I Introduction
Ia Background and motivation
Energy storage is becoming a crucial element to ensure the stable and efficient operation of the newgeneration of power systems. The benefits of the energy storage at the grid side have been wellrecognized (e.g., for generation backup, transmission support, voltage control, and frequency regulation) [2]. Recently, there has also been an increasing interest in leveraging energy storage for end users (e.g., by harvesting distributed generations, and cutting electrical bill) [2]. However, deploying energy storage at the enduser side also faces challenges. On one hand, the current commercial storage products for end users often have high price tags.^{1}^{1}1A Tesla Powerwall storage with a capacity of 13.5 kWh costs $6200[3]. Further, since a storage product lasts for years, it is challenging for a user to decide the storage size due to the uncertainty of future energy demand. In fact, the Tesla Powerwall only provides one or two choices of storage size for users. Both of these factors can discourage users from purchasing such storage products and enjoying the benefits. On the other hand, if many users invest in energy storage, it is possible for them to cooperate and share the benefits of storage due to complementary charge and discharge needs. The above considerations motivate us to study the following problem in the paper: what would be a good business model that promotes users’ more efficient use of energy storage?
In our work, we develop a novel business model to virtualize and allocate central energy storage resources to end users through a pricing mechanism. This is analogous to the practice of cloud service providers, who set prices for virtualized computing resources shared by end users [4]. In the power system, we can also envision that a storage aggregator invests in a central physical storage unit and then virtualizes it into separable virtual storage capacities that are sold to end users at a suitable price. Users purchase the virtual storage to reduce the energy cost.
One key advantage of our storage virtualization framework is the ability to leverage users’ complementary charge and discharge profiles. Note that the aggregator only cares about the net power flowing in and out the storage. As some users may choose to charge while others choose to discharge in the same time slot, some requests will cancel out at the aggregated level. This suggests that even if all the users are fully utilizing their virtual storage capacity, it is possible to support users’ needs by using a smaller central storage comparing with the total virtual storage capacities sold to users. Such complementary charge and discharge profiles can arise in practice due to the diverse load and renewable generation profiles of end users. Specifically, as the most promising sources of clean and sustainable energy, solar and wind energy have both been increasingly adopted by households, commercial buildings, and residential communities [5][6]. Studies in [7, 8, 9] showed that solar and wind energy exhibit diverse and locationaldependent generation profiles. Similarly, end users’ load profiles can also be significantly diverse even in a localized region [10].^{2}^{2}2We show the diversity of users’ load profiles in the online Appendix O of the technical report [11] based on the data from[10].
Another key advantage of storage virtualization is that a user can flexibly change the amount of virtual capacity to purchase over time based on his varying demand. Such flexibility is difficult to realize if the user owns physical storage by himself, and encourages the users to take advantage of the energy storage. The above key advantages can further increase users’ demand for the storage and reduce the aggregator’s investment cost, which can increase the aggregator’s profit.
To rigorously study such benefits of storage virtualization, in this paper, we consider two possible types of aggregators. The first possibility is a profitseeking aggregator. In a deregulated energy market, the profitseeking storage aggregator can decide whether or not and how much storage capacity to invest in, so as to maximize her profits. Such deregulated markets can be found in the U.S. and many European countries, and third parties are encouraged to participate in the market to provide different services for the grid and end users [12][13]. The second possibility is that the aggregator is regulated by the system operators or regulatory agents, which may have the goal of maximizing the benefit of end users subject to a nonnegative profit.
IB Main results and contributions
To the best of our knowledge, our paper is the first work that develops a pricing mechanism for the storage virtualization and sharing. In such a framework, a storage aggregator invests in a central physical storage unit and then virtualizes it into separable virtual storage capacities that are sold to end users at a suitable price. Users purchase the virtual storage to reduce the energy cost.
