# Development of Building Design Optimization Methodology: Residential Building Applications

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## Abstract

**:**

## 1. Introduction

^{2}) that indicates how well the data fit the model; however, the heating regression model has a low value for R

^{2}from 0.498 to 0.816. Ghiaus [9] used the concept of balance point temperature to construct a regression model. Hygh et al. [11] used a fixed 18 °C balance temperature to predict energy consumption, and Krarti [12] estimated the exact balance temperature. Eisenhower [13] developed an analytical metamodel that fit the building simulation data and then performed the optimization. Although a simplified modeling approach is capable of carrying out the building design optimization process with little computational effort, it essentially requires a tremendous number of pre-energy simulation results to obtain a reliable model, which is the major disadvantage of this approach. Consequently, regenerating the database is necessary when changing the design variables to create a simplified model.

## 2. Overall Methodology

## 3. Variable Selection Algorithm and Optimization Process

#### 3.1. Variable Selection Algorithm Using Singular Value Decomposition

#### 3.2. Application to an Optimization Problem

#### 3.3. Simplified Energy Consumption Modeling Approach

- The actual degree day varies with the chosen design variables. Consequently, the balance temperature must be calculated for each design combination;
- The modified ANAGRAM method is used to shift the diagonal linear regression models to the origin based on daily average degree days to separately predict heating, cooling, and fan energy consumption;
- To find an appropriate number of data samples to yield an accurate regression model, the recursive least square (RLS) algorithm is used;
- A simplified model is constructed specifically to replace energy simulation software during the variable selection phase. This model is employed for the sole purpose of identifying which variables significantly influence the output. Therefore, the model must be sufficiently accurate to discern these influential trends.

**.**After the simulation studies for residential buildings, a day is selected as $P$ (i.e., $P=1$ day). A natural question regards the selection of an appropriate sample size to accurately estimate the parameters; that is, the question of how many daily simulations are needed to accurately estimate $\mathsf{\alpha}$ and ${T}_{\mathrm{b}}$. This problem can be handled by employing the RLS method because it calculates confidence intervals for estimated parameters as each sample is updated. Because the original model, Equation (8), is a nonlinear function with respect to the parameters, it was reformulated as follows:

**,**two simplified models at two design points of $(x,x+\Delta {x}_{i})$ are constructed. The process is repeated for each variable and each data point to construct a series of Jacobian matrices (shown in Equation (4)).

#### 3.4. Optimization with the Significant Variables and Sequential Search Method

## 4. Case Studies

#### 4.1. Descriptions of Case Study Buildings

^{2}with two different spaces, which include a living space and attic. The living space is the only conditioned zone, and the net conditioned area is 223.1 m

^{2}. The entry doors are located on the south and north sides of the building, and two windows are placed on each side of the building. The house is assumed to have three bedrooms and three bathrooms. The second house has a gross floor area of 446.1 m

^{2}. The house has a living space, attic, and heated basement. The prototype complies with the 2012 International Energy Conservation Code with modified construction layers for optimization purposes. The living space is the only conditioned zone, and the net conditioned area is 334.6 m

^{2}. The house has four bedrooms and four bathrooms, and the basement is placed completely below the ground level.

#### 4.2. Defining Design Variables for Optimization

#### 4.3. Optimization Objective Function

- $LCC$—incremental LCC of given building system;
- ${I}_{f}$—influence factor of the location;
- ${C}_{\mathrm{C}\mathrm{o}\mathrm{n}}$—construction cost;
- ${C}_{\mathrm{H}\mathrm{V}\mathrm{A}\mathrm{C}}$—HVAC equipment cost;
- ${C}_{\mathrm{E}\mathrm{l}\mathrm{e}\mathrm{c}}$—electricity cost;
- ${C}_{\mathrm{N}\mathrm{G}}$—natural gas cost;
- $UP{V}_{\mathrm{E}\mathrm{l}\mathrm{e}\mathrm{c}}^{\mathrm{*}}$—UPV* factor for electricity cost;
- $UP{V}_{\mathrm{N}\mathrm{G}}^{\mathrm{*}}$—UPV* factor for natural gas cost.

