Shared Energy Storage Capacity Configuration of a Distribution Network System with Multiple Microgrids Based on a Stackelberg Game

Abstract

With the ongoing development of new power systems, the integration of new energy sources is facing increasingly daunting challenges. The collaborative operation of shared energy storage systems with distribution networks and microgrids can effectively leverage the complementary nature of various energy sources and loads, enhancing energy absorption capacity. To address this, a shared energy storage capacity allocation method based on a Stackelberg game is proposed, considering the integration of wind and solar energy into distribution networks and microgrids. In this approach, a third-party shared energy storage investor acts as the leader, while distribution networks and microgrids serve as followers. The shared energy storage operator aims to maximize annual revenue, plan shared energy storage capacity, and set unit capacity leasing fees. Upon receiving pricing, distribution networks and microgrids aim to minimize annual operating costs, determine leased energy storage capacity, and develop operational plans based on typical daily scenarios. Distribution networks and microgrids report leasing capacity, and shared energy storage adjusts leasing prices, accordingly, forming a Stackelberg game. In the case study results, the annual cost of MGs decreased by 29.63%, the annual operating cost of the ADN decreased by 11.25%, the cost of abandoned light decreased by 60.77%, and the cost of abandoned wind decreased by 27.79% to achieve the collaborative optimization of operations. It is proven that this strategy can improve the economic benefits of all parties and has a positive impact on the integration of new energy.

1. Introduction

In the context of the “dual carbon” goals and the development of new power systems, new opportunities have emerged for the development of new energy [1], leading to the larger-scale development of distributed new energy. In order to enhance the integration capacity of new energy sources, microgrids (MGs) have gradually attracted attention from the academic community [2], becoming a research hotspot in power systems. The large-scale connection of MGs has become an important form supporting the utilization of distributed new energy on the distribution side [3]; however, at the same time, the integration of new energy sources and the imbalance between the time and space of supply and demand are becoming increasingly serious. The integration of a large number of MGs and new energy sources into the grid has changed the network’s power flow and voltage distribution, increasing the difficulty of the operation and management of an active distribution network (ADN). An ADN refers to a modernized electricity distribution system that integrates advanced technologies for the dynamic monitoring, control, and management of electricity flow. This term distinguishes ADNs from MGs by emphasizing their broader scope within conventional distribution systems, supporting grid-wide optimization and resilience. Energy storage technology plays an important role in the integration and consumption of new energy. By incorporating energy storage systems, the intermittent and uncertain issues of new energy sources in the power system can be effectively alleviated, thereby improving the stability and reliability of the grid [4].

At present, extensive research has been conducted by scholars at home and abroad on the optimization of new energy storage configurations. In reference [5], focusing on the planning problem of the energy storage capacity for a power system with high renewable energy penetration, distributed robust planning for the energy storage capacity is proposed, constructing a model for optimizing the configuration of mixed-energy storage capacity in wind power clusters. In reference [6], based on the characteristics and critical clearing time of an FSIG-based wind farm, a novel capacity configuration method for a hybrid wind farm is proposed to calculate the required reactive power of an FSIG-based wind farm for an LVRT operation. Reference [7] establishes an energy storage model by considering the absorption capacity of a wind–solar ADN and uses optimization theory to find the optimal solution. The study in [8] simultaneously considers the interests of MGs and the ADN, optimizing the energy storage capacity of MGs to minimize the costs. The study in [9] addresses the uncertainty of renewable energy generation in microgrids by proposing an optimized configuration of energy storage capacity. The study in [10] discusses a tri-layer non-cooperative energy trading approach among multiple grid-tied multi-energy microgrids (MEMGs) in the restructured integrated energy market. With the continuous development of the energy internet, there has been a rapid growth in the demand for distributed energy storage [11]; however, the high cost of energy storage equipment currently limits the widespread application of distributed energy storage [12]. To promote the popularization of energy storage applications, the study in [13] conducts research and analysis on the application of a sharing economy model in energy storage systems, providing new business models and ideas for the development of future energy storage systems.

In the coming years, shared energy storage (SES) is expected to experience rapid growth [14]. Introducing a sharing model for energy exchange among multiple MGs and fully leveraging the collaborative advantages of MGs is an effective means with which to improve the energy utilization efficiency of energy storage systems [15]. Utilizing SES can enhance the operational flexibility, security, and power quality of renewable energy-based MGs [16,17]. The study in [18] introduced application scenarios, such as a reduction in power output fluctuations, agreement to an output plan on the renewable energy generation side, and a distributed as well as mobile energy storage system on the power distribution side. Meanwhile, in the SES market, both suppliers and operators need to determine the required energy storage capacity in order to develop investment and operation plans [19]. The study in [20] proposed an optimization method for the configuration of SES with an MG group, aiming to enhance power supply reliability and reduce operational costs by adjusting the energy storage capacity. In the optimization and scheduling of SES, it is necessary to consider not only the scheduling of a user’s energy storage demand [21] but also the actual investment and profit situation of a shared energy storage operator (SESO). The study in [22] represented a Shared-ESS real-time pricing model, which achieves an amazing tradeoff of the interests between service providers and end users. The study in [23] further considered the complementary charging and discharging behaviors within MGs as well as complementary energy production and consumption under uncertain scenarios, solving the coordinated scheduling problem between MGs and SES.

The Stackelberg game is an important concept in game theory, referring to a leader–follower game model, and has been widely referenced in the field of shared energy storage configuration. Lu, J. et al. propose a strategy that involves the leasing of SES to establish a collaborative microgrid coalition (MGCO), enabling active participation in the dispatching operations of ADNs [24]. Lu, Liu, N. et al. propose a Stackelberg game approach for ESM, where the MGCO acts as the leader and all participating prosumers are considered the followers [25]. Özge, E. et al. focus on the energy-sharing management of a microgrid including photovoltaic–wind turbine prosumers with energy storage systems and plug-in electric vehicle charging stations [26]. The bi-level trading problem is formulated as a Stackelberg game considering integrating the energy market and the “cap-and-trade” carbon market mechanism [27]. In these studies, they basically used MGCOs as investors in shared energy storage. However, the prerequisite for interconnected sharing is that each participant must possess independently configured energy storage equipment capable of bidirectional energy transfer. This sharing mode is particularly suitable for entities already equipped with distributed energy storage systems [28]. Nonetheless, it exhibits notable practical limitations.

