Optimal Community Energy Storage System Operation in a Multi-Power Consumer System: A Stackelberg Game Theory Approach

Abstract

The proliferation of community energy storage systems (CESSs) necessitates effective energy management to address financial concerns. This paper presents an efficient energy management scheme for heterogeneous power consumers by analyzing various cost factors relevant to the power system. We propose an authority transaction model based on a multi-leader multi-follower Stackelberg game, demonstrating the existence of a unique Stackelberg equilibrium to determine optimal bidding prices and allocate authority transactions. Our model shows that implementing a CESS can reduce total electricity costs by 16% compared to the conventional case that does not account for authority transactions among CESS users, highlighting its effectiveness in practical power systems.

1. Introduction

With the introduction of distributed power resources driving many changes over the past several decades, the energy industry has been undergoing continuous transformation [1]. Since traditional fossil fuels generate toxic emissions and cause environmental harm, numerous efforts have been made to reduce their usage by incorporating renewable energy sources. However, the instability of power generation has emerged as a significant challenge in renewable energy supply [2,3,4]. To address this issue, system operators often struggle to secure additional reserve power. Given the limitations of operating a supplier-centered power system, such as the high costs and inefficiencies associated with constructing additional power facilities, research on ancillary services and controllable power sources is actively being conducted to enhance power reliability [5].

Compared to conventional centralized power systems, the development of distributed energy generation and energy storage technologies has made it necessary to control power at the distribution level [6,7,8]. Additionally, as local communities increasingly rely on renewable energy, flexible energy management and load balancing techniques have become essential. Furthermore, the introduction of an energy storage system (ESS) helps coordinate energy production and consumption, thereby improving power reliability and meeting user needs [9,10,11]. Nevertheless, due to the high cost of energy storage devices, various studies on energy management systems are being conducted to explore system control while considering storage costs, rather than solely focusing on managing the variability of renewable energy [11,12,13].

To address these economic and operational limitations, several approaches have been developed to maximize the profitability of facilities [14,15,16,17]. For example, Zhang et al. [14] proposed a simplified nonlinear battery cycle aging cost model using an actual dataset, demonstrating that this approach could minimize overall facility costs. Another study by Maheshwari et al. [15] mathematically expressed the degradation ratio cost of batteries, developing a scheduling model suited to the time constraints of the market model using experimental aging data from commercial batteries. Despite these advances, the construction of an ESS still incurs high costs. Moreover, previous studies have primarily focused on battery aging costs without considering other operational costs, such as maintenance, operation, and environmental factors. Consequently, studies using recycled batteries are being actively conducted to achieve further cost reductions. Ziyou et al. [16] utilized a dynamic aging model of a lithium-ion battery to compare the benefits of recycled batteries versus new ones. The authors evaluated the impact of installing a recycled battery on a wind farm to address scheduling issues. However, the conventional operation models are often context-specific, meaning they may not be generalizable across different environments to guarantee financial benefits in all situations. Additionally, the cost of building an ESS remains prohibitive for many users.

To solve the limitations of the single use of an ESS, the community energy storage system (CESS) emerged as a solution to minimize overall costs and stabilize power supply. Built through the cooperation of local communities, a CESS allows users to share energy storage, thereby increasing energy efficiency in facilities. Consequently, studies on CESS operation have been conducted from various perspectives [18,19,20,21]. Hernan et al. [18] introduced an approach to integrate CESSs into low- and medium-voltage grids to provide ancillary services that improve energy quality for end users. Although their study presented an innovative energy operation algorithm to stabilize the system, it did not consider aspects such as variability in renewable energy and load prediction. To address these challenges, Shen et al. [19] proposed a hybrid energy storage system using a two-stage stochastic planning model, demonstrating that operational flexibility could be enhanced with minimal investment. Next, Tom et al. [20] explored different CESS ownership scenarios and demonstrated the economic and environmental performance of each using their proposed energy management system. Despite this progress, limited research has focused on the equitable allocation of CESS resources among multiple users. When specific facilities use CESS excessively, other users may lose access to their fair share, and a third-party ownership model that allocates operating ranges can further decrease the efficiency of the storage system.

To address these challenges, we propose an environment in which users of multiple facilities share a community energy storage system (CESS) to maximize their profits by establishing a solution for trading the rights to use CESS (RUC) among themselves. We introduce a capacity transaction mechanism that enables users to trade their CESS operating authority based on a day-ahead analysis of their utility. This game-theoretic framework is built on Stackelberg game theory, which has been successfully applied to resource allocation problems in energy systems [22,23,24,25]. For instance, Lee et al. [22] applied a Stackelberg game model to distributed energy trading among microgrids, achieving a unique equilibrium that maximized economic benefits for all participants. In another study, Reka et al. [23] used game-theoretic methods to optimize real-time demand response in smart grids, improving energy efficiency and system reliability. Building on these studies, our approach introduces a solution for trading the rights to use a CESS through a capacity transaction mechanism based on Stackelberg game theory. This allows users to trade their CESS operating authority through day-ahead utility analysis, thereby preventing penalties from excessive power consumption. Moreover, unlike existing methods that often assume centralized control or homogeneous user characteristics, this approach operates in a decentralized framework and accommodates heterogeneous users, enhancing scalability and applicability to real-world settings. Additionally, users can generate profits by selling surplus RUC, enhancing the overall operational efficiency of the system. Overall, the contributions of this paper can be summarized as follows:

• Conventional studies focus on optimizing CESS usage to maximize community profits. In contrast, this paper introduces a solution not only for efficient CESS operation but also for trading RUC among users. This approach allows users to avoid penalties from excessive power consumption and generate profits by selling surplus RUC.
• The proposed RUC trading scheme is formulated based on the Stackelberg game model, where sellers are considered leaders and buyers are followers. By demonstrating the existence of a unique Nash equilibrium, we prove that it is possible to maximize the profits of all participants.
• The proposed CESS operating scheme considers heterogeneous power consumers, each with distinct power consumption patterns. We mathematically model the revenue structure in alignment with actual electricity pricing environments, enabling reasonable buy-and-sell authority among users. The effectiveness of this model is demonstrated through the analysis of profits based on real electricity tariffs.
• By analyzing the revenue of users with diverse power consumption and profit structures, we prove the potential of integrating CESSs into actual power systems. We also demonstrate the practicality of the proposed scheme through an analysis of individual users’ profit structures and power reliability in the system.

The rest of this paper is organized as follows. Section 2 introduces the proposed RUC trading model along with the details of the utility function for power consumers in the system. In Section 3, we present an approach for RUC transactions using heterogeneous Stackelberg game theory. Section 4 analyzes the performance of the proposed CESS operating scheme. Finally, Section 5 provides the conclusion of this paper.

2. System Model

In this paper, we address a scenario where several facilities share a community energy storage system (CESS), as illustrated in Figure 1. Each facility participating in the market has a pre-allocated capacity to use a CESS, referred to as Right to Use CESS (RUC) in this paper. Since the proposed model focuses on CESS sharing among adjacent power consumers, we assume an environment with a relatively short grid length and do not consider distribution loss. With the pre-allocated CESS capacity, users calculate their operating schedules a day in advance and determine the transaction quantity of their RUC. Buyers determine their bidding price based on the potential peak penalty that may be incurred if the daily peak value exceeds a specified threshold. Sellers are participants who own additional RUC and determine the transaction capacity based on their own profit considerations. Since the buyers’ strategies are influenced by the sellers’ decisions (i.e., the total trading capacity available in the market), we formulated the model using Stackelberg game theory [22,26].

2.1. First Stage: Day-Ahead Energy Scheduling

As depicted in Figure 2, the proposed CESS operation technique is divided into two main stages. In the first stage, users analyze the power consumption patterns of their facilities to determine whether they need to purchase additional RUC in the market. In the second stage, each participant aims to maximize their profit by trading RUC based on the strategy associated with their role. With the newly determined RUC value, users can schedule CESS operations in a manner that minimizes electricity costs.

Initially, the objective is to achieve the optimal charging/discharging ratio and determine the requirement for additional RUC. This process is repeated daily to ensure consistent system operation. We denote $P_{d,i}\left(t\right)$ as the estimated load profile for user i at time slot t, with T representing the total number of time slots in a day. Here, users are charged electricity costs based on the pricing announced by the main grid, which includes time-of-use pricing, $C_{ToU}$ (i.e., the electricity cost based on power consumption per hour), and baseline cost, $P_{CPP}$ (i.e., the penalty cost incurred when peak power usage exceeds the contracted peak value). These parameters are constant values set according to data provided by the power utility in South Korea [27,28]. Given this environment, we can mathematically formulate the energy scheduling as follows:

$$\begin{array}{cc}\hfill & \underset{x\in \mathcal{F}}{min}(\sum _{t=1}^T{C}_{ToU}\left(t\right)(P_{d,i}\left(t\right)+\eta r{x}_i\left(t\right))+\sum _{t=1}^T(\xi x_i^2\left(t\right)+\kappa )\hfill \\ \hfill & +P_{CPP}·ma{x}_{t\in \mathcal{T}}(P_{d,i}\left(t\right)+\eta r{x}_i\left(t\right)-P_{th,i},0)),\hfill \end{array}$$

(1)

where $\mathcal{F}$ represents the feasible set of x, which is the decision variable in the problem. Here, $x_i>0$ indicates battery charging, while $x_i<0$ indicates battery discharging. Furthermore, $\eta$ represents the battery efficiency of the CESS, and r is the maximum operating value of the battery. The first term of the formula represents the electricity cost that user i incurs through the use of the CESS. The second term represents the battery wear-out cost borne by user i due to CESS usage [29]. Here, $\xi (\xi >0)$ and $\kappa (\kappa >0)$ refer to operational costs associated with CESS operation. Specifically, $\xi$ is a slope coefficient determined by the type of battery and $\kappa$ is a fixed cost linked to the initial energy level of the battery. The last term of the formula refers to the penalty cost incurred when the peak power of consumer i exceeds the predetermined threshold $P_{th,i}$. This term is included to enhance the stability of the power system by discouraging sudden spikes in user power consumption [28].