A new question for this storage virtualization model is how the aggregator’s investment and pricing decisions are coupled with the users’ purchase and storage operation decisions. To answer this question, we formulate a twostage optimization problem for the interactions between the aggregator and users at two different horizons: the investment horizon divided into many operational horizons. Over the investment horizon (e.g., 15 years), the aggregator determines the size of the physical storage for virtualization and the price of the virtual storage. At the beginning of each operational horizon (e.g., one day), each user determines the virtual capacity to purchase as well as the charge and discharge decision. The aggregator chooses a price of the virtual storage to balance her profit and users’ benefits. For a profitseeking aggregator, we aim to find the optimalprofit price to maximize her profit. For an aggregator that is regulated by the system operator or regulatory agents, we aim to find the lowestnonnegativeprofit price, which can give the most benefits to users while maintaining a nonnegative profit for the aggregator. We demonstrate that such a virtualization leads to more efficient use of the physical energy storage, compared with the case where each user acquires his own physical storage.
The main contributions of this paper are as follows:

Storage virtualization framework: In Section II, we develop a storage virtualization and sharing framework. To the best of our knowledge, this is the first work that develops a pricing mechanism for storage virtualization and sharing.

Pricingbased virtual capacity allocation: In Section III, we formulate a twostage optimization problem between the aggregator and users. In Stage 1, the aggregator determines the pricing and investment of the storage. In Stage 2, each user decides his purchase decision and the storage schedule. We consider two pricing strategies for the aggregator: one maximizes the aggregator’s profit while the other gives the highest benefits to users.

Thresholdbased search algorithm: In Section IV, we resolve a multioptima issue of Stage 2 by introducing a penalty on users’ charge and discharge power. As the penalty approaches zero, we characterize a stepwise structure of users’ optimal solutions, and show a piecewise linear structure of the aggregator’s optimal profit with respect to the virtual storage price. This structure then allows us to iteratively search for nearoptimal investment and pricing strategies within an arbitrary precision.

Realisticdata simulations: In Section V, we conduct the simulation using realistic load data from PG&E Corporation and meteorology data from Hong Kong Observatory. We show that our model enables the aggregator to save the physical storage investment cost by 54.3% and the users to reduce energy costs by 34.7%, compared with the case where users acquire their own physical storage.
IC Related works
There have been several studies on the deployment of energy storage at the enduser side[14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. In [14, 15, 16, 17], each user only utilizes his own energy storage units for demand management without mutual sharing, which may lead to inefficient use of the storage. In contrast, the works in [18, 19, 20] considered user sharing of a central storage without considering the investment issue of the central storage and the potential impact on the users. In [21] and [22], end users share the energy storage with a third party. However, both works allocate the physical storage capacities (instead of virtual capacities) to the users, which does not take advantage of the complementariness of users’ profiles.
The work that is most closely related to ours is [23], in which the authors proposed a business model to enable users to share the central storage. Our work differs from [23] in several crucial ways. First, in [23] the storage sizing decisions are not coordinated between the storage aggregator and users. More specifically, the aggregator needs to invest in a sufficiently large capacity to satisfy users’ needs, which is not costefficient. In contrast, in our work, the aggregator can adjust the price of virtual storage to influence the demand of storage, and effectively coordinate the benefit sharing of virtual storage between the aggregator and users. Second, the model in [23] assumes that a user’s purchased virtual storage cannot change on a daily basis. In contrast, our model allows a user to flexibly choose the amount of virtual storage to purchase every day, depending on his daily renewable generation and load demand. This additional level of flexibility further explores the potential of storage virtualization and reduces users’ cost.
Our work models the virtual storage sharing framework as a twostage optimization problem. Such multistage problems have been studied in smart grid systems (e.g.,[22, 24, 7, 25]). The work [22] built a twostage optimization problem for the sharing of a central storage unit between a distribution company and customers. The work [24] proposed a twostage model for the energy pricing and dispatch problem of the electricity retailers. Both works [22] and [24] solved the twostage problem by constructing a single optimization problem, which requires the operator to know all the users’ private information. Compared with [22] and [24] that require complete network information, our work designs a distributed algorithm based on the information exchange between users and the aggregator.