The aforementioned literature has developed shared energy optimization economic strategies and operational plans tailored to leasing objects and demands, optimizing shared energy capacity configurations to enhance the consumption of distributed new energy. This has improved the economic efficiency of MG operation and increased the utilization rate of system assets; however, the following issues persist: most of the existing literature focuses on energy storage sharing among multiple individual entities (such as multiple industrial users, multiple MGs, etc.). From the perspective of the sharing mode, current research about SES can be classified into three categories, i.e., the co-construction sharing mode, the interconnected sharing mode, and the energy storage operator leasing mode. Different from the first two modes, the operator leasing mode of SES is more flexible in practice [28]. Under this mode, demanders can flexibly obtain the use rights of energy storage resources according to their short-term demands from an independent SES aggregator under clear leasing rules, without considering the negotiation and coordination challenges that need to be encountered in the cooperation.

The introduction of a SESO transforms the shared energy capacity configuration problem into a decision-making problem involving multiple interacting and coordinated objects; therefore, achieving friendly interaction among multiple entities is an urgent issue to be addressed in the shared energy planning of an ADN and MGs. In addition, due to the integration of new energy sources, the leasing capacity demands of the ADN and MGs exhibit significant differences at different time periods. Therefore, when a SESO provides leasing services to MGs and the ADN, coordination between MGs and the ADN is required to develop charging and discharging strategies for different time periods to meet leasing demands.

In response to the aforementioned challenges, this study proposes an SES capacity allocation model for an ADN system with multiple MGs, based on a Stackelberg game framework. Within the MGs connected to an ADN, SES is invested in and deployed by a third-party shared SESO. By considering the energy storage capacity requirements and operational strategies of the MGs within the ADN, the optimal unit energy storage leasing fee is determined. The MGs and an ADN engage in reciprocal electricity transactions to mitigate the variability of wind as well as solar curtailment rates and tie-line power. Compared with existing studies, the main contributions and innovations of this paper include the following:

(1)

Based on typical daily scenarios, we constructed a Stackelberg game optimization framework with a SESO as the master and an ADN and MGs as the slaves. In this framework, the master adjusts the unit SES capacity leasing price to maximize annual revenue, while the slaves adjust their own leasing capacity and operating plans to minimize annual operating costs based on the received unit capacity leasing fees. In solving the Nash equilibrium, we obtained the optimal SESO energy storage capacity, unit capacity leasing fee, and the leasing energy storage capacity and operating strategy for the ADN and MGs, thereby improving the operational efficiency of all parties involved.

(2)

Between the ADN and MGs, internal power purchasing and selling strategies are considered to increase the penetration rate of wind and solar power, enhance energy storage utilization, and reduce their own operating costs. In the ADN, strategies are considered to balance the power deficit from the upper grid by sharing energy storage, thereby balancing the power purchased from the upper grid and reducing the fluctuation of tie-line power.

(3)

The output-sharing optimal capacity and unit capacity leasing fee are determined using a nested interior point method with a genetic algorithm. This approach enhances both the global and local search capabilities of the algorithm, ensuring that the computation does not become stuck in local optima, thus accelerating the solution process.

The rest of this paper is organized as follows. Section 2 introduces the interaction framework for SESO to lease SES to MGs and the ADN. Based on this, Section 3 proposes a pricing and capacity planning model for SES leasing based on a Stackelberg game. Section 4 presents a solution method using a genetic algorithm nested with the interior point method to solve the model. Finally, Section 5 conducts case studies and analysis, and Section 6 summarizes the work, describing its limitations and possible improvements.

2. The Stackelberg Game Framework of SES Accessing Multiple MGs and the ADN

As the main investor in SES, the SESO aims to maximize its return on investment while satisfying the energy storage capacity requirements of MGs. Its objective is to maximize operational efficiency by investing in SES stations. Meanwhile, MG and ADN operators seek to improve the integration rate of solar as well as wind power and reduce their own operating costs by leasing SES; therefore, a Stackelberg game relationship is formed among the SES investors, the ADN, and MGs. This game relationship will play an important role in the operation of the SES market and will influence the investment and operation strategies of SES.

The framework of the SES capacity allocation method is illustrated in Figure 1. Initially, MGs receive the unit energy storage leasing price issued by a SESO. With the aim of optimizing their own operational economics, MGs preliminarily plan the required energy storage capacity and derive corresponding charging/discharging strategies and electricity purchasing/selling statuses. Based on the behavior of MGs, a SESO plans its own capacity allocation and adjusts leasing prices accordingly. This process forms a Stackelberg game model. Through multiple iterations, the system ultimately reaches a Nash equilibrium.

A Stackelberg game model for the ADN, MGs, and a SESO is constructed as shown in Figure 2. Initially, time-of-use electricity prices are disseminated based on peak shaving requirements. MGs and the ADN optimize their operational strategies in response to the electricity prices, while a SESO utilizes its remaining capacity to assist in peak shaving. This Stackelberg game process is repeated until a Nash equilibrium is reached, yielding optimal operational strategies and costs for all participants.

During the optimization phase, a SESO and MGs represent different interest entities, with the SESO acting as the leader while the ADN and MGs are followers, thus constructing a leader–follower game model. The SESO adjusts the leasing prices of power and energy capacity based on the total power generation and consumption curve reported by the followers, guiding the ADN and MGs to adjust the leasing capacity of energy storage to increase the profit from energy storage leasing. MGs utilize their idle capacity to interact with the ADN for energy arbitrage based on time-of-use electricity prices. The ADN and MGs respond to the leasing prices of energy storage, optimizing their own energy storage leasing capacity and operation plans, and their decisions, in turn, affect the SESO’s pricing strategy. The decisions of the leader and followers influence each other, eventually reaching a Nash equilibrium.