To implement CESSs in real-world environments, it is crucial to ensure that the physical characteristics of the battery and the operating range for users, based on their allocated capacity, are met. This requirement leads to the following constraint:

$$\begin{array}{c}\hfill B_{i,min}\le B_{i,init}+\sum _{k=1}^t{r}_i{x}_i\left(t\right)\le B_{i,max},\forall t\in \mathcal{T},\end{array}$$

(2)

where $B_{i,min}$ and $B_{i,max}$ define the operating range of the battery. Here, $B_{i,init}$ represents the initial state of the battery, and $B_{i,init}+{\sum }_{t=1}^T{r}_i{x}_i\left(t\right)$ represents the state of charge (SoC) of the battery at time T. Therefore, Equation (2) ensures that the SoC for user i remains within a specified range at all times. Moreover, to enable the sale of authority, the SoC level at the start of each day must return to its initial value. This requirement imposes an additional constraint that the total sum of the charging and discharging activities should be zero by the end of the day:

$$\begin{array}{c}\hfill \sum _{k=1}^T{x}_i\left(k\right)=0.\end{array}$$

(3)

Lastly, the target market prohibits the sale of charged energy in the battery to others. Therefore, we should impose the following constraint:

$$\begin{array}{c}\hfill P_{d,i}\left(t\right)+\eta r_i{x}_i\left(t\right)\ge 0\forall t\in \mathcal{T}.\end{array}$$

(4)

In such scenarios, if the peak power in the facility exceeds the penalty threshold, expressed as $P_{d,i}\left(t\right)+\eta r_i{x}_i\left(t\right)-P_{th,i}\ge 0$, the user is classified as a buyer who needs to acquire additional usage rights to reduce their peak value. Conversely, if the peak power is below the penalty threshold, expressed as $P_{d,i}\left(t\right)+\eta r_i{x}_i\left(t\right)-P_{th,i}\le 0$, and the operating range of the battery is within its maximum capacity, $ma{x}_{t\in \mathcal{T}}(B_{i,0}+{\sum }_{k=1}^t{r}_i{x}_i\left(k\right))\le B_{i,max}$, the user is classified as a seller.

2.2. Second Stage: Authority Trading

With the given electricity cost, participants are divided into buyers and sellers. We define $\mathcal{I}$ as the set of buyers and $\mathcal{J}$ as the set of sellers, where these are disjoint sets.

For each $j\in \mathcal{J}$, let ${\widehat{B}}_j$ represent the RUC value that seller j prefers to sell in the market. Let ${\omega }_j$ be the portion of ${\widehat{B}}_j$ that seller j decides to sell, where ${\omega }_j$ ranges between $[0,1]$. Thus, the amount of RUC sold by seller j is given by ${\widehat{B}}_j{\omega }_j$, while the amount of energy retained by seller j is ${\widehat{B}}_j(1-{\omega }_j)$. Here, $\omega ={\left(\omega \right)}_{j\in \mathcal{J}}$ represents the set of strategies for all sellers. Let ${\omega }_{-j}$ denote the profile of strategies for all sellers except seller j. Therefore, we have $\omega =({\omega }_j,{\omega }_{-j})$. Since ${\widehat{B}}_j{\omega }_j$ is the RUC value that seller j sells in the market, the total amount of RUC available in the market can be defined as

$$\begin{array}{c}\hfill \widehat{B}=\sum _{j\in \mathcal{J}}{\widehat{B}}_j{\omega }_j.\end{array}$$

(5)

For each $i\in \mathcal{I}$, let $B_i$ denote the required RUC value for buyer i, and let $b_i$ represent the bidding price of buyer i to purchase additional RUC in the market. The strategy set for buyer i is denoted as ${\mathbf{b}}_i$, and it lies between $[b_{i,min},b_{i,max}]$. Considering the market policy, we assume that the penalty for exceeding peak power, $P_{CPP}$, is higher than the maximum allowable bidding price for additional RUC. Thus, we have the condition $P_{CPP}\ge b_{i,max}$ for all $i\in \mathcal{I}$. Let $\mathbf{b}={\left(b_i\right)}_{i\in \mathcal{I}}$ represent the set of strategies for all buyers, and let ${\mathbf{b}}_{-i}$ denote the strategies of all buyers except i. Therefore, we have $\mathbf{b}=(b_i,{\mathbf{b}}_{-i})$. Overall, RUC are allocated to buyers based on their relative bidding prices. Thus, the amount of authority allocated to buyer i can be expressed as

$$\begin{array}{c}\hfill B_i=\widehat{B}{\displaystyle \frac{b_i}{{\sum }_{i\in \mathcal{I}}{b}_i}}.\end{array}$$

(6)

2.2.1. Utility Function of Seller

In this paper, the seller aims to earn monetary profit by selling RUC to other users. The utility function of the seller can be represented using two terms. The first terms is a satisfaction factor derived from storing RUC, which helps prevent situations where the seller uses more energy than predicted. The second terms represents the monetary profit that seller j attains by selling RUC to buyers. Thus, for each $j\in \mathcal{J}$, the utility function of seller j is denoted as

$$\begin{array}{cc}\hfill {\widehat{U}}_j({\omega }_j,{\mathbf{\omega }}_{-j},\mathbf{c})& =ln[1+{\widehat{B}}_j(1-{\omega }_j)]+{\alpha }_j\sum _{i\in \mathcal{I}}{B}_i{b}_i{\displaystyle \frac{{\widehat{B}}_j{\omega }_j}{{\sum }_{j\in \mathcal{J}}{\widehat{B}}_j{\omega }_j}},\hfill \end{array}$$

(7)

where ${\alpha }_j$ is the weighting factor for the satisfaction value. Additionally, ${\widehat{B}}_j$ is the amount of RUC that can be sold, determined as follows:

$$\begin{array}{c}\hfill {\widehat{B}}_j=B_{j,max}-\left(ma{x}_{t\in \mathcal{T}}(B_{j,0}+\sum _{k=1}^{t}r{x}_j\left(k\right))\right),\end{array}$$

(8)

where $B_{j,max}$ represents the maximum capacity of seller j in the market, while $ma{x}_{t\in \mathcal{T}}(B_{j,0}+{\sum }_{k=1}^{t}r{x}_j\left(k\right))$ represents the maximum required amount of CESS per day. By calculating the difference between these two factors, we determine the amount of CESS authority that the seller does not use during the day.

2.2.2. Utility Function of Buyer

For each $i\in \mathcal{I}$, the buyer aims to earn a monetary benefit by purchasing additional RUC from the market. The buyer’s utility function can be simply represented as

$$\begin{array}{c}\hfill U_i(b_i,{\mathbf{b}}_{-i},\mathbf{\omega })=B_i{P}_{CPP}-B_i{b}_i,\end{array}$$

(9)

where $B_i$ represents the required battery usage authority, while $P_{CPP}$ denotes the baseline cost charged to buyers. Thus, $B_i{P}_{CPP}$ reflects the reduction in peak costs through RUC transactions. Additionally, $B_i{b}_i$ represents the cost incurred when buyers acquire additional RUC from the market. In this way, the equation shows the monetary profit that buyers achieve through authority transactions.

Next, $B_{i,req}$ is determined by the day-ahead energy scheduling process, which can be mathematically formulated as follows:

$$\begin{array}{c}\hfill B_{i,req}=\sum _{t=1}^T\left(ma{x}_{t\in \mathcal{T}}(P_{d,i}\left(t\right)+\eta r_i{x}_i-P_{th,i},0)\right).\end{array}$$

(10)

In Equation (10), the amount of energy required by buyer i is calculated based on the amount of power consumed the previous day. Here, $B_{i,req}$ is predetermined to prevent the case in which buyer i purchases unnecessary RUC in the market.

3. Game-Theoretic Analysis

RUC trading among power consumers can be modeled as a hierarchical non-cooperative decision-making problem, which is effectively analyzed using a multi-leader multi-follower Stackelberg game framework. This section provides a comprehensive analysis of the game-theoretic model, including assumptions related to the participants’ categorization and the consistency of their situations.

3.1. Game-Theoretic Model and Assumptions

The Stackelberg game is a well-known non-cooperative game model in which participants are classified as leaders or followers based on their strategic roles and utilities. Originally introduced by H. von Stackelberg [30], the model describes a scenario where a leading firm (leader) makes strategic decisions first, and the following firm (follower) responds accordingly after observing the leader’s actions. This hierarchical structure is particularly suitable for modeling interactions between sellers with surplus RUC (leaders) and buyers requiring additional RUC (followers) in CESS scenarios.

