Determination of the Optimal Order of Grey-Box Models for Short-Time Prediction of Buildings’ Thermal Behavior
Abstract
:1. Introduction
2. Methodology
2.1. Grey Box Modeling
2.2. Parameters Estimation
3. Experimental Data
- -
- Experiment A: The room was submitted only to external thermal condition without any indoor heating power.
- -
- Experiment B: The room was submitted to the external thermal conditions as well as to an indoor 900 W heating power.
- -
- Experiment C: The room was submitted to the external thermal conditions as well as to an indoor 1500 W heating power.
4. Results and Discussion
4.1. Sensitivity Analysis
4.2. Experiment A (Heating Power = 0)
4.3. Experiment B (Heating Power = 900 W)
4.4. Experiment C (Heating Power = 1500 W)
5. Conclusions
Author Contributions
Acknowledgments
Conflicts of Interest
Nomenclature
X(t) | State vector of the dynamic system, temperature of building’s components |
U(t) | Vector of measured inputs (outdoor temperature, sun radiation and heating power). |
W | Random function of time (Wiener process). |
Y(t) | Measured output. |
Measurement error. | |
θ | Estimated parameter |
Ti | Indoor air temperature, |
Tf | Temperature of building envelope |
Tfe | Temperature of the external building façade |
Tfi | Temperature of the internal building façade |
Tm | Temperature of internal wall |
R: | Resistance between indoor and outdoor medium |
Re | Convection resistance of outdoor air |
Ri, Rm | Convection resistance of indoor air |
Rf: | Conduction resistance of the façade |
C | Equivalent mass capacity for building |
Ci | Air mass capacity, |
Cf | Envelope mass capacity |
Cfe | External capacity of the façade |
Cfi | Internal capacity of the façade |
Cm | Mass capacity of internal walls |
Te | Outdoor temperature |
Qs | Solar energy gain |
Qh | Heating energy gain |
RMSE | Root-mean-square error |
FPE | Final prediction error |
FIT | Level of fit |
NRMSE | Normalized root mean square error |
e | Error |
STi | Total Sobol index |
N | Number of samples |
Yb, Ya | Vectors of output data in which all parameters vary |
Yci | Output vector in which all parameters vary except the ith |
yi | Predicted temperature |
Reference temperature | |
n | Number of samples. |
Appendix A
- -
- Year of construction or renovation
- -
- Type of use (offices, shops, etc.)
- -
- Heated surface (Sh)
- -
- Surface of vertical walls (Sm)
- -
- External exchange surface (Sext)
- -
- Internal exchange surface (Sint)
- -
- Indoor air volume (Vint)
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- Coefficients of internal convection (hint) and external (hext), supposed constant.
- -
- -
- The impact of the furniture on the air capacity (Mob = 20 kJ / K.m2 for non-empty buildings and zero otherwise).
- -
- Conductivity of the outer walls: "Uwall", "Uslab" and "Uroof", depending on the year of construction (Table A3).
Low Floor | High Floor | Vertical Wall | Inertia Class |
---|---|---|---|
heavy | heavy | heavy | very heavy |
- | heavy | heavy | heavy |
heavy | - | heavy | heavy |
heavy | heavy | - | heavy |
- | - | heavy | average |
- | heavy | - | average |
Daily Inertia Class | Daily Capacity Cm (KJ/K) | Exchange Surface Am(m2) |
---|---|---|
Very heavy | 80 × Abuild | 2.5 × Abuild |
light | 110 × Abuild | 2.5 × Abuild |
average | 165 × Abuild | 2.5 × Abuild |
heavy | 260 × Abuild | 3 × Abuild |
very light | 370 × Abuild | 3 × Abuild |
Construction Date | H1 | H2 | H3 | |||
---|---|---|---|---|---|---|
Joule Effect | Other | Joule Effect | Other | Joule Effect | Other | |
