## 5 Results
### Scenario 2--False Preference Declaration by One of the Drivers  
Scenario 2 is similar to the previous one. However, there is one difference. This is that the driver of the EV connected to the fourth terminal does not declare their preferences truthfully in this scenario, trying to increase their utility (decrease cost). The driver of this EV declares \(\theta^{3}\) to be equal to 1 instead of 0.5 (Table 1). So, they report that the price per discharge is more important to them than it is. The results of this scenario are presented in Figures 5-8.  
Figure 5 presents the SoC curves of each connected EV. Since the EV driver connected to the fourth terminal declared that the price per discharge is more important to them, their EV discharges in this scenario. Due to the discharging of EVs connected to the fourth terminal, the EVs connected to the second and third terminals are charging more rapidly than in the previous case.  
Figure 6 shows the sum results over the whole period for each EV driver's cost function (10) and final cost function involving the incentives as well (23) in this scenario. As shown in the figure, the driver of an EV connected to the fourth terminal would decrease its cost over the simulation time (blue bar) with its untruthful report if there was no incentive mechanism. However, due to the defined mechanism, their final cost increases (orange), and it is even higher than in the case of reporting true preferences in Scenario 1, while  
Figure 4: Charging power vs. output of the PV power plant in Scenario 1.  
The comparison of the final costs of all participants (charging station--\(i=0\)) in Scenarios 1 and 2 is presented in Figure 7. As noted above, the drivers that report their preferences truthfully in both cases increase their final cost in Scenario 2 in contrast with the EV driver connected to the fourth terminal. This is the result of the mechanism design feature that the truthful report of preferences is the optimal strategy for each agent at every time \(t\) regardless of the reports of the others (Proposition 1).  
Since no EV is discharging in Scenario 1, the charging station operator just earned money from selling the energy. The sum of its cost function over simulation is therefore negative and equal to \(-4.33\). However, the charging station pays for discharging the EV connected to the fourth charging terminal instead of utilizing the power produced from its PV power plant in Scenario 2. Since not all power of the photovoltaic power plant is delivered and sold to the EVs and the charging station operator needs to pay the EV connected to the fourth terminal for discharging, the operator's utility decreases or their cost increases to \(-0.69\). This shows that truthful declaration of the preferences by EV drivers is also important for the charging station, and the presented incentive system is advantageous for the charging station operator as well. So, the defined incentive mechanism is important to meet the social welfare of all system participants, i. e. EV drivers and the charging station operator too.  
Figure 8 presents the generated and consumed power at the charging station in Scenario 2. In contrast with Scenario 1, the curve of the consumed power (P\({}_{\text{C}}\)) by EVs does not strictly follow the output of the photovoltaic power plant in this scenario. It is caused by the discharging of the EV connected to the fourth charging terminal based on its driver's untruthful report of preferences. This issue is also reflected in the increase in the final cost of the charging station, as mentioned above.