An enhanced linear regression-based building energy model (LRBEM + ) for early design

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1 Journal of Building Performance Simulation, An enhanced linear regression-based building energy model (LRBEM + ) for early design Maged Al Gharably a, Joseph F. DeCarolis b and S. Ranji Ranjithan b a PPMG Consultants, Cloverland Ave, Baton Rouge, LA 70809, USA; b Department of Civil, Construction, and Environmental Engineering, North Carolina State University, Raleigh, NC 27695, USA (Received 13 June 2014; accepted 30 December 2014) The design community lacks simple, data-driven energy assessment tools to explore energy-efficient alternatives during the early stages of building design. A promising option is to utilize a whole building energy simulation engine (e.g. EnergyPlus) within a Monte Carlo simulation framework to develop a linear regression-based building energy model (LRBEM) that can predict idealized heating and cooling loads based on parameters relevant to early design. Previous work was limited to medium-sized US commercial office buildings with rectangular geometries. A key limitation is addressed in this paper by considering complex geometries. A reformulated model, LRBEM +, is developed and tested with a suite of building geometries that represent limiting cases. The resultant relative error between LRBEM + and EnergyPlus is generally less than 10%. Furthermore, LRBEM + correctly predicts the direction and magnitude of changes in heating and cooling loads in response to changes in the most influential early design parameters. Keywords: EnergyPlus; Monte Carlo simulation; multivariate regression; building geometry 1. Introduction In 2011, approximately 41% of total US energy consumption (equivalent to 42 EJ) was consumed in buildings (EIA 2013a). For perspective, US building energy consumption is approximately 2.4 times the total energy consumed by the African continent in 2010 (EIA 2013b). Significant potential for energy savings exists by optimizing new building design in a way that minimizes the thermal cooling and heating loads, which constitute a major portion of the total energy consumed by buildings. Formal consideration of energy performance has typically been addressed by engineers in the later stages of the design process when much of the design that affects building energy load is already fixed. Thus, building energy performance is typically viewed as a design outcome rather than as a design target. The American Institute of Architects (AIA) describes Building Energy Modeling (BEM) as a tool to predict the anticipated building energy usage and to capture the corresponding energy savings compared to a baseline (AIA 2012). However, utilizing the available suite of building energy simulation tools requires substantial time, buildingspecific information, and technical expertise, all of which make effective use of BEM difficult in the early design stages. With increasing attention focused on energy performance modelling, design professionals are responding to contractual arrangements by conducting energy simulations towards the end of the design phase using tools such as the U.S. Department of Energy s DOE-2, EnergyPlus, IES Virtual Environment, and Carrier s Hourly Analysis Program (HAP) (Maile, Fischer, and Bazjanac 2007). However, the design community still lacks practical tools to perform a rigorous assessment of building energy performance during the conceptual design phase. Consequently, most design decisions that greatly influence building energy performance are frequently made in the absence of model-based energy estimates. AIA (2012) indicates that most architects acknowledge the importance of interoperability between energy simulations and design. The AIA study calls for simplified simulation methods or improved user interfaces for complex building energy simulation engines that are appropriately responsive to key early design variables. Such efforts could improve building energy performance by integrating energy analysis into all stages of design, from schematic design to construction documents. Some notable efforts such as ISO (Nielsen 2005), MIT Design Advisor (Urban and Glicksman 2007), and Sefeira (2014) have attempted to reduce the number of required inputs and to simplify the underlying model physics. While software tools such as OpenStudio (NREL 2014) and Ecotect (Autodesk 2014) provide a graphical user interface for energy simulation, they often do not meet the expectations of architects who quickly iterate through alternative building schemes in the early design stages. A more recent and fundamentally different approach developed by Hygh et al. (2012) utilizes EnergyPlus, an existing whole building *Corresponding author. jdecarolis@ncsu.edu 2015 International Building Performance Simulation Association (IBPSA)

