## 3 Modelling
### Low-Pressure (LP)-TES: Regenerator  
The time-varying temperature field during thermal cyclic operation is calculated with a thermal model considering the storage media as a heterogeneous porous medium [39]. Based on this formulation, a simplified thermal model can be derived with well-justified negligence of the storage capacity term of the gaseous medium and of the axial conduction terms due to low heat conductivities of both media and to thermal losses for large-scale dimensions. Assuming a one-dimensional axial model and normalization in time t* and space z*, the heterogeneous model can be expressed as:  
\[\frac{\partial\mathrm{T_{S}}}{\partial\mathrm{t^{*}}}=\Pi(\mathrm{T_{F}}- \mathrm{T_{S}})\quad(1)\quad\mathrm{with:}\ \Pi=\frac{\tau(\mathrm{kO})_{LP-TES}}{\mathrm{m_{SC_{S}}}} \tag{1}\]  
\[\frac{\partial\mathrm{T_{F}}}{\partial\mathrm{z^{*}}}=\Lambda(\mathrm{T_{S}}- \mathrm{T_{F}})\quad(2)\quad\mathrm{with:}\ \Lambda=\frac{(\mathrm{kO})_{LP-TES}}{\mathrm{m_{F}c_{F}}} \tag{2}\]  
This formulation describes the thermal behavior of the regenerator-based thermal energy storage with only two dimensionless parameters, the reduced period duration \(\Pi\) and the reduced regenerator length \(\Lambda\).  
Here, \(\mathrm{T_{S}}\) and \(\mathrm{T_{F}}\) represent the solid medium and the heat-transfer fluid temperature, \(\mathrm{O}\) the heat-transfer surface, \(\tau\) the charge/discharge duration, \(\mathrm{m_{S}}\) the total storage mass, \(\mathrm{m_{F}}\) the mass flow rate, and \(\mathrm{c_{S}}\) and \(\mathrm{c_{F}}\) the specific heat capacities of the solid and fluid, respectively. The total heat transfer coefficient k regarding the thermal resistance of the inventory option is defined as in references [40; 41]. Inside the simulation studies, a packed bed is assumed as an inventory option with a heat-transfer coefficient as defined by reference [42].  
The expression in Equations (1) and (2) are solved numerically by using a backward finite-difference method in space for the fluid phase. Subsequently, the resulting set of differential-algebraic equations is solved in time with a commercial simulation tool (Matlab) to obtain the required temporal fluid outlet temperatures during the cyclic operation.  
For a simplified implementation of the resulting time-varying regenerator fluid outlet temperatures in the A-CAES system simulations, a temporal averaged fluid outlet temperature (\(\overline{\mathrm{T}}_{\mathrm{F,out}}\)) and a regenerator efficiency \(\eta_{\mathrm{ILP-TES}}\) are used, expressed by the cyclic inlet fluid temperatures (\(T_{F,in}^{charge}\), \(T_{F,in}^{discharge}\)):  
\[\overline{\mathrm{T}}_{\mathrm{F,out}}^{\mathrm{discharge}}=\eta_{\mathrm{ILP- TES}}\Big{(}\mathrm{T}_{\mathrm{F,in}}^{\mathrm{charge}}-\mathrm{T}_{\mathrm{F,in}}^{ \mathrm{discharge}}\Big{)}+\mathrm{T}_{\mathrm{F,in}}^{\mathrm{discharge}} \tag{3}\]  
\[\overline{\mathrm{T}}_{\mathrm{F,out}}^{\mathrm{charge}}=(1-\eta_{\mathrm{ILP- TES}})\Big{(}\mathrm{T}_{\mathrm{F,in}}^{\mathrm{charge}}-\mathrm{T}_{\mathrm{F,in}}^{ \mathrm{discharge}}\Big{)}+\mathrm{T}_{\mathrm{F,in}}^{\mathrm{discharge}} \tag{4}\]  
Assuming equal charging-discharging duration, only one regenerator efficiency is required. Because of the dependency of the regenerator efficiency on the dimensionless parameters \(\Pi\) and \(\Lambda\) given in Equations (1) and (2) and on systemic factors (mass flow rate and inlet temperatures), the LP-TES storage mass/volume is calculated. For that, additional specifications regarding the inventory option and the bed diameter are need (see Section 4 ).