3. Model Construction

3.1. SES Model

3.1.1. The Objective Function of SESO

As the energy storage investment entity, the SESO aims to fully utilize the constructed energy storage capacity and achieve favorable returns. Therefore, its objective is to maximize annual revenue.

$$W_{\mathrm{share}}=C_{\mathrm{MG},\mathrm{rent}}+C_{\mathrm{ADN},\mathrm{rent}}-C_{\mathrm{share},\mathrm{INV}}-C_{\mathrm{share},\mathrm{OM}},$$

(1)

where $W_{\mathrm{share}}$ represents the total annual revenue of the SESO, including $C_{\mathrm{MG},\mathrm{rent}}$, which is the annual capacity leasing fee paid by MGs to the SES investor; $C_{\mathrm{ADN},\mathrm{rent}}$ is the annual capacity leasing fee paid by the ADN; $C_{\mathrm{share},\mathrm{INV}}$ is the annual investment cost of SES construction; and $C_{\mathrm{share},\mathrm{OM}}$ is the annual operating cost of SES.

$$C_{\mathrm{share},\mathrm{INV}}={\theta }_{\mathrm{share}}\times C_{\mathsf{\theta },\mathrm{share}}+P_{\mathrm{share}}\times C_{\mathrm{p},\mathrm{share}}\times \frac{r{\left(1+r\right)}^y}{{\left(1+r\right)}^y-1},$$

(2)

where in the annual investment and construction cost, $C_{\mathrm{share},\mathrm{INV}}$, of SES, ${\theta }_{\mathrm{share}}$ represents the rated capacity of SES, $C_{\mathsf{\theta },\mathrm{share}}$ denotes the unit capacity cost of SES, $P_{\mathrm{share}}$ indicates the rated power of SES, $C_{\mathrm{p},\mathrm{share}}$ is the unit power cost of SES, $r$ represents the discount rate, and $y$ represents the service life of SES.

$$C_{\mathrm{share},\mathrm{OM}}=365\times \gamma {\displaystyle \sum }_{t=1}^T\left(D_{\mathrm{share}\_\mathrm{C}}^t{P}_{\mathrm{share}\_\mathrm{C}}^t+D_{\mathrm{share}\_\mathrm{D}}^t{P}_{\mathrm{share}\_\mathrm{D}}^t\right),$$

(3)

where the variable $\gamma$ represents the unit charging/discharging power cost of SES. $D_{\mathrm{share}\_\mathrm{C}}^t$ and $D_{\mathrm{share}\_\mathrm{D}}^t$ are binary variables taking values of 0 or 1, and their values, respectively, represent the charging and discharging operations of SES in time period t. $P_{\mathrm{share}\_\mathrm{C}}^t$ and $P_{\mathrm{share}\_\mathrm{D}}^t$, respectively, represent the charging and discharging power of SES in time period t.

The annual capacity leasing fees paid by MGs and the ADN are as follows:

$$C_{\mathrm{MG},\mathrm{rent}}=365\times \alpha {\theta }_{\mathrm{MG},\mathrm{rent}},$$

(4)

$$C_{\mathrm{ADN},\mathrm{rent}}=365\times \alpha {\theta }_{\mathrm{ADN}.\mathrm{rent}},$$

(5)

where ${\theta }_{\mathrm{MG},\mathrm{rent}}$ represents the capacity leased by MGs from SES, ${\theta }_{\mathrm{ADN}.\mathrm{rent}}$ represents the capacity leased by the ADN from SES, and $\alpha$ is the unit capacity leasing fee of SES.

Set the capacity and rated power of SES to be proportional, as expressed below:

$$P_{\mathrm{share}}=\mu {\theta }_{\mathrm{share}},$$

(6)

where $\mu$ is the scaling factor between the rated power and rated capacity of SES.

3.1.2. The Constraint Conditions of SESO

(1)

SES charging and discharging power constraints are as follows:

$$0⩽P_{\mathrm{share}\_\mathrm{C}}^t,P_{\mathrm{share}\_\mathrm{D}}^t⩽P_{\mathrm{share}},$$

(7)

$$D_{\mathrm{share}\_\mathrm{C}}^t{D}_{\mathrm{share}\_\mathrm{D}}^t=0,$$

(8)

where $P_{\mathrm{share}\_\mathrm{C}}^t$ and $P_{\mathrm{share}\_\mathrm{D}}^t$, respectively, represent the charging and discharging power of SES in time period t, subject to the maximum power-sharing limit, $P_{\mathrm{share}}$.

(2)

Constraints on the upper and lower bounds of SES capacity are as follows:

$$S_{\mathrm{SOC}}^{\min }{\theta }_{\mathrm{share}}⩽E_{\mathrm{share}}^t⩽S_{\mathrm{SOC}}^{\max }{\theta }_{\mathrm{share}},$$

(9)

where the internal energy of SES, $E_{\mathrm{share}}^{\mathrm{t}}$, is subject to the constraints imposed at time t by the minimum SES state of charge, $S_{\mathrm{SOC}}^{\min }{\theta }_{\mathrm{share}}$, and the maximum SES state of charge, $S_{\mathrm{SOC}}^{\max }{\theta }_{\mathrm{share}}$.

(3)

The relationship between the energy storage levels in adjacent time periods is as follows:

$$E_{\mathrm{share}}^t=E_{\mathrm{share}}^{t-1}+\Delta t\left({\eta }_{\mathrm{share}}^{\mathrm{C}}{P}_{\mathrm{share}\_\mathrm{C}}^t-P_{\mathrm{share}\_\mathrm{D}}^t/{\eta }_{\mathrm{share}}^D\right),$$

(10)

where $E_{\mathrm{share}}^t$ represents the internal energy of SES at time t, $E_{\mathrm{share}}^{t-1}$ represents the internal energy of SES at time t − 1, $\Delta t$ is the time interval (taken as 1 h), and ${\eta }_{\mathrm{share}}^{\mathrm{C}}$ and ${\eta }_{\mathrm{share}}^D$ are the charging and discharging efficiencies of SES (taken as 0.95), respectively. The energy level of SES is determined via the energy level at the previous time step and the charging or discharging during the time interval.

