## 2 Formulation of the 2030 Optimization Model
### Constraints  
(1) The heating technologies considered were combined heat and power (CHP) installations, gas and boiler, direct electric heaters, ground source heat pumps (GSHP), and air source heat pumps (ASHP). The contributions of heating technologies in heat supply were taken from [17; 22] and correlated with the daily heat demand by Equation (2). It is expected that nearly 42% of GB's total heat demand by 2030 shall be provided via electricity. \[H_{t}=\sum\
olimits_{j}\frac{H_{t}\times Share_{j}}{100}=\sum\
olimits_{j}h_{ j,t},\ t=1,\ldots,48\ \text{and}\ j:\ \text{type of heating device} \tag{2}\]  
In Equation (2), \(H_{t}\): total half-hourly heat demand (MW); \(j\): type of heating device (CHP, gas boilers, electric heaters, GSHPs, ASHPs, and oil boilers); and \(h_{j,t}\): heat from heating device \(j\) at time \(t\) (MW). The CHP units' power output at each half-hour of a certain day can be obtained from Equation (3). In Equation (3), \(\alpha_{t}\) was used to implicitly take into account the thermal energy storage units, and therefore levels out the intra-day heat demand variations. \(h_{\text{CHP},t}\) is the \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Power Production Technology (\(i\))**} & \multicolumn{2}{c}{**Marginal Generation Cost**} \\  & **(\(\pounds\)/MWh)** & **Emission 1** \\ \hline
**(1) Combined cycle gas turbine without CCS** & 39 & 16.7–30.0 \\
**(2) Combined cycle gas turbine with CCS** & 39 & 1.8–3.3 \\
**(3) CCS equipped coal-fired plants** & 10 & 4.8–8.6 \\
**(4) Nuclear plants** & 10 & 0 \\
**(5) Hydropower** & 0 & 0 \\
**(6) Cross-border interconnect 1 (IC1)** & 20–100 & 0 \\
**(7) biomass -fired** & 73 & 0 \\
**(8) GB-France interconnector (IC2)** & \(\min(\pi_{FR,t},\pi_{GB,t})\) & 0 \\
**(9) Solar photovoltaics (PVs)** & 0 & 0 \\
**(10) Wind farms** & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Paraments of generators [17]. CHP units' heat output at time \(t\). In Equation (3), \(P_{t}^{\text{CHP}}\) is the power output (MW) of CHPs, \(h_{\text{CHP},t}\) is the heat output (MW) of CHPs, \(ED_{t}\) stands for the electricity demand (MW) at \(t\) of the day and the 45/30 multiplier represents the ratio of thermal efficiency to electrical efficiency. Note that the calculated \(\alpha_{t}\) of a certain day can be different from other days, because the daily electricity demand profiles are not the same. \[P_{t}^{\text{CHP}}\times\frac{45}{30}\leq\alpha_{t}\times h_{\text{CHP},t},\ \alpha_{t}=\frac{ED_{t}}{\sum_{t=1}^{48}ED_{t}}\,\ t=1,\ldots,48 \tag{3}\]  
The electricity consumption by direct electric heaters, ASHP, and GSHP at each half-hour of the day is a function of their heat output based on Equation (4). \(SPF_{j}\), the seasonal performance factors of electric heater, GSHP, and ASHP, were taken from [17,19]. Equation (4) states that intra-day high electricity demands are captured by direct electric heaters, ASHPs, and GSHPs. \[E_{k,t}=\alpha_{t}\sum_{t=1}^{48}\frac{h_{k,t}}{SPF_{k}},\ t=1,\ldots,48,\ k= \text{direct electric heaters, ASHPs, and GSHP} \tag{4}\]  
The historical solar and wind power generation data over 2010 are available in [25]. These historical data were upscaled to calculate the available electricity from wind farms (\(A_{t}^{\text{Wind}}\)) and solar PVs (\(A_{t}^{\text{Solar}}\)) for each half-hour of 2030. (2) At each half-hour of a certain day, \(t\), the power injection to the electricity grid from solar (\(P_{t}^{\text{Solar}}\)) and wind (\(P_{t}^{\text{Wind}}\)) sites should be less than or equal to the available power (\(A_{t}^{\text{Solar}}\) and \(A_{t}^{\text{Wind}}\)); see Equation (5). These inequality constraints state that wind and solar curtailment may occur at any time. \[P_{t}^{\text{Solar}}\leq A_{t}^{\text{Solar}}\ \text{and}\ P_{t}^{\text{Wind}} \leq A_{t}^{\text{Wind}} \tag{5}\]  
(3) Equation (6) states that the change in the power outturn of technology \(i\) at \(t\) and \(t-1\) of a certain day should be less than or equal to its ramp rate (\(R_{i}\)).