## 4 Single Level Equivalent Linearization
### Simulations and Discussion
It has to be mentioned that the strong shift from DA into the balancing market is because of the assumption of full insight of balancing market prices within the 24 h.  
Figure 8: Variation of expected profit and conditional value at risk with varying \(\beta\). Figure 7: Cumulative distribution functions with varying \(\beta\) at hour 4:00. When future markets and contracts are not considered, the aggregator can control risk only by participation in the DA and balancing markets. It should be noted that DA procurements are made under uncertainty, while balancing purchases are made with awareness of prices. Also, when the aggregator takes part in the balancing market, it has a perfect knowledge of EVs' energy requests and their availability. For a detailed discussion and elaboration of the DA and balancing purchases, the expected procurements during the whole day are assessed. Figures 10-12 illustrate the expected DA, positive and negative supplements at each hour during a given day, respectively. During midnight and early in the morning, EV owners usually connect their vehicles to the network to charge them for the next work-day hours. Since, at this time the DA prices are very low, the aggregator purchases a high amount of DA energy. Accordingly, less energy is purchased in the positive balancing market, since the aggregator bought high quantities of energy in the DA market, so it is not required to purchase a high bulk of energy from positive balancing. Also, it can be a good source of revenue for the aggregator to participate in negative balancing market such that it encourages the EV owners to decrease their demand, (i.e., unplug their vehicles or shift the charging of their EVs to other hours). From 7:00 to 21:00, when the predicted EVs' demand is lower than the other hours, the aggregator decides to participate in the positive balancing market to supply EVs during this operating period. Moreover, the aggregator can deal economically with this low amount of demand in joint DA and positive markets. However, from 19:00 to 21:00, the EVs' demand increases, so the aggregator purchases more energy from the positive market to meet their energy requirements. During these hours, it participates in negative balancing to obtain some profits from load decrements. After 22:00, there are many EV owners who want to charge their EVs for the next day; so the aggregator participates in the DA market to buy energy. In this regard, it purchases less energy from the positive balancing market. Also, the aggregator is paid based on the negative balancing price to decrease its demand. It should be mentioned that  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{\(\beta\)} & \(\sum\limits_{\mathbf{t}\in\mathbf{T}}\mathbf{p}(\mathbf{\omega})\mathbf{E}_{\mathbf{t},\mathbf{\omega}}^{ \text{DA}}\) & \(\sum\limits_{\mathbf{t}\in\mathbf{T}}\mathbf{p}(\mathbf{\omega})\mathbf{E}_{\mathbf{t},\mathbf{\omega}}^{ +,B}\) & \  
negative prices fluctuate more than the positive ones, because the aggregators should be encouraged to decrease their demand based on that. Generally, the expected DA and balancing procurements during the day also demonstrate that increment of risk factor leads to lower participation in the DA and negative balancing energy trading floor and higher positive balancing purchases at each hour. Figure 13 shows the expected values of the selling prices offered by the rivals with spotted lines and the price signal offered by the understudy aggregator with a solid line at each hour of the day. The rivals' prices shown in this figure are the expected values of the considered scenarios that are generated based on normal PDF as explained in Section 2.2. Moreover, the price signal offered by the understudy aggregator is obtained based on the optimization problem. As can be seen, from 1:00 to 2:00, because the demand is high and the DA and positive balancing prices are relatively high, the aggregator buys a low amount of energy from these markets.