## 3 Optimization Problem  
The goal of the optimization problem is to optimize the hourly scheduling of battery charge/discharge and to optimize the power and time required to charge the EV. The problem is treated as a multicriterial because three criteria will be simultaneously taken into account. The optimization is based on the minimization of supply costs, maximization of renewable energy for the EV charging and the minimization of the battery wear. The scheduling is optimized using 24 h-ahead forecasts of PV plant production, load demand of other consumers, and expected arrivals of EVs. The first criterion is the maximization of renewable (solar) energy charging the EV. \[\max(F_{1})=\sum_{i=1}^{N_{EV}}\sum_{t=1}^{T}\left(P_{EVS,i,t}+P_{EVBS,i,t} \right), \tag{13}\]  
The second criterion is the minimization of battery installation and operation and energy supply costs. Assuming that the energy is traded at the spot market with hourly prices \(c_{t}\), the criterion can be mathematically represented in (14). \(c_{B}\) represents the levelized daily costs of battery installation and operation (\(\epsilon\)/kWh\(\cdot\)day). \[\min(F_{2})=c_{B}\cdot C+\sum_{t=1}^{T}c_{t}\cdot P_{G,t}, \tag{14}\]The third criterion refers to the aging of the batteries (15). \[\min(F_{3})=\sum_{i=1}^{T}B_{1}e^{B_{2}\cdot I_{net\cdot i}}.Ah_{i}, \tag{15}\]  
Optimization process constraints are related to the active and reactive power balance in the network Equations (7)-(11), minimal and maximal battery state of charge (16), minimal and maximal allowed charge/discharge rate (17) maximal rated power of EV charger (18) and capacity of the grid connection supply feeder (19). \[SOC_{\min}\leq SOC_{G,I}\leq SOC_{\max}, \tag{16}\]  
\[P_{B}^{\min}\leq P_{B,t},P_{BS,t},P_{BG,t}\leq P_{B}^{\max}, \tag{17}\]  
\[P_{EV,t}\leq P_{EV}^{\max}, \tag{18}\]  
\[P_{G,t}\leq P_{G}^{\max}, \tag{19}\]  
Multi-attribute utility theory (MAUT) defines the utilities of multiple-attribute outcomes as a function of the utilities of each attribute taken individually. The theory specifies several possible functions and the conditions to be met under which each of these functions (multiplicative, additive and multilinear) would be appropriate. The multi-attribute utility function is of the following form if mutual utility independence exists (20). \[U(x_{1},x_{2},\ldots,x_{n})=\frac{\prod\limits_{i}\left(1+Kk_{i}u_{i}(x_{i}) \right)-1}{K}, \tag{20}\]  
\(u_{i}(x_{i})=\) the single-attribute utility value for attribute \(i\) with value \(x_{i}\) (ranges from 0 to 1); \(k_{i}=\) a parameter from the tradeoffs for component \(i\); \(K=\) a normalization constant. The utility function (20) reflects different types of interactions between individual criteria. In the compensatory case, the performance of one criterion compensates for the lack of performance by the other one (21). In the additive case, the performance of one criterion does not interact with the value of the other ones (22). In the complementary case, the balanced performance of all criteria is more important than the individual criteria fulfillment (23). \[\sum_{i=1}^{n}k_{i}>1\;then\;-1\;<K<\;0, \tag{21}\]  
\[\sum_{i=1}^{n}k_{i}=\;1\;then\;K\;=\;0 \tag{22}\]  
\[\min(F_{2})=c_{B}\cdot C+\sum_{t=1}^{T}c_{t}\cdot P_{G,t}, \tag{14}\]0  
If additive independence exists, the multi-attribute utility function is also additive (24):  
\[\min(F_{2})=c_{B}\cdot C+\sum_{t=1}^{T}c_{t}\cdot P_{G,t}, \tag{14}\]1  
The multicriteria optimization used in this paper is based on the methodology applied for the EV charging fleet schedule presented in [17]. In this methodology, the first step in scheduling optimization is the single objective optimization of all of the previously explained objectives (13)-(15). The outputs of this optimization are utility functions based on maximal renewable energy (\(U_{res}\)), minimal battery and energy costs (\(U_{lc}\)), and minimal battery degradation (\(U_{bd}\)). In the second step, the multiobjective optimization is performedapplying MAUT for the utility aggregation (20).