating an output distribution and (e) assessing the influence of each input parameter on the output variable [7]. The sampling based method, as basis for both analyses, consists in the repeated execution of the model from the combination of input parameters sampled with some probability distribution. In this context, factorial design appears as the complete solution, since it considers all combination of samples from the input parameters. However, it is evident that a large number of parameters generate a tremendous number of model runs. De Wit (2001) proposes that the economy of the method - in terms of the number of model evaluations to obtain a sufficiently accurate result - represents an important criterion for the selection of a technique to assessing uncertainties [6]. According to literature, the most widely applicable and easy to implement method for this purpose is Monte Carlo simulation [3] [6]. Monte Carlo Analysis (MCA) performs multiple evaluations with randomly selected input parameters, generating input and output distributions useful in assessing model and parameter uncertainties in a global sense. The range of each variable is divided into N non-overlapping intervals of equal probability 1/N [2]. MCA also is a black-box approach, where the only parameters that can be influenced are contained in the data model and consequently can be relatively easy to implement to any desired tool Application of Factorial design and Monte Carlo Analysis EN ISO 13790:2004 was chosen as the calculation algorithm for implementation of routines for uncertainty analysis. This standard gives a simplified calculation method for assessment of the annual energy use for space heating of a residential or a non-residential building [8]. Table 1 present the 9 considered input parameters, which are related to building envelope definition, architectural design criteria and other aspects that belong to the range of decisions from the real estate market (e.g. layout and orientation). Weather data is defined for Santiago de Chile and considered as fixed scenario for all simulations. As can be observed, an amount between 2 and 4 variables were defined by each parameter (see details in the table), which were established using a triangular probability density function. This kind of distribution is generally used as a subjective description of a population for which there is limited sample data or where the relationship of variables is roughly known but data is scarce. Table 1: Input parameters Type of parameter Input parameters Unit Variables by parameter Description U-value of walls W/m²K 3 3 levels of thermal insulation in concrete walls (without insulation, 10 and 20 mm EPS 15 kg/m 3 ) U-value of windows W/m²K 3 3 levels of thermal insulation corresponding to single glazing, double glazing and low emissivity double glazing 3 levels of thermal insulation in roofs (80, 100 and 120 U-value of roofs W/m²K 3 Independent mm EPS 15 kg/m 3 ) Internal gains W/m² 2 Low (5 W/m²) and high (15 W/m²) internal gains Shading correction (0-1) 2 Average shaded fraction of the area of opening for windows: with and without balcony Frame factor % 2 Transparent fraction of the area of opening for windows not occupied by a frame (9 and 93% respectively) Architectural layout types of architectural layout: 1 bedroom, 1 bathroom; 2 bedrooms, 2 bathrooms; 3 bedrooms, 2 bathrooms Covariant Ratio of the glazed area to the total area of the exposed Glazing ratio % 3 facade (3, and 5) Orientation 4 0 = North; 90 = East; 180 = South; 270 = West Table 2: Values and probability distribution for the U-value of walls parameter Period TOTAL Description Walls of 200 mm concrete 2400 kg/m 3 Without thermal insulation 10 mm EPS 15 kg/m³ 20 mm EPS 15 kg/m³ TOTAL U-value 3.42 W/m²K 1.74 W/m²K 1.22 W/m²K Number of units 15, ,813 % of the total Number of units 445 5,339 3,114 8,898 % of the total Number of units 16,417 6,179 3,114 25,711 % of the total 63.9% %

3 These parameters and their distributions were applied with the aim to obtain the annual heating demand for a sample of cases that can be representative of the real estate market of apartments in Santiago de Chile. For this reason, some assumptions, especially the covariant parameters, were based on information of the Portal Inmobiliario database, which includes the most important real estate attributes of 25,711 apartments for the period [9]. Table 2 presents the 3 values and their probability distribution of the U-value of walls input parameter. It can be observed that 2 periods were considered ( and ), which correspond to the scenarios of before and after of the entry into force of the national thermal regulation (second stage), respectively. This energy building code introduced a maximum admissible U-value for external walls in residential buildings. Cumulative frequency Cumulative frequency Level 2 (Middle-floor