## 5. Case Study Results

#### 5.1. Validation Methodology

#### 5.2. Variable Selection Results

#### 5.3. Simplified Energy Consumption Model

#### 5.4. Optimization Result Comparison

## 6. Conclusions

- Using the variable selection process, significant variables (5 out of 12) that demonstrate the strongest contribution to the optimization study are identified;
- The proposed methodology significantly shortens the time requirement for the optimization process in the two case studies of 74% and 84%, and the optimized LCC is within 0.05% and 0.06%, respectively, of the optimum point.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

^{5}and ${p}_{\mathrm{t}\mathrm{h}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{h}}$ as 10

^{−1}based on a simulation study.

## Appendix B

Case Study Building 1 | Full Optimization | Optimization with the Significant Variables |

Roofing Material | F12 asphalt shingles | F12 asphalt shingles |

Roof Eave Overhang Depth | 305 mm (12 in.) | - |

Attic Insulation Material | Attic loose fill—R3.3 (IP-R19) | - |

External Wall Siding Material | Vinyl siding | Vinyl siding |

External Foam Board | 12.7 mm (0.5 in.) | - |

Wall Core | 38 × 89 mm (2 × 4 in.) studs at 400 mm (16 in.) on center filled with loose fill cellulose insulation | 38 × 89 mm (2 × 4 in.) studs at 400 mm (16 in.) on center filled with loose fill cellulose insulation |

Foundation Wall Insulation | Extruded polystyrene 25.4 mm (1 in.) | - |

Window Type | Double-pane window | - |

Air Conditioner Speed | Single-speed | Single-speed |

Heat Recovery Type | None | None |

Seasonal Coefficient of Performance (Air Conditioner SEER) | 3.81 (SEER 13) | - |

Natural Gas Furnace Efficiency | 0.80 | - |

Case Study Building 2 | Full Optimization | Optimization with the Significant Variables |

Roofing Material | F12 asphalt shingles | F12 asphalt shingles |

Roof Eave Overhang Depth | 305 mm (12 in.) | - |

Attic Insulation Material | Attic loose fill—R3.3 (IP-R19) | - |

External Wall Siding Material | Vinyl siding | Vinyl siding |

External Foam Board | 12.7 mm (0.5 in.) | - |

Wall Core | 38 × 89 mm (2 × 4 in.) studs at 400 mm (16 in.) on center filled with loose fill cellulose insulation | |

Under-Floor Insulation | Extruded polystyrene 25.4 mm (1 in.) | - |

Window Type | Double-pane window | - |

Air Conditioner Speed | Single-speed DX | Single-speed DX |

Heat Recovery Type | None | None |

Seasonal Coefficient of Performance (Air Conditioner SEER) | 3.81 (SEER 13) | - |

Natural Gas Furnace Efficiency | 0.80 | - |

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**Figure 4.**(

**a**) Result of singular values in case study 1. (

**b**) Result of singular values in case study 2.

Design Variables (Number of Options) | Available Values |
---|---|

Roofing Material (3) | Asphalt shingles, metal surface, concrete tile roof |

Roof Eave Overhang Depth (3) | 305 mm (12 in.), 457 mm (18 in.), 610 mm (24 in.) |

Attic Insulation Material (12) | Loose fill cellulose: R3.3 (IP ^{1}-R19), R4.4 (IP-R25), R5.3 (IP-R30), R6.7 (IP-R38), R8.6 (IP-R49), R10.6 (IP-R60) |

Fiberglass batting: R3.3 (IP-R19), R4.4 (IP-R25), R5.3 (IP-R30), R6.7 (IP-R38), R8.6 (IP-R49), R10.6 (IP-R60) | |

External Wall Siding Material (4) | Vinyl siding, wood siding, fiber cement siding, brick |

Wall Core (16) | Stud: 38 × 89 mm (2 × 4 in.) studs at 400 mm (16 in.), 38 × 140 mm (2 × 6 in.) studs at 600 mm (24 in.) on center, 38 × 184 mm (2 × 8 in.) studs at 600 mm (24 in.) on center Insulation: filled with fiberglass batting insulation, sprayed-on foam insulation, loose-fill cellulose insulation |

Structural insulated panels: 114 mm (3 5/8 in.), 165 mm (5 5/8 in.), 210 mm (7 3/8 in.), 260 mm (9 3/8 in.) | |

