*5.1. Case\_1—Inflexible*

In this case study, the electric vehicle aggregator has no power over the behavior of the vehicle, i.e., owners of vehicles are inflexible. The aggregator has power only when electric vehicles are not moving. Thus, through the communication system between the aggregator and the owners, the latter sends the predictable data on their behavior. The owner of the vehicle has the possibility to send 10 scenarios of driving requirements, according to the data in Appendix A, Table A1. The data show that vehicles between hour 1 and hour 7 are parked. In addition, the data show that the vehicles are, not with certainty, parked in periods of likely high day-ahead market prices, namely, hours 10, 13, and 22, and are explicitly driving in hour 21. Consequently, the vehicles are, not with certainty, able to charge or discharge the batteries in those periods. Based on these data, at the time of decision making, the aggregator defines the optimal values and the times for charging and discharging the electric vehicles, respectively, for purchase offers and sales offers, to present optimal offers in the electricity market. So, the aggregator has no opportunity to present offers in periods that have the potential for economic advantage. Purchase offering and sale curves are in Figures 2 and 3, respectively.

**Figure 2.** Purchase offering curves: **left**, hour 6; **right**, hour 11.

**Figure 3.** Sale offering curves: **left**, hour 12; **right**, hour 24.

Figure 2 shows the offering curves for purchase offers for hour 6 and hour 11. The offering curves for purchase offers decrease monotonically, as imposed by (10). At hour 6, as the day-ahead market price is one of the lowest, the aggregator takes the opportunity to charge the battery of electric vehicles. However, the aggregator is only willing to buy energy at a price below 45 €/MWh. So, when the price is higher than this value, the offer has a value of 0 MWh. The aggregator is willing to charge the battery of electric vehicles if the purchase price is around 30 €/MWh. At hour 11, as the day-ahead market price is already higher, the aggregator chooses to buy less energy. Thus, the aggregator only buys around 3.5 MWh of energy. To buy this amount of energy, the aggregator is willing to pay a maximum of 65 €/MWh. Above that value, the best option for the aggregator is not to buy energy in the day-ahead market. Figure 3 shows the offering curves for sale offers for hour 12 and hour 24. The offering curve for sale offers increases monotonically, as imposed by (9). At hour 12, one of the periods of the day with the best day-ahead market price, the aggregator is only available to sell energy at a price above 69 €/MWh. Below this value, the offer has a value of 0 MWh. At hour 24, the day-ahead market price is higher than some periods of the time horizon. Then, the aggregator is willing to receive anything between 44 €/MWh and 68 €/MWh for 5.5 MWh of energy. This behavior is the behavior of a perfectly inelastic supply curve. The approach proposed in this paper makes it possible to acquire the offering curves, which allow the aggregator to present bids for offer blocks at the lowest possible price when purchasing energy, and bids for offer blocks at the highest possible price when selling energy. In addition, this approach allows studying strategies to allocate part of the augmented profit among the owners of vehicles, thus allowing for flexibility.