2 2 M. Al Gharably et al. energy simulation engine, within a Monte Carlo simulation framework to develop a linear regression-based building energy model (LRBEM) that is based on a set of building parameters relevant during the early design stages. The resultant LRBEM accurately predicts the energy performance of medium-sized, rectangular office buildings within four different US climate zones represented by Miami, Winston-Salem, Albuquerque, and Minneapolis. LRBEM can serve as the basis for a tool that can provide real-time feedback to designers as they modify basic building design elements. This paper enhances LRBEM (Hygh et al. 2012) by extending and testing its functionality to handle nonrectangular building geometries. We hypothesize that nonrectangular geometries can increase building surface area, produce self-shading effects, and affect thermal zoning, all of which may exert a strong influence on building heating and cooling loads. If so, LRBEM may in turn yield inaccurate estimates of building loads for non-rectangular building geometries. To rigorously test LRBEM s accuracy for non-rectangular geometries, this paper describes three sets of experiments. First, we test the existing LRBEM s accuracy in predicting heating and cooling loads for nonrectangular building geometries. Second, we modify the LRBEM input parameter set to better capture variation in building geometry, resulting in a refined model referred to as LRBEM +. Third, we test the accuracy of LRBEM + in capturing the effect of changes in early design parameters on heating and cooling loads for buildings with non-rectangular geometries. The paper is organized as follows. Section 2 briefly summarizes the previous work by Hygh et al. (2012) and outlines experiments conducted with the original, unmodified LRBEM. Section 3 describes the development of LRBEM + and the experiments conducted to test its performance. Section 4 presents the results, and Section 5 outlines the insights and conclusions drawn from the analysis and provides recommendations for applying LRBEM Prior model development During LRBEM development (Hygh et al. 2012), 27 early design parameters were identified (Table 1) based on their architectural relevance and potential effect on building energy consumption. Hygh et al. (2012) used a set of exogenous assumptions about prevailing climate as well as building size, type, and geometry. Separate LRBEMs were developed for each of the four different US climate zones as defined by ASHRAE (2007), which were selected to capture climatic extremes. Weather files from the following representative cities were utilized: Miami, FL (Zone 1A), Winston-Salem, NC (Zone 4A), Albuquerque, NM (Zone 4B), and Minneapolis, MN (Zone 6A). In addition, while building size is an explanatory variable in LRBEM, it is limited to the range associated with the US mediumsized commercial office building type drawn from the DOE Commercial Reference Building Models (Deru et al. 2011). This particular building type was chosen based on its ubiquity in the US commercial building market. Building geometry was limited to rectangular shapes, defined by building depth and aspect ratio, the latter defined as the length of the longer side of the rectangular floor area divided by that of the shorter side (depth). Fixed internal loads were modelled under the following assumptions: 13.9 m 2 /person, 10.8 W/m 2 for artificial lighting, 10.0 W/m 2 for electrical equipment, and operational schedules based on ASHRAE 90.1 (Hygh 2011). For simplicity, Hygh et al. (2012) assumed that the artificial lighting requirement was fixed and did not respond to variations in the amount of daylighting. As a result, variations in artificial lighting density and associated heat gain in response to changes in daylighting were not captured. With these assumptions, Hygh et al. (2012) developed a base EnergyPlus model to represent a medium-sized commercial office building in each of the four specified climate zones. The 27 early design parameters were assigned numerical ranges (Table 1), which, when taken collectively, spanned a broad segment of the design space. Several additional parameters (derived from the original 27 parameters) were developed to increase the accuracy of the resultant LRBEM by representing interactive effects. For example, the total solar gain through windows, expressed as the product of the solar heat gain coefficient (SHGC) and window area, was utilized to potentially explain some of the variance in building energy performance. Next, a Monte Carlo simulation was performed in which random draws from a uniform distribution associated with each parameter range were made to create 20,000 instances of the commercial building model. Each building model instance was analysed using EnergyPlus to produce estimates of idealized heating and cooling loads. The original 27 early building design parameters and derived parameters were treated as potential explanatory variables in a stepwise regression model based on the results from 16,000 model instances. The stepwise regression procedure retained the design parameters that reduced the residual error in the load prediction at a 5% level of significance. Eight separate models were developed to predict heating and cooling loads in each of the four climate zones. Building design parameters and the additional derived parameters became explanatory variables in the LRBEM if retained by the stepwise regression. Parameters included and excluded by the stepwise regression procedure are provided in the appendix to Hygh et al. (2012). The remaining 4000 model instances were used to validate LRBEM Applicability of LRBEM to new building geometries We first pursued simple extensions to the original LRBEM. Our aim was not to uniquely capture the specific geometry

3 Journal of Building Performance Simulation 3 Table 1. Building design parameters included in the LRBEM developed by Hygh et al. (2012). Parameter Unit Minimum Maximum Total building area a m Number of stories 3 6 Aspect ratio b Length/depth 1 10 Orientation (rotation) Degrees Roof R-value m 2 K/W Roof colour Solar absorptance Roof emissivity Emissivity Window R-value(N,S,E,andW) m 2 K/W Window SHGC (N, S, E, and W) SHGC Wall R-value(N,S,E,andW) m 2 K/W Shading projection factor (N, S, E, and W) Per cent of window height Window-to-wall ratio (N, S, E, and W) % 2 90 a The area range is chosen to represent a typical US commercial office building. For reference, a building area of m 2 corresponds to 20,000 1,000,000 ft 2. b Note that depth is included as a design parameter in Hygh et al. (2012), but it is fully determined by the total building area, number of stories, and aspect ratio, and is therefore omitted here. Table 2. Methods used to estimate heating and cooling loads using the LRBEM developed by Hygh et al. (2012). Relative error (%) a [LRBEM EnergyPlus]/ EnergyPlus Method Brief description of the estimation procedure Heating Cooling Combinations of rectangles Floor area preserved Perimeter preserved Bigger rectangle with area factor b LRBEM + Predict energy performance for several constituent rectangular buildings that collectively represent the floor area associated with the non-rectangular building shape Predict energy performance for a single rectangular building that has the same floor area as the non-rectangular building shape Predict energy performance for a single rectangular building that has the same perimeter as the non-rectangular building shape Predict the energy performance for a single rectangular building that can enclose the non-rectangular building, but is scaled by the ratio of the rectangular floor area to the true non-rectangular floor area Revised LRBEM discussed in Section 3; includes wall area by cardinal direction and roof area as early design parameters a These errors are based on testing the building depicted in Figure 1 in the Climate Zone 4A (Winston-Salem). b While the relative error associated with this method is small for the non-rectangular building depicted in Figure 1, higher errors were observed when this method was applied to other non-rectangular forms. of a given building, but rather to generalize the geometric properties of non-rectangular buildings such that the model can capture the associated thermal effects with sufficient accuracy. Several procedures to apply LRBEM to non-rectangular buildings were tested. We began by testing non-rectangular, rectilinear building shapes. This was the simplest approach because the energy loads for any rectilinear building can be approximated by the energy loads associated with one or more constituent rectangular forms, which can be estimated with the original LRBEM. These procedures are summarized in Table 2, which also includes the relative error between the LRBEM and EnergyPlus predictions for the building shown in Figure 1. Several different building instances were also tested across the same climate zones utilized by Hygh et al. (2012). Overall, these procedures did not yield prediction performance comparable to the energy load prediction accuracy, especially for heating, reported by Hygh et al. (2012). 3. Development and application of LRBEM + Although we did not develop a sufficiently accurate procedure for applying LRBEM to non-rectangular buildings, we hypothesized that the heating and cooling load prediction performance might be improved by incorporating building wall area by cardinal direction and roof area as potential explanatory variables in the regression. These

