## 4 Single Level Equivalent Linearization
### Simulations and Discussion  
In this study, risk aversion is considered in the form of CVaR weights. Figure 7 shows the cumulative distribution functions of the scenario profits plotted for different levels of \(\beta\). In order to prevent data crowding in the figure, only four \(\beta\)s are selected to show the cumulative distribution function of profits for all scenarios at hour 4:00. At this time the DA prices are too low and as a result, the aggregator can economically supply the EVs. It is observed that with increasing risk aversion (increasing \(\beta\)), the worst scenarios are eliminated. Also, it shows that with increasing risk aversion, the profits which are far from the mean profit are ignored. This implies that when the CVaR concept is modeled, lower-profit scenarios with a low probability are not considered in the decision-making process. In fact, CVaR results in an average value with a higher probability and a lower variance with respect to the cases with the lowest risk aversion factor (\(\beta\) = 0.1). It can be observed that with increasing \(\beta\) and improvements in the lower tail of the distribution and omitting the worst scenarios with lower probabilities, the cumulative distribution function has higher values. Also, with increasing \(\beta\), the upper profits with high probabilities are also omitted and the scenarios which are near the expected profit would remain. That is why the cumulative function for higher risk aversion factor reaches 1 before the other curves. Figure 5: Scenarios and mean of negative imbalance price. Figure 6: Scenarios of Electric vehcles’ demand. Figure 8 depicts the variation of expected profits for the whole day against their corresponding CVaR for different values of \(\beta\). The efficient frontier illustrates how the risk averse EV aggregator can trade off in the electricity market for different levels of \(\beta\). It is observed that with increasing the risk aversion weighting factor, the expected profit decreases and CVaR increases. This means that as the aggregator becomes more risk averse, its expected profit decreases. Also it shows that beyond a critical \(\beta\) (i.e., beyond \(\beta=10\)), the expected profit decreases significantly. Moreover, negative CVaR indicates there is negative profit in some of the scenarios. With increasing risk aversion factor, these undesired scenarios are ignored. Figure 7 shows that with increasing values of \(\beta\), negative scenarios are eliminated; this means that with increasing \(\beta\), profits in the scenarios become more positive, because when it hedges against volatilities, the profits that are far from the mean profit would be ignored. In other words, lower profit scenarios with low probabilities would be eliminated. Therefore, the expected profit for the all hours of the day would be reduced. Figure 9 illustrates the expected profit versus its standard deviation for different values of \(\beta\). It is observed that with increasing \(\beta\), since low profit scenarios are not considered, the profit volatility decreases. So, the standard deviation associated with profit scenarios reduces. In other words, a conservative aggregator prefers minimizing risk; therefore, it chooses a large value of \(\beta\) to increase the weight of the risk aversion in the objective function. In contrast, a non-conservative aggregator might be willing to obtain a higher profit, hence it selects values for \(\beta\) close to 0. In order to assess the effect of risk aversion on the purchased power by the aggregator, Table 1 represents the expected DA, positive and negative procurements in different values of \(\beta\). It should be mentioned that a higher risk aversion weighting factor leads to a decrease in the DA purchases such that less negative and more positive balancing is provided. These results are consistent with the results provided in [4,18].