## 2 Model and Methodology
### Model Structure
#### 2.2.3. II Step: Calculation of Daily Charging Events on Publicly Available Infrastructure  
In this step of the model the average number of daily charging events related to the BEV and PHEV circulating fleet is calculated, together with the corresponding energy requests. To do so, the average number of daily charging vehicles and their SOC at the beginning of the charge have to be estimated. BEV and PHEV have different electric driving ranges, thus they are separately evaluated within the model; anyway, since the operations are conceptually identical, in the followings of this section we would simply refer to xEV without losing generality. A discrete-time Markov chain (DTMC), applied to a countable, finite state-space was used to obtain the estimate of the average daily distribution of the SOC levels within the xEV circulating fleet. The DTMC is defined by the transition matrix \([\mathbf{A}]\), which values are obtained by applying the charging events probability distribution \(p\) over the specific driving range considered, with a 1-day timestep:  
\[[\mathbf{A}]=\left( \begin{array}{ccccc}p_{1}&(1-p_{1})&0&\cdots&0\\ p_{2}&0&(1-p_{2})&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ p_{n-1}&0&0&\cdots&(1-p_{n-1})\\ p_{n}=1&0&0&\cdots&0\end{array} \right). \tag{6}\]  
The transition matrix defines the charging probability of a xEV for each day of the driving range; once the driving range \(\left[(avr_{V})\right]_{A}\), is defined, also the dimension of the state-space and of the transition matrix \([\mathbf{A}]\) are set accordingly. Figure 4 shows an example of DTMC graph, applied to a 4-days driving range scenario. The evolution of the DTMC is calculated with an iterative process--highlighted by Figure 5--that usually reached convergence within 15 steps. The final stationary probability distribution describes the average distribution of SOC levels within the xEV circulating fleet. The application of the transition  
Figure 4: Discrete-time Markov chain (DTMC) plot example, showing the transition probability over a 4-days driving range scenario. matrix \([A]\) to this stationary distribution, finally allows us to stochastically define the average number of xEV daily charging, for each SOC level. The horizontal axis represents timesteps \"a\", which are inscribed within the interval \(\left[0,\left[(av_{V})_{A}\right]_{Y}\right]\), while the vertical axis represents the iteration steps. A different SOC is associated to every timestep \"a\", thus a different charging energy request; taking into account the hypothesis of only complete charges its value \(E_{a}\) can be calculated as:  
\[\left[(E_{a})_{V}\right]_{y}=\left[(av_{CV})_{A}*av_{A}\right]_{y}*a. \tag{7}\]  
The output is then a series of couples \((xEV_{a},E_{a})\), dividing the charging xEV in different groups in terms of energy requests. In order to check the stability of the iterative process and the quality of the results, two control methods were implemented:  
* The sum of the elements \(a_{i,j}\) for each row of the transition matrix \([A]\) must be equal to one, since they define all the possible transition events completely: \[\sum_{j=1}^{j}a_{i,j}=1.\] (8)
* The total energy recharged by infrastructure (after convergence) must be equal to the average energy consumed by the xEV fleet using public charging infrastructure (Figure 6): \[\left[(xEV_{P})_{A}\right]_{y}*\left[(av_{CV})_{A}*av_{A}\right]_{y}=\sum_{a=1 }^{\left[(am_{xEV})_{A}\right]_{y}}xEV_{a,t}*E_{a},\;\forall t>\bar{t},\] (9) where \(\bar{t}\) is the convergence iteration step. Figure 5: DTMC iterative process graph, showing the probability distribution across driving range days (axis “a”) and time (days, axis “t”). .2.4. III Step: Publicly Available Infrastructure Solutions' Space Definition and Costs Minimization  
Single event charging energy requests \(E_{a}\) are related to specific consumption \(\left[(avc_{V})_{A}\right]_{y}\) and to battery capacity \(\left(batt_{V}\right)_{y}\), thus they are different between BEV and PHEV; they also evolve during the time period.