4 4 M. Al Gharably et al. Figure 1. An example of a rectilinear building with uniform floor heights used to test the predictability of the LRBEM +. Detailed specifications for this building are provided in Table A1 in the appendix. additional parameters are responsive to building geometry, and taken together, capture total building surface area, which has a direct effect on heat lost or gained by the building and therefore the overall heating and cooling loads. In the LRBEM developed by Hygh et al. (2012), the constant floor-to-floor height of 4.57 m and the rectangular building shape enabled the wall and roof area to be derived from building depth and aspect ratio. In addition, the wall area was assumed to be the same in opposing directions (i.e. east west and north south) since all building instances conformed to a rectangular shape. These observations led to a reexamination of the explanatory variables used to predict the building energy loads, which constitute the LRBEM model structure. The following section describes the development and testing of an improved LRBEM Development of LRBEM + The simulation data used to develop the original LRBEM were used to develop a revised regression model, LRBEM +, which included five new parameters: wall area by cardinal direction and roof area. This approach enabled the development of a revised model without redoing the Monte Carlo simulation with EnergyPlus conducted by Hygh et al. (2012). For heating and cooling loads within each of the four climate zones, the stepwise linear regression procedure (using the stepwise-fit function in MAT- LAB (Mathworks 2013)) was applied to the simulation data set consisting of 16,000 realizations, chosen at random from the complete set of 20,000 building realizations. Parameters were retained in the stepwise regression at a significance level (p-value) of.05. The remaining 4000 realizations were used to conduct the model validation exercise. The stepwise regression was initiated with 30 design parameters as well as an additional set of 48 terms derived from the design parameters, as potential explanatory variables in the linear regression model (Appendix, Table A2). The 30 building design parameters include 25 of the original 27 parameters from Hygh et al. (2012): aspect ratio and number of stories were excluded, but roof area and wall areas by cardinal direction were added. Details regarding changes to the potential explanatory variables in LRBEM + relative to LRBEM are provided in Table 3. The coefficient of determination (R 2 ), coefficient of variation of the root mean squared error CV(RMSE), and the mean bias error (MBE) were used to assess LRBEM + s goodness of fit. Because our purpose is to accurately predict EnergyPlus outputs, we report R 2 and CV(RMSE) to quantify how well LRBEM + predicts the output from the EnergyPlus validation set. Because MBE accounts for the direction and magnitude of errors, it can help determine whether LRBEM + exhibits a systematically high or low bias relative to the EnergyPlus results. Use of these metrics is consistent with prior work aimed at calibrating different building models to observed data (e.g. Haberl and Bou- Saada 1998; Reddy and Claridge 2000; Rafferty, Keane, and O Donnell 2011). Extended testing was conducted to assess the performance of LRBEM +. While LRBEM + was derived from a Monte Carlo simulation based on rectangular buildings, it was applied to several different non-rectangular building forms. Several limiting cases were designed to test key building shape attributes that may affect the heating or cooling loads. In addition, we tested a building that combines all of these limiting cases into a single form and used it to perform a sensitivity analysis that tests the accuracy of LRBEM +. The following subsections describe the application of LRBEM + to each of these cases. Since these tests represent one-to-one comparisons between LRBEM + and EnergyPlus for a given building form, relative error (i.e. x LRBEM + x EnergyPlus /x EnergyPlus ) is used to assess the difference between the two estimates Buildings with a change in a single shape attribute LRBEM + was used to systematically test deviations from rectangular building instances. The following cases include fundamental deviations from a purely rectangular building form that will affect the building s thermal loads: Case A buildings with courtyards in the middle; Case B buildings with non-right-angled corners; Case C buildings with non-uniform roof heights; and Case D buildings with non-uniform floor-to-floor heights. These deviations were chosen because they increase the surface area, change the thermal zoning and solar gain, and can produce self-shading effects that affect heating and cooling loads. Renderings of the buildings associated with each case are shown in Figure 2. All tests described in the following subsections were performed, as an illustration, for the Winston-Salem climate zone, which has a relatively moderate climate among the four study locations. Cases A1 and A2 represent buildings with interior courtyards that account for 3.9% and 15% of the ground floor area, respectively. This deviation from a purely rectangular building leads to different amounts of building self-shading and surface area, which affects the heating and cooling loads. Cases B1 and B2 represent two buildings with non-right-angled corners; Case B1 includes both

5 Journal of Building Performance Simulation 5 Table 3. Changes to the set of design parameters considered in the new regression model, LRBEM +. Parameter Original LRBEM a LRBEM + b,c Aspect ratio Length Depth Excluded Number of stories Integer Excluded Wall areas (N, S, E, and W) Excluded Long sides: Short sides: ( ) ( Floor area Floor area ) Stories/Depth Stories/(Depth Aspect ratio) 4.57 Stories 4.57 Stories Roof area Excluded Floor area Stories a Note that although wall and roof areas were not considered independent parameters in the original LRBEM, they were used in some of the derived parameters, as noted in the appendix to Hygh et al. (2012). b Although depth and aspect ratio are not considered explanatory variables in LRBEM +, they are used to calculate the wall area since all 20,000 building instances generated through the Monte Carlo simulation conducted by Hygh et al. (2012) are rectangular. c 4.57 represents the metric floor to floor height (15 feet) that was used in the baseline model of the Monte Carlo simulation framework. A1 B1 C1 D1, D2 Figure 2. Limiting cases, each with a single shape attribute deviation from a rectangular building. Cases A1 and A2 (top row) include courtyards to test the effect of self-shading. Cases B1 and B2 (second row) include buildings with non-right-angled corners. Cases C1 and C2 (third row) includes buildings with non-uniform roof heights. Case D (bottom row) includes non-uniform floor-to-floor heights. Detailed specifications for these buildings are provided in Table A3 in the appendix. acute and obtuse-angled corners, and Case B2 includes obtuse-angled corners only. These cases have more than four corners, unlike the rectangular building that was used to generate the simulation data for LRBEM +. Changing the corner angles will affect the amount of solar gain per cardinal direction. Cases C1 and C2 represent buildings with different roof heights, which affect surface area, solar gain, and thermal zoning. In Cases D1 and D2, the floor-to-floor height is varied from the default value of 4.57 m (15 ft) used in the data set to develop LRBEM + to 3.35 m (11 ft) and 5.49 m (18 ft), respectively, which A2 B2 C2 affects thermal zoning. Although floor-to-floor height is not an explicit design parameter within the LRBEM +,itis represented by scaling the wall area per cardinal direction. Cases A D each entail a change to a single-building shape attribute, which was mapped to early design parameter values (Table A3) that fall within the range specified in Table 1. This approach enables us to isolate the effect of a specific non-rectangular shape attribute on heating and cooling loads while remaining within the parameter value ranges used to build LRBEM and LRBEM A building with a combination of non-rectangular shape attributes To assess the predictive performance of LRBEM + for a building with multiple non-rectangular shape attributes, a building design that includes all of the variations reflected in Cases A, B, C, and D was incorporated into Case E (Figure 3). It includes the self-shading effect due to the presence of a courtyard. Non-uniform roof heights are incorporated by having only 41.4% of the first floor covered by a second floor. This building also includes varying floor-to-floor heights: the first floor height is 5.18 m (17 ft), while the other floors are 4.57 m (15 ft) high. The shape of the building was changed by including two right-angled corners, two corners at 102.9, and another at on the first floor. As with Cases A D, we calculated the relative error between the LRBEM + prediction and equivalent EnergyPlus simulation for Case E. Furthermore, because LRBEM + is designed to provide feedback in the early design stages, we wanted to test its ability to capture the predicted change in heating and cooling loads in response to changes in design parameters. To this end, we utilized Case E to conduct a sensitivity analysis that systematically tested LRBEM + performance against the equivalent

6 6 M. Al Gharably et al. Figure 3. Case E: A building with all of the non-rectangular shape attributes present in Cases A D. Detailed specifications for this building are provided in Table A3 in the appendix. EnergyPlus model. First, each of the early design parameters selected by the stepwise regression and included in LRBEM + was incremented from its Case E baseline value (Table 4) by an amount equivalent to 25% of the plausible range used in the Monte Carlo simulation (Table 1). For example, the floor area was 3724 m 2 in the default Case E representation, and was incremented by 25% of the range used in the Monte Carlo simulation ( m 2 ) to 5574 m 2 (Table 4). Scaling each parameter by an amount proportional to its plausible range ensures that no single parameter has an undue influence on the results. Second, an EnergyPlus model corresponding to each individual parameter change was developed, and served as the basis for comparison with LRBEM + results. While conducting the sensitivity analysis, we only considered the early design parameters and omitted the cross-terms to (1) simplify the results by focusing on tangible parameters of interest to architects during early design, and (2) rank the early design parameters by their standardized regression coefficients (SRCs), which require independence among the considered parameters. This sensitivity analysis on Case E was conducted for each of the four locations used to generate the data. 4. Results The results are organized into subsections, with a parallel structure to that of Section 3. Section 4.1 presents the results from the stepwise regression described in Section 3.1, Section 4.2 presents the results associated with nonrectangular geometry Cases A D, as shown in Figure 2 and described in Section 3.2. Section 4.3 presents the results for the limiting Case E, as shown in Figure 3 and described in Section Results from stepwise regression used to create LRBEM + For each climate zone, the stepwise regression was conducted with a random selection of 16,000 data points from the full set of 20,000. The resultant stepwise regression model consisted of a reduced set of explanatory variables identified by the stepwise regression process: 48 explanatory variables were included in the final model for heating load and 35 explanatory variables in the model for cooling load. All 30 early design parameters were selected by the stepwise regression used to create LRBEM +, either as a stand-alone parameter (e.g. floor area) or as part of a cross-term (e.g. west SHGC window area). See Tables A4 and A5 in the Appendix for the resulting sets of explanatory variables and their corresponding coefficients for heating and cooling load in each climate zone. In LRBEM +, the addition of derived parameters and the replacement of some of the design parameters from the original LRBEM indicate that the new interactions captured among wall, window, and roof areas, along with their material properties, are significant. Most notably, the building wall areas by cardinal direction were included in all the regression models. Roof area was only included through derived parameters in both the heating and cooling models, as floor area adequately captures the horizontal surface area. The test data set consisting of the remaining 4000 energy simulations not used in LRBEM + development was first used to evaluate the ability of LRBEM + to predict the energy loads for rectangular buildings. The energy loads predicted with LRBEM + were compared against the corresponding EnergyPlus simulation results for the validation set. The LRBEM + predictions for heating energy load in each climate zone yielded an R 2 value between and 0.965, and for cooling an R 2 value between and Figure 4 shows for each location the linear fit and R 2 values for the heating and cooling loads predicted by LRBEM + versus the EnergyPlus simulations. In addition, Table 5 presents a systematic comparison between LRBEM and LRBEM + results to assess the relative performance of the revised model compared to the original one. Table 5 includes the coefficient of variance of the root mean square error (CV(RMSE)), average per cent error, MBE, and R 2. In general, the errors are higher for LRBEM +, which indicates that the original LRBEM has better skill at predicting heating and cooling loads for rectangular buildings. With the exception of Miami heating, the MBE for the LRBEM + compared to that of LRBEM ranges from 1.5% lower to 0.8% higher, indicating no significant change in model bias. Similarly, the change in average per cent error (9% lower to 4% higher) and CV(RMSE) (12.9% lower to 3.5% higher) indicates no significant change in average error across the validation set. Overall, the prediction performance of LRBEM + is comparable to the performance of the original LRBEM presented in Hygh et al. (2012) for rectangular-shaped buildings. Both models have difficulty predicting the heating load in Miami, which has a small magnitude and is therefore

7 Journal of Building Performance Simulation 7 Table 4. Building parameter values for the baseline and the incremented value used to test the LRBEM + energy prediction for Case E. Baseline 25% increase based on Monte Carlo range Building floor area (m 2 ) Orientation (degrees) Roof emissivity Roof solar absorptance Roof R-value Window SHGC (N, S, E, and W) Window R-value (N, S, E, and W) Wall R-value (N, S, E, and W) Shading projection factor (N, S, E, and W) Window-to-wall ratio (N, S, E, and W) Figure 4. Scatterplots demonstrating the linear relationship between predictions from LRBEM + (horizontal axis) and EnergyPlus (vertical axis) for cooling (left column) and heating loads (right column). The rows represent the results from each of the four selected locations. prone to high errors. LRBEM + tends to over-predict low values, as indicated by the large positive value for MBE. While this strong positive bias for Miami results requires further investigation, the focus of LRBEM + is to predict relative changes in heating and cooling load accurately rather than the magnitude.

