## 2 Methodology
### Advanced Exergoeconomic (AdvExEcon) Analysis  
Total operating cost \(\Big{(}\dot{C}_{tot,k}\Big{)}\) for the system and its components in a traditional economic (TradEcon) analysis is given as follows:  
\[\dot{C}_{tot,k}=\dot{C}_{D,k}+\dot{Z}_{k} \tag{8}\]  
In exergoeconomic analysis, an AdvEx analysis, which is similar to splitting exergy destruction, can be broken into parts, such as by splitting total operating costs (exergy destruction-related cost--\(\dot{C}_{D,k}\) and investment cost--\(\dot{Z}_{k}\)) into endogenous/exogenous [35; 36] and avoidable/unavoidable parts [15; 28].  
While, all other components work theoretically, and the \(k\)th component works with its real state, only within the \(k\)th component is the endogenous part of exergy destruction-related \(\Big{(}\dot{C}^{EN}_{D,k}\Big{)}\) and investment \(\Big{(}\dot{Z}^{EN}_{k}\Big{)}\) costs incurred. The system exergy efficiency of the overall system remains constant in all estimates [37; 38]. As such, the following can be observed:  
\[\dot{C}^{EN}_{D,k}=C^{R}_{F,k}\dot{E}^{EN}_{D,k} \tag{9}\]  
\[\dot{Z}^{EN}_{k}=\dot{E}^{EN}_{P,k}\Bigg{(}\frac{\dot{Z}}{E_{P}}\Bigg{)}^{R}_{k} \tag{10}\]The exogenous part of the exergy destruction-related \(\left(\dot{\dot{C}}^{EN}_{D,k}\right)\) and investment \(\left(\dot{\dot{Z}}^{EN}_{k}\right)\) costs is verified by subtracting the endogenous cost from the real cost. Their formulas are given below:  
\[\dot{\dot{C}}^{EX}_{D,k}=\dot{\dot{C}}^{R}_{D,k}-\dot{\dot{C}}^{EN}_{D,k} \tag{11}\]  
\[\dot{\dot{Z}}^{EX}_{k}=\dot{Z}^{R}_{k}-\dot{\dot{Z}}^{EN}_{k} \tag{12}\]  
The inevitable part is obtained under extremely inefficient conditions. In practical applications, \(\left(\frac{\dot{Z}}{\dot{E}_{P}}\right)^{UN}_{k}\), a set of thermodynamic parameters for the \(k\)th component leading to a highly inefficient solution, is determined by choosing and estimating the investment cost arbitrarily for this solution [15; 28]. Thus, exergy destruction-related \(\left(\dot{\dot{C}}^{UN}_{D,k}\right)\) and investment \(\dot{\dot{Z}}^{UN}_{k}\) costs are inevitable parts determined by the following expressions, respectively:  
\[\dot{\dot{C}}^{UN}_{D,k}=\dot{C}^{R}_{F,k}\dot{\dot{E}}^{UN}_{D,k} \tag{13}\]  
\[\dot{Z}^{UN}_{k}=\dot{E}^{R}_{P,k}\left(\frac{\dot{Z}}{\dot{E}_{P}}\right)^{ UN}_{k} \tag{14}\]  
Once the inevitable exergy destruction-related and investment costs are determined, it can be calculated in available parts, as follows:  
\[\dot{\dot{C}}^{AV}_{D,k}=\dot{\dot{C}}^{R}_{D,k}-\dot{\dot{C}}^{UN}_{D,k} \tag{15}\]  
\[\dot{Z}^{AV}_{k}=\dot{Z}^{R}_{k}-\dot{Z}^{UN}_{k} \tag{16}\]  
To explain the \(k\)th component more, the potential for economic improvement is associated with the avoidable and unavoidable exergy destruction and investment costs that are endogenous, and it is decomposed once again into exogenous sections. The sections, separated further below, are presented, respectively:  
\[\dot{\dot{C}}^{UN,EN}_{D,k}=\dot{C}^{R}_{F,k}\dot{\dot{E}}^{UN,EN}_{D,k} \tag{17}\]  
\[\dot{\dot{Z}}^{UN,EN}_{k}=\dot{\dot{E}}^{EN}_{P,k}\left(\frac{\dot{Z}}{\dot{E }_{P}}\right)^{UN}_{k} \tag{18}\]  
\[\dot{\dot{C}}^{UN,EX}_{D,k}=\dot{\dot{C}}^{UN}_{D,k}-\dot{\dot{C}}^{UN,EN}_{D,k} \tag{19}\]  
\[\dot{\dot{Z}}^{UN,EX}_{k}=\dot{Z}^{UN}_{k}-\dot{Z}^{UN,EN}_{k} \tag{20}\]  
\[\dot{\dot{C}}^{AV,EN}_{D,k}=\dot{\dot{C}}^{EN}_{D,k}-\dot{\dot{C}}^{UN,EN}_{D,k} \tag{21}\]  
\[\dot{Z}^{AV,EN}_{k}=\dot{Z}^{EN}_{k}-\dot{Z}^{UN,EX}_{k} \tag{22}\]  
\[\dot{\dot{C}}^{AV,EX}_{D,k}=\dot{\dot{C}}^{EX}_{D,k}-\dot{\dot{C}}^{UN,EX}_{D,k} \tag{23}\]  
\[\dot{\dot{Z}}^{AV,EX}_{k}=\dot{\dot{Z}}^{EX}_{k}-\dot{\dot{Z}}^{UN,EX}_{k} \tag{24}\]Some markers are needed to evaluate the AdvExEcon analysis. For the TradExEcon and AdvExEcon analysis, in this study, the performance indicators of the system are listed in Table 3.