## 2 Problem Description
### Uncertainty Characterization  
When the aggregator wants to participate in a short term trading structure, it may face three sources of uncertainty including: DA prices, positive/negative balancing prices and EVs' demand. These uncertainties are discretized with scenarios and the vector of stochastic processes is represented as follows: \(\textit{Scenario}\,\omega\,=\,\left\{\,\lambda_{t,\omega}^{DA}\,,\,\,\lambda_{t, \omega}^{+,B}\,,\,\lambda_{t,\omega}^{-B},E_{t,\omega}^{D}\,\right\}_{t\in T, \quad\omega\in\Omega}\) where, \(\lambda_{t,\omega}^{DA}\), \(\lambda_{t,\omega}^{+,B}\), \(\lambda_{t,\omega}^{-B}\) and \(E_{t,\omega}^{D}\) are DA, positive/negative balancing prices and EVs' demand at time \(t\) and scenario \(\omega\). Each scenario has a probability of occurrence \(\omega\), such that the sum of the probabilities over all scenarios is unity: \(\sum\limits_{\omega\in\Omega}p(\omega)=1\). Moreover, it is assumed that the aggregator has a forecast of the expected demand of EVs. In addition, it is considered that EVs' demand is correlated to DA prices. Thus, it is possible to generate EVs demand scenarios as follow [26]:  
\[E_{t,\omega}^{D}=\hat{E}_{t}^{D}\Bigg{[}1+\Delta(\frac{\lambda_{t,\omega}^{DA} -\hat{\lambda}_{t}^{DA}}{\hat{\lambda}_{t}^{DA}})\Bigg{]}\qquad\forall t\in T,\quad\forall\omega\in\Omega \tag{1}\]  
where, \(\hat{\lambda}_{t}^{DA}\) is the expected DA price in period t and \(\Delta\) is a parameter representing the relationship between DA price and the EVs' demand in each scenario \(\omega\) that can be assumed as \(\Delta=0.2\)[26]. Based on the above equation, EVs' demand at time \(t\) and scenario \(w\) is equal to the sum of the forecasted demand and an additional value that is variable with the DA price. Therefore, EVs' demand at each scenario \(w\) is determined based on the DA price at that scenario. Here, historical data is used to generate DA and balancing price scenarios based on time series. The EVs' demand is also obtained from (1). Then, for the sake of tractability of the problem, the size of the set of DA and positive/negative balancing price scenarios are reduced by scenario-reduction techniques [27]. Similarly, rival aggregators' prices are also uncertain sources. The prices offered by rivals are considered as random variables, \(\lambda_{s}(\xi)\), for a finite set of scenarios \(\Xi\) where \(\sum\limits_{\xi\in\Xi}v(\xi)=1\). \[\textit{Scenario}\,\xi=\{\lambda_{s1}(\xi),\lambda_{s2}(\xi),\ldots,\lambda_ {s}(\xi)\},s\in S\]  
Here, the uncertainty associated with the prices offered by the rival aggregators is modeled with Probability Distribution Function (PDF) with the related standard deviation that is used to generate scenarios based on historical forecasting errors. In order to model the forecast inaccuracies stemming from the uncertain nature of offered prices by the rival aggregators, normal PDF is used [25]. In this case, the mean values are equivalent to the forecasted values of prices. Then the PDFs are divided into three discrete intervals with different probability levels as illustrated in Figure 2. The forecasted errors which correspond to the mentioned uncertain resources are obtained based on historical forecasting errors. These errors are considered within intervals equal to the standard deviation. The price forecasting state is centered as the base case and its forecasted errors as the standard deviations are considered with the other intervals. In order to model each probability level associated with the selling prices offered by the rival aggregators, the Roulette Wheel Mechanism (RWM) is used. Each interval throughout the RWM represents a specific forecasting error corresponding to the offered prices by the rivals. Then, the probabilities of the intervals on the Roulette Wheel are normalized such that their summation equals unity. To generate a scenario corresponding to each forecast error of the offered prices, first Monte Carlo simulation is used to generate a random number in the range of [0-1].