Abstract
This paper presents grey-box modeling of the heat dynamics of an apartment in a highly insulated test building located in the Arctic. Data from a 16-day-long experiment is analyzed and used to fit lumped parameter models formulated as coupled stochastic differential equations. The output of the models is the measured indoor air temperature, and the models are fitted using maximum likelihood techniques with the software CTSM-R. Models are compared using likelihood-ratio tests and validated considering autocorrelation and periodograms of residuals. The fitted models facilitate description of both the fast responses to mechanical ventilation and solar radiation through a large window facade, and the slow responses to floor heating and outdoor temperature. To successfully describe the dynamics of the system, solar radiation is given special attention in modeling of both the physical system and the observational noise. The estimated physical parameters which include UA-value, total heat capacity, and time constants for the apartment are discussed. Simulations are performed to illustrate step and impulse responses of inputs.
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Abbreviations
- A :
-
system matrix in a linear system
- A s :
-
area with which the global horizontal solar radiation is scaled (W/m2)
- A w :
-
area with which the projected solar radiation (P s,win) is scaled (W/m2)
- A noise :
-
scaling of projected solar radiation in a system noise process ((K·m2·s1/2)/J)
- B :
-
matrix describing how inputs enter in a linear system
- C i :
-
thermal capacity of state i (or e, f) (J/K)
- P el :
-
electric power (W)
- P h :
-
floor heating power (W)
- P s :
-
global horizontal solar radiation (W/m2)
- P s,win :
-
global horizontal solar radiation projected onto the window side of the building (W/m2)
- P v :
-
estimated ventilation heating to one apartment (W)
- R ij :
-
thermal resistance between states or inputs I and j (W/K)
- SACF:
-
sample autocorrelation function
- T a :
-
outdoor temperature (°C)
- T e :
-
temperature of the building envelope (°C)
- T i :
-
indoor air and surface temperature (in three-state model) (°C)
- T f :
-
floor heating system temperature (°C)
- UA:
-
common UA-value for the building envelope (W/K)
- W s :
-
wind speed (m/s)
- c v :
-
scaling of ventilation heating signal
- c W :
-
a constant related to the effect of wind speed ((K·s1/2)/m)
- e k :
-
observational noise at discrete time k (°C)
- k :
-
discrete time
- n :
-
number of parameters in a model
- p :
-
p-value of a hypothesis test
- r k :
-
one-step-ahead prediction uncertainty of the indoor air temperature measurement at time k (°C)
- t :
-
continuous time
- Y k :
-
indoor temperature measurement at time t k (°C). When referring to particular observations, y k is used
- ∈k :
-
residual, i.e. one-step-ahead prediction error of the indoor temperature at time k (°C)
- \(\tilde \in _k \) :
-
standardized residual at time k, \(\tilde \in _k = \in _k /r_k \)
- ω i :
-
Wiener processes in state i (or e, f))(s1/2)
- σ i :
-
scaling of Wiener process in state i (or e, f) (K/s1/2)
- σ o :
-
standard deviation of observational noise (°C)
- ℓ :
-
log-likelihood value
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Andersen, P.D., Jiménez, M.J., Madsen, H. et al. Characterization of heat dynamics of an arctic low-energy house with floor heating. Build. Simul. 7, 595–614 (2014). https://doi.org/10.1007/s12273-014-0185-4
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DOI: https://doi.org/10.1007/s12273-014-0185-4