Insulated concrete forms: 228 mm (9 in.), 278 mm (11 in.), 328 mm (13 in.) | |

External Foam Board Insulation (6) | Board insulation: 12.7 mm (0.5 in.), 25.4 mm (1 in.), 38.1 mm (1.5 in.), 50.8 mm (2 in.), 63.5 mm (2.5 in.), 76.2 mm (3 in.) |

Under-Floor Insulation or Foundation Wall Insulation (8) | Extruded polystyrene: 25.4 mm (1 in.), 50.8 mm (2 in.), 76.2 mm (3 in.), 101.6 mm (4 in.) |

Expanded polystyrene: 25.4 mm (1 in.), 50.8 mm (2 in.), 76.2 mm (3 in.), 101.6 mm (4 in.) | |

Window Type (2) | Double-pane window, triple-pane window |

Air Conditioner Speed (2) | Single-speed, multispeed |

Heat Recovery Type (2) | None, sensible heat recovery |

Seasonal Coefficient of Performance (Air Conditioner SEER ^{2}) (6) | 3.81 (SEER 13), 4.10 (SEER 14), 4.40 (SEER 15), 4.69 (SEER 16), 4.98 (SEER 17), 5.28 (SEER18) |

Natural Gas Furnace Efficiency (4) | 80%, 85%, 90%, 95% |

^{1}IP stands for imperial units;

^{2}SEER represents Seasonal Energy Efficiency Ratio.

Case Study Building 1 | Coordinate 1 | Coordinate 2 | Coordinate 3 | Coordinate 4 |

Roofing Material | −0.012 | 0.024 | 0.401 | −0.910 |

Roof Eave Overhang Depth | 0.000 | −0.001 | −0.030 | 0.003 |

Attic Insulation Material | −0.006 | −0.006 | 0.014 | 0.101 |

External Wall Siding Material | 0.999 | 0.028 | 0.017 | −0.005 |

External Foam Board | 0.000 | −0.005 | −0.041 | −0.010 |

Wall Core | −0.028 | 0.999 | −0.008 | 0.023 |

Foundation Wall Insulation | −0.003 | −0.009 | −0.021 | −0.069 |

Window Type | −0.002 | 0.001 | 0.125 | 0.053 |

AC Speed | −0.004 | −0.001 | 0.274 | 0.118 |

Heat Recovery Type | −0.012 | −0.002 | 0.847 | 0.367 |

Air Conditioner Seasonal Coefficient of Performance | −0.001 | 0.000 | 0.041 | 0.018 |

Natural Gas Furnace Efficiency | −0.003 | −0.002 | 0.163 | 0.071 |

Case Study Building 2 | Coordinate 1 | Coordinate 2 | Coordinate 3 | Coordinate 4 |

Roofing material | −0.012 | 0.024 | 0.416 | −0.903 |

Roof Eave Overhang Depth | 0.000 | −0.001 | −0.030 | 0.003 |

Attic Insulation Material | −0.006 | −0.006 | 0.013 | 0.101 |

External Wall Siding Material | 0.999 | 0.028 | 0.017 | −0.004 |

External Foam Board | 0.000 | −0.006 | −0.041 | −0.011 |

Wall Core | −0.028 | 0.999 | −0.009 | 0.023 |

Under-floor Wall Insulation | −0.002 | −0.009 | −0.023 | −0.074 |

Window Type | −0.002 | 0.001 | 0.130 | 0.057 |

AC Speed | −0.004 | −0.001 | 0.271 | 0.122 |

Heat Recovery Type | −0.012 | −0.002 | 0.837 | 0.379 |

Air Conditioner Seasonal Coefficient of Performance | −0.001 | 0.000 | 0.040 | 0.018 |

Natural Gas Furnace Efficiency | −0.003 | −0.002 | 0.176 | 0.080 |

^{3}Significant variables in both case studies are highlighted in blue in the first column.

**Table 3.**Correlation matrices between singular vectors obtained from detailed and simplified models

^{4}.

Case 1 | Case 2 | ||||||
---|---|---|---|---|---|---|---|

−1.00 | −0.08 | −0.02 | 0.02 | −0.99 | −0.11 | −0.05 | 0.02 |

−0.08 | 0.99 | 0.05 | 0.04 | −0.11 | 0.99 | −0.01 | −0.04 |

−0.02 | −0.05 | 1.00 | −0.07 | 0.05 | 0.00 | −0.99 | 0.14 |

−0.02 | 0.04 | −0.07 | −0.99 | −0.01 | −0.04 | −0.14 | −0.97 |

^{4}Diagonal elements of ${V}_{\mathrm{s}}^{T}{V}_{\mathrm{f}}$ are shown in grey.