In the RUC trading model, participants are dynamically classified as leaders or followers based on their energy consumption patterns and initial RUC allocations. Specifically, users whose estimated peak power demand exceeds their allocated RUC are categorized as buyers seeking additional RUC to avoid penalties, while those with surplus RUC are classified as sellers offering their excess. Each participant’s primary objective is to maximize their own profit, and they independently determine optimal strategies according to their utility function. This dynamic classification reflects realistic, fluctuating conditions within the system and highlights the inherent diversity among participants in terms of energy demands, utility functions, and strategic responses. By capturing this diversity, the RUC trading model can effectively represent a wide range of real-world scenarios, making it adaptable to various situations. Here, the RUC trading model operates under the following key assumptions:

• Rationality: All participants act rationally to maximize their individual utility functions.
• Information Structure: Each participant has complete knowledge of the system model, including utility functions and strategy options. Sellers (leaders) have visibility into the buyers’ (followers’) optimized responses based on backward induction, allowing leaders to incorporate this into their own strategies.
• Decision Sequence: While Stackelberg games generally solve for leaders’ strategies first, this study adopts a backward induction method. By solving for followers’ optimal strategies initially, leaders can then formulate their strategies based on anticipated follower responses, enhancing both stability and predictability within the system.
• Non-Cooperative Behavior: Participants operate independently, without forming coalitions or collaboration, and base their decisions solely on individual utility maximization.

In the system, buyers and sellers aim to maximize their utility functions by determining bidding prices $\mathbf{b}$ and transaction proportion $\mathbf{\omega }$, respectively. To reach equilibrium, we apply the backward induction technique. Therefore, we first calculate the best response of buyers given the sellers’ strategies, and then incorporate these into the sellers’ utility functions to optimize their strategies accordingly.

3.2. Non-Cooperative Game of Followers—The Case of Buyers

Definition 1.

Given the strategies of the other buyers, ${\mathbf{b}}_{-i}$, and the sellers’ strategies, ω, the best response function of buyer i can be expressed as

$$\begin{array}{c}\hfill B_i({\mathbf{b}}_{-i},\mathbf{\omega })=arg\; max{\widehat{U}}_j({\omega }_j,{\mathbf{\omega }}_{-j},\mathbf{c})\forall i\in \mathcal{I}\end{array}$$

(11)

Definition 2.

A Nash equilibrium of the non-cooperative game among the buyers is a profile of strategies ${\mathbf{b}}^{\ast }={\left(b_i^{\ast }\right)}_{(i\in \mathcal{I})}$. Here, the optimal strategy of buyer i is denoted as

$$\begin{array}{c}\hfill b_i^{\ast }=U(b_{-i}^{\ast },\mathbf{\omega }).\end{array}$$

(12)

Lemma 1.

The utility function of buyer i is strictly concave, ensuring that a global optimal value can be achieved.

Proof. 

Given Equation (9), we have

$$\begin{array}{c}\hfill U_i(b_i,{\mathbf{b}}_{-i},\omega )=B_i{P}_{CPP}-B_i{b}_i.\end{array}$$

(13)

Taking the first and second derivatives of $U_i(b_i,{\mathit{b}}_{-i},\mathbf{\omega })$ with respect to $b_i$, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial U_i(b_i,{\mathit{b}}_{-i},\mathbf{\omega })}{\partial b_i}}& ={\displaystyle \frac{\widehat{B}}{{\left({\sum }_{i\in \mathcal{I}}{b}_i\right)}^2}}\left(-b_i^2-2\sum _{i\in \mathcal{I}\setminus i}{b}_i·b_i+P_{CPP}\sum _{i\in \mathcal{I}\setminus i}{b}_i\right),\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^2{U}_i(b_i,{\mathit{b}}_{-i},\mathbf{\omega })}{\partial b_i^2}}& =-{\displaystyle \frac{2\widehat{B}{\sum }_{i\in \mathcal{I}\setminus i}{b}_i}{{\left({\sum }_{i\in \mathcal{I}}{b}_i\right)}^3}}\left(\sum _{i\in \mathcal{I}\setminus i}{b}_i+P_{CPP}\right).\hfill \end{array}$$

(15)

Since ${\sum }_{i\in \mathcal{I}\setminus i}{b}_i$ and $P_{CPP}$ are both greater than 0, the second derivative in Equation (15) always has a value less than 0. Therefore, we conclude that the utility function of buyer i is strictly concave with respect to its bidding price $b_i$. After proving that the utility function is concave, we can solve the problem as follows:

• Derive the maximum value using the first derivative.
• Check that the optimal value is within constraints.

To determine the best response function of buyer i, we initially set the right-hand side of Equation (14) equal to zero:

$$\begin{array}{c}\hfill {\displaystyle \frac{\widehat{B}}{{\left({\sum }_{i\in \mathcal{I}}{b}_i\right)}^2}}\left(-b_i^2-2\sum _{i\in \mathcal{I}\setminus i}{b}_i·b_i+P_{CPP}\sum _{i\in \mathcal{I}\setminus i}{b}_i\right)=0.\end{array}$$

(16)

Then, the bidding strategy for buyer i can be obtained as

$$\begin{array}{c}\hfill B_i({\mathbf{b}}_{-i},\mathbf{\omega })=\sqrt{{\left(\sum _{i\in \mathcal{I}\setminus i}{b}_i\right)}^2+P_{CPP}\sum _{i\in \mathcal{I}\setminus i}{b}_i}-\sum _{i\in \mathcal{I}\setminus i}{b}_i.\end{array}$$

(17)

Herein, we finally demonstrate that the non-cooperative game among buyers results in a unique Nash equilibrium. □

Proposition 1.

