# Method for Scalable and Automatised Thermal Building Performance Documentation and Screening

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## Abstract

**:**

## 1. Introduction

#### Motivation

## 2. Method

#### 2.1. Heat Consumption Models

#### 2.1.1. Heat Consumption Models for Periods with Heat Demand

#### ${f}_{0}$: Fixed Base Temperature

#### ${f}_{1}$: Free Base Temperature

#### ${f}_{2}$: Convection and Infiltration

#### ${f}_{3}$: Solar Gain

#### ${f}_{4}$: Thermal Long Wave Radiation

#### 2.2. Smooth Maximum Approximation with LogSumExp

#### 2.3. Transition Interval

#### 2.4. Heat Consumption Modelling with LogSumExp

#### 2.5. Unmodelled Dynamics

#### 2.6. Thermal Performance Evaluation

## 3. Case Study

#### 3.1. Data

#### 3.2. Software

#### 3.3. Model Validation

#### 3.4. Residuals

#### 3.5. Parameter Sensitivity

#### 3.6. Thermal Performance Characterisation

## 4. Discussion

- A
- Unintended occupants’ related differences in the energy consumption can be estimated as the difference between the estimated user-related heat gain ${\Phi}_{\mathrm{x},t}$ and the user-related heat gain assumed in the design phase, ${\Phi}_{\mathrm{x},t}^{*}$.Assuming that the indoor temperature is equal to the actual temperature, the user related heat gain assumed in the design phase is$$\begin{array}{cc}\hfill {\Phi}_{\mathrm{x},t}^{*}& ={\Phi}_{\mathrm{int}}^{*}-{\Phi}_{\mathrm{vent}}^{*}\phantom{\rule{0.166667em}{0ex}},\hfill \end{array}$$Based on Equation (34), the difference between the assumed heat gain caused by the occupants and the actual heat gain (see A in Figure 7) can be estimated by$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\Phi}_{\mathrm{user}}& ={\mathrm{UA}}_{0}\phantom{\rule{0.166667em}{0ex}}({T}_{\mathrm{i}}-{T}_{\mathrm{b},t})-{\Phi}_{\mathrm{x},t}^{*}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& ={\Phi}_{\mathrm{x},t}-{\Phi}_{\mathrm{x},t}^{*}\phantom{\rule{0.166667em}{0ex}},\hfill \end{array}\end{array}$$The higher ${\Phi}_{\mathrm{x},t}|{T}_{\mathrm{i}}={T}_{\mathrm{i}}^{*}$ is, the more occupant-related heat is required to bring the building in thermal balance. This means, if ${\Phi}_{\mathrm{x},t}>{\Phi}_{\mathrm{user}}$, the internal heat gains are higher than expected in the design phase, the ventilation loss is lower than expected in the design phase, the indoor temperature is lower than the design temperature or a combination.On the other hand, if ${\Phi}_{\mathrm{x},t}<{\Phi}_{\mathrm{user}}$ the opposite is true.In this scenario, it is assumed that ventilation and internal gains are independent of the weather. In reality, this might be violated, and Equation (36) must by altered to account for that.
- B
- Weather-related differences in the energy use can be estimated by comparing the predicted energy use with the actual weather conditions, and the predicted energy use with the outdoor temperature, wind speed and global solar irradiation used in the design phase. Model M3 in Table 1 is used for prediction.
- C
- Building envelope-related differences in the energy use can be estimated as the difference between the predicted energy use obtained using model M3 in Table 1 and the occupants and weather corrected energy use obtained from points A and B, above.

