## 1 Introduction  
As the distributed power generation is growing, owing to the advent of renewable energy generation, power system operation is becoming challenging because the frequency variation and duration of disruptions that indicate power system reliability are increasing [1]. An energy storage system (ESS) is an essential system to ensure the continuity of power or energy to the customers [2]. ESSs are used in different ways, from utility-scale applications, including black start [3], voltage regulation [4], frequency regulation [5], grid fluctuation suppression [6], and spinning reserve [7], to demand-scale applications, including demand flexibility management [8] and energy bill reduction [9]. To meet the net zero emissions objective of the International Energy Agency (IEA) by 2050, the share of energy system flexibility from ESS is expected to increase from 0.2% in 2020 to 6% in 2030 [10]. However, the high cost of ESS acts as a barrier to its usage, although the cost is reducing today due to technological developments [11]. Recently, virtual ESSs (VESSs) have been introduced, and their concept is to allow logical sharing of a physical ESS by multiple participants [12]. A VESS is an energy sharing method that is being considered in energy systems. Another representative energy sharing method is the peer-to-peer (P2P) energy transaction [13], which involves trading excess energy produced between peers to improve benefits. However, similar to the cloud data storage services [14], a VESS is a shared pool of energy storage resources that provides storage services to small participants [15]. Installation of a VESS, which offers more capacity than individual ESS installations, achieves economies of scale. It allows participants to use the ESS service at a lower price compared with that of installing and using an individual ESS. In a study, a business model was presented based on the shared ESS considering the business cases in 2016 and 2025 [16]. The results indicate that ESS sharing leads to additional profits. VESS structures have been proposed considering different scenarios, such as heat storage capability of a building [17], flexible demand and flywheel ESS [18], residential demand with air conditioners [19], and electric vehicles combined with air conditioning load [20]. In particular, the main topic in the area of VESS research is the energy management of the VESS by applying methodologies such as optimization [21; 22] and game theory [23; 24; 25]. [PERSON] et al. [21] proposed a pricing-based virtual storage sharing scheme formulating a two-stage optimization problem for the interaction between the aggregator and participants. They highlighted total cost reduction for the participants by using VESS. [PERSON] et al. [22] developed a real-time energy management system for an energy storage sharing system to minimize the time average system cost. Their method was based on the current system states, without having to predict the future uncertain system states. [PERSON] et al. [23] proposed a combinatorial auction mechanism to obtain the desired ESS capacity using a VESS. To determine the solution of the auction, an algorithm that combines a genetic algorithm with particle swarm optimization was used. [PERSON] et al. [24] designed a modified auction-based mechanism to determine the cost of usage and allocation of VESS capacity, and they applied a non-cooperative Stackelberg game between the participants and auctioneer to determine the solution. [PERSON] et al. [25] formulated an ESS sharing problem as a cooperative game having two cases, (a) cost minimization with already procured storage elements and (b) expected cost minimization for joint storage investment. They showed that ESS sharing in a cooperative manner is an effective way to amortize storage costs and increase the system's utilization. All these studies show that VESS is an effective way to lower the cost barrier of ESS usage, and participants can achieve profits by way of savings on electricity bills. Research on VESS is summarized in Table 1.