**Appendix A. Energy System Optimisation Model (ESOM) Calculations**

Fabric heat loss is calculated as in the following equation:

$$L\_F = \sum (A \times \mathcal{U}) \tag{A1}$$

where *LF* is fabric heat loss (W/K), *A* is the area of component (m2) and *U* is the thermal transmittance of the component (W/m2K). Thermal bridges are calculated as in the following equation:

$$L\_{TB} = y \times A\_{exp} \tag{A2}$$

where *LTB* is losses from thermal bridging (W/K), *Aexp* is the total area of exposed surfaces to the external environment and *y* is the thermal bridging factor (W/m2K). Ventilation heat loss is calculated as in the following equation:

$$L\_V = 0.33 \times n \times V \tag{A3}$$

where *LV* is ventilation heat loss (W/K), *n* is air change rate (ach) and *V* is the volume of heated space (m3). Total heat loss is calculated as in the following equation:

$$\sum \mathcal{L}\_{\text{Total}} = \left( L\_F + L\_{TB} + L\_V \right) \times T\_h - T\_c \tag{A4}$$

where *LTotal* is total heat loss (W), *Th* is heating setpoint temperature, *Te* is external temperature. Solar gains are calculated as in the following equation:

$$\mathbf{G\_S} = 0.9 \times Aw \times \mathbf{S} \times \mathbf{g} \times FF \times Z \tag{A5}$$

where 0.9 represents typical average transmittance, *Aw* is the window area (m2), *S* is the solar flux (W/m2), *g* is the transmittance factor of the glazing at normal incidence, *FF* is the frame factor (fraction of the glazed area) and *Z* is the solar access factor. Internal gains are calculated as in the following equation:

$$\mathbf{G}\_{\mathbf{I}} = \mathbf{A} \times \mathbf{f} \tag{A6}$$

where *A* is the gross floor area (m2) and *f* is the internal gain factor (W/m2). Total gains are calculated as in the following equation:

$$
\sum G\_{Total} = G\_{\mathbb{S}} + G\_{I} \tag{A7}
$$

The temperature reduction from setpoint temperature depends on the thermal properties of building materials used in the dwelling calculated as in the following equations:

$$HLP = \frac{\sum L\_{Total}}{AGF} \tag{A8}$$

$$TMP = \frac{\sum \mathbb{C} \times A}{AGF} \tag{A9}$$

$$\text{Tau} = \text{TMP} \;/\; \text{(3.6} \times \text{HLP)}\tag{A10}$$

$$a = 1 + \text{Tau} / 15\tag{A11}$$

$$T\mathbf{c} = \mathbf{4} + 0.25 \times \mathbf{T}u\mathbf{u} \tag{A12}$$

$$\chi = \sum G\_{\text{Total}} / \left(\sum L\_{\text{Total}} \times \left(T\_h - T\_c\right)\right) \tag{A13}$$

$$\begin{array}{c} \text{if } \mathbf{y} > 0 \text{ and } \mathbf{y} \neq 1: \\ \qquad \qquad \qquad \qquad \qquad \qquad \eta = \frac{1-\gamma^{a}}{1-\gamma^{a+1}} \\ \qquad \qquad \qquad \qquad \qquad \eta = \frac{a}{a+1} \\ \qquad \qquad \qquad \qquad \qquad \qquad \eta = 1 \end{array} \tag{A14}$$

$$\text{Tsc} = (1 - R) \times (Th - 2) + R \times (Tc + \eta \times G\_{Total} / L\_{Total}) \tag{A15}$$

$$
\mu = 0.5 \times \left( T\_h - T\_{\text{sc}} \right) / T\_{\text{c}} \tag{A16}
$$

where *HLP* is the heat loss parameter (W/m2K), *LTotal* is total heat losses (W/m2K), *AGF* is the gross floor area of the building (m2), *TMP* is the thermal mass parameter (kJ/m2K), *C* is the specific heat capacity of building materials, *A* is the area of building materials, *Tau* is a time constant, A is a constant, *Tc* is a time constant, Gis a constant, *GTotal* is total gains, *LTotal* is total losses, *Th* is heating setpoint temperature, *Te* is external temperature, ï is utilisation factor, *Tsc* is internal temperature without heating, *R* is the responsiveness of the heating system, *u* is temperature reduction.
