## 2 Virtual Energy Storage Systems for Smart Energy Communities  
Three prime components of the VESS for SECs are depicted in Figure 1. A VESS can be configured by either a smart energy service provider (SESP) or a third-party VESS service provider. In this study, it is considered that the VESS is configured by the SESP and thus it is operated in a non-profit manner, where the implementation cost is recovered through the service cost. The SEC consists of community members who are both residential or commercial units, and the SESP, which is the demand aggregator for the community members. The SESP manages the energy transactions between the members and energy producers, such as between the VESS and utility grid. Considering the service price of the VESS, the community members participate in the VESS service. The SESP logically decides individual ESS operation actions for each participant according to the objective and constraints of the VESS operation. The determined actions are combined into one action at each operation time and are used to operate the VESS. From the viewpoint of the SESP, a VESS is a physical ESS. However, it is logically shared among multiple participants according to the management of the SESP. For energy balance, the SEC is also connected to the utility grid. Effectively serving the demand of the SEC using the resources in the VESS is the main challenge of this system. The VESS serves according to the capacity limit of the resources, which is determined by the size of the resource that the VESS has implemented and how the resource of the VESS is allocated. If the capacity is insufficient, it will not be able to provide the adequate service, and if there is too much capacity, the burden of the operating cost will increase. The VESS cost is also related to the cost of the physical ESS capacity. By increasing the capacity, the unit cost of the VESS is reduced by the economics of scale. If \(c_{0}\) is the VESS capacity for charging and discharging operations, the unit cost of the VESS related to the ESS is modeled as [27]  
\[p_{C}(c_{0})=\alpha_{1}\exp(-\alpha_{2}c_{0})+\alpha_{3}\;[\text{USD}/\text{ kWh}], \tag{1}\]  
where \(\alpha_{1}\), \(\alpha_{2}\), and \(\alpha_{3}\) are the ESS cost parameters based on the ESS characteristics. \(\alpha_{1}\) and \(\alpha_{2}\) represent the variable costs according to the capacity, and \(\alpha_{3}\) is the fixed cost, such as the operation and maintenance costs. \(c_{0}\) (kWh) is the energy capacity of the VESS. Assuming a service time of 2 h, the power capacity of the VESS is determined as \(c_{0}/2\) h (kWh) [27]. One characteristic parameter of the ESS is service time. To simplify the problem, in this study, the service time is set to 2 h. This is the value used in other studies such as [27] and in actual products such as Tesla's powerpack [30]. As mentioned above, in this study, the SESP installs and operates the VESS in a non-profit manner, and so, the VESS cost is recovered through the service price. Therefore, the service cost of the VESS is equal to its unit cost, and it is given by:  
\[p_{V}=p_{C}(c_{0}). \tag{2}\]  
Figure 1: Constitution of a virtual energy storage system (VESS) for smart energy community (SEC). A unit, which is a community member, decides to participate in the VESS service by considering its service cost. When the demand of unit \(i\) at time \(t\) is \(d_{t}^{i}\), the electricity bill of unit \(i\) is measured as:  
\[B_{i}(\cdot)=p_{0}\underset{t\in\mathcal{T}}{\max}\Big{(}d_{t}^{i}\Big{)}+\sum _{t\in\mathcal{T}}p_{t}d_{t}^{i}, \tag{3}\]  
where \(p_{0}\) is the demand price for the billing period, \(p_{t}\) is the energy price at time \(t\), and \(\mathcal{T}\) is the billing period, for example, one month.