## 3 Parametric Study  
### Pressure and Temperature in the Stored Compressed Air  
The temperature (\(T\)) and pressure (\(P\)) in the stored compressed air can be determined using the thermodynamic process of CAES by inputting available energy for storage (\(\dot{w}_{in}\)) and available volumes (\(V_{s}\)) for storage following Option 2 discussed in Section 2.1. According to [PERSON] et al. [7], the temperature (\(T\)) in the stored compressed air can be calculated by solving an implicit equation [Equation (1)]. The pressure (\(P\)) in the stored compressed air can then be calculated using Equation (2).  
\[V_{s}\bigg{(}\frac{T-\eta_{2}T_{1}}{1-\eta_{2}}\bigg{)}^{5/2}\bigg{/}C^{7/2} = \dot{w}_{in}\eta_{1}t_{in}\bigg{/}3.5R\bigg{(}\frac{T-\eta_{2}T_{1 }}{1-\eta_{2}}-T_{1}\bigg{)}+\rho_{i}V_{s}/\mu \tag{1}\]  
\[P = RT\bigg{(}\frac{T-\eta_{2}T_{1}}{1-\eta_{2}}\bigg{)}^{5/2}\bigg{/}C^{7/2} \tag{2}\]  
where, \(R\) is the universal gas constant; \(T_{1}\) is the ambient air temperature; \(C\) is a constant taken as 10.89; \(t_{in}\) is the time for compressing air; \(\rho_{i}\) and \(\mu\) are the initial density and molar mass of the air taken, 1.2 kg/m\({}^{3}\) and 0.029 kg/mol, respectively; \(\eta_{1}\) and \(\eta_{2}\) is the efficiency for the compression and the heat exchange, respectively.  
The available energy for storage can be determined as differences between energy supplies from solar panels and energy demand from the end-users. The available volume for storage for each column is dependent on the length of the pile, the number of the piles, and the inner diameter (\(d_{i}\)) of the pile, which can be originated from the geotechnical design. Typical residential buildings have been selected to determine the available energy and volume for storage for different numbers of stories (\(N\)) ranged from 2 to 10, different column spacing (5 \(\times\) 5 m, 6 \(\times\) 6 m, and 7 \(\times\) 7 m), and different inner diameter (\(d_{i}\)) of the pile ranged from 200 mm to 800 mm. Figure 3 shows the final calculated available energy and volume for storage per each column inside the building. The detailed calculations are discussed in [PERSON] et al. [7].  
Figure 4 shows the temperature (\(T\)) and pressure (\(P\)) in the stored compressed air for a one-day (24 h) cycle for the case: 2-story with \(7\times 7\) m column spacing by solving Equations (1) and (2) with inputs presented in Figure 3. The temperature and pressure start to increase when renewable energy becomes available for storage at 7:00 and reach a peak at 19:00 when renewable energy becomes unavailable. Afterward, the temperature and pressure will decrease as the compressed air is released to generate electricity. As the inner diameter of the pile foundation increases, the temperature and pressure in the stored compressed air reduce because the available volume for storage increases.  
The maximum temperature (\(T_{max}\)) in the stored compressed air is shown in Figure 5 for all other cases by solving Equations (1) and (2) with inputs presented in Figure 3. The maximum temperature is insensitive with different column spacing and numbers of stories because both available energy and volume for storage increase as the column spacing and the number of stories increase (refer to Figure 2). On the other hand, it varies significantly with the changing of the inner diameter (\(d_{i}\)) of the pile foundation. Therefore, this paper focuses on the most critical case (2-story with \(7\times 7\) m column spacing) but considers different \(d_{i}\).  
Figure 4: One day energy storage cycle for a 2-story with \(7\times 7\) m column spacing: (**a**) temperature; (**b**) pressure.  
Figure 3: Available energy and volume for storage per each column: (**a**) total available storage energy for a single day; (**b**) volume for \(d_{i}=200\) mm; (**c**) volume for \(d_{i}=800\) mm (modified from [8]).