(4)

The constraint on energy storage with equal energy levels at the beginning and end of the day is as follows:

$$E_{\mathrm{share}}^0=E_{\mathrm{share}}^T,$$

(11)

where T represents the scheduling period (taken as 24 h). Within one scheduling period, $E_{\mathrm{share}}^0$ represents the initial energy level of SES and $E_{\mathrm{share}}^T$ represents the final one. In other words, the initial and final energy levels of SES within one day must be equal.

(5)

Constraints on the leased capacity of SES are as follows:

$${\theta }_{\mathrm{MG},\mathrm{rent}}+{\theta }_{\mathrm{ADN},\mathrm{rent}}⩽{\theta }_{\mathrm{share}},$$

(12)

where the total leased capacity of the ADN (${\theta }_{\mathrm{ADN},\mathrm{rent}}$) and MGs (${\theta }_{\mathrm{MG},\mathrm{rent}}$) is subject to the capacity limit of the SES operator (${\theta }_{\mathrm{share}}$).

3.2. The MGs Model

3.2.1. The Objective Function of MG

The MG adopts a wind–solar–battery hybrid MG, consisting of wind and solar power generation units, micro gas turbines, internal energy storage, and loads. As the operating entity, the MG aims to improve its operational efficiency and reduce curtailed renewable energy by leasing SES. Therefore, the objective of the MG is to minimize its annual operating costs:

$$C_{\mathrm{MG}}=C_{\mathrm{omt}}+C_{\mathrm{oess}}+C_{\mathrm{afd}}+C_{\mathrm{th},\mathrm{PV}}+C_{\mathrm{MG},\mathrm{rent}},$$

(13)

where $C_{\mathrm{MG}}$ represents the annual operating cost of the MG, $C_{\mathrm{omt}}$ represents the annual operation and maintenance cost of the micro gas turbine within the MG, $C_{\mathrm{oess}}$ represents the annual operation and maintenance cost of the energy storage system within the MG, $C_{\mathrm{afd}}$ represents the annual cost of purchasing or selling electricity between the MGs and an ADN, $C_{\mathrm{th},\mathrm{PV}}$ represents the annual cost of curtailed solar energy within the MG, and $C_{\mathrm{MG},\mathrm{rent}}$ represents the annual leasing fee for energy storage capacity paid by the MG.

$$C_{\mathrm{omt}}=365\times {\displaystyle \sum }_{t=1}^T{P}_{\mathrm{b}}^t\times K_{\mathrm{ON}},$$

(14)

where $P_{\mathrm{b}}^t$ represents the output power of the micro gas turbine at time t and $K_{\mathrm{ON}}$ represents the cost coefficient of fuel and carbon emissions at time t.

$$C_{\mathrm{oess}}=365\times {\displaystyle \sum }_{t=1}^T\left(D_{\mathrm{i}\_\mathrm{C}}^t{P}_{\mathrm{i}\_\mathrm{C}}^t+D_{\mathrm{i}\_\mathrm{D}}^t{P}_{\mathrm{i}\_\mathrm{D}}^t\right)\times K_{\mathrm{OM}},$$

(15)

where $D_{\mathrm{i}\_\mathrm{C}}^t$ and $D_{\mathrm{i}\_\mathrm{D}}^t$ are binary 0–1 variables. When their value is 0 or 1, they, respectively, indicate whether the energy storage system of the MG is charging or discharging during time period t; $P_{\mathrm{i}\_\mathrm{C}}^t$ and $P_{\mathrm{i}\_\mathrm{D}}^t$ represent the charging and discharging power of the energy storage system of the MG during time period t, respectively; $K_{\mathrm{OM}}$ is the cost coefficient of the energy storage system of the MG for charging or discharging during time period t.

$$C_{\mathrm{afd}}=365\times {\displaystyle \sum }_{t=1}^T\left(P_{\mathrm{d}}^t\times s^t\right),$$

(16)

where $P_{\mathrm{d}}^t$ represents the power purchased or sold by the MG to the ADN at time t; positive values indicate purchasing power, while negative values indicate selling power. $s^t$ represents the price of electricity purchased or sold by the MG to the ADN at time t.

$$C_{\mathrm{th},\mathrm{PV}}=365\times {\displaystyle \sum }_{t=1}^T\left(P_{\mathrm{th},\mathrm{PV}}^t\times K_{\mathrm{PV}}\right),$$

(17)

where $P_{\mathrm{th},\mathrm{PV}}^t$ represents the curtailed solar energy of the MG at time t and $K_{\mathrm{PV}}$ is the cost of curtailed solar energy per unit.

3.2.2. The Constraint Conditions of MG

(1)

Power balance constraints are as follows:

$$P_{\mathrm{PV}}^t+D_{\mathrm{i}\_\mathrm{D}}^t{P}_{\mathrm{i}\_\mathrm{D}}^t-D_{\mathrm{i}\_\mathrm{C}}^t{P}_{\mathrm{i}\_\mathrm{C}}^t+P_{\mathrm{b}}^t+P_{\mathrm{d}}^t+P_{\mathrm{share},\mathrm{MG}}^t-P_{\mathrm{th},\mathrm{PV}}^t-P_{\mathrm{load}}^t=0,$$

(18)

where $P_{\mathrm{PV}}^t$ is the solar power generation of the MG, $P_{\mathrm{share},\mathrm{MG}}^t$ is the power exchanged between SES and the MG, and $P_{\mathrm{load}}^t$ is the power consumption of the MG.

(2)

Energy storage charging and discharging constraints are as follows:

$$0⩽P_{\mathrm{i}\_\mathrm{C}}^t,P_{\mathrm{i}\_\mathrm{D}}^t⩽P_{\mathrm{imax}},$$

(19)

$$D_{\mathrm{i}\_\mathrm{C}}^t{D}_{\mathrm{i}\_\mathrm{D}}^t=0,$$

(20)

where $P_{\mathrm{imax}}$ represents the rated charging and discharging power of the energy storage system in the MG.

