## 2 Methods  
### Boundary Conditions and Assumptions  
To maximize the efficiency and benefits of a Brayton battery, the following options are considered in the present study:  
* Pure electricity generation;
* Pure electricity generation with waste heat integration;
* Coupled generation of electricity and heat;
* Coupled generation of electricity and heat with waste heat integration;
* Coupled generation of electricity and cooling;
* Coupled generation of electricity and cooling with waste heat integration;
* Coupled generation of electricity, heat, and cooling. The system requirements for each purpose of the plant are defined below. This focuses on the necessary parameters, namely the duration of electricity, heat, or cooling supply, time-of-day restrictions, and temperature levels. Furthermore, a distinction is made between the summer half-year (see Table 1) and the winter half-year (see Table 2). \begin{table}
\begin{tabular}{c l l} \hline \hline  & **Input/Consumption** & **Output/Supply** \\ \hline \multirow{2}{*}{Electricity} & Duration: 8 h & Duration: 16 h \\  & Daytime restrictions: 8–16 h & Daytime restrictions: 16–8 h \\ \hline \multirow{4}{*}{Heat} & Waste heat & Process heat \\  & Duration: 24 h & Duration: 24 h \\ \cline{1-1}  & Time of day restrictions: none & Time of day restrictions: none \\ \cline{1-1}  & Temperature level: 90 \({}^{\circ}\)C & Temperature level: 250 \({}^{\circ}\)C \\ \hline \multirow{4}{*}{Cold} & & Air conditioning \\  & & Duration: 12 h \\ \cline{1-1}  & & Daytime restrictions: 8–20 h \\ \cline{1-1}  & & Temperature level: 6 \({}^{\circ}\)C (Supply)\(\rightarrow\) 16 \({}^{\circ}\)C (Return) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: System requirements (summer half-year). \begin{table}
\begin{tabular}{c l l} \hline \hline  & **Input/Integration** & **Output/Delivery** \\ \hline \multirow{2}{*}{Electricity} & Duration: 8 h & Duration: 16 h \\  & Daytime restrictions: 22–6 h & Daytime restrictions: 6–22 h \\ \hline \multirow{4}{*}{Heat} & Waste heat & Process heat \\  & Duration: 24 h & Duration: 24 h \\ \cline{1-1}  & Time of day restrictions: none & Time of day restrictions: none \\ \cline{1-1}  & Temperature level: 90 \({}^{\circ}\)C & Temperature level: 250 \({}^{\circ}\)C \\ \hline \hline \end{tabular}
\end{table} 
Table 2: System requirements (winter half-year). These system requirements are based on the respective needs or external constraints:  
* During the summer half-year, the electricity input results from the high solar share between 8 am and 4 pm [34], while in the winter half-year, it is due to the nighttime reduction in electricity demand between 10 pm and 6 am. * Process waste heat from industry occurs around the clock--both in winter and summer. Here, a temperature level of 90 \({}^{\circ}\)C is deliberately chosen, which is typically not utilized because there is no recipient available. Sources include cooling circuits from compressed air generation systems, for example. But also, alkaline electrolysis could be a possible source. [35]
* Process heat also needs to be supplied around the clock. The temperature level of 250 \({}^{\circ}\)C is deliberately chosen to be ambitiously high. * For the supply of cooling, air conditioning of shopping centers and office buildings is chosen as the sink. The time restrictions arise from the usual opening and office hours. The temperature levels for supply and return are taken from [36]. For the intended concept comparison, it is necessary to define a basic topology. This is already given by the flow diagram in Figure 1a, which, however, is supplemented by optional heat exchangers for heat injection and extraction between the obligatory components (turbomachinery and heat storage). Also necessary for a consistent comparison is the specification of assumptions and a uniform parameter set for the stationary system simulations.