## 3 System Model
### The Discomfort of EV Drivers  
When determining the discomfort of EV drivers during charging or discharging sessions, it is necessary to define the attributes that affect the final discomfort experienced by the EV drivers during these operation modes. These attributes were determined from an SP survey in which the participants chose between charging and V2G services [27; 28]. The statistically significant attributes influencing the decision of electric vehicle drivers between these alternatives were state of charge, measure of uncertainty, reward per 1 kWh of the sold energy, and the amount of required energy [29].  
In [27], the price for charging was not statistically significant. However, it will be considered as a factor affecting the discomfort of EV drivers since the high \(p\)-value of this parameter could be caused by the small sample in the survey. Despite the results in [27; 28], the measure of uncertainty is not considered in the model since it is highly subjective, and sometimes EV drivers are unaware of or cannot quantify this measure. To simplify the cost function and decrease the amount of the required private information, the SoC and amount of energy required for discharging attributes are mixed and represented just by the change of the SoC during the charging or discharging session [27; 28; 29].  
Taking into account previous assumptions, the discomfort of the EV drivers connected to the \(i\)th charging terminal at time \(t\) is presented using the following cost function:  
\[c_{it}(\mathbf{\theta}_{it},\mathbf{a}_{it})=\theta_{it}^{1}\frac{1}{c_{i}}\cdot(u_{it} +v_{it})+\theta_{it}^{2}\cdot p_{ch}\cdot u_{it}+\theta_{it}^{3}\cdot p_{d} \cdot v_{it} \tag{10}\]  
where \(\theta_{it}^{1}\) is the weight of the energy gain or drop in the battery of the vehicle connected to the \(i\)th charging terminal, \(\theta_{it}^{2}\) is a parameter of the attribute price of charging (\(p_{ch}\)) and \(\theta_{it}^{3}\) is a weight of the reward for discharging (\(p_{d}\)) attribute. The signs in front of parameters are based on results in [27; 28].  
For shorter notation, the vector \(\mathbf{a}_{it}\) is used to define the control policy of the charging station for EV connected to its \(i\)th terminal and so:  
\[\mathbf{a}_{it}\triangleq\begin{bmatrix}u_{it}\\ v_{it}\\ z_{it}\end{bmatrix}, \tag{11}\]  
where \(z_{it}\) is an integer variable defining the operation mode of the EV at the \(i\)th charging station terminal (\(z_{it}=1\rightarrow\) charging, \(z_{it}=0\rightarrow\) discharging).  
As mentioned above, it is assumed that the experienced discomfort of EV drivers represented by the cost function in (10) is known, but some parameters of this function representing preferences of EV drivers are privately held information of each EV driver. The preferences reported by the drivers are denoted as \(\mathbf{\theta}_{t}=\left\{\mathbf{\theta}_{it}\right\}_{i\in I}\), where:  
\[\mathbf{\theta}_{it}\triangleq\begin{bmatrix}\theta_{it}^{1}\\ \theta_{it}^{2}\\ \theta_{it}^{3}\end{bmatrix} \tag{12}\]  
**Assumption 2**.: _The cost function (10) of the electric vehicle drivers is common knowledge. However, the specific preferences \(\mathbf{\theta}_{it}\) are unknown._  
**Assumption 3**.: _Electric vehicle drivers do not communicate with each other, and their preferences \(\mathbf{\theta}_{t}\) are privately held information._