## 2 Model and Methodology
### Model Structure
#### 2.2.5 IV Step: Output Definition  
This final step is designed to aggregate the outputs coming from steps II and IV, and to post-process them with some of the inputs in order to obtain the other PI for the specific charging infrastructure. In this section the operations needed to accomplish the first goal will be described, while the results of the latter will be discussed in the next section.  
BEV and PHEV daily charging on publicly available infrastructure:  
Figure 7: An example of the space of solutions as calculated by the model with highlighted boundaries.  
The total number of xEV daily using the charging infrastructure during y-th year can be obtained by summation of the values related to every j-th level:  
\[\left\{\begin{array}{c}\left(BEV_{P}\right)_{y}=\sum\limits_{j=1}^{10}\left( BEV_{j}\right)_{y}\\ \left(PHEV_{P}\right)_{y}=\sum\limits_{j=1}^{10}\left(PHEV_{j}\right)_{y}\\ \end{array}\right.. \tag{14}\]  
Energy and power demand from publicly available charging infrastructure:  
Couples of values \(\left(E_{j},BEV_{j}\right)_{y}\) and \(\left(E_{j},PHEV_{j}\right)_{y}\) are sufficient for the calculation of total energy provided by charging infrastructure during y-th year:  
\[\left(E_{TOT}\right)_{y}=\sum\limits_{j=1}^{10}\left(E_{j}\ast BEV_{j}\right) _{y}+\left(E_{j}\ast PHEV_{j}\right)_{y}. \tag{15}\]  
In order to calculate the energy provided by each power level of the infrastructure, each of the j-th level contributions must be evaluated separately and finally summed:  
\[\left[\left(E_{TOT}\right)_{k}\right]_{y}=\sum\limits_{j=1}^{10}k_{MIN,j}\ast E _{j}\ast r_{k,j},k\in(s,m,f). \tag{16}\]  
Publicly available charging infrastructure composition and cost:  
The number of charging stations for the various CS power levels and their cost are provided as output for each y-th year and each j-th energy class by the equations state da the end of Section 2.2.4; total yearly values are obtained by a sum in j and total global values are the obtained by another sum in i:  
\[\left\{\begin{array}{c}\left(C_{MIN}\right)=\sum\limits_{i}\sum\limits_{j=1 }^{10}\left[\left(C_{MIN}\right)_{j}\right]_{y}\\ \left(s_{MIN},m_{MIN},f_{MIN}\right)=\sum\limits_{i}\sum\limits_{j=1}^{10} \left[\left(s_{MIN},m_{MIN},f_{MIN}\right)_{j}\right]_{y}\\ \end{array}\right.. \tag{17}\]  
Energy and power demand from private residential charging infrastructure  
The basic assumption regarding the use of private residential charging infrastructure is that each vehicle is assumed to be used and charged every day. This simplifying assumption is related to the fact that, being the stall private and related to the vehicle, this one will be parked there at least once in the day, ready to be charged. The equation describing the total average daily energy request is:  
\[\left[\left(E_{r}\right)_{A}\right]_{y}=\sum\limits_{V}\left\{\left[1-\left( \%xEV_{P}\right)_{y}\right]\ast\left(xEV\right)_{y}\ast\left(xcv\right)_{y} \ast\left.avd_{A}\right\}. \tag{18}\]  
Private residential charging infrastructure cost:  
The least cost for the private residential charging infrastructure is obtained using the following equation:  
\[\left[\left(C_{r}\right)_{A}\right]_{y}=\sum\limits_{i=1}^{y}\sum\limits_{V} \left\{\left[1-\left(\%xEV_{P}\right)_{y}\right]\ast\left(xEV\right)_{y} \ast\left(C_{r}\right)_{y}\right\}. \tag{19}\]  
The least cost assumption derives from the fact that no overlap in the use of publicly accessible and private residential charging infrastructure is modeled, while a certain share of it is expected.