(3)

Constraints on the upper and lower bounds of energy storage capacity are as follows:

$$S_{\mathrm{SOC}}^{\min }{\theta }_{\mathrm{MG}}⩽E_{\mathrm{MG}}^t⩽S_{\mathrm{SOC}}^{\max }{\theta }_{\mathrm{MG}},$$

(21)

where ${\theta }_{\mathrm{MG}}$ represents the energy storage capacity of the MG, $E_{\mathrm{MG}}^t$ represents the energy level of the MG energy storage at time t, and $S_{\mathrm{SOC}}^{\min }$ and $S_{\mathrm{SOC}}^{\max }$ represent the minimum and maximum state of charge of the MG energy storage, respectively.

(4)

The relationship between the energy levels of the MG’s own energy storage in adjacent time periods is as follows:

$$E_{\mathrm{MG}}^t=E_{\mathrm{MG}}^{t-1}+\Delta t\left({\eta }_{\mathrm{MG}}^{\mathrm{C}}{P}_{\mathrm{i}\_\mathrm{C}}^t-P_{\mathrm{i}\_\mathrm{D}}^t/{\eta }_{\mathrm{MG}}^{\mathrm{D}}\right),$$

(22)

where $E_{\mathrm{MG}}^t$ represents the internal energy of the MG’s energy storage at time t, $E_{\mathrm{MG}}^{t-1}$ represents the internal energy of the MG’s energy storage at time t – 1, and ${\eta }_{\mathrm{MG}}^{\mathrm{C}}$ and ${\eta }_{\mathrm{MG}}^{\mathrm{D}}$ are the charging and discharging efficiencies of the MG’s energy storage, respectively. The energy level of the MG’s energy storage at time t is determined by the energy level at the previous time step and the charging or discharging during the time interval.

(5)

Constraints on the MG’s own energy storage having equal energy levels at the beginning and end of the day are as follows:

$$E_{\mathrm{MG}}^0=E_{\mathrm{MG}}^T,$$

(23)

where T is the scheduling period, which is 24 h. $E_{\mathrm{MG}}^0$ and $E_{\mathrm{MG}}^T$ represent the initial and final energy levels of the MG’s energy storage within one scheduling period, respectively.

(6)

Constraints for the micro gas turbine are as follows:

$$\left\{\begin{array}{c}0⩽P_{\mathrm{b}}^t⩽P_{\mathrm{bmax}}\\ P_{\mathrm{b}}^t-P_{\mathrm{b}}^{t-1}⩽R_{\mathrm{up}}\\ P_{\mathrm{b}}^{t-1}-P_{\mathrm{b}}^t⩽R_{\mathrm{down}}\end{array}\right.,$$

(24)

where $P_{\mathrm{bmax}}$ is the rated power of the micro gas turbine, $P_{\mathrm{b}}^{t-1}$ is the output power of the micro gas turbine at time t – 1, and $R_{\mathrm{up}}$ and $R_{\mathrm{down}}$ are the limits for increasing and decreasing the output power of the micro gas turbine, respectively.

(7)

Power constraints of the interconnection line between the MG and ADN are as follows:

$$-P_{\mathrm{dmax}}⩽P_{\mathrm{d}}^t⩽P_{\mathrm{dmax}},$$

(25)

where $P_{\mathrm{dmax}}$ is the maximum transmission power of the interconnection line.

3.3. The ADN Model

3.3.1. The Objective Function of the ADN

As the operating entity, the ADN aims to improve the integration of renewable energy, reduce network losses, and decrease the cost of purchasing electricity from the upper-level grid by leasing an appropriate amount of SES. Utilizing the spatiotemporal transfer characteristics of energy storage ensures the safe and stable operation of the grid:

$$C_{\mathrm{ADN}}=C_{\mathrm{th},\mathrm{wind}}-C_{\mathrm{afd}}+C_{\mathrm{buy}}+C_{\mathrm{loss}}+C_{\mathrm{ADN},\mathrm{rent}},$$

(26)

where $C_{\mathrm{ADN}}$ represents the annual cost of curtailed wind energy for the ADN, $C_{\mathrm{afd}}$ represents the annual cost of purchasing or selling electricity between the MG and the ADN, $C_{\mathrm{buy}}$ represents the annual cost of purchasing electricity from the upper-level grid for the ADN, and $C_{\mathrm{loss}}$ represents the annual cost of network losses for the ADN.

$$C_{\mathrm{th},\mathrm{wind}}=365\times {\displaystyle \sum }_{t=1}^T\left(P_{\mathrm{th},\mathrm{wind}}^t\times K_{\mathrm{wind}}\right),$$

(27)

where $P_{\mathrm{th},\mathrm{wind}}^t$ represents the curtailed wind power of the ADN at time t and $K_{\mathrm{wind}}$ represents the cost of curtailed wind power per unit. Due to the randomness and fluctuations in the output of distributed energy sources, excessive output may cause overvoltage at nodes and power congestion in branches. To ensure the safe operation of the network, it is necessary to curtail some wind power at certain times.

$$C_{\mathrm{buy}}=365\times K_{\mathrm{buy}}^t\times {\displaystyle \sum }_{t=1}^T{P}_{\mathrm{ADN},\mathrm{buy}}^t,$$

(28)

distributed energy sources, the MG, and SES provide power support to the ADN load; however, in the case of a power deficit, the ADN needs to purchase electricity from the upper-level grid. In the annual purchasing cost, $C_{\mathrm{buy}}$, of the ADN from the upper-level grid, $K_{\mathrm{buy}}^t$ represents the unit price of electricity purchased by the ADN from the upper-level grid at time t, and $P_{\mathrm{ADN},\mathrm{buy}}^t$ represents the power purchased by the ADN from the upper-level grid at time t.

$$C_{\mathrm{loss}}=365\times K_{\mathrm{buy}}^t\times {\displaystyle \sum }_{t=1}^T{P}_{\mathrm{loss}}^t,$$

(29)

where $P_{\mathrm{loss}}^t$ represents the network loss of the ADN at time t, and its value is measured by the unit price, $K_{\mathrm{buy}}^t$, of electricity purchased by the ADN from the upper-level grid at time t.