The works [7] and [25] designed distributed algorithms based on the information exchange with an aggregator to coordinate the decisions among different users or microgrids. Such distributed algorithms can be used to realize the concept of transactive energy in the smart grid, which can achieve an equilibrium by exchanging valuebased information [26]. In such a transactive energy framework, the agents of mid or smallsized energy resources can automatically negotiate with each other as well as exchange information with the main grid through advanced energy management and control system. In our work, users can actively and automatically respond to the price signal from the aggregator, which is supported by the transactive energy framework. Compared with [7] and [25], our work focuses on the pricing mechanism of the aggregator who sells the virtual storage capacities to users and seeks the profits, while the works [7] and [25] focused on the coordination between users (or microgrids) under the help of the aggregator to reach the social optima or consensus.
Ii System Overview
Figure 1 illustrates the system model, where a community of users are connected with a central storage unit and the main grid (and with each other) through power and communication infrastructures.^{3}^{3}3We assume that the grid constraints are not stringent, so that we can focus on how the aggregator sets the price of the virtual storage and how storage virtualization reduces the requirement of physical storage and the users’ costs. We will consider the grid constraints in the future work. Each user has his load demand and may also own some local renewables. An aggregator invests and operates the central storage unit. Next, we introduce the models of the users and the aggregator in more details.
Iia Users
We consider a set of users whose energy load profiles can be different. Users may own renewables of solar and wind energy. To satisfy the demand, a user can use the locally generated renewable energy, purchase energy from the main grid, or use the energy from the energy storage. Next, we first introduce the user’s electricity bill, and then discuss how storage can be used to reduce the electricity bill.
We adopt a peakbased demand charge tariff for the electricity bill. Peakbased demand charge has been widely adopted for commercial and industrial consumers in order to reduce the system peak and recover grid costs. Thanks to the increasingly more advanced metering infrastructure, such a demand charge scheme has also been offered to residential customers by some utilities in the United States [27] [28]. For a billing cycle of time slots, if user consumes electricity from the grid in time slot , his electricity bill[29] in is calculated by:
(1) 
where is the unit energy price and is the unit price for peak consumption in the billing cycle. To reduce users’ peak demand, the utility usually sets much higher than [28].
Demand charge tariff in (1) provides a strong incentive for users to utilize energy storage to shave their peak loads. Specifically, users can proactively charge their storage using energy from the grid, and discharge to meet the peak load so as to reduce the electricity bill. Furthermore, if a user owns the renewables, he can store excessive renewable energy in the storage for later use. We assume that users can sell back renewable energy to the grid and the unit feedin price satisfies , such that users prefer to first use the locally generated renewable energy to serve their loads rather than to directly sell to the grid.^{4}^{4}4It is common that the renewable feedin tariff is lower than the consumption tariff, for example, in Germany and some states of the U.S.[30].
IiB Storage aggregator
The aggregator invests and operates the central physical storage. She virtualizes the physical storage into separable virtual capacities and sells them to users. Since users can’t control the central storage directly, they report their charge and discharge decisions to the aggregator, and the aggregator dispatches the central storage on behalf of users accordingly. Further, the aggregator can coordinate users’ charge and discharge decisions by setting the price of virtual storage, which will ensure that users’ charge and discharge decisions are well accommodated by the physical storage.
We assume that there is a billing arrangement among the utility, the storage aggregator and users such that when the utility calculates users’ electricity bill, it will count both the physical load and the virtual storage charge/discharge. Thus, even though users don’t own and operate their physical storage, users can use the virtual storage to achieve a peak load reduction and reduce the electricity bill.
As we have discussed in Section I, our storage virtualization model can lead to a more efficient use of physical storage due to two reasons: (i) the complementarity of different users’ charge and discharge decisions, and (ii) the flexibility in purchasing different amounts of virtual capacities on different days. The aggregator’s investment in the physical storage will take advantage of these aspects while satisfying users’ demand. However, occasionally there can be very high aggregate demand from users. Satisfying such demand with a fixed physical storage investment will lead to low efficiency due to either overinvestment or overpricing. Thus, we further generalize our model by allowing the aggregator to use additional energy resources other than the physical storage to meet users’ demand. For example, the aggregator can contract with other generators (or consumers) to purchase additional energy (or sell surplus energy) to serve users’ demand.^{5}^{5}5Such a generalization can be supported by the works [31] and [32], which propose a control framework for the general energy storage system by aggregating other energy resources, e.g., demand responses in addition to the physical energy storage.
Iii Twostage Formulation
Figure 2 illustrates two timescales of decision making in our model. Figure 3 illustrates a twostage problem for the interactions between the aggregator and users. In Stage 1, at the beginning of an investment horizon of days (e.g., corresponding to many years), the aggregator determines the size of the physical storage and the unit price of the virtual storage. The investment horizon is divided into many operational horizons, i.e., each corresponds to one operational horizon, which is further divided into many time slots (e.g., 24 time slots corresponding to 24 hours). In Stage 2, at the beginning of each operational horizon, given the unit price of virtual storage, each user decides the optimal capacity to purchase and the corresponding charge and discharge profiles over the operational horizon, based on the prediction of their loads and renewable generations.^{6}^{6}6Since the focus of our work is on the design of the virtual storage sharing framework, we have initially chosen to assume that users can perfectly predict their renewable generations and loads. We include the discussions about the impact of uncertainties on the users’ decisions in the online Appendix N [11]. Then, the aggregator operates the physical storage by aggregating all the users’ charge and discharge decisions. Note that we consider a daily operation of virtual storage sharing as well as the daily demand charge tariff for users’ electricity bills, because users’ electricity loads reflect their activities which are often periodic on a daily basis (see, e.g., extensive studies in[33] and[34]). Furthermore, users’ load profiles can differ from one day to another (e.g., the differences between weekdays and weekends). The operation and billing cycle on a daily basis can leverage the diversity in users’ loads and provide flexibility to users, such that users can purchase a different amount of virtual storage on different days to minimize their costs.
In order to solve the aggregator’s investment and pricing problem over the investment horizon, the aggregator needs to incorporate users’ responses across different operational horizons in the entire investment horizon. Since users’ responses depend on different operational conditions (e.g., local renewable generations and loads), we use historical data to build a set of scenarios
that empirically models the joint distribution of all users’ load and renewable generation profiles. For each operational horizon
, scenariooccurs with a probability
. In scenario , we denote user ’s load profile as and his renewable profile as . Each user will report a set of the threshold decisions (explained in detail later in Algorithm 1) in each operational scenario to the aggregator. Based on users’ reported information, the aggregator makes the investment and pricing decisions over the investment horizon by considering the expected profit over the scenarios.Furthermore, note that the aggregator’s decision and users’ decisions in two stages are coupled. On the one hand, the aggregator’s virtual storage pricing will affect the users’ decisions of virtual storage, and the aggregator’s invested physical storage size will constraint the aggregated charge and discharge decisions of users. On the other hand, the aggregated charge and discharge decisions of users will determine the aggregator’s operation of the physical storage. Such a coupled twostage problem needs to be solved through backward induction. Thus, in the next two subsections, we will first explain users’ model in Stage 2 and then explain the aggregator’s model in Stage 1.
Iiia Stage 2: User’s model
IiiA1 User’s power scheduling in each operational horizon
Given the price of the virtual capacity, user decides the virtual capacity and the corresponding power scheduling (as illustrated in Figure 4). We then explain the power scheduling in time slot . Assume that user has locally generated renewable energy . He decides the amount of selfused renewable energy that will serve his load or charge into his virtual storage. He sells back the remaining renewable energy to the grid.^{7}^{7}7We assume that users only feed in the unused locallygenerated renewable energy to the grid. We do not consider the feedin power from the storage to the grid, which may complicate users’ decisions. User can purchase the amount of energy from the grid. Part of energy may serve his own load, and the remaining part will be charged into his virtual storage.^{8}^{8}8We assume that users can have an extra meter and wires to connect the renewable generators to the grid. Hence, it is technically feasible for a user to sell back the renewable energy to the grid while consuming energy from the grid. Finally, user can discharge the amount of energy from his virtual storage to serve his load. To balance the power, we can express user ’s energy purchase from the grid in time slot as follows:
(2) 
If a user has no renewables, the charged energy is only from the purchase from the grid, i.e., in (2). We denote , , , and .