From 1948 to 1974 | 2.5 | 2.5 | 2.5 | |||
From 1975 to 1977 | 1 | 1.05 | 1.11 | |||
From 1978 to 1982 | 0.8 | 1 | 0.84 | 1.05 | 0.89 | 1.11 |
From 1983 to 1988 | 0.7 | 0.8 | 0.74 | 0.84 | 0.78 | 0.89 |
From 1989 to 2000 | 0.45 | 0.5 | 0.47 | 0.53 | 0.5 | 0.56 |
From 2001 to 2005 | 0.4 | 0.4 | 0.47 | |||
From 2006 | 0.36 | 0.36 | 0.4 |
Wall Position | Emissivity | hint | hext | ||
---|---|---|---|---|---|
Normal | Sheltered | Severe | |||
Vertical | 0.9 | 8.13 | 18.2 | 12.5 | 33.3 |
Vertical | 0 | 3.29 | 14.9 | 9.1 | 33.3 |
External ceiling | 0.9 | 9.43 | 22.2 | 14.3 | 50 |
External ceiling | 0 | 4.59 | 18.9 | 11.1 | 50 |
External floor | 0.9 | 6.67 | 20 | 20 | 20 |
External floor | 0 | 1.78 | 20 | 20 | 20 |
Internal horizontal | 0.9 | 8 | - | - | - |
Internal horizontal | 0 | 3 | - | - | - |
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Estimated Parameter | Value |
---|---|
Ci (J/K) | 1.47 × 105 |
Cfe (J/K) | 1.77 × 108 |
Cfi (J/K) | 9.36 × 106 |
Cm (J/K) | 4.54 × 106 |
Ri, Rm (K/W) | 1.82 × 10−2 |
Re (K/W) | 3 × 10−3 |
Rf (K/W) | 1.1 × 10−1 |
Experiment | Indoor Heating Power (W) |
---|---|
A | 0 |
B | 900 |
C | 1500 |
Result | Free-Floating (Test A) | Heating - 900W (Test B) | Heating - 1500W (Test C) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Parameter | C | R | Ci | Cfe | Cfi | Ri | Re | Rf | Ci | Cf | Ri | Re |
STi | 0.98 | 0.99 | 0.07 | 0.09 | 0.24 | 0.60 | 0.12 | 0.18 | 0.62 | 0.85 | 0.72 | 0.59 |
Result | 1R1C | 2R2C | 3R3C | 4R4C |
---|---|---|---|---|
Fit percent | 95.11 | 95.35 | 95.12 | 10.09 |
RMSE | 0.0656 | 0.0616 | 0.0648 | 1.2004 |
Result | 1R1C | 2R2C | 3R3C | 4R4C |
---|---|---|---|---|
Fit percent | 91.86 | 92.91 | 91.97 | - |
RMSE | 0.1086 | 0.0949 | 0.1072 | - |
Result | 1R1C | 2R2C | 3R3C | 4R4C |
---|---|---|---|---|
Fit percent | 87.83 | 90.24 | 88.05 | - |
RMSE | 0.1625 | 0.1304 | 0.1594 | - |
Result | 1R1C | 2R2C | 3R3C | 4R4C |
---|---|---|---|---|
Fit percent | - | 93.97 | 95.43 | 44.15 |
RMSE | - | 0.1204 | 0.0917 | 1.1170 |
Result | 1R1C | 2R2C | 3R3C | 4R4C |
---|---|---|---|---|
Fit percent | - | 87.6 | 92.98 | 36.52 |
RMSE | - | 0.2480 | 0.1404 | 1.2697 |
Result | 1R1C | 2R2C | 3R3C | 4R4C |
---|---|---|---|---|
Fit percent | - | 80.65 | 90.71 | 31.25 |
RMSE | - | 0.3869 | 0.1857 | 1.3751 |
Result | 1R1C | 2R2C | 3R3C | 4R4C |
---|---|---|---|---|
Fit percent | 93.36 | 97.2 | 95.7 | 96.18 |
RMSE | 0.2349 | 0.0990 | 0.1523 | 0.1353 |
Result | 1R1C | 2R2C | 3R3C | 4R4C |
---|---|---|---|---|
Fit percent | 83.34 | 95.56 | 90.7 | 92.17 |
RMSE | 0.5895 | 0.1572 | 0.3291 | 0.2769 |
Result | 1R1C | 2R2C | 3R3C | 4R4C |
---|---|---|---|---|
Fit percent | 71.49 | 93.54 | 84.71 | 87.59 |
RMSE | 1.0090 | 0.2285 | 0.5410 | 0.4392 |
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Attoue, N.; Shahrour, I.; Mroueh, H.; Younes, R. Determination of the Optimal Order of Grey-Box Models for Short-Time Prediction of Buildings’ Thermal Behavior. Buildings 2019, 9, 198. https://doi.org/10.3390/buildings9090198
Attoue N, Shahrour I, Mroueh H, Younes R. Determination of the Optimal Order of Grey-Box Models for Short-Time Prediction of Buildings’ Thermal Behavior. Buildings. 2019; 9(9):198. https://doi.org/10.3390/buildings9090198
Chicago/Turabian StyleAttoue, Nivine, Isam Shahrour, Hussein Mroueh, and Rafic Younes. 2019. "Determination of the Optimal Order of Grey-Box Models for Short-Time Prediction of Buildings’ Thermal Behavior" Buildings 9, no. 9: 198. https://doi.org/10.3390/buildings9090198