## 4 Formulation of the Advanced Global Efficiency Index, AGEI
### Example 1  
The first case refers to a comparison between the performances of a hybrid configuration and an integrated machine.  
The hybrid configuration consists of two machines. Machine 1 provides two useful effects simultaneously: heating energy \(Q_{h}\) = 3 kJ and cooling energy \(Q_{c}\) = 3 kJ, while consuming energy, \(en\) = 1 kJ, while Machine 2 produces only water from the air. Its production is expressed in terms of water condensation energy, \(Q_{w}\) = 2 kJ.  
Remembering that  
\[\text{COP}=Q_{h}/en \tag{18}\]  
\[\text{EER}=Q_{c}/en \tag{19}\]\[\text{WET}=Q_{w}/en \tag{20}\]  
it follows that the efficiency terms of Machine 1 are expressed by COP = 3 and EER = 3, while for Machine 2, the efficiency is expressed by WET = 2.  
The integrated machine consists of a single equipment unit that provides simultaneously all the aforementioned useful effects, with \(Q_{h}+Q_{c}+Q_{w}=8\) kJ in total, while consuming only \(en\) = 1 kJ of energy. Also in this case, COP and EER are both equal to 3, and WET is 2.  
Table 2 summarises the various useful effects and consumption data, along with the corresponding coefficient of performance values.  
In summary, the hybrid configuration, composed of Machine 1 and Machine 2, provides a total of \(Q_{h}+Q_{c}+Q_{w}=8\) kJ of useful effects while consuming \(en\) = 2 kJ.  
The integrated machine delivers the same 8 kJ of useful effects but consumes only 1 kJ.  
Clearly, the energy efficiency of the integrated machine is exactly double that of the hybrid configuration. Therefore, it is expected that the global evaluation index values reflect this difference in efficiency. In other words, the index value for the integrated machine should be twice that of the hybrid configuration.  
In the following, both GEIeq and AGEI are applied in order to understand which of them is more aligned with the expected result.  
Considering the definitions reported in Section WET and GEI, Equation (14) provides the expression of GEIeq suitable for the hybrid configuration composed of the two machines. Noting that \(n_{1}=2\) and \(n_{2}=1\), and referring to Table 1, the calculation is as follows:  
\[\text{GEIeq}=n_{1}\cdot(\text{COP}+\text{EER})+n_{2}\text{WET}=2\cdot(3+3)+2=14\]  
Applying (1) to the integrated machine, noting that \(n\) = 3  
\[\text{GEI}=n\cdot(\text{COP}+\text{EER}+\text{WET})=3\cdot(3+3+2)=24\]  
Comparing the two results, it is clear that GEIeq provides a result not far from the expected value of 12. Nevertheless, it does not exactly reflect the real efficiency ratio between the two configurations. In fact, in this case, GEIeq overestimates the efficiency of the hybrid system, giving a result that is 8.3% higher than the expected one--the ratio between the efficiencies is 58% instead of 50%.  
Applying, instead, the AGEI Equation (16), noting that \(n\) is equal to 3 and \(m\) is equal to 2 for the hybrid system, it follows that  
\[\text{AGEI}=\frac{3}{2}\cdot(\text{COP}+\text{EER}+\text{WET})=\frac{3}{2} \cdot(3+3+2)=12\]  
Recalling that, for integrated configurations, AGEI is equal to GEI, the AGEI value for the integrated machine is equal to 24. The efficiency proportion, applying AGEI, is perfectly maintained: the value of the index for the hybrid system is exactly half of the value reached by the integrated machine.  
\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**] & & & \\ \hline Machine1 & 3 & 3 & - & 1 & 3 & 3 & - \\ Machine 2 & - & - & 2 & 1 & & 2 \\ Integrated machine & 3 & 3 & 2 & 1 & 3 & 3 & 2 \\ \hline \hline \end{tabular}
\end{table}
Table 2: First case: summary of useful effects, energy consumption, and indicator values for each machine.