User ’s charge and discharge decision should satisfy the constraint of the virtual capacity:
(3)  
(4)  
(5) 
We let denote the energy level in the storage over the operational horizon, where and denotes the initial energy level. Since the user’s storage is virtual, we allow user to optimize in each operational horizon. We let and denote the virtual charge and discharge efficiency rate respectively. We assume that the aggregator enforces the same virtual efficiency rate as the physical one. We model the charge and discharge efficiencies for the physical storage (e.g., Liion batteries) as constant values. Such an assumption has been widely used in the literature (e.g., [2][14]) and can capture the key characteristics of energy loss during the charging and discharging process. Constraint (5) ensures the independent operation of the virtual storage across operational horizons [35]. Other powerrelated variables are constrained as follows:
(6)  
(7) 
IiiA2 User’s net cost in each operational horizon
Each user minimizes his net cost in each operational horizon. The cost includes the payment for the virtual capacity and the electricity bill. The revenue is from selling back the renewable energy.
Specifically, given the virtual storage price , over the operational horizon of scenario , user ’s payment to the aggregator for purchasing the capacity is The electricity bill for the consumption from the grid is
(8) 
and we can substitute the variable from (2) and denote the electricity bill as . User ’s revenue of selling back renewable energy is
(9) 
Thus, the net cost in the operational horizon of scenario is
(10) 
We then formulate user ’s Problem that minimizes the net cost in the operational horizon of scenario as follows.
Stage 2: User optimization problem
s.t.  
var: 
where the price is determined by the aggregator in Stage 1. We denote the optimal solution to Problem by . Note that the aggregator sets the same price for all users without any price discrimination, and each user makes his own purchase decision to minimize his cost by solving Problem . Therefore, if the virtual storage can bring higher revenues to some users (e.g., those who have more renewable energy), then these users have higher demands and purchase more virtual capacities (than the users who benefit less using the storage). In this sense, our business model is fair for all users.
IiiB Stage 1: Aggregator’s model
In Stage 1, at the investment phase, the aggregator decides the unit price of the virtual capacity, as well as the capacity and power rating of the physical storage.
IiiB1 Aggregator’s power scheduling over each operational horizon
Each user decides the charge and discharge profiles (as in Stage 2) and then reports them to the aggregator. The aggregator aggregates these charge and discharge decisions and obtains the net charge and discharge as follows:
(11)  
(12) 
, where we define . Note that (11) and (12) ensure that and cannot be positive at the same time, i.e., the physical storage cannot be charged and discharged simultaneously.
The aggregator can use the physical storage and additional resources to satisfy users’ requirement. We denote the charge and discharge requirement served by the physical storage as , . Similarly, we denote the charge discharge and requirement supported by the additional resources as , . They satisfy the following constraints for users’ demand:
(13)  
(14) 
The aggregator’s charge and discharge scheduling is constrained by the physical storage size as follows:
(15)  
(16)  
(17)  
(18) 
We let denote the energy level in the physical storage. We let and denote the charge and discharge efficiency rate respectively. The fraction coefficients and correspond to the minimum and maximum energy levels that can be stored in the storage, respectively. Constraint (18) ensures the independent operation of storage in each operational horizon.^{9}^{9}9We assume that the aggregator can adjust the initial energy level of the storage by purchasing or selling the energy with the same price through an external market. Since the storage operational constraint in (18) restricts the terminal level to be equal to the initial level, the total adjustment and its cost is negligible in the long run. In addition, the case where the initial energy level is fixed can be viewed as a special case of our problem.
IiiB2 Aggregator’s profit over the investment phase (scaled in one operational horizon)
Over the investment phase, the aggregator bears the storage capital cost which includes the capacity cost and power rating cost [2]. In each operational horizon, the aggregator obtains the revenue from selling the virtual capacity but also bears the operational cost of the physical storage and additional resources.
