## 3 Dynamic VESS Operation Strategy  
The main objective of the VESS operation for SECs is to maximize social welfare, which is the total net benefit. The objective function is expressed as follows:  
\[O_{B}(\mathbf{c})=\sum_{i\in\mathcal{I}}N_{i}(c_{i}), \tag{7}\]  
where \(\mathcal{I}\) is the participant set and \(\mathbf{c}\) is the allocated capacity set for the participants.  
The conventional VESS only considers cost reduction according to the economics of scale, and the VESS operation is performed in a same manner as that of the individual ESS [27]. Therefore, the VESS operation in the existing study is formulated as:  
\[\begin{array}{ll}\text{P0}:&\underset{\mathbf{c},\mathbf{a}}{\max}&O_{B}( \mathbf{c})\\ &\text{subject to}&\sum\limits_{i\in\mathcal{I}}c_{i}\leq c_{0},\\ &-c_{i}/2\leq a_{t}^{i}\leq c_{i}/2,\ \ \forall i\in\mathcal{I},t\in \mathcal{T},\\ &0\leq a_{0}^{i}+\sum\  
is considered. However, additional capacity restrictions could be considered depending on the DoD.  
Units participating in the VESS service operated in the form of Equation (8) can benefit from cost reduction only. However, the allocated capacity is fixed during the service period, and thus, it is difficult to obtain operational benefit from the VESS service compared with the individual ESS operation.  
To achieve operational benefit, the proposed dynamic VESS operation relaxes the restrictions on the fixed allocation of the VESS capacity and a method of guaranteeing usage is incorporated instead of guaranteeing the allocated capacity method as follows:  
\[\frac{24}{2T}\sum_{t\in\mathcal{T}}\left|a_{t}^{i}\right|\leq\beta\,c_{i},\ \ \forall i\in\mathcal{I}, \tag{9}\]  
where \(\beta\) is the usage factor that ensures the VESS usage of the participant. For example, \(\beta=1\) means that the participant can use one charge and discharge quantity for the allocated capacity for an average of 24 h.  
Because there is no limit on the individually allocated capacity, the VESS operates in an aggregated form for all the participants. The VESS operation at time \(t\) is expressed as:  
\[a_{t}^{Total}=\sum_{i\in\mathcal{I}}a_{t}^{i}. \tag{10}\]  
Using Equations (9) and (10), the proposed VESS operation can be expressed as follows:  
\[\begin{array}{ll}\text{P1}:&\underset{\text{c.a}}{\text{max}}&O_{B}(\text{ c})\\ &\text{subject to}&\sum\limits_{i\in\mathcal{I}}c_{i}\leq c_{0},\\ &&-c_{0}/2\leq a_{t}^{Total}\leq c_{0}/2,\ \ \forall t\in\mathcal{T},\\ &&0\leq a_{0}^{Total}+\sum\limits_{j=1}^{t}a_{j}^{Total}\leq c_{0},\ \ \forall t\in\mathcal{T},\\ &&\frac{24}{2T}\sum\limits_{t\in\mathcal{T}}\left|a_{t}^{i}\right|\leq\beta\,c _{i},\ \ \forall i\in\mathcal{I}.\end{array} \tag{11}\]  
In **P1**, similar to **P0**, the allocated VESS capacity to the participants should be less than the implemented VESS capacity. However, the individual operation in **P0** is reconstructed as an aggregated operation, as shown by the second and third constraints in Equation (11). The fourth constraint is considered to guarantee the VESS usage of each participant. **P1** is a mixed-integer linear programming problem that satisfies convexity [31], and it can be effectively solved by applying gradient methods and dual decomposition [32].  
The VESS operations of **P0** and **P1** can be compared schematically, as shown in Figure 2. In the cases of Figure 2a,b, the participants used the same amount of charging and discharging quantities. The VESS for participant 2 performs the same operation. However, the VESS for participant 1 operates differently based on **P0** and **P1** in Section A. In Figure 2a, the VESS operation for participant 1 is limited by the maximum capacity allocated to the participant. However, the VESS for the same participant in Figure 2b operates over the allocated capacity. This is because the VESS by **P1** is operated by the usage limit rather than the capacity limit. Therefore, the VESS by **P1** can operate more dynamically to maximize the net benefit to the participants compared to the VESS operation by **P0**.