8 8 M. Al Gharably et al. Table 5. LRBEM+ performance in predicting heating and cooling loads for buildings in four climate zones; the errors (expressed in %) represent the difference between LRBEM + predictions and EnergyPlus simulations across the 4000 runs in the validation set. Location Miami Winston-Salem Albuquerque Minneapolis Climate zone 1A 4A 4B 6A LRBEM + Heating Average per cent error CV (RMSE) MBE R Cooling Average per cent error CV (RMSE) MBE R Original LRBEM Heating Average per cent error CV (RMSE) MBE R Cooling Average per cent error CV (RMSE) MBE R As a first step to assess LRBEM + performance on non-rectangular building geometry, we used it to predict heating and cooling loads for the building shown in Figure 1 and compared the results to those obtained using the four methods based on the original LRBEM. Among the five tested methods, LRBEM + produced the lowest relative error in the heating load prediction, and the second highest relative error in the cooling load prediction (Table 2). While the cooling error was high, the total spread in cooling relative error among the five tested methods was only 4.5%. While these results are encouraging, further tests were performed to conduct a more rigorous comparison, as described in Section Results for Cases A D Table 6 presents the relative error between LRBEM + and EnergyPlus when predicting the heating and cooling loads for Cases A D in all four climate zones. The relative errors between LRBEM + and EnergyPlus are generally less than 10% across Cases A D, though higher errors are observed in heating load predictions for Miami and Winston-Salem under Cases B2, C1, and C2 and in heating and cooling load predictions for Albuquerque under case B1. In Cases B2, C1, and C2, the varying roof heights simultaneously increase building surface area, change thermal zoning, and create self-shading effects. We speculate that the high errors in Winston-Salem heating load predictions occur because the magnitude of the heating load is relatively small: the Winston-Salem heating load is smaller in magnitude than for cooling and represents the second lowest heating load among the four locations. The high error in Albuquerque heating and cooling load predictions for Case B1 is likely due to the effect of non-right-angled building corners on the amount of solar gain estimated per cardinal direction, which is a key factor in arid regions. The results for Cases A1 and A2 suggest that LRBEM + accurately predicts the energy loads for designs with a courtyard configuration that produces a shading effect. We also tested all five methods summarized in Table 2 LRBEM + and the four methods based on the original LRBEM against Cases A C. Since a rigorous comparison between LRBEM and LRBEM + prediction errors for rectangular buildings is made in Section 4.1, Case D is omitted from this comparison as it is also based on a rectangular building. A total of 48 tests were conducted, based on separate heating and cooling load predictions across all four locations and all six building cases. The LRBEM + produced the lowest relative error among the five methods in 37 of the 48 separate tests conducted (77%). Figure A1 presents the relative errors associated with each method and test Results for Case E Case E (Figure 3) has a building geometry that combines the four cases (A D) tested earlier. We describe the results for Winston-Salem in this section, and the equivalent results for the other three locations are provided in the Appendix (Figures A2 A4). The predicted baseline heating loads for Case E in Winston-Salem are 204 and 230 GJ/yr for LRBEM + and EnergyPlus, respectively, resulting in a relative error of 11%. Similarly, the cooling loads predicted by LRBEM + and EnergyPlus are 274 and 256 GJ/yr, respectively, resulting in a relative error of 7.0%. Relative errors for all four locations are provided in Table 6.

9 Journal of Building Performance Simulation 9 Table 6. Prediction of relative error (%) between LRBEM + and EnergyPlus for heating and cooling in the four climate regions for the different shape-factor Cases A E. Case Miami Winston-Salem Albuquerque Minneapolis Cooling Heating a Cooling Heating Cooling Heating Cooling Heating A A B B C C D D E a We did not report the relative errors in Miami heating for Cases A1, A2, and B1 given their very high values ( > 25%). Figure 5. Change in cooling (top) and heating (bottom) loads for Winston-Salem predicted by EnergyPlus (black) and LRBEM + (grey) in response to the incremental change in each building parameter (Table 4). Parameters on the x-axis are ranked by level of importance in LRBEM + based on their SRCs (left is most influential). Figure 5 summarizes the results of the Case E sensitivity analysis and shows the relative change in energy loads predicted with EnergyPlus and LRBEM +. The bars associated with each parameter in Figure 5 are arranged highest to lowest (left to right) based on the value of the SRCs obtained from LRBEM +. Because the SRCs normalize the raw regression coefficients by the ratio of standard deviation in the parameter input to the standard deviation in