apartments) Factorial Monte Carlo Level 3 (Top-floor apartments) Factorial Monte Carlo Heating demand [kwh/m²/y] Figure 1: Cumulative frequency for heating demand in both sampling models for the levels 2 (above) and 3 (below) From the given number of variables by parameter, all the possible combinations of them produce a total of 7,776 cases (where = 3*3*3*2*2*2*3*3*4), which correspond to the complete sample of the factorial design. On the contrary, the second model based on the MCA contains only 88 cases, which were obtained from the square root of the complete sample (factorial design). This estimation is based on the concept that MCA relies on the central limit theorem and consequently, an increment in the number of trails above the limit of only produces a marginal gain in accuracy [1]. Output samples for both methods (factorial and Monte Carlo) were generated for three different situations depending on if the cases are situated in the ground-floor, in one of the middle-floors (above the ground floor and below the top floor) or in the top-floor, defined as levels 1, 2 and 3 respectively. The differences between the output results of the 3 levels justify this selective treatment. Figure 1 presents the cumulative frequency for the output samples of Factorial and Monte Carlo models applied to the levels 2 and 3. As it was explained, factorial design shows the complete solution through all the combinations, while MCA is based on random trials where several cases of the complete sample are missing due to their low probability. To determine the level of statistic similarity between the two samples, the t-test for unequal sample sizes was applied Application of t-test for unequal sample sizes to analyze similarity between models Table 3 presents the results of the parametric t-test for both models in the three different levels. This test assesses the statistically significant difference between the means of two conditions (in this case, unpaired samples). The procedure basically consists in the estimation of the t-value (a ratio between a measure of the between-groups variance and the within-groups variance) and the probability of obtaining this statistic by sampling error ( t-critical in the table). This probability determines the null hypothesis (H 0 ) of the test. For the three levels, the obtained t-value is higher than t critical and consequently, the null hypothesis (H 0 ) is rejected. At the light of these results, it is possible to conclude that the difference between the means of both samples is no significant and therefore both distributions can be considered as statistically similar. Table 3: T-test for Factorial and Monte Carlo models in the three different levels Level 1 Level 2 Level 3 Monte Carlo Factorial Monte Carlo Factorial Monte Carlo Factorial Mean Variance Observations Degrees of freedom t-value P(T<=t) E E-12 t-critical

4 3. RESULTS 3.1. Uncertainty and sensitivity analysis In the literature, when referring to the degree to which an input parameter affects the model output, the terms sensitive or important are used. However, a distinction must be made regarding these two concepts. The parameters which have a significant influence on assessment results are termed as sensitive in the model and the important parameters are those whose uncertainty contributes substantially to the uncertainty of the outputs [7]. This difference between important and sensitive parameters also leads a distinction in the type analysis being conducted: UA (parameter importance) or SA (parameter sensitivity). The 9 parameters of Table 2 were separated in two groups: parameters that depend on and parameters that do not depend on the building quality, with the aim of identifying situations that latterly may be proposed as building improvements. Figure 2 presents the propagation of the uncertainty through the MCA model for the input parameters that depend on the building quality. Each parameter suggests its level of importance based on the contribution to its specific uncertainty on the heating demand. At the light of these graphs, U value of both walls and windows show that are very important parameters due to the range of variability of the outputs. On the contrary, input parameters of glazing ratio and frame factor show a moderate and a very low level of importance respectively. Additionally, the series of graphs in Figure 2 suggest that the U-value of windows besides being the most important parameter is also the most sensitive, since is the only parameter that explains the obtainment of low heating demands in the range between kwh/m²/y. This situation supports the concept of Hamby (1994), where an important parameter is always sensitive because parameter variability will not appear in the output unless the model is sensitive to the input [7]. Finally, Table 4 presents the Pearson s correlation coefficient (r) for all the input