## 4 Proposed Control and Incentive Mechanism
### Mechanism of the Incentives  
As noted above, the presented optimal control problem is solved by the charging station. It determines the operation modes and corresponding charging or discharging rates for each connected EV socially based on the defined preferences of the EV drivers. However, the reported preferences need not be truthful and can be purposefully misreported by some EV drivers to gain monetary profits or lower discomfort, as shown in [29; 35]. Thus, the proper incentive mechanism is of significant importance to meet the minimal discomfort and so the concept of social welfare maximization. Therefore, the charging station also computes an incentive for each EV driver connected to its terminals. This incentive is denoted by a payment policy \(p_{it}(\mathbf{b}_{t},\mathbf{a}_{t}^{*})\) that is a function of the reported preferences of all EV drivers connected to the charging station [35]. Since these incentives will be paid to the drivers, they will decrease the discomfort caused to the EV drivers by controlling their charging or even their discharging. So, the final cost for the EV driver is noted as follows [35]:  
\[l_{it}(\mathbf{\theta}_{it},\mathbf{a}_{it},\mathbf{b}_{t})=c_{it}(\mathbf{\theta}_{it},\mathbf{a} _{it}^{*})-p_{it}(\mathbf{b}_{t},\mathbf{a}_{t}^{*}) \tag{22}\]  
We assume that each EV driver strategically chooses their preference report to minimize their discomfort and payment at each step \(t\). **Assumption 5**. : \(EV\) _drivers are strategic agents_. So, the optimal cost for the EV driver connected to the \(i\)th terminal of the charging station can be denoted as [35]:  
\[L_{it}(\mathbf{\theta}_{it},\mathbf{a}_{it},\mathbf{b}_{-i,t})=\inf_{\mathbf{b}_{it}\in\mathbf{ \Theta}_{it}}l_{it}(\mathbf{\theta}_{it},\mathbf{a}_{it},\mathbf{b}_{t}) \tag{23}\]  
The defined system also assumes that the EV drivers cannot communicate (Assumption 3), so they cannot cooperate strategically with their reports to increase their gains. To define the incentives, a proper mechanism needs to be defined. This mechanism will design the set of rules for the payment of the incentives. The mechanism is represented by a set of charging station control and payment policies \(\{\mathbf{a}_{t}\,\ \mathbf{p}_{t}\}\)[37]. For the defined incentive mechanism, we focus on the direct mechanism that asks every agent (EV drivers) to report their preferences every time \(t\) and supports the truthful implementation of the socially efficient policy \(\mathbf{a}_{t}^{*}\) by defining the payment policies. The mechanism is defined with the following stated properties. **Definition 1**. : _A mechanism \(\mathcal{M}(\mathbf{a}_{t}^{*}\,\mathbf{p}_{t}^{*})\):_  
* _is efficient, meaning that_ \(\mathbf{a}_{t}^{*}\) _is a socially efficient policy,_
* _is incentive compatible, meaning that_ \(\mathbf{p}_{t}^{*}\) _is such that the truthful report of preferences is an optimal strategy for each agent at every time_ \(t\) _regardless of the reports of the others,_
* _is individually rational, meaning that the total discomfort experienced by agent_ \(i\) _(_22_) is smaller than the cost experienced under the socially efficient control policy_ \(\mathbf{a}_{-i,t}^{*}\) _of the mechanism in their absence,_
* _satisfies the efficient exit condition, meaning that if agents are irrelevant for the mechanism in time_ \(t\) _then these agents do not receive payment at that time step_ _[_35_]__._  
As defined in the previous subsection, the socially efficient control policy \(\mathbf{a}_{t}^{*}\) is obtained by minimizing the social cost function (20). Then the optimal social program is defined in (21).