## 4 Formulation of the Advanced Global Efficiency Index, AGEI
### Example 2  
The second case presents another comparison between a hybrid configuration and an integrated machine. In this case, the hybrid configuration is composed of other two machines, Machine 3 and Machine 4, which operate differently from the ones described in the first case. Machine 3 provides simultaneously three useful effects: heating energy \(Q_{h}\) = 3 kJ, cooling energy \(Q_{c}\) = 3 kJ, and water condensation energy \(Q_{w1}\) = 2 kJ  
Machine 4 contributes only to water generation, providing \(Q_{w2}\) = 2 kJ, which, along with \(Q_{w1}\), satisfies the whole needs of \(Q_{w}\)= 4 kJ. Each machine consumes an energy amount \(en\)= 1 kJ. Therefore, Machine 3 has COP = EER = 3 and WET = 2. Machine 4 also has WET = 2. The integrated machine, as in the first case, meets all the required useful effects (\(Q_{h}\) = 3 kJ, \(Q_{c}\) = 3 kJ, \(Q_{w}\) = 4 kJ) consuming only \(en\) = 1 kJ. Therefore, COP = EER = 3, while WET = 4. A summary of the aforementioned data is given in Table 3.  
Upon analysing the data above, it is clear that the integrated machine is once again twice as efficient as the hybrid configuration. In fact, the sum of the useful effects is equal to 10 kJ for both systems, but the energy consumption of the hybrid configuration is twice that of the integrated machine. Also for this second case, the evaluation metric results are expected to align with the actual efficiency. Therefore, the value for the integrated machine should be twice that of the hybrid system. Similarly to the first case, GELeq and AGEI are applied in the following analysis to determine which is more aligned with the expected results. Given that in the hybrid configuration of this second case, Machine 3 and Machine 4 contribute to cover the same useful effect \(Q_{w}\), in compliance with the definitions reported in Section WET and GEI, the applicable formulation of GELeq is given by Equation (15). Observing that \(m\) is equal to 2 and that \(en_{3}/en\) is equal to 0.5 as well as \(en_{4}/en\), it follows that  
\[ \begin{split}\text{GEI}_{\text{eq}}=0.5\cdot 3\cdot(\text{COP}+\text{EER}+\text{ WET}_{3})+0.5\cdot\text{WET}_{4}\\ \text{GEI}_{\text{eq}}=0.5\cdot 3\cdot(3+3+2)+0.5\cdot 2\\ \text{GEI}_{\text{eq}}=13\end{split} \]  
Applying (1) to the integrated machine, noting that \(n\) = 3, it follows that  
\[ \begin{split}\text{GEI}=n\,\cdot(\text{COP}+\text{EER}+\text{ WET})=3\cdot(3+3+4)\\ \text{GEI}=30\end{split} \]  
Once again, the GELeq provides a result close to the expected value of 15, but it does not perfectly align with it. In this case, the index underestimates the efficiency of the hybrid configuration by approximately 6.7%--the ratio between the efficiencies is about 43.3% instead of 50%. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(Q_{h}\) & \(Q_{c}\) & \(Q_{w}\) & \(en\) & COP & EER & WET \\ \hline
**[ kJ]** & [ kJ] & [ kJ] & [ kJ] & [ kJ] & & & \\ \hline Machine 3 & 3 & 3 & 2 & 1 & 3 & 3 & 2 \\ Machine 4 & - & - & 2 & 1 & & & 2 \\ Integrated machine & 3 & 3 & 4 & 1 & 3 & 3 & 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Second case: summary of the useful effects, energy consumption, and indicator values for each machine. Now, applying AGEI Equation (16), noting that \(n\) is equal to 3 and \(m\) is equal to 2 for the hybrid system, it follows that  
\[\text{AGEI}=\frac{3}{2}\cdot(\text{COP}+\text{EER}+WET_{3}+WET_{4})=\frac{3}{2} \cdot(3+3+2+2)=15\]  
As previously stated, AGEI reverts to GEI for the integrated machine, thus AGEI = 30. It is evident that, even in this type of hybrid scenario, AGEI values are consistent with the real efficiency ratio, as the index result of the hybrid configuration is half that of the integrated machine. The two examples above, constructed with straightforward numbers, show the higher suitability of AGEI in cases of hybrid configurations in comparison to \(\text{GEI}_{\text{eq}}\).