parameters with respect to the output model (annual heating demand) by level. This index represents a quantitative estimate of linear correlation between the input and output values. The larger the absolute value of r the stronger the degree of linear relationship between them. At the same time, a negative value of r indicates that the output is inversely related to the input. The table shows that in general, parameters that depend on the building quality are most sensitive than the others that do not depend on that. Between them, U-value of windows and walls are first and second in the rank respectively, according to their contribution to uncertainty prediction. On the contrary, the contribution of frame factor and internal gains are negligible, since their correlations are not statistically significant W/m²K 1.74 W/m²K 3.42 W/m²K 2.80 W/m²K 5.80 W/m²K U-value of walls U-value of windows Glazing ratio Frame factor Heating demand [kwh/m²/y] Figure 2: Cumulative frequency for heating demand with respect to the parameters that depend on the building quality (from top to bottom) for level 2: U-value of both walls and windows, glazing ratio and frame factor

5 Table 4: Pearson s correlation coefficient for the different input parameter with respect to the heating demand Level 1 Level 2 Level 3 Parameters that depend on building quality U-value of walls U-value of windows U-value of roofs Glazing ratio Frame factor Parameters that do not depend on building quality FA/CA ratio* bedroom, 1 bathroom bedrooms, 2 bathrooms bedrooms, 2 bathrooms Internal gains North orientation East orientation South orientation West orientation Shading correction factor (*) Facade area / conditioned area ratio Correlation is significant at the 0.05 level (2 tailed) 4. DISCUSSION 4.1. Explanatory analysis by means of a Principal Component Analysis (PCA) Principal Component Analysis (PCA) is a multivariate analysis technique that it is used to reduce the dimensions of a large set of observed variables. The obtained components are a linear combination of variables that allow identifying the subjacent dimensions from a correlation matrix. In that sense, it is possible to affirm that this technique is exploratory in nature [10]. Table 5 presents the rotated component loadings, which give information about the strength of the relationships between the different variables (parameters) and their corresponding components (C1, C2,... C7). Table 5: Principal component analysis Rotated component matrix by means of VARIMAX Classification Parameters that depend on building quality Parameters that do not depend on building quality According to the Kaiser s criterion, 7 components were extracted, which account for 78.9% of variance. Other 5 components were dismissed since they presents Eigen values below 1. Components loadings are expressed in terms of correlation coefficients. In this case, coefficients above 0.60 were considered as significant. As the significance of factorial loadings depends on the size of the sample, this reference value was taken for a sample of 85 observations [11]. At the same time, the parameter of internal gains was dismissed because of its communality (sum of the squared loadings) is less than 0.50 and in consequence, it does not have enough explanation for its variance. This is consistent with the correlation coefficients of Table 4, where internal gains were not statistically significant in any level. The rotated matrix of Table 5 can be interpreted as function of the parameters. The sign of the coefficients positive or negative - indicates if they are positively or negatively correlated with respect to their corresponding component, respectively. Therefore, the 7 components can be defined as: C1: Facade area/conditioned ratio is positively correlated and 1 bedroom / 1 bathroom layout is negatively correlated C2: South orientation and both 2 bedrooms / 2 bathrooms and 3 bedrooms / 2 bathrooms layouts, which are positively correlated C3: North and west orientations, both are negatively correlated C4: East orientation, which is negatively correlated C5: U-values of walls and glazing ratio, both are positively correlated C6: Shading correction factor, positively correlated C7: This component can be understood with respect to window properties, since considers both U-value of windows and frame factor, which are positively correlated Finally, Table 6 presents the Pearson s correlation coefficients for the 7 components with respect to the model output (annual heating demand). Both C5 and C7 present the highest correlations. Parameters Components C1 C2 C3 C4 C5 C6 C7 Communalities U value of walls U value of windows Glazing ratio Frame factor FA/CA ratio* North orientation East orientation South orientation West orientation bedroom / 1 bathroom bedrooms / 2 bathrooms bedrooms / 2 bathrooms Internal gains Shading correction factor (*) Facade area / conditioned area ratio Significant variables per each component (factorial loadings > 0.60)