10 10 M. Al Gharably et al. heating or cooling load, they provide a convenient metric to assess the relative effect of a given regression parameter on the resultant heating or cooling load. While roof area and wall area by cardinal direction were included in the LRBEM + for both heating and cooling loads, they were omitted from the Case E sensitivity analysis (and therefore Figure 5), because it is not possible to isolate the effect of these parameters individually. For example, scaling the west wall area would necessarily require the wall areas in other directions to be scaled proportionally, perhaps along with the floor and roof area, depending on the desired building geometry. The floor area parameter, which exhibits the largest effect on heating and cooling loads, implicitly includes the effect of roof area and wall areas in all orientations (N, S, E, and W) since those parameters must be scaled in proportion with the floor area. In Winston-Salem, the incremental changes in heating and cooling loads predicted by LRBEM + and Energy- Plus generally match in direction, with exceptions related to roof and wall R-values. Figures A2 A4 similarly show a match in the direction of the incremental change with exceptions related primarily to roof emissivity as well as window and wall R-values. Given the 25 early design parameters tested across the four sites for both heating and cooling loads, there are a total of 200 separate predictions within the Case E sensitivity analysis. Only 33 of the 200 LRBEM + predictions (17%) predict a load change in a direction opposite to that predicted by EnergyPlus. Notably, the associated EnergyPlus results for these 33 parameters indicate an incremental change in heating or cooling loads smaller than 5% despite a 25% change in each input parameter value. This result suggests that LRBEM + predicts the incremental changes in energy loads correctly in scenarios where the magnitude of the energy load change is significant (i.e. at least a 5% change). 5. Conclusions The work presented in this paper represents a significant extension of Hygh et al. (2012) by generalizing the prediction capability of LRBEM to medium-sized commercial buildings with geometries that are more complex than a simple rectangular shape. The updated model, referred to as LRBEM +, represents a revised multivariate linear regression equation based on 30 early building design parameters using the results from the original 20,000 Monte Carlo realizations developed by Hygh et al. (2012). LRBEM + performs well compared to the original LRBEM for rectangular buildings, and can accurately predict the heating and cooling loads associated with more complex building geometries, including those with the following features: courtyards, non-rectangular forms, various floor-to-floor heights, and non-uniform roof heights. Moreover, LRBEM + was able to predict the direction and approximate magnitude of changes in the Case E heating and cooling loads associated with incremental changes in most of the early building design parameters considered across the four sites. Only a small subset of the LRBEM + predictions (17%) incorrectly predict the sign of the heating or cooling load change relative to EnergyPlus simulations. This error is not significant since the changes in energy loads predicted by EnergyPlus for this small subset of parameters are less than 5%, even though the input parameter values changed by 25%. These errors tend to focus on roof emissivity as well as window and wall R-values, which suggest that the interaction of these material properties with building geometry is not well captured by LRBEM +. Given the unresolved uncertainties regarding building form and function during the early design stages, the purpose of the tool is not to precisely predict the EnergyPlus output, but rather to match the direction and approximate magnitude of changes in heating and cooling loads when the input parameters are varied by the designer. With further development, LRBEM + could serve as a practical substitute for building energy simulation engines in early building design stages by providing instantaneous feedback on energy performance through the specification of a limited number of early design parameters. Like LRBEM, the revised LRBEM + is currently limited to four climate zones in the USA represented by the cities Winston-Salem, Miami, Albuquerque, and Minneapolis, which are meant to span the climatic extremes within the continental USA. Future work could extend LRBEM + development beyond these four climate zones. The model framework could be expanded in several additional ways. First, the model could be extended to include responsive estimates of lighting and plug loads and to allow variation in internal load assumptions, enabling feedback that links building design attributes to internal load estimates. In our work, thus far, we have held internal loads at fixed values as described in Section 2; however, internal loads play a critical role in determining building energy consumption and can be influential in the early design process. A high priority for future work is to make the artificial lighting requirement responsive to buildingspecific daylighting, which will affect the heating, cooling, and lighting loads. Second, the framework could be extended to predict different building categories, such as large- and smallsized office buildings, hotels, high- and low-rise residential, warehouses, malls, and retail stores. Incorporating these other building types would involve conducting the Monte Carlo simulation with a different base EnergyPlus model for each building type. The results would lead to different stepwise regression equations, and the relative influence of different design parameters could vary significantly between building types. For example, consideration