Specifically, the capital cost over the investment phase (scaled into one operational horizon) is:
(19) 
where the coefficient and are the unit cost of the capacity and power rating over the investment phase, respectively.^{10}^{10}10Here we consider a single type of storage technology for the central storage. We can easily generalize this model to incorporate multiple types. The coefficient is the scaling factor that is illustrated in the online Appendix L [11].
The expectation of the revenue of selling the virtual storage capacity over all scenarios is
(20) 
The expectation of the storage operational cost over all scenarios is
(21) 
where is the unit cost of the charge and discharge, which models the cost of degradation of the storage. We adopt the linear cost model that is widely used in the literature[35][36].
The expected operational cost of additional resources over all scenarios is
(22) 
where is the unit cost of absorbing users’ charge demand and is the unit cost of acquiring the energy to support users’ discharge demand.
Thus, the aggregator’s expected profit over the investment phase, scaled into one operational horizon, is
(23) 
The aggregator’s power scheduling is determined by price and investment decisions and . Thus we denote the operational cost of storage and additional resources as functions of respectively, i.e., and .
IiiB3 Aggregator’s pricing
We consider two pricing strategies for the aggregator: the optimalprofit price (OP price) and the lowestnonnegativeprofit price (LNP price). The OP price maximizes the aggregator’s profit, and the LNP price is the minimal price that keeps the aggregator’s profit nonnegative. The latter is reasonable when the aggregator is regulated and required to provide the most benefit for users. We formulate Problem AOP to obtain the OP price as follows. Due to the page limit, we enclose all the discussions about the problem of the LNP price in the online Appendix J[11].
Stage 1: Aggregator’s Optimalprofit Price Problem (AOP)
s.t.  
var: 
Iv Solving Twostage Problem
The twostage problem is challenging to solve due to its nonconvex nature. To solve the problem, we first characterize the properties of each user’s optimal solution (in Stage 2) under a fixed price in Proposition 1 and Theorem 1, and then incorporate users’ decisions into Stage 1 to characterize the properties of the aggregator’s profit in Proposition 3. Based on the properties of Stage 2 (in Proposition 1 and Theorem 1) and Stage 1 (in Proposition 3), we propose Algorithm 1 to solve the twostage problem, which determines the aggregator’s optimal pricing and investment decisions.
Iva Solution of Stage 2
For Stage 2, we first prove that the optimal capacity that user purchases is stepwise in price . Then we add a small penalty on the user’s cost to solve the issue of multioptima of the charge and discharge decision. We show that the user’s optimal decision has a simple stepwise structure over price as the penalty approaches zero.
IvA1 Stepwise structure of
The optimal capacity is stepwise over price as shown in Proposition 1:
Proposition 1 (Stepwise property of virtual capacity).
The optimal capacity of Problem is a nonincreasing and stepwise correspondence of the price . Specifically, there exists the set of threshold prices and , such that is given by
(24) 
where . For any threshold price , can be any value in . For , can achieve any value in . We denote user ’s optimal capacity set as .
We illustrate Proposition 1 in Figure 5(a). As price increases, the optimal capacity that a user purchases decreases. If the price is higher than the threshold , the user will purchase none. Between two adjacent threshold prices, the optimal capacity remains the same. In the online Appendix C [11], we prove Proposition 1. In the Appendix D [11], we present Algorithm 3 for computing the sets and .
IvA2 Solving multioptima problem
Although we have characterized a user ’s optimal capacity decision in Proposition 1, we still face the difficulty of multioptima. More specifically, even at a fixed optimal capacity , there may still be multiple virtual charge and discharge solutions (the set of which is denoted as ) that lead to the same net cost to the user. As a result, the aggregator’s cost is not welldefined, because the cost is due to users’ charge and discharge demand. To address this difficulty, we introduce a small positive penalty coefficient on the user’s net cost in Problem as follows:
(25) 
After including penalty (25) to the objective function of Problem , we obtain a new modified problem denoted as , which is a quadratic programming problem. We denote the optimal solution to Problem as for any price and penalty . Proposition 2 below shows such a solution of Problem is unique, which is proved in Appendix E[11]. To ensure users’ unique decisions, we let the aggregator choose and .^{11}^{11}11Note that when , even when we set , users can purchase an arbitrarily large capacity beyond their minimum needs because they bear no capacity costs.