3.3.2. The Constraint Conditions of the ADN

(1)

The voltage constraint at node k of the ADN is as follows:

$$U_k^{\min }⩽U_k⩽U_k^{\max },$$

(30)

where $U_k$ is the actual voltage magnitude at node k. $U_k^{\min }$ and $U_k^{\max }$ are the lower and upper limits of the voltage at node k, respectively.

(2)

The power balance constraint for the ADN is as follows:

$$\left\{\begin{array}{c}{P}_i-U_i{\displaystyle {\displaystyle \sum }_{j\in i}}{U}_j\left(G_{ij}cos{\delta }_{ij}-B_{ij}sin{\delta }_{ij}\right)=0\\ Q_i-U_i{\displaystyle {\displaystyle \sum }_{j\in i}}{U}_j\left(G_{ij}sin{\delta }_{ij}-B_{ij}cos{\delta }_{ij}\right)=0\end{array}\right.,$$

(31)

where $P_i$ and $Q_i$ are the active and reactive power injections at node i, representing all nodes connected to node i, $G_{ij}$ and $B_{ij}$ are the conductance and susceptance of the node admittance matrix, respectively, and ${\delta }_{ij}$ is the voltage phase angle difference between nodes i and j.

(3)

The branch power constraint is as follows:

$$\left|S_{\mathrm{l}}^t\right|⩽S_l^{\max },$$

(32)

where $S_{\mathrm{l}}^t$ and $S_l^{\max }$ represent the apparent power and maximum apparent power of line l at time t, respectively, serving as the power constraint for line l.

4. Model Solution

4.1. Nested Interior Point Method with a Genetic Algorithm

4.1.1. The Stackelberg Game

The game model consists of three elements: the participant set, the strategy set, and the benefit set. The established master–slave game model is as follows:

$$\mathrm{S}=\left\{\begin{array}{c}\left\{\mathrm{SESO}\cup \mathrm{MG}\cup \mathrm{ADN}\right\}\\ \left\{\begin{array}{c}\left\{\alpha ,\beta \right\}\\ \left\{{\theta }_{\mathrm{MG},\mathrm{rent}},{\theta }_{\mathrm{AND},\mathrm{rent}}\right\}\end{array}\right\}\\ \left\{{\mathrm{W}}_{\mathrm{share}},{\mathrm{C}}_{\mathrm{MG}},{\mathrm{C}}_{\mathrm{ADN}}\right\}\end{array}\right\},$$

(33)

(1)

Participant Set: Let $\left\{\mathrm{SESO}\cup \mathrm{MG}\cup \mathrm{ADN}\right\}$ denote the set of participants, where SESO acts as the leader while MGs and an ADN represent the followers.

(2)

The leader’s strategy involves adjusting the leasing capacity and capacity prices of the SES, denoted as $\left\{\alpha ,\beta \right\}$. The followers’ strategy involves continuously adjusting the leasing capacity plan, represented as $\left\{{\theta }_{\mathrm{MG},\mathrm{rent}},P_{\mathrm{MG},\mathrm{rent}},{\theta }_{\mathrm{AND},\mathrm{rent}},P_{\mathrm{AND},\mathrm{rent}}\right\}$, where ${\theta }_{\mathrm{MG},\mathrm{rent}}$ denotes the energy storage capacity leased by MGs and ${\theta }_{\mathrm{AND},\mathrm{rent}}$ represents the energy storage capacity and capacity power leased by the ADN.

(3)

Benefit Set: The benefit of each participant is their objective function: $\left\{{\mathrm{W}}_{\mathrm{share}},{\mathrm{C}}_{\mathrm{MG}},{\mathrm{C}}_{\mathrm{ADN}}\right\}$. Nash equilibrium is reached when no participant can achieve greater profit for themselves by changing their strategy. At this point, the strategy set becomes the equilibrium solution.

4.1.2. Nested Interior Point Method with a Genetic Algorithm

The basic idea of using the nested interior point method with a genetic algorithm in order to solve the optimal capacity and unit capacity lease fee for energy storage sharing is to use the genetic algorithm as a global search algorithm. After each update, each offspring obtained through selection, crossover, and mutation contains a solution vector. This solution vector is used as the initial value for the interior point method, which quickly obtains a locally optimal initial value.

By combining the genetic algorithm with the interior point method, the nested interior point method with a genetic algorithm can fully utilize the advantages of both algorithms, achieving efficient solutions to nonlinear optimization problems. The genetic algorithm ensures the global search capability of the algorithm, making it less prone to local optima, while the interior point method enhances the local search capability, speeding up the solution process.

4.2. Solving the Stackelberg Game Model Based on the Nonlinear Interior Point Genetic Algorithm

This paper combines the characteristics of the interior point method and a genetic algorithm to adopt a hybrid algorithm for solving the optimal capacity and unit capacity leasing fee for energy storage sharing, as well as determining the optimal operation strategy for MGs and ADNs. Specifically, a genetic algorithm nested with the interior point method is used to solve the shared energy storage leasing price scheme within a Stackelberg game optimization model, while the operation strategy is determined using the interior point method. The SESO’s profits are calculated through the energy storage leasing optimization model for both MGs and the ADN, with the shared energy storage capacity optimization model for MGs and the ADN solved using a genetic algorithm nested with the interior point method. The specific steps are shown in Figure 3.

Optimization Model for SES Capacity between MGs and an ADN:

(1)

Initialize the population representing the plan by setting the initial values for the unit leasing capacity prices.

(2)

Import grid data and initialize the population representing an ADN and MGs by setting their initial leasing plans.

(3)

Use the operational cost as the fitness for the MGs and ADN particles. Apply the genetic algorithm to solve their operational and leasing costs and use the interior point method to calculate the operational plan.

