Theory
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This subsection describes the formulation of a physics-based model to simulate the thermal behaviour of the building using a resistor-capacitor (RC) model. This is based on an electrical analogy corresponding to the equivalent thermal physics. The model, shown in Figure 1, consists of one internal thermal capacitance, and five thermal resistances. This is also known as a 5R1C model and is based on the ISO 13790 standard.
Denoting by Tm, the temperature of the thermal mass in the room, the differential equation for the circuit in Figure 5 is given by
The value φmtot represents an equivalent thermal heat flux based on the solar heat gains, internal heat gains, external air temperature and the thermal conductances of the building elements.
where Cm is the thermal capacitance of the room, Te is the external air temperature, Tsup is the conditioned air supply temperature. The solar heat gains φsol, and internal heat gains φint are represented by three equivalent heat fluxes φio, φst and φm which correspond to a heat exchange to the air Tair, internal room surface Ts, and thermal mass Tm respectively. The heating and cooling heat flux is represented by φHC. The five thermal conductances H are represented by three equivalent conductances
The internal heat flow rates due to internal gains and solar sources are divided between the thermal nodes by
Equation 1 is discretised using the Crank-Nicolson method so it can be solved numerically as,
where the subscripts k and k + 1 refer to the timesteps of length ∆t.
The heating or cooling demand for each time step is determined to ensure that the temperature Tmk+1 is within the set thermal set points for occupant comfort. We assume that there is no restriction of the heating or cooling system. This means that the heating or cooling supply will always meet the calculated demand. The thermal demand is converted to electrical energy based on an average coefficient of performance (COP) of the heating or cooling system.