Proposition 2 (Uniqueness).
For any and any , the optimal solution to Problem is unique.
IvA3 Asymptotic solution
Later in Section B3, we will choose a small to obtain a nearoptimal solution for Stage 1. Intuitively, when approaches zero, one would expect that the solution to Problem approaches the solution to Problem . Since the optimal charge and discharge decision of Problem has multiple optima, it would seem that the same difficulty will persist as approaches zero. Surprisingly, we show below that the limit of the solution to Problem exists and can be uniquely determined as approaches zero.
Theorem 1 (Asymptotic solution).
For any price (as defined in Proposition 1), as approaches zero,
(a) there is a unique limit of the optimal charge and discharge decision . This limit belongs to the set , i.e.,
and it remains constant for all prices within each threshold price interval , where ;
(b) the optimal capacity approaches the optimal solution (given in Proposition 1) with , i.e.,
Theorem 1(a) is highly nontrivial due to multiple optimal solutions of Problem . We prove Theorem 1(a) based on the Maximum Theorem [37] by showing the uniqueness of the optimal objective value of Problem , and prove Theorem 1(b) by showing the uniqueness of . We present the detailed proof in Appendix G[11].
Based on Theorem 1(b), we can compute the limit by the optimal solution of Problem , which takes discrete values from the set defined in Proposition 1. Then, given each element in , we can solve optimization problems (by Algorithm 4 of Appendix H[11]) that minimize the penalty term to obtain the limiting optimal charge and discharge decision as approaches zero. In other words, the limiting optimal charge and discharge solution is a function of each element in . The above results show that as approaches zero, the user’s solution in Stage 2 has a stepwise structure, which can be efficiently computed.
IvB Solution of Stage 1
Based on the asymptotic structure of the users’ decisions in stage 2 as shown in Theorem 1, we further analyze the structure of the aggregator’s optimal profit as a function of price as approaches zero. We propose an algorithm to derive the nearoptimal profit price by considering that is small enough. We use a similar method to compute the LNP price and present it in Appendix J[11].
IvB1 The optimal profit given
Given , the aggregator can vary the investment and to optimize her profit as follows. First, the aggregator can compute the revenue as in (20) with replaced by . Second, given users’ optimal charge and discharge decision , the aggregator can solve Problem CO to compute her optimal cost as follows.
s.t.  
var: 
This is a linear programming problem, which can be solved efficiently using the simplex method
[38]. Finally, we compute the optimal profit for any given by(26) 
IvB2 The limit of the optimal profit as approaches zero
In Proposition 3 below, as approaches zero, we first show in part (a) that the revenue approaches a limit as in (20) without . We further show in (b) that the optimal cost approaches a limit , which is stepwise over the threshold price set . Finally, we show in (c) that the optimal profit approaches denoted as as approaches zero. We present the proof of Proposition 3 in Appendix I[11].
Proposition 3 (Asymptotic profit).
For any , we have
where denotes the optimal value of Problem CO given the limit of each user ’s optimal charge and discharge decision ,
Based on Proposition 1, the revenue is piecewise linear over the threshold price set as depicted in Figure 5(b): The revenue increases linearly from each threshold price and then decreases vertically at the next adjacent threshold price. The slopes of over different price intervals are determined by . The limiting cost is a stepwise function over the threshold price set as shown in Figure 6. Hence the limiting profit is a piecewise linear function of price as shown in Figure 6 with multiple local optima.^{12}^{12}12Although in Proposition 3 these limiting values are only defined for , we can use the lefthanded limits of as the function values for . In this way, the function is welldefined for all . Finally, based on each user ’s limiting decision and in Theorem 1, we can compute the limiting revenue , the limiting cost , and thus the limiting profit as in Lines 28 in Algorithm 1 below.