## 5. Conclusions

- In the present paper, only 24 h average values were used with the argument that the effects of the heat capacities were averaged out as stated in [22]. Several tests on parameter sensitivity could be done with the input variables averaged over longer and shorter periods than 24 $\mathrm{h}$.Furthermore, the heat capacity could be modelled to account for potential dynamics related to the heat capacities of the building. Residuals (${e}_{t}$) with no cross-correlation with the differentiated outdoor temperature ($\mathrm{corr}({e}_{t},\phantom{\rule{0.166667em}{0ex}}{T}_{a,t}-{T}_{a,t-1})\approx 0$) indicate that no thermal dynamics are left unmodelled. However, autocorrelation in the residuals might still appear, which indicates time correlated building use, e.g., the building use in one time step is correlated with the next.
- In Figure 5, it was shown that the wind speed had a tremendous effect on the estimated heat loss (UA value). Even though the model predictions improved and the parameters that describe the wind sensitivity are significant for all 16 houses (see Table 2), it might be worth investigating other ways of modelling the wind’s effect on the heat consumption. As the effects are highly dependent on surroundings, building geometry and other unknown factors, it is suggested to model the wind dependence by means of nonparametric methods such as kernel or splines estimation.
- As the variance of the model residuals is highly dependent on the outdoor temperature, they are seemingly heteroskedastic, i.e., not constant. The implication of heteroskedastic residuals is that the standard errors of the model parameters are biased. To correct it, the model should be formulated as a weighted least square problem where the weights are the inverse of the error variance.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Conceptual diagram of transition interval definition. The thick black and red lines in the lower plot show the smooth maximum approximation of $f\left(x\right)=-x$ and $g\left(x\right)=0$ obtained by the LogSumExp function. The upper plot shows the slope (i.e., the logistic function) of the LogSumExp function. From the logistic function the start and end of the transition interval between $f\left(x\right)$ and $g\left(x\right)$ is obtained by $ub$ (upper bound) and $lb$ (lower bound) indicated by the horizontal dotted lines and the intersection with the logistic function.

**Figure 2.**Slope chart of root mean squared error (RMSE) of models M0–M4 tested on 16 houses. The RMSE is based on a 20% cross-validation data set. Notice that the logarithmic scale is used, meaning that the slopes between the points indicates the relative change in RMSE.

**Figure 3.**Forward model selection for House 6. The figure shows the effect of a particular model extension, starting with the model residuals of Model M0 in the top-left corner. The figure is read row-wise and each row represents a model extension. The model formulations of the model names in the top-left corner of the individual plot can be found in Table 1. Notice that the x-axis of the residuals plots changes depending on the model extension.

**Figure 4.**Plot of residuals (${e}_{t}$), time-varying heat balance contribution ($\Delta {\Phi}_{\mathrm{x},t}$), constant base temperature (${T}_{\mathrm{b}}$) and time-varying base temperature (${T}_{\mathrm{b},t}$). The figure shows how the time-varying heat balance contribution, $\Delta {\Phi}_{\mathrm{x},t}$, is estimated from the model residuals from model M3 applied on data from House 6. From the estimate of $\Delta {\Phi}_{\mathrm{x},t}$, the time-varying base temperature is estimated.

**Figure 5.**Slope chart of UA value (

**left**) and base temperature ${T}_{\mathrm{b}}$ (

**right**) of models M0–M4 obtained from 16 different houses. The left-side plot shows the UA values (i.e., the heat loss coefficients) under wind speed conditions corresponding to the mean wind speed observed in the measurement period (per-mode=symbol $2.5$ $\mathrm{m}$/$\mathrm{s}$). The right-side plot shows the base temperature obtained by the different models for each of the 16 houses.

**Figure 6.**Plot of ${\mathrm{U}}_{0}$ and ${\mathrm{U}}_{\mathrm{W}}$ per heated floor area as function of construction year. The black dots (left axis) show a clear increment in insulation level as the year of construction becomes more recent. The red dots (right axis) show the sensitivity to wind, e.g., the level of air leakages.

**Figure 7.**Conceptual illustration of performance gap and three causes of the discrepancy. Points A–C are further explained in the following list. Each column correspond to the total heat demand, and the hatched part illustrates the user related heat heat contribution to the total demand.