11 Journal of Building Performance Simulation 11 of high-rise buildings with proportionally smaller footprints may diminish the importance of roof area and its associated thermal properties. Future work may also consider a reformulation of LRBEM + in which the Monte Carlo simulation that serves as the basis for the stepwise regression includes non-rectangular buildings. Trying to capture the thermal effects of non-rectangular geometry within the Monte Carlo simulation may further improve its skill, but would require the manual or programmatic specification of geometry for approximately 1000 building instances, which would represent a new and significant undertaking. Third, formal search techniques can be applied to LRBEM + to help identify optimal design solutions. Such solutions could be reached by allowing users to optimize a subset of their design decisions to minimize heating and cooling loads. Availability of LRBEM + is significant, because it can serve as a practical decision support tool for architects during the early design stages, when building design decisions have strong implications on realized energy performance, yet tools to assess energy in early design are most lacking. By targeting the early design phase, the intention of LRBEM + is to help transform building energy performance from a design outcome to a design target. Acknowledgements The authors would like to thank Soolyeon Cho, Travis Stratakes, and Janelle Griffin for the valuable feedback they provided during the course of this research. References AIA An Architects Guide to Intergrating Energy Modeling in the Design Process. American Institute of Architects, AIA Energy Modeling Working Group. Accessed June 12, ASHRAE Energy Standard for Buildings Except Low- Rise Residential Buildings, ASHRAE American Society of Heating, Refrigeration, and Air-Conditioning Engineers, Atlanta, GA. Autodesk Autodesk Ecotect Analysis. Accessed September 29, sis/. Deru, M., K. Field, D. Studer, K. Benne, B. Griffith, P. Torcellini, B. Liu, et al U.S. Department of Energy Commercial Reference Building Models of the National Building Stock. National Renewable Energy Laboratory, Golden, CO, NREL/TP EIA. 2013a. Annual Energy Review U.S. Energy Information Administration. Washington, DC, DOE/ EIA0384(2013). EIA. 2013b. International Energy Statistics. US Energy Information Administration. Accessed July 22, Haberl, J. S., and T. E. Bou-Saada Procedures for Calibrating Hourly Simulation Models to Measured Building Energy and Environmental Data. Journal of Solar Energy Engineering 120: doi: / Hygh, J. S Implementing Energy Simulation as a Design Tool in Conceptual Building Design with Regression Analysis. Master of Science thesis in Civil Engineering, NC State University. Accessed November 26, Hygh, J. S., J. F. DeCarolis, D. B. Hill, and S. R. Ranjithan Multivariate Regression as an Energy Assessment Tool in Early Building Design. Building and Environment 57: /j.buildenv Maile, T., M. Fischer, and V. Bazjanac Building Energy Performance Simulation Tools a Life cycle and Interoperable Perspective. White paper; Center for Integrated Facility Engineering, Stanford University. Accessed June 12, MathWorks Stepwisefit: Stepwise regression. Math- Works Documentation Center. Accessed July 22, Nielsen, T. R Simple Tool to Evaluate Energy Demand and Indoor Environment in the Early Stages of Building Design. Solar Energy 78: doi: /j.solener NREL Commercial Buildings Research and Development: OpenStudio. US Department of Energy, National Renewable Energy Laboratory (NREL). Accessed September 29, Rafferty, P., M. Keane, and J. O Donnell Calibrating Whole Building Energy Models: An Evidence-based Methodology. Energy and Buildings 43: doi: /j.enbuild Reddy, T. A., and D. E. Claridge Uncertainty of Measured Energy Savings from Statistical Baseline Models. HVAC&R Research 6: doi: / Sefeira Software Accessed September 29, Urban, B. J., and L. R. Glicksman A Simplified Rapid Energy Model and Interface for Nontechnical Users. Paper presented at the Buildings X Conference, ASHRAE, Clearwater, FL, December 2 7. Appendix Table A1. Assumed parameter values for the building shown in Figure 1. Parameter Value Total floor area (m 2 ) 8926 Envelop area/floor area 0.89 Orientation 0 Stories 3 Window/wall ratio (N, S, E, and W) (%) 49 Shading projection factor (N, S, E, and W) (%) 50 Window SHGC (N, S, E, and W) 0.35 Window R-value (N, S, E, and W) (m 2 K/W) 0.39 Wall R-value(N,S,E,andW)(m 2 K/W) 3.2 Roof emissivity 0.9 Roof colour (solar absorptance) 0.7 Roof R-value (m 2 K/W) 4.5

12 12 M. Al Gharably et al. Table A2. List of original building parameters and derived cross-terms, which served as input to the stepwise regression used to formulate LRBEM +. Building parameter 1 Building floor area 2 Wall area west 3 Wall area south 4 Wall area east 5 Wall area north 6 Roof area 7 Orientation 8 Roof emissivity 9 Roof solar absorptance 10 Roof R-value 11 Window SHGC west 12 Window SHGC south 13 Window SHGC east 14 Window SHGC north 15 Window R-value west 16 Window R-value south 17 Window R-value east 18 Window R-value north 19 Wall R-value west 20 Wall R-value south 21 Wall R-value east 22 Wall R-value north 23 Shading projection west 24 Shading projection south 25 Shading projection east 26 Shading projection north 27 Window-to-wall ratio west 28 Window-to-wall ratio south 29 Window-to-wall ratio east 30 Window-to-wall ratio north Derived parameters cross-terms 31 Window area west 32 Window area south 33 Window area east 34 Window area north 35 West window area cos(rotation) 36 South window area cos(rotation) 37 East window area cos(rotation) 38 North window area cos(rotation) 39 West window R-value window area 40 South window R-value window area 41 East window R-value window area 42 North window R-value window area 43 Roof emissivity roof area 44 Roof solar absorptance roof area 45 Roof R-value roof area 46 Roof emissivity roof area/number of stories 47 Roof solar absorptance roof area/number of stories 48 Roof R-value roof area/number of stories (Continued) Table A2. Continued. 49 West SHGC window area 50 South SHGC window area 51 East SHGC window area 52 North SHGC window area 53 West shading project factor window-to-wall ratio 54 South shading project factor window-to-wall ratio 55 East shading project factor window-to-wall ratio 56 North shading project factor window-to-wall ratio 57 (North window area)2 58 (South window area)2 59 sin(orientation) 60 cos(orientation) 61 abs(cos(orientation)) 62 sin(orientation) squared 63 cos(orientation) squared 64 sin(orientation) + abs(cos(orientation)) 65 West R-value window area sin(o) 66 South R-value window area sin(o) 67 East R-value window area sin(o) 68 North R-value window area sin(o) 69 3-[sin(orientation) + abs(cos(orientation))] 70 West window area [sin(o) ] 71 South window area [sin(o) ] 72 East window area [sin(o) ] 73 North window area [sin(o) ] 74 West wall R-value opaque wall area 75 South wall R-value opaque wall area 76 East wall R-value opaque wall area 77 North wall R-value opaque wall area 78 West SHGC window area sin(o) 79 South SHGC window area sin(o) 80 East SHGC window area sin(o) 81 North SHGC window area sin(o) 82 West shading project factor window-to-wall ratio sin(o) 83 South shading project factor window-to-wall ratio sin(o) 84 East shading project factor window-to-wall ratio sin(o) 85 North shading project factor window-to-wall ratio sin(o)

13 Journal of Building Performance Simulation 13 Table A3. Assumed parameter values for the buildings shown in Figures 2 and 3. Building case A1 A2 B1 B2 C1 C2 D1 D2 E Building floor area (m 2 ) Roof area (m 2 ) Orientation West wall area (m 2 ) South wall area (m 2 ) East wall area (m 2 ) North wall area (m 2 ) Roof emissivity Roof solar absorptance Roof R-value (m 2 K/W) West window SHGC South window SHGC East window SHGC North window SHGC West window R value (m2 K/W) South window R value (m 2 K/W) East window R value (m 2 K/W) North window R value (m 2 K/W) West wall R-value (m 2 K/W) South wall R-value (m 2 K/W) East wall R-value (m 2 K/W) North wall R-value (m2 K/W) Shading projection factor west (% of window height) Shading projection factor south (% of window height) Shading projection factor east (% of window height) Shading projection factor north (% of window height) Window-to-wall ratio west Window-to-wall ratio south Window-to-wall ratio east Window-to-wall ratio north