The non-cooperative game among buyers has a Nash equilibrium given by

$$\begin{array}{c}\hfill b_i^{\ast }=P_{CPP}{\displaystyle \frac{N-1}{2N-1}}\forall i\in \mathcal{I}.\end{array}$$

(18)

Proof. 

The best response function for buyer i is identical for all buyers. According to Definition 2, we have

$$\begin{array}{cc}\hfill b_i^{\ast }& =\sqrt{{(\sum _{i\in \mathcal{I}\setminus i}{b}_i^{\ast })}^2+P_{CPP}\sum _{i\in \mathcal{I}\setminus i}{b}_i^{\ast }}-\sum _{i\in \mathcal{I}\setminus i}{b}_i^{\ast }\hfill \\ \hfill & =\sqrt{{\left((\mathcal{I}-1)b_i^{\ast }\right)}^2+P_{CPP}(\mathcal{I}-1)b_i^{\ast }}-(\mathcal{I}-1)b_i^{\ast }.\hfill \\ \hfill \therefore b_i^{\ast }& =P_{CPP}·{\displaystyle \frac{N-1}{2N-1}}\forall i\in \mathcal{I}.\hfill \end{array}$$

(19)

□

3.3. Non-Cooperative Game of Leaders—The Case of Sellers

By substituting the total transaction RUC $\widehat{B}$ from Equation (6) into Equation (7), we can reconstruct the seller’s utility function as follows:

$$\begin{array}{cc}\hfill {\widehat{U}}_j({\omega }_j,{\mathbf{\omega }}_{-j},\mathbf{c})& =ln[1+{\widehat{B}}_j(1-{\omega }_j)]+{\alpha }_j\sum _{i\in \mathcal{I}}{B}_i{b}_i{\displaystyle \frac{{\widehat{B}}_j{\omega }_j}{{\sum }_{j\in \mathcal{J}}{\widehat{B}}_j{\omega }_j}},\hfill \\ \hfill & =ln[1+{\widehat{B}}_j(1-{\omega }_j)]+{\alpha }_j{\widehat{B}}_j{\omega }_j{\displaystyle \frac{{\sum }_{i\in \mathcal{I}}{B}_i{b}_i}{\widehat{B}}},\hfill \\ \hfill & =ln[1+{\widehat{B}}_j(1-{\omega }_j)]+{\alpha }_j{\widehat{B}}_j{\omega }_j{\displaystyle \frac{{\sum }_{i\in \mathcal{I}}{\left(b_i\right)}^2}{{\sum }_{i\in \mathcal{I}}\left(b_i\right)}},\hfill \\ \hfill & =ln[1+{\widehat{B}}_j(1-{\omega }_j)]+{\alpha }_j{\widehat{B}}_j{\omega }_j{b}_N.\hfill \end{array}$$

(20)

The bidding price of buyer i is determined by the baseline cost $P_{CPP}$ announced by the main grid and the number of buyers N in the market. Therefore, we replace ${\displaystyle \frac{{\sum }_{i\in \mathcal{I}}{b}_i^2}{{\sum }_{i\in \mathcal{I}}{b}_i}}$ with $b_N$, an auxiliary variable introduced in this paper. The seller’s utility function can be expressed in terms of the transaction proportion ${\omega }_j$, and the optimal transaction proportion ${\omega }_j^{\ast }$ is derived by taking the first derivative of the utility function, as shown in Equation (21).

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial {\widehat{U}}_j({\omega }_j,{\mathbf{\omega }}_{-j},\mathbf{c})}{\partial {\omega }_j}}& ={\alpha }_j{\widehat{B}}_j{b}_N-{\displaystyle \frac{{\widehat{B}}_j}{1+{\widehat{B}}_j(1-{\omega }_j)}},\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^2{\widehat{U}}_j({\omega }_j,{\mathbf{\omega }}_{-j},\mathbf{c})}{\partial {\omega }_j^2}}& ={\displaystyle \frac{{\left({\widehat{B}}_j\right)}^2}{{(1+{\widehat{B}}_j(1-{\omega }_j))}^2}}.\hfill \end{array}$$

(22)

Since the second derivative in Equation (22) is always greater than zero, the seller’s utility function is convex with respect to ${\omega }_j$. Therefore, we find the optimal transaction proportion ${\omega }_j^{\ast }$ by setting the first derivative to zero:

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial {\widehat{U}}_j({\omega }_j,{\mathbf{\omega }}_{-j},\mathbf{c})}{\partial {\omega }_j}}& ={\alpha }_j{\widehat{B}}_j{b}_N-{\displaystyle \frac{{\widehat{B}}_j}{1+{\widehat{B}}_j(1-{\omega }_j)}}=0.\hfill \\ \hfill \therefore {\omega }_j^{\ast }& =1+{\displaystyle \frac{1}{{\widehat{B}}_j}}(1-{\displaystyle \frac{1}{{\alpha }_j{b}_N}}).\hfill \end{array}$$

(23)

Here, ${\omega }_j^{\ast }$ represents the transaction proportion of seller j, which lies between 0 and 1. If ${\omega }_j^{\ast }$ falls outside this range, it is adjusted to either 0 or 1 to ensure it stays within the feasible boundary.

4. Numerical Results

To demonstrate the effectiveness of the proposed energy management scheme using CESS, we present several simulation results based on data obtained from a real energy environment [28,31,32]. More specifically, power consumption data from factories in South Korea were used as the primary energy input for the simulation.

This dataset, recorded by the Advanced Metering Infrastructure (AMI) of the Korea Electric Power Corporation (KEPCO), covers the period from 1 January 2012 to 31 December 2012, with hourly intervals. The details of the simulation environment are fully summarized in

Table 1. Simulation parameters to operate CESSs in the systems.

Simulation Parameters	Symbols	Values
Time interval	-	1 h
Time slot	t	1–24 (based on 24 h)
Time-of-use pricing [31]	$C_{ToU}$	USD 0.05–0.20\kWh
Penalty for high peak power [31]	$P_{CPP}$	USD 8.3
Maximum RUC transaction price	$b_{max}$	USD 0.4
Minimum SoC battery level	$s_{min}$	0.1
Maximum SoC battery level	$s_{max}$	1.0
CESS operating efficiency [28]	$\eta$	0.9
Maximum CESS operation [28]	$\gamma$	150 kWh
CESS capacity	$B_{max}$	300 kWh
Number of users	N	10
Coefficients for sale price	$\alpha$	0.1

.

4.1. Description of Power Consumption Data

To validate the proposed energy management scheme, we utilized data from three representative types of factories, as depicted in Figure 3. The figure displays normalized values to identify typical power consumption patterns across different factory types, allowing us to understand their distinct energy usage characteristics.

The first dataset represents a Manufacturing factory, where workloads increase during late-night hours, resulting in a peak during that time. The second dataset pertains to an Automobile assembly factory, which is busiest during daytime hours, with peak power consumption occurring between 10:00 a.m. and 4:00 p.m. This factory type has one or two distinct peaks, and the difference between peak and non-peak consumption is significant. Finally, the third dataset features a Batch processing factory, characterized by an evenly distributed workload, which causes power consumption to follow a consistent six-hour cycle. Each dataset reflects distinct operational characteristics, highlighting the need for an effective energy management approach capable of addressing diverse power demands.

4.2. Nash Equilibrium of Stackelberg Game Theory

The strategic decisions of users in Nash equilibrium represent the optimal choices given the current conditions, making the changes in these values highly significant for this study. As outlined in Equation (17), the determination of Nash equilibrium is primarily influenced by two variables. The first is $P_{CPP}$, the price announced according to the main grid policy, which generally remains stable. The second variable, however, is the number of buyers, which can vary depending on users’ power consumption patterns. Therefore, it is essential to analyze user strategies and outcomes relative to changes in the number of buyers in the system.

Figure 4 illustrates the strategic behavior of participants as the number of buyers needing additional RUC increases. As the number of buyers grows, the bidding price also tends to rise, eventually stabilizing at approximately one-quarter of the baseline cost, $P_{CPP}$. Conversely, the authority allocated to each buyer decreases rapidly. This trend aligns with the principle of supply and demand, particularly in conditions of limited supply. It indicates that buyer strategies directly influence sellers’ decisions regarding how much authority to sell in the market. Next, as expressed in Equation (23), sellers tend to increase the portion of their surplus RUC for sale in response to rising buyer bids. This behavior is depicted in Figure 5, which shows strategies of two sellers with differing levels of surplus authority. The figure demonstrates that as the number of buyers increases, sellers become more inclined to sell their surplus rights. This behavior suggests that sellers prioritize immediate profits from selling their RUC over concerns regarding future uncertainties or risks.

To conclude, the analysis of Nash equilibrium in the Stackelberg game setting reveals the dynamic interplay between buyers and sellers within the RUC trading market. Buyers adjust their bidding strategies based on the fluctuating number of participants, while sellers adapt by increasing their willingness to sell surplus capacity as bids rise.

4.3. Effectiveness of the Proposed Model

This section evaluates the effectiveness of the proposed model by comparing the electricity prices under the proposed RUC transaction scheme with those of a conventional CESS. To fully determine the electricity price, we consider two primary indicators: time-of-use (ToU) pricing ($C_{ToU}$) and CPP ($P_{CPP}$), as outlined in Equation (1). We also take into account each user’s additional income and costs incurred during RUC trading. Under these parameters, we analyzed daily energy management for a buyer purchasing additional authority through the CESS transaction.

Figure 6 illustrates the energy management results for a buyer who obtains additional RUC through the proposed Stackelberg game model. In this figure, the conventional method represents the scenario where a buyer manages energy consumption without trading additional RUC. Under this conventional approach, the peak power usage reached $64.3676$ kWh. In contrast, the proposed method illustrates the scenario in which a user purchases additional CESS operating authority from the market. The peak value in this case is $61.6432$ kWh, representing a reduction of $2.7244$ kWh compared to the conventional approach.

By analyzing users’ power consumption throughout the day, we confirmed that the proposed method effectively reduces peak power consumption. Additionally, we extended our analysis to one year of operation to assess the benefits in terms of overall cost savings achieved by employing the proposed RUC transaction scheme with a CESS.

Table 2. Comparison of annual operation results.

	ToU Pricing	Peak Penalty	Total Cost
Conventional method	USD 359,910	USD 110,530	USD 470,450
Proposed method	USD 349,450	USD 44,310	USD 397,510

presents a comparison of the annual operational results between the two techniques. In terms of ToU pricing, the proposed method achieves an annual cost reduction of USD 10,000. However, this represents only a 2.9% decrease compared to the conventional approach, suggesting a relatively modest impact. More significantly, the proposed model reduces peak penalty occurrences by up to 59.9%, illustrating its effectiveness in minimizing penalties associated with peak power consumption. These findings indicate that different users require varying levels of authority to effectively use the CESS. By establishing an ancillary market for RUC trading, our model demonstrates the potential to reduce costs from the consumer’s perspective while simultaneously enhancing power reliability for the provider.

In the given simulation environment, we analyzed each participant’s revenue, considering the application of the proposed model from both the buyers’ and sellers’ perspectives. Figure 7 illustrates the revenue outcomes with and without the proposed model. In the figure, Revenue represents the cost gap arising from the implementation of the proposed transaction model. Users 3 and 10 represent individuals with excess CESS authority, while the remaining users are buyers seeking additional rights. The profit earned by sellers is directly derived from the sale of RUC, demonstrating the benefits for users who participate in the system by selling their RUC on the market.

For buyers, additional expenditures are incurred when purchasing RUC, but these costs are offset by a reduction in penalties through the decrease in peak power consumption. Based on the results, we can confirm that the proposed scheme increases overall benefit by at least 16% compared to the conventional CESS operating scheme. Moreover, the consistent improvement in performance across users with diverse power consumption patterns, including those from manufacturing, automobile assembly, and batch processing factories, demonstrates the versatility and applicability of the proposed scheme across a broad spectrum of consumers. This underscores the practicality and effectiveness of the proposed model in decentralized energy systems, where users exhibit diverse consumption behaviors and operate independently.

5. Conclusions

This paper has proposed a two-stage operating scheme for a community energy storage system (CESS) that introduces a rights to use CESS (RUC) transaction mechanism among CESS users. In the first stage, we propose a scheme to determine the additional authority requirements or trading quantities by mathematically modeling the cost structures of users in the system. In the second stage, we employ Stackelberg game theory to structure RUC transactions among the users, where sellers act as leaders and buyers as followers. By utilizing actual electricity pricing and real-world power consumption data in our simulations, we have demonstrated the efficiency and practicality of the proposed method. The results show that the proposed scheme effectively reduces total electricity cost by approximately 16% compared to conventional CESS operating schemes, while ensuring that all participants benefit financially from engaging in the market.

The proposed energy management scheme not only guarantees the profitability of using CESS in systems with clustered consumers but also enhances overall system efficiency and stability. The proposed model also demonstrates strong scalability, effectively accommodating a growing number of users with diverse consumption patterns. This scalability is achieved by dynamically categorizing participants and leveraging the flexible structure of the Stackelberg game framework, enabling efficient management of both small-scale and large-scale implementations within decentralized energy systems. Such adaptability ensures that the scheme remains effective as the number of participants increases, making it suitable for widespread adoption in various real-word settings.

To further enhance the practical applicability of the proposed energy management scheme, future research could incorporate battery degradation modeling into the framework. However, this integration could result in a complex, non-convex optimization problem, presenting significant challenges for game-theoretic approaches and increasing computational demands as the system scales. Advanced techniques such as reinforcement learning techniques may offer effective alternatives for efficiently managing these complex, nonlinear systems. Additionally, while this paper has focused on a non-cooperative environment, exploring cooperative game-theoretic models could facilitate user collaboration, potentially improving overall system efficiency and stability. Furthermore, while the proposed method assumes deterministic system inputs and user power demands, future work could consider uncertainty modeling by incorporating stochastic elements or employing robust prediction techniques to enhance the feasibility and resilience of the proposed scheme.