6 This is highly consistent with the previous analyses, since they are components based on the parameters that depend on the building quality (Uvalue of both walls and windows, glazing ration and frame factor). On the contrary, r coefficients of C1, C3 and C4 are not statistically significant with respect to the heating demand, as they are based on real estate market attributes, such as layout or orientation. Table 6: Pearson s correlation coefficients for all the components of the PCA with respect to heating demand Components r C C C C C C C CONCLUSIONS Correlation is significant at the 0.05 level (2 tailed) The premise underlying this research is that introducing uncertainty considerations into building performance simulation models will improve the certainty level with respect to aspects such as material definition of the envelope or the glazing ratio and that this, in turn, will help to improve designer confidence in these numeric simulations. Firstly, Monte Carlo Analysis (MCA) appears as an appropriate tool to propagate uncertainty of the input parameters on the model output in the context of the energy performance assessment (using the ISO 7730 standard as calculation algorithm). This was observed by means of the comparison between a Factorial design with 7776 cases and a MCA with 88 cases (n= 7776), where both sample distributions are statistically similar. Secondly, uncertainty analysis (UA) and sensitivity analysis (SA) were used to investigate the uncertainty of the predicted thermal performance in a group of cases, which are representative of the real estate market of apartments in Santiago de Chile for the period The U-value of both windows and walls are the most important and sensitive input parameters (in that order) since they have the largest impact on the heating demand. Other parameters with a moderate influence on the assessment results are the glazing ratio and shading correction factor, while frame factor and internal gains were considered as negligible. At the light of these results, some recommendations can be done with the aim of obtaining a better thermal performance in future real estate developments in Santiago. Clearly, improvements in the building envelope (especially in windows) appear as priority. However, further research related to ventilation regimens and other occupant behaviour patterns is required in order to refine the predictions of thermal performance based on this probabilistic approach. 6. ACKNOWLEDGEMENTS This study was carried out as a part of a PHD Thesis at the Architecture et Climat research centre from the Université catholique de Louvain in Belgium. The study is funded by the Bourse de coopération au développement scholarship from the same university. The authors would like to thank to Portal Inmobiliario.com for providing the information about the real estate market that makes possible this study. 7. REFERENCES [1] De Wit, S & Augenbroe, G 2002, Analysis of uncertainty in building design evaluations and its implications, Energy and Buildings, no. 34, pp [2] Breesch, H & Janssens, A. 2005, Building simulation to predict the performances of natural night ventilation: uncertainty and sensitivity analysis, Proceedings of the 9th International. IBPSA Conference, International Building Performance Simulation Association, Montreal, August. [3] Hopfe, C, Hensen, J & Plokker, W 2006, Introducing uncertainty and sensitivity analysis in non-modifiable building performance software Proceedings of the 1st IBPSA Germany/Austria Conference BauSIM, International Building Performance Simulation Association, Munich, 9-11 October. [4] Struck, C, Hensen, J 2006, Uncertainty analysis for conceptual building design - a review of input data, Proceedings of the 1st IBPSA Germany/Austria Conference BauSIM, International Building Performance Simulation Association, Munich, 9-11 October. [5] Macdonald, IA, Clarke, JA & Strachan, PA 1999, Assessing uncertainties in building simulation, Proceedings of Building Simulation 1999, Paper B-21, Kyoto, Japan. [6] De Wit, S 2001, Uncertainty in predictions of thermal comfort in buildings, PhD Tesis, Delf University of Technology, The Netherlands. [7] Hamby, DM 1994, A review of techniques for parameter sensitivity analysis of environmental models, Environmental Monitoring and Assessment, no. 32, pp [8] CEN 2004, EN ISO 13790:2004 Thermal performance of buildings Calculation of energy use for space heating, European Committee for Standardization (CEN), Brussels. [9] Portal Inmobiliario 2010, Portal Inmobiliario.com, < [10] Vivanco, M 1999, Análisis estadístico multivariante. Teoría y practica, Editorial Universitaria, Santiago [11] Hair, J, Anderson, R, Tatham, R & Black, W 2005, Análisis multivariante (Quinta edición), Pearson Educación, S.A., Madrid