# Risk-Based Virtual Energy Storage System Service Strategy for Prosumers
## 2 VESS Service Model
### VESS SP Model  
As the SP installs and operates the VESS, it incurs cost and receives compensation for this cost through a service charge. The SP may be an independent SP for VESS services, or a utility may operate as a non-profit to increase the stability of the distributed grid operation.  
The main cost of the VESS service is in installing a VESS. Let \(c_{0}\) be the VESS capacity for charging and discharging the energy service. The VESS cost related to the capacity is modeled as [18]  
\[p_{C}(c_{0})=\alpha_{1}\exp(-\alpha_{2}\,c_{o})+\alpha_{3}[\$/\text{kWh}], \tag{1}\]  
where \(c_{0}\) [kWh] is the energy capacity of the VESS. The power capacity of the VESS is determined as \(c_{0}/2\) h [kW], assuming a maximum service time of 2 h. \(\alpha_{1}\), \(\alpha_{2}\), and \(\alpha_{3}\) are the ESS cost parameters according to the ESS characteristics. Here, \(\alpha\)1 and \(\alpha\)2 indicate the variable costs according to ESS size, such as the manufacturing cost, and \(\alpha\)3 indicate fixed cost parameters such as the operation and management cost. In (1), the unit VESS cost is exponentially reduced by increasing the VESS capacity. This is because by increasing VESS capacity, the VESS equipment and maintenance costs are reduced [28]. This is a basic advantage of economies of scale.  
Assuming the unit service price of the VESS is \(p_{V}\) [$/kWh], the benefit to the SP is measured as  
\[B_{0}(c_{0},p_{V})=p_{V}\,\sum\limits_{i\in\mathcal{I}}c_{i}-p_{C}(c_{0})\,c_{0}, \tag{2}\]  
where \(i\) is the unit index for participating in the VESS service, and \(\mathcal{I}=\{1,\cdots,i,\cdots,1\}\) is the set of participants. \(c_{i}\) is the ESS service capacity purchased by unit \(i\).  
The prior research for VESS reviewed in the Introduction section shows that the SP achieves benefit due to the price difference between \(p_{V}\) and \(p_{C}\). Moreover, in the service problem formulation, there is the constraint that the serving capacity, \(\sum\limits_{i\in\mathcal{I}}c_{i}\), must be less than or equal to the VESS capacity, \(c_{0}\). This is a measure of service stability. However, the proposed SP model considers installing a smaller VESS capacity than the serving capacity. This model expects higher benefits, but there are service risks. Therefore, in the proposed VESS service model, the benefit of the SP, including risks, can be rewritten as  
\[B_{0}^{V}(c_{0},p_{V})=p_{V}\,\sum\limits_{i\in\mathcal{I}}c_{i}-p_{C}(c_{0}) \,c_{0}-R(c_{0},c_{i}), \tag{3}\]  
where \(R(c_{0},c_{i})\) is the value at risk (VaR) according to the VESS capacity and serving capacity as described in Section 3.2. To measure the VaR, the VESS capacity and serving capacity should be expected. The VESS capacity is installed by the SP. The serving capacity depends on the service price and the stochastic characteristics of the participant. The service price is  
Figure 1: Virtual energy storage system service model for prosumers.  
determined by the SP and the stochastic characteristics can be obtained through the historic data of the participant. A more detailed description is expressed in Section 3.2.