14 14 M. Al Gharably et al. Table A4. Parameter coefficients used in the LRBEM + heating model, by location. 1A Miami 4A Winston-Salem 4B Albuquerque 6A Minneapolis Parameter a 4.99E E E E + 10 Intercept 2.17E E E E + 07 Building floor area 1.75E E E E + 08 Wall area W 4.65E E E E + 08 Wall area S 1.75E E E E + 08 Wall area E 4.65E E E E + 08 Wall area N 7.76E E E E + 11 Wall U-value west 9.08E E E E + 09 Wall U-value south 7.78E E E E + 09 Wall U-value east 2.89E E E E + 09 Wall U-value north 1.19E E E E + 10 Window-to-wall ratio west 1.21E E E E + 10 Window-to-wall ratio south 1.90E E E E + 10 Window-to-wall ratio east 1.03E E E E + 10 Window-to-wall ratio north 3.30E E E E + 07 Window area west 1.17E E E E + 07 Window area south 1.81E E E E + 08 Window area east 8.18E E E E + 07 Window area north 1.23E E E E + 08 West window U-value window area 8.80E E E E + 08 South window U-value window area 6.88E E E E + 08 East window U-value window area 7.65E E E E + 08 North window U-value window area 9.39E E E E + 08 Roof emissivity roof area/number of stories 1.42E E E E + 07 Roof solar absorptance roof area 7.57E E E E + 08 Roof R-value roof area/number of stories 2.91E E E E + 08 West SHGC window area 2.96E E E E + 08 South SHGC window area 2.31E E E E + 08 East SHGC window area 8.34E E E E + 08 North SHGC window area 2.06E E E E + 10 West shading project factor window-towall ratio 1.01E E E E + 11 South shading project factor windowto-wall ratio 1.07E E E E + 10 East shading project factor window-towall ratio 9.87E E E E + 11 North shading project factor windowto-wall ratio 1.91E E E E + 09 sin(orientation) + abs(cos(orientation)) 1.28E E E E + 06 West window U-value window area sin(o) 8.81E E E E + 06 South window U-value window area sin(o) 5.41E E E E + 05 East window U-value window area sin(o) 4.72E E E E + 05 North window U-value window area sin(o) 9.05E E E E + 08 West wall U-value opaque wall area 7.48E E E E + 07 South wall U-value opaque wall area 3.97E E E E + 06 East wall U-value opaque wall area 5.67E E E E + 07 North wall U-value opaque wall area 3.63E E E E + 08 West SHGC window area sin(o) 5.51E E E E + 05 South SHGC window area sin(o) 3.63E E E E + 08 East SHGC window area sin(o) 1.59E E E E + 07 North SHGC window area sin(o) a Wall and window R-values are converted to U-values before performing the Monte Carlo simulation with EnergyPlus.

15 Journal of Building Performance Simulation 15 Table A5. Parameter coefficients used in the LRBEM + cooling model, by location. 1A Miami 4A Winston-Salem 4B Albuquerque 6A Minneapolis Parameter a 2.71E E E E + 10 Intercept 1.38E E E E + 07 Building floor area 4.76E E E E + 06 Wall area W 3.94E E E E + 06 Wall area S 4.76E E E E + 06 Wall area E 3.94E E E E + 06 Wall area N 2.53E E E E + 10 Window-to-wall ratio west 8.52E E E E + 10 Window-to-wall ratio south 4.22E E E E + 10 Window-to-wall ratio east 9.70E E E E + 10 Window-to-wall ratio north 6.63E E E E + 07 Window area west 6.33E E E E + 07 Window area south 5.82E E E E + 07 Window area east 7.35E E E E + 07 Window area north 3.58E E E E + 06 West window U-value window area 4.63E E E E + 06 South window U-value window area 6.07E E E E + 06 East window U-value window area 5.17E E E E + 06 North window U-value window area 5.44E E E E + 07 Roof emissivity roof area 1.54E E E E + 06 Roof solar absorptance roof area 1.31E E E E + 06 Roof R-value roof area 5.77E E E E + 08 West SHGC window area 4.44E E E E + 08 South SHGC window area 4.25E E E E + 08 East SHGC window area 4.46E E E E + 08 North SHGC window area 3.98E E E E + 10 West shading project factor window-to-wall ratio 1.49E E E E + 10 South shading project factor window-to-wall ratio 8.04E E E E + 10 East shading project factor window-to-sall ratio 1.36E E E E + 10 North shading project factor window-to-wall ratio 8.29E E E E + 07 sin(orientation) 4.09E E E E + 10 sin(orientation) squared 1.30E E E E + 07 West wall U-value opaque wall area 1.02E E E E + 06 South wall U-value opaque wall area 2.21E E E E + 06 East wall U-value opaque wall area 4.65E E E E + 05 North wall U-value opaque wall area 1.45E E E E + 07 West SHGC window area sin(o) 3.42E E E E + 07 South SHGC window area sin(o) 9.38E E E E + 07 East SHGC window area sin(o) 3.15E E E E + 07 North SHGC window area sin(o) a Wall and window R-values are converted to U-values before performing the Monte Carlo simulation with EnergyPlus.

16 16 M. Al Gharably et al. A1 A2 B1 C1 Figure A1. Comparison of relative errors between the methods listed in Table 2 and EnergyPlus, across building Cases A C, and all four tested locations. Cooling ( C ) and heating ( H ) loads are presented separately. Note that the combination of rectangles approach was not applied to Cases B or C, since non-rectilinear building forms cannot easily be represented with that method. B2 C2