*5.3. Case\_3—Flexible*

In this case study, the aggregator has full power over the batteries of electric vehicles, and the aggregator decides the charging and discharging periods throughout the day, i.e., by assumption, the owners of the electric vehicle do not use the vehicle during the entire time horizon under study. Purchase offering and sale curves are in Figures 6 and 7, respectively.

**Figure 6.** Purchase offering curves: **left**, hour 5; **right**, hour 11.

**Figure 7.** Sale offering curves: **left**, hour 13; **right**, hour 23.

Figure 6 shows that compared to Case 2, the optimum values suggest that the batteries of electric vehicles be charged at hour 5, an hour with an average market price slightly lower than that of hour 6. Thus, at hour 5, the aggregator is only available to buy energy below 48 €/MWh. Above this value, the offer is 0 MWh. The aggregator is available to buy 10 MWh for values between €29 and €48. Compared to hour 6 in Case 2, the aggregator can buy large quantities of energy at lower prices. At hour 11, the aggregator is only available to buy energy below 56 €/MWh. Above this value, the offer is 0 MWh. Compared to hour 11 in Case 2, the aggregator can buy larger quantities of energy for the same prices. For example, for a price of €44, the aggregator in Case 3 buys 10 MWh, while, in Case 2, the aggregator buys 9.8 MWh, and for a price of €48, the aggregator buys 9.7 MWh in Case 3 and buys 7.6 MWh in Case 2. Figure 7 shows that at hour 13, the aggregator is only available to sell energy above 69.6 €/MWh. Below this value, the offer is 0 MWh. Compared to hour 12 in Case 2, the aggregator can sell larger quantities of energy at better market prices. For example, for a price of €69 at hour 12 of Case 2 and a price of €69.6 for hour 13 of Case 3, the aggregator sells 8.9 MWh and 10 MWh, respectively. Similarly, for a price of €71.5, the aggregator sells 10 MWh, while for €70.9, the aggregator sells only 8.9 MWh at hour 12 of Case 2. Likewise, at hour 23 of Case 3, compared to hour 22 of Case 2, the aggregator can sell larger quantities of energy at better market prices. For example, for a price of €68.6 at hour 12 of Case 2 and a price of €69.5 for hour 23 of Case 3, the aggregator sells 5.47 MWh and 10 MWh, respectively. Similarly, for a price of €71.9, the aggregator sells 10 MWh, while, for €70.3, the aggregator buys only 5.47 MWh at hour 22 of Case 2. The possibility of total control over the batteries allows the aggregator to make better decisions through the knowledge that the aggregator has about the market. The comparison between the cases is in Table 1.


**Table 1.** Comparison between Case 1, Case 2, and Case 3.

Table 1 shows that although the cost of degradation increases, the expected profit in Case 2 and Case 3 is higher than the expected profit in Case 1. The expected profit for Case 2 is 32% higher than the profit for Case 1, and the expected profit for Case 3 is more than 100% higher than the profit for Case 1. Thus, the formulation developed offers aggregator support for the management of the charge/discharge of electricity in the vehicles to improve the expected profit. With full control over electric vehicles, owners of electric vehicles are expected to receive a higher reward for being compliant with flexibility, allowing the aggregator to have a further economic advantage. Note that the V2G tariff is important for an aggregator in electricity markets due to the high costs of battery degradation. Hence, the aggregator can make a profit only with adequate values of V2G tariffs.

## **6. Conclusions**

Although the conventional power system perceives the integration of electric vehicles as only a further new load to be satisfied in due time, these vehicles are new sources of energy and opportunities for business in the electricity market. Nevertheless, the owners of electric vehicles cannot participate in electricity markets due to conditions of minimal power requirements imposed in these markets. Thus, these owners can only through an aggregator achieve sufficient conditions to participate in those markets. The aggregator is the intermediary entity between vehicle owners and electricity markets submitting blocks of offers to buy and sell energy to the market and wanting the achieve the most profitable operation, subject to operational and technical constraints. However, a set of electric vehicles is a fleet of non-stationary energy storage devices not necessarily under the total control of the aggregator. The aggregator is only able to manage charge/discharge of the energy storage devices in periods of consent by the owners of the vehicles. This consent is upon agreement with the owners and, in general, must take into consideration the cost due to the incurred degradation of the energy storage devices due to extra usage. So, a support decision for the fleet of vehicles is a crucial aid to the most favorable management regarding participation in the market, and this paper is a contribution in this regard. The support decision for the fleet of vehicles proposed is a formulation based in a mathematical programming problem written as a maximization of the expected profit in a stochastic programming framework, considering the uncertainty in day-ahead market prices and the driving requirements of electric vehicles.

The consent of owners is the level of flexibility regarding electricity usage for charge/discharge to a practice stated by the aggregator, allowing the aggregator to have some control, which is an opportunity to achieve a higher expected profit than the one in the case of inflexibility. As expected, the assumed level of flexibility accepted by the owners has repercussions in the aggregator management, as shown by a comparison of the addressed case studies assuming that the owners are inflexible, partially inflexible, or flexible.

The application proposed in this paper offers the aggregator support for the management of the charge/discharge of electricity in the vehicles to improve the expected profit. Quantitively identifying the augmented profit is a result of having the flexibility stated by the owners to allow the schedule of charge/discharge of vehicles by the aggregator and allowing for simulating the studies of strategies of persuasion for further consent.

**Author Contributions:** I.G., R.M. and V.M. conceived the formulation, performed the simulations, analyzed the simulation results, and wrote the paper. All authors have read and agreed to the published version of the manuscript. **Funding:** This research received no external funding.

**Acknowledgments:** This work is funded by Bolsas Camões, IP/Millennium BCP Foundation and funded by: European Union through the European Regional Development Fund, included in the COMPETE 2020 (Operational Program Competitiveness and Internationalization); Foundation for Science and Technology (FCT) under the ICT (Institute of Earth Sciences) project UIDB/04683/2020; Portuguese Funds through the Foundation for Science and Technology (FCT) under the LAETA project UIDB/50022/2020; Portuguese Foundation for Science and Technology (FCT) under the CISE Projects UIDB/04131/2020 and UIDP/04131/2020.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **List of Symbols**