IvB3 The algorithm to compute the nearOP price
The above results demonstrate important solution structures of Stage 2 and Stage 1 as goes to zero. Unfortunately, the aggregator cannot choose (which has the multioptima problem as mentioned in Section A). Thus, below we propose an iterative procedure in Algorithm 1 (i.e., Lines 916) for computing a sufficiently small so that (i) it is close enough to the above limit, and (ii) the nearOP price can be efficiently computed. Specifically, first, the aggregator computes the lefthanded limit of profit at each threshold price , and selects the price that achieves the global optimum (Line 9). Second, since profit is linearly increasing over each threshold price interval, the aggregator chooses a nearOP price slightly lower than such that the profit approximates the maximum profit within the given accuracy (Lines 1011). Finally, the aggregator chooses a small in an iterative fashion so that the profit approximates within the given accuracy (Lines 1216), where each user solves the quadratic programming problem in each iteration.^{13}^{13}13The quadratic programming problem can be efficiently solved by the interior point method[39]. After computing the nearOP price and , we can obtain the nearoptimal investment decision by Subroutine 1 accordingly.
Algorithm 1 is scalable in terms of the number of users and scenarios. On the users’ side, they compute their decisions in parallel, which will not increase the execution time as the number of users increases. On the aggregator’s side, she searches the threshold price set for the maximum profit, where the number of threshold prices is simply linear in the product of the number of users and the number of scenarios. Therefore, the execution time of Algorithm 1 is proportional to the number of users and scenarios, and thus scales well.
V Numerical Study
We simulate a system based on realistic data where users have diverse load and renewable profiles. We demonstrate that, as long as there are groups of users with diverse load and renewable profiles, our virtualization model can promote more efficient use of the physical storage, offer flexibility to users in the choices of virtual capacities, and reduce users’ costs compared with the case where users acquire their own physical storage.
Va Simulation setup
VA1 Parameters
We consider the lithiumion battery as the energy storage technology. We use realistic load data from PG&E Corporation in 2012 [40] to characterize users’ load profiles, and we use wind speed and solar radiation data from Hong Kong Observatory [41] to characterize users’ renewable generations. To illustrate how our storage virtualization model works, we simulate a system as in Figure 1 with three users of different types. Type1 user owns the local wind generation, while Type2 and Type3 users own solar panels (with the same capacity).^{14}^{14}14Though the installation of wind turbines faces geographical restrictions, technological innovations (such as vertical axis wind turbines) have promoted the adoption of wind energy for households, commercial building, and residential communities [6][42][43]. Therefore, to capture a more complete picture of renewable energy deployment in practice, we consider both solar energy and wind energy in our simulation. We also conduct the simulations where Type1 user has no renewable energy while Type2 and Type3 users have solar energy (with details in Appendix M[11]), which shows that our framework still works well for significantly reducing the invested physical storage capacity and the cost of users with solar energy.
We obtain the oneyear historical data, namely 366 scenarios for users’ load and renewable profile jointly. As a large number of scenarios lead to a high computational complexity, we choose a smaller subset of 7 typical scenarios that can approximate the original scenario set as shown in Figure 7: Type1 user has the peak load around noon (typical for some commercial users) and produces more wind power at night; Type2 and Type3 users have the peak load in the morning and evening (typical for residential users), and their solar power reaches peak supply around noon. For more details regarding the data of renewable generation and load, we include them in the online dataset [44]. For other parameters of electricity bill and energy storage, we present them in Appendix K[11]. Note that we do not make any assumptions on the correlation between users’ load profiles and renewable generations; we just select an example with three types of users for illustration. Furthermore, since our framework for virtual energy storage sharing is general, interested readers can also use their own data as inputs to examine the performance of the framework.
VA2 Benchmark
To show the performance of our virtualization model, we consider a benchmark where each user invests in a physical storage product (e.g., Tesla Powerwall) by himself. He will optimize the (fixed) capacity of the physical storage, and can only use his own storage over the investment phase. Apart from the capacity cost, each user also bears the power rating cost and the operational cost by himself. Each user ’ solves an optimization problem
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