(4)

If the convergence condition is met, report the required leasing capacity to the SESO; if not, perform selection, crossover, and mutation on the population to form a new population. Then, for each individual, calculate the fitness for the MGs and ADN again. Repeat step 3. until the convergence condition is met.

A Stackelberg Game Model:

(5)

The SESO establishes the initial leasing price scheme and generates the initial population.

(6)

Use the SESO’s total revenue function as the fitness. Receive the capacity leasing plans from the MGs and ADN and calculate the SESO’s leasing profits and operational costs.

(7)

If the convergence condition is met, output the optimal leasing capacity plan and unit leasing price; if not, perform selection, crossover, and mutation on the population to form a new population. Then, for each individual, calculate the SESO’s fitness again. Repeat step (3) until the convergence condition is met.

5. Results

5.1. Description of the Application Scenario

This paper focuses on a modified IEEE-33 node system as the research object, whose system structure is illustrated in Figure 4. The model setup parameters are as shown in

Table 1. Setting of model parameters.

Parameters	Numeric Values	Unit of Parameter
${\theta }_{\mathrm{MG},\mathrm{own}}$	100	kW·h
$P_{\mathrm{MG},\mathrm{own}}$	200	kW
$\eta$	0.95
$P_{\mathrm{bmax}}$	300	kW
$P_{\mathrm{dmax}}$	150	kW
$K_{\mathrm{wind}}$	3.3	CNY/(kW·h)
$K_{\mathrm{buy}}^t$	0.65	CNY/(kW·h)
$K_{\mathrm{PV}}$	3.3	CNY/(kW·h)
$K_{\mathrm{OM}}$	0.1542	CNY/(kW·h)
$\gamma$	0.1542	CNY/(kW·h)
$C_{\mathsf{\theta },\mathrm{share}}$	1100	CNY/(kW·h)
$C_{\mathrm{p},\mathrm{share}}$	1000	CNY/kW
$r$	0.08
$y$	15	year
$K_{\mathrm{ON}}$	0.128	CNY/(kW·h)
$SO{C}_{\mathrm{ini}}$	0.5
$\delta$	0.1542	CNY/(kW·h)

Where ${\theta }_{\mathrm{MG},\mathrm{own}}$ is an MG’s internal energy storage capacity, $P_{\mathrm{MG},\mathrm{own}}$ is the maximum power of the MG internal energy storage, $SO{C}_{\mathrm{ini}}$ is the initial value of SOC over a 24 h period, and $\delta$ is the unit power operation cost of energy storage.

. An MG is connected at node 13, consisting of wind and solar power generation stations, energy storage systems, micro gas turbines, and loads. The wind power generation unit is connected to node 16 as a distributed renewable energy station, where an SES station is also established.

The model setup parameters are as shown in Table 1. The typical day is illustrated in

Table A1. Typical daily data.

Time/h	Photovoltaic Power/kW	Load/kW	Wind Power/kW
1	0	85.55	65.85934574
2	0	79.5	69.52278845
3	0	73.65	109.3087072
4	0	72.1	99.800368
5	0	73.65	107.9724857
6	102	79.35	120.5800556
7	164.22	88.8	133.0136941
8	255.54	120.8	46.35347706
9	300.3	150.95	109.1667316
10	355.68	178.95	105.1321635
11	367.86	192	111.6689321
12	375.81	147.85	102.1278672
13	374.1	139.4	103.1997468
14	370.38	129.35	124.572721
15	355.77	120.55	98.16952669
16	305.7	134.75	102.6855052
17	255.15	150	111.1108952
18	200.49	155.05	77.2132968
19	96	179.7	91.88930118
20	0	193.8	90.43364187
21	0	179.35	71.56817707
22	0	151.9	64.77175064
23	0	106.75	83.96377487
24	0	90.2	89.48485631

in Appendix A. Time-of-use electricity prices are provided in

Table A2. Time-of-use electricity pricing.

Time/h	Time-of-Use Electricity Pricing/CNY	Time/h	Time-of-Use Electricity Pricing/CNY
1	0.39	13	0.78
2	0.39	14	0.78
3	0.39	15	1.29
4	0.39	16	1.29
5	0.39	17	1.29
6	0.39	18	0.78
7	0.39	19	0.78
8	0.39	20	1.29
9	0.78	21	1.29
10	0.78	22	1.29
11	0.78	23	0.78
12	0.78	24	0.78

in Appendix A.

5.2. Analysis of SESO Benefit

After the implementation of SES, the energy storage leasing situation is as summarized in

Table 2. SES Leasing Status.

Capacity Type	Capacity/(kWh)
SES capacity	1255.52
MG leasing capacity	551.48
ADN leasing capacity	704.04

, and the profit analysis of the SES investor is presented in

Table 3. Operation Status of SES Investors.

Price Parameter Type	Price (CNY/kWh)
Unit capacity leasing fee	4.41
Annual MG leasing fee	887,671.15
Annual ADN leasing fee	1,133,220.63
SESO annual revenue	262,763.7

.

From Table 2 and Table 3, we can see that a SESO, as a third-party investment institution, constructs an energy storage capacity of 1255.52 kWh. The unit capacity leasing fee set by SESO is CNY 4.41/(kW·h). The total SES leased by the MG is 551.48 kWh, incurring an annual leasing fee of CNY 887,700. The SES leased by the ADN is 704.04 kW·h, incurring an annual leasing fee of CNY 1,133,200. Therefore, the SESO’s annual income is CNY 262,800.

The SESO constructs an appropriate capacity with which to provide energy storage capacity leasing services to MGs and the ADN. Setting a reasonable unit capacity leasing fee helps MGs and the ADN reduce wind and solar curtailment while also lowering their operating costs. The SESO charges the MG and ADN corresponding capacity leasing fees, thereby generating revenue.

5.3. Analysis of MG Benefits

The operating cost of MGs consists of the operating costs of the energy storage system and micro gas turbines, the cost of purchasing and selling electricity to the ADN, and the cost of curtailed power generation. A comparison of the internal operating costs of the MGs before and after the introduction of SES is shown in

Table 4. Comparison of MGs Operating Costs Before and After Setting up SES.