**Table 1.**Overview of full models. The model names are stated in the leftmost column as M0, M1 up to M4. The bullets (•) indicate if the given inputs are included in the specific model. Model M0 distinguishes itself from the remaining models, as the base temperature ${T}_{\mathrm{b}}$ is fixed at 17 ${}^{\circ}$C.

Name | Functions | Input | Model Formulation | ||||
---|---|---|---|---|---|---|---|

$\mathit{f}(\phantom{\rule{0.166667em}{0ex}}\mathbf{\xb7}\phantom{\rule{0.166667em}{0ex}})$ | $\mathit{g}(\phantom{\rule{0.166667em}{0ex}}\mathbf{\xb7}\phantom{\rule{0.166667em}{0ex}})$ | ${\mathit{T}}_{\mathbf{a}}$ | ${\mathit{W}}_{\mathbf{s}}$ | ${\mathit{I}}_{\mathbf{g}}$ | ${\mathit{T}}_{\mathbf{sky}}$ | ${\mathit{\Phi}}_{\mathbf{heat}}=\mathbf{LSE}[\mathit{f}\left(\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\right),\phantom{\rule{0.166667em}{0ex}}\mathit{g}\left(\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\right)]$ | |

M0 | ${f}_{0}$ | g | • | – | – | – | ${\Phi}_{\mathrm{heat}}=\mathrm{LSE}[\mathrm{UA}\phantom{\rule{0.166667em}{0ex}}({T}_{\mathrm{b}}-{T}_{\mathrm{a}})+{\Phi}_{0},\phantom{\rule{5.0pt}{0ex}}{\Phi}_{0}]+e$ |

M1 | ${f}_{1}$ | g | • | – | – | – | ${\Phi}_{\mathrm{heat}}=\mathrm{LSE}[\mathrm{UA}\phantom{\rule{0.166667em}{0ex}}({T}_{{\mathrm{b}}_{0}}-{T}_{\mathrm{a}}),\phantom{\rule{5.0pt}{0ex}}{\Phi}_{0}]+e$ |

M2 | ${f}_{2}$ | g | • | • | – | – | ${\Phi}_{\mathrm{heat}}=\mathrm{LSE}[({\mathrm{UA}}_{0}+{W}_{\mathrm{s}}\phantom{\rule{0.166667em}{0ex}}{\mathrm{UA}}_{\mathrm{W}})\phantom{\rule{0.166667em}{0ex}}({T}_{{\mathrm{b}}_{0}}-{T}_{\mathrm{a}}),\phantom{\rule{5.0pt}{0ex}}{\Phi}_{0}]+e$ |

M3 | ${f}_{3}$ | g | • | • | • | – | ${\Phi}_{\mathrm{heat}}=\mathrm{LSE}[({\mathrm{UA}}_{0}+{W}_{\mathrm{s}}\phantom{\rule{0.166667em}{0ex}}{\mathrm{UA}}_{\mathrm{W}})\phantom{\rule{0.166667em}{0ex}}({T}_{{\mathrm{b}}_{0}}-{T}_{\mathrm{a}})-\mathrm{gA}\phantom{\rule{0.166667em}{0ex}}{I}_{\mathrm{g}},\phantom{\rule{5.0pt}{0ex}}{\Phi}_{0}]+e$ |

M4 | ${f}_{4}$ | g | • | • | • | • | ${\Phi}_{\mathrm{heat}}=\mathrm{LSE}[({\mathrm{UA}}_{0}+{W}_{\mathrm{s}}\phantom{\rule{0.166667em}{0ex}}{\mathrm{UA}}_{\mathrm{W}})\phantom{\rule{0.166667em}{0ex}}({T}_{{\mathrm{b}}_{0}}-{T}_{\mathrm{a}})-\mathrm{gA}\phantom{\rule{0.166667em}{0ex}}{I}_{\mathrm{g}}+{\gamma}_{1}({T}_{\mathrm{sky}}^{4}-{T}_{\mathrm{a}}^{4}),\phantom{\rule{5.0pt}{0ex}}{\Phi}_{0}]+e$ |

**Table 2.**Estimated performance parameters for 16 houses in Denmark. The number in parentheses states the standard error of the parameters. ${\mathrm{U}}_{0}$ and ${\mathrm{UA}}_{0}$ are the heat loss coefficients under wind-free conditions, and ${\mathrm{UA}}_{\mathrm{W}}$ is the wind-dependent increment in the UA value. ${T}_{\mathrm{transition}}$ states the range of outdoor temperatures at which the building is in transition from heating to non-heating period, given no wind and solar irradiation. Finally, ${\overline{\Phi}}_{\mathrm{x},t}|{T}_{\mathrm{i}}=20\text{}{}^{\circ}\mathrm{C}$ and ${\sigma}_{{\Phi}_{\mathrm{x},t}}$ state the mean and the standard deviation, respectively, of the estimated time-varying heat balance contribution required to obtain thermal balance with an indoor temperature of 20 ${}^{\circ}\mathrm{C}$.

House | Year | Floor Area | ${\mathbf{U}}_{0}$ | ${\mathbf{UA}}_{0}$ | ${\mathbf{UA}}_{\mathbf{W}}$ | $\mathbf{gA}$ | ${\mathit{\Phi}}_{0}$ | ${\mathit{T}}_{\mathbf{b}}$ | ${\mathit{T}}_{\mathbf{transition}}$ | ${\overline{\mathit{\Phi}}}_{\mathbf{x},\mathit{t}}|{\mathit{T}}_{\mathbf{i}}=$20 ${}^{\circ}\mathbf{C}$ | ${\mathit{\sigma}}_{{\mathit{\Phi}}_{\mathbf{x},\mathit{t}}}$ | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

[${\mathbf{m}}^{2}$] | [$\mathbf{W}$/$(K\text{}{m}^{2})$] | [$\mathbf{W}$/$\mathbf{K}$] | [$\mathbf{W}$/$\mathbf{K}$ per $\mathbf{m}$/$\mathbf{s}$] | [${\mathbf{m}}^{2}$] | [$\mathbf{W}$] | [${}^{\circ}\mathbf{C}$] | [${}^{\circ}\mathbf{C}$] | [$\mathbf{W}$] | [$\mathbf{W}$] | ||||||||||||||

1 | 1970 | 151 | 1.25 | (0.03) | * | 189 | (4) | * | 58 | (7) | * | 2.5 | (0.3) | * | 676 | (84) | * | 16.5 | (0.5) | 12.1–21.0 | 702 | 157 | |

2 | 1969 | 163 | 1.25 | (0.02) | * | 204 | (4) | * | 39 | (8) | * | 3.7 | (0.3) | * | 340 | (47) | * | 14.2 | (0.4) | * | 9.5–18.9 | 1246 | 194 |

3 | 1963 | 140 | 1.28 | (0.02) | * | 179 | (2) | * | 32 | (5) | * | 2.5 | (0.1) | * | 141 | (30) | * | 15.7 | (0.2) | * | 11.9–19.5 | 810 | 103 |

4 | 1952 | 86 | 1.45 | (0.03) | * | 125 | (2) | * | 41 | (5) | * | 1.5 | (0.2) | * | 215 | (19) | * | 12.8 | (0.3) | * | 10.2–15.4 | 971 | 118 |

5 | 1966 | 111 | 1.54 | (0.03) | * | 171 | (3) | * | 61 | (7) | * | 1.6 | (0.2) | * | 110 | (63) | 16.6 | (0.3) | 9.6–23.6 | 643 | 155 | ||

6 | 1963 | 119 | 0.97 | (0.02) | * | 115 | (2) | * | 65 | (6) | * | 2.8 | (0.2) | * | 47 | (19) | * | 13.3 | (0.3) | * | 10.2–16.4 | 880 | 129 |

7 | 1947 | 119 | 2.17 | (0.04) | * | 258 | (5) | * | 72 | (13) | * | 1.2 | (0.4) | * | 6 | (50) | 13.5 | (0.3) | * | 6.9–20.0 | 1810 | 243 | |

8 | 1965 | 160 | 1.24 | (0.04) | * | 199 | (6) | * | 57 | (14) | * | 2.2 | (0.4) | * | 376 | (45) | * | 12.6 | (0.5) | * | 8.9–16.4 | 1569 | 258 |

9 | 1965 | 173 | 1.21 | (0.02) | * | 210 | (3) | * | 42 | (6) | * | 1.2 | (0.2) | * | 523 | (62) | * | 18.2 | (0.3) | * | 15.8–20.6 | 389 | 275 |

10 | 1996 | 135 | 0.90 | (0.02) | * | 121 | (2) | * | 51 | (6) | * | 2.5 | (0.2) | * | 106 | (25) | * | 14.1 | (0.4) | * | 10.2–18.0 | 786 | 193 |

11 | 1966 | 122 | 1.09 | (0.04) | * | 133 | (4) | * | 31 | (11) | * | 1.2 | (0.3) | * | 108 | (46) | * | 14.7 | (0.5) | * | 10.5–18.9 | 751 | 96 |

12 | 1975 | 136 | 1.05 | (0.02) | * | 143 | (2) | * | 31 | (4) | * | 1.9 | (0.1) | * | 644 | (17) | * | 13.4 | (0.3) | * | 11.3–15.4 | 1001 | 94 |

13 | 1937 | 86 | 2.67 | (0.06) | * | 229 | (5) | * | 92 | (14) | * | 4.4 | (0.4) | * | 45 | (31) | 11.2 | (0.3) | * | 7.6–14.8 | 2227 | 431 | |

14 | 1965 | 123 | 1.36 | (0.02) | * | 167 | (2) | * | 57 | (6) | * | 2.4 | (0.2) | * | 356 | (22) | * | 14.1 | (0.3) | * | 11.8–16.4 | 1068 | 203 |

15 | 1953 | 127 | 1.65 | (0.03) | * | 209 | (4) | * | 80 | (10) | * | 3.1 | (0.3) | * | 166 | (35) | * | 13.0 | (0.3) | * | 7.0–19.1 | 1593 | 210 |

16 | 1967 | 137 | 1.22 | (0.02) | * | 167 | (3) | * | 34 | (7) | * | 1.3 | (0.2) | * | 193 | (26) | * | 13.5 | (0.3) | * | 8.1–18.9 | 1137 | 143 |

${H}_{0}:$ | ${\mathrm{U}}_{0}=0$ | ${\mathrm{UA}}_{0}=0$ | ${\mathrm{UA}}_{\mathrm{W}}=0$ | $\mathrm{gA}=0$ | ${\Phi}_{0}=0$ | ${T}_{\mathrm{b}}=17$ | |||||||||||||||||

Significance code ‘*’: $p-\mathrm{value}<0.05$. |

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## Share and Cite

**MDPI and ACS Style**

Rasmussen, C.; Bacher, P.; Calì, D.; Nielsen, H.A.; Madsen, H.
Method for Scalable and Automatised Thermal Building Performance Documentation and Screening. *Energies* **2020**, *13*, 3866.
https://doi.org/10.3390/en13153866

**AMA Style**

Rasmussen C, Bacher P, Calì D, Nielsen HA, Madsen H.
Method for Scalable and Automatised Thermal Building Performance Documentation and Screening. *Energies*. 2020; 13(15):3866.
https://doi.org/10.3390/en13153866

**Chicago/Turabian Style**

Rasmussen, Christoffer, Peder Bacher, Davide Calì, Henrik Aalborg Nielsen, and Henrik Madsen.
2020. "Method for Scalable and Automatised Thermal Building Performance Documentation and Screening" *Energies* 13, no. 15: 3866.
https://doi.org/10.3390/en13153866