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Research Papers

Analytical Solution Strategy for Building Energy Dynamics With Stochastic Thermal Gains and External Temperature

[+] Author and Article Information
Zili Zhang

Department of Engineering,
Aarhus University,
Aarhus C 8000, Denmark

Biswajit Basu

School of Engineering,
Trinity College Dublin,
Dublin 2, Ireland
e-mail: basub@tcd.ie

Søren R. K. Nielsen

Department of Civil Engineering,
Aalborg University,
Aalborg 9000, Denmark

1Corresponding author.

Manuscript received July 31, 2016; final manuscript received March 15, 2017; published online June 13, 2017. Assoc. Editor: Siu-Kui Au.

ASME J. Risk Uncertainty Part B 3(4), 041005 (Jun 13, 2017) (8 pages) Paper No: RISK-16-1104; doi: 10.1115/1.4036310 History: Received July 31, 2016; Revised March 15, 2017

Energy dynamics in buildings are inherently stochastic in nature due to random fluctuations from various factors such as heat gain (including the solar) and ambient temperature. This paper proposes a theoretical framework for stochastic modeling of building thermal dynamics as well as its analytical solution strategies. Both the external temperature and the heat gain are modeled as stochastic processes, composed of a periodic (daily) mean-value function and a zero-mean deviation process obtained as the output process of a unit Gaussian white noise passing through a rational filter. Based on the measured climate data, the indicated mean-value functions and rational filters have been identified for different months of a year. Stochastic differential equations in the state vector form driven by white noise processes have been established, and analytical solutions for the mean-value function and covariance matrix of the state vector are obtained. This framework would allow a simple and efficient way to carry out predictions and parametric studies on energy dynamics of buildings with random and uncertain climate effects. It would also provide a basis for the robust design of energy efficient buildings with predictive controllers.

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References

Figures

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Fig. 1

Model of the Bettina-Iain house, Findhorn Eco-Village

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Fig. 2

Radio controlled (RC) schematic of the simplified building energy model

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Fig. 3

Zero-mean deviation process obtained as the output of a unit white noise passing through a rational shaping filter

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Fig. 4

Comparison of zone air temperatures between beps and ROM

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Fig. 5

Validation of the results (zone air temperatures) from the calibrated ROM with that from beps

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Fig. 6

Definition of the broken line process

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Fig. 7

Simulated equivalent white noise WT(t): (a) a sample curve of WT(t) and (b) the theoretical autospectral density function by Eq. (37) and the averaged autospectral density function from 50 simulated sample curves of WT(t)

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Fig. 8

Simulated sample curves of the deviation processes VT(t) and VQ(t), July

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Fig. 9

Mean-value functions μT(t) and μQ(t), July

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Fig. 10

Mean-value functions of the state vector, July: (a) mean-value function for the zone air temperature, (b) mean-value function for the internal surface temperature, (c) mean-value function for the external surface temperature, and (d) mean-value function for the temperature of the internal structure, floors, and furniture

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Fig. 11

Evolution of the variances of the state variables, July: (a) Variance of the zone air temperature, (b) variance of the internal surface temperature, (c) variance of the external surface temperature, and (d) variance of the temperature of the internal structure, floors, and furniture

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Fig. 12

Sample curves of the state variables together with the corresponding analytical solution of mean ± standard deviation (STD), July: (a) Tz(t): zone air temperature, (b) Tsi(t): internal surface temperature, (c) Tse(t): external surface temperature, and (d) Tf(t): temperature of the internal structure, floors, and furniture

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Fig. 13

Sample curves of the state variables together with the corresponding analytical solution of mean ± STD, January: (a) Tz(t): zone air temperature, (b) Tsi(t): internal surface temperature, (c) Tse(t): external surface temperature, and (d) Tf(t): temperature of the internal structure, floors, and furniture

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