Case Study Building 1 | Optimized LCC (USD) | Percentage Difference (%) |

Optimization with the Significant Variables | 53,062 | 4.19 |

Optimization with the Significant Variables and Sequential Search | 50,956 | 0.05 |

Optimization with All Design Variables | 50,929 | - |

Case Study Building 2 | Optimized LCC (USD) | Percentage Difference (%) |

Optimization with the Significant Variables | 57,818 | 4.60 |

Optimization with the Significant Variables and S Search | 55,308 | 0.06 |

Optimization with All Design Variables | 55,273 | - |

^{5}This cost does not include construction and HVAC costs in Equation (11) that are not associated with design variables.

Case Study Building 1 | Optimal Design with Full Optimization | Optimal Design with the Proposed Method |

Roofing Material | F12 asphalt shingles | F12 asphalt shingles |

Roof Eave Overhang Depth | 305 mm (12 in.) | 457 mm (18 in.) |

Attic Insulation Material | Attic loose fill—R3.3 (IP-R19) | Attic loose fill—R4.4 (IP-R25) |

External Wall Siding Material | F11 wood siding | F11 wood siding |

External Foam Board | 12.7 mm (0.5 in.) | 12.7 mm (0.5 in.) |

Wall Core | 38 × 140 mm (2 × 6 in.) studs at 600 mm (24 in.) on center filled with loose fill cellulose insulation | 38 × 140 mm (2 × 6 in.) studs at 600 mm (24 in.) on center filled with loose fill cellulose insulation |

Foundation Wall Insulation | Expanded polystyrene 50.8 mm (2 in.) | Expanded polystyrene 50.8 mm (2 in.) |

Window Type | Triple-pane window | Triple-pane window |

AC Speed | Multispeed | Multispeed |

Heat Recovery Type | Sensible | Sensible |

Seasonal Coefficient of Performance (Air Conditioner SEER) | 5.28 (SEER 18) | 5.28 (SEER 18) |

Natural Gas Furnace Efficiency | 0.95 | 0.95 |

Case Study Building 2 | Optimal Design with Full Optimization | Optimal Design with the Proposed Method |

Roofing Material | F12 asphalt shingles | F12 asphalt shingles |

Roof Eave Overhang Depth | 305 mm (12 in.) | 457 mm (18 in.) |

Attic Insulation Material | Attic loose fill—R4.4 (IP-R25) | Attic loose fill—R4.4 (IP-R25) |

External Wall Siding Material | F11 wood siding | F11 wood siding |

External Foam Board | 12.7 mm (0.5 in.) | 12.7 mm (0.5 in.) |

Wall Core | 38 × 140 mm (2 × 6 in.) studs at 600 mm (24 in.) on center filled with loose fill cellulose insulation | |

Under-Floor Insulation | Expanded polystyrene 25.4 mm (1 in.) | Expanded polystyrene 25.4 mm (1 in.) |

Window Type | Triple-pane window | Triple-pane window |

AC Speed | Multispeed | Multispeed |

Heat Recovery Type | Sensible | Sensible |

Seasonal Coefficient of Performance (Air Conditioner SEER) | 4.98 (SEER 17) | 5.28 (SEER 18) |

Natural Gas Furnace Efficiency | 0.95 | 0.95 |

^{6}The differences are shown in blue rows in the table.

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## Share and Cite

**MDPI and ACS Style**

Bae, Y.; Kim, D.; Horton, W.T.
Development of Building Design Optimization Methodology: Residential Building Applications. *Buildings* **2024**, *14*, 107.
https://doi.org/10.3390/buildings14010107

**AMA Style**

Bae Y, Kim D, Horton WT.
Development of Building Design Optimization Methodology: Residential Building Applications. *Buildings*. 2024; 14(1):107.
https://doi.org/10.3390/buildings14010107

**Chicago/Turabian Style**

Bae, Yeonjin, Donghun Kim, and William Travis Horton.
2024. "Development of Building Design Optimization Methodology: Residential Building Applications" *Buildings* 14, no. 1: 107.
https://doi.org/10.3390/buildings14010107