Cost Type	Non-SES	SES	Difference
MGs annual cost/(CNY·104)	75.9362	53.4351	22.5011
Energy storage annual cost/(CNY·104)	2.5692	1.8902	0.6790
Diesel generator annual cost/(CNY·104)	3.4007	9.6911	−6.2904
Cost of purchasing and selling electricity to ADN/(CNY·104)	−31.4508	−86.6962	55.2454
Cost of curtailed solar power/(CNY·104)	101.4171	39.7830	61.6341
Rental fee/(CNY·104)	0	88.7671	−88.7671

.

From Table 4, it can be observed that, after the introduction of SES by the third-party investment institution, the annual cost of the MG has significantly decreased, with a reduction ratio of 29.63%. There are two main reasons for this decrease:

(1)

The MG has significantly reduced curtailed power generation costs by leasing SES, with a reduction ratio of 60.77%, thus increasing the proportion of photovoltaic new energy absorption within the MG.

(2)

After leasing SES, the MG can optimize the scheduling of internal micro gas turbines and energy storage, among other flexible resources, based on the time-of-use electricity price information provided by the ADN. This optimization allows the MG to significantly increase its revenue from selling electricity to the ADN.

The power exchange between the MG and the ADN is shown in Figure 5. Negative power indicates that the MG is purchasing electricity from the ADN, while positive power indicates that the MG is selling electricity to the ADN. It can be observed that, after leasing SES, the MG as a whole sells electricity to the ADN to generate revenue. Particularly during peak electricity price periods, from 15:00 to 17:00 and from 20:00 to 22:00, when there is no SES, the MG needs to purchase electricity from the ADN; however, after leasing SES, the MG sells electricity to the ADN during these periods, significantly improving the operational efficiency of the MG.

5.4. Analysis of ADN Benefits

The comparison of ADN operating costs before and after the introduction of SES is shown in

Table 5. Comparison of the ADN Operating Costs Before and After Setting up SES.

Cost Type	Non-SES	SES	Difference
Annual cost of ADN/(CNY·104)	2969.98	2635.94	334.04
Loss cost/(CNY·104)	137.43	116.26	21.17
The cost of purchasing electricity from the higher power grid/(CNY·104)	2668.14	2223.65	444.49
The cost of buying and selling electricity to MGs/(CNY·104)	31.45	86.7	−55.25
The cost of abandoning wind/(CNY·104)	132.96	96.01	36.95
Rental fee/(CNY·104)	0	113.32	−113.32

. It can be seen that, after leasing SES, the annual operating costs of the ADN decreased by 11.25%. This reduction is mainly due to three reasons:

(1)

By leasing SES, the ADN has increased its absorption of wind power, leading to a significant decrease in curtailed wind power generation costs.

(2)

The increase in electricity sales from the MG to the ADN has reduced the ADN’s purchasing of electricity from the higher-level power grid, significantly reducing its purchasing costs.

(3)

The ADN fully utilizes the flexibility of SES to improve its operational efficiency, reduce network losses, and decrease network loss costs. Although the cost of purchasing and selling electricity between the ADN and MG has increased, and the ADN has also paid a leasing fee for SES capacity, the significant reduction in operating costs of the ADN outweighs these costs.

The network losses of the ADN before and after the introduction of SES are shown in Figure 6. From the figure, it can be seen that, after leasing SES, the optimization of ADN operations through SES scheduling significantly reduces network losses.

From Figure 7, it can be seen that, after leasing SES, the curtailed wind power of the ADN significantly decreases. During peak wind power generation periods, such as 4:00–5:00 and 18:00–19:00, a portion of wind power can be stored through SES, effectively smoothing out the peaks and troughs, improving the ADN’s ability to absorb wind power, and reducing curtailed wind power.

Analyzing energy dispatching using a typical day as an example, within the MGs, the micro gas turbine maintains a stable output. From midnight to 05:00, the solar power output is zero, and the electricity price is low. During this period, the internal energy storage of the MGs remains in a charging state. The leased energy storage switches from discharging to charging, while the wind turbine output supports load demand and peak shaving. The surplus electricity is purchased from the distribution grid at a lower price to achieve power balance. From 07:00 to 14:00, the output of wind and solar power is less than the load demand. During this time, the output of renewable energy and leased energy storage discharge prioritizes meeting the load demand, while some excess electricity is used for valley filling. The remaining electricity is sold to the distribution grid at peak electricity prices to make a profit.

5.5. Operation Plans and Analysis of MGs and the ADN under Typical Scenarios

From the perspective of Figure 8, during the time period from 00:00 to 11:00, the distribution grid prioritizes the consumption of its own leased capacity. At the peak electricity demand times of 11:00 and 20:00, SES maintains sufficient capacity. Despite fulfilling the load demand for the remainder of the day, surplus electricity remains available. During this interval, energy storage prioritizes the absorption of surplus electricity, preparing for potential power shortages. Consequently, at 22:00, the distribution grid procures electricity from the MGs to elevate the SOC of the leased energy storage to 0.894. The internal energy storage system of the MGs exhibits several significant “M” shapes overall, following the fluctuations in wind and solar energy. Specifically, between 05:00 and 15:00, it demonstrates a notable trend of rapid charge and discharge, frequently reaching both the upper limit value of 0.9 and the lower limit value of 0.1, indicating pronounced fluctuations between these limits. Conversely, the leased energy storage smoothly provides power externally. During the period from 15:00 to 17:00, characterized by high electricity prices, the MGs sell electricity to the distribution grid. In the low-price period from 18:00 to 19:00, rapid charging is observed to supply electricity during the peak period from 20:00 to 22:00, enabling arbitrage. Throughout the entire duration, the SOC of SES exhibits a complete cycle, ranging from the initial value of 0.5 at time 0 to 0.5 at time 24, facilitating subsequent power interactions with various entities. SES not only enhances the MG’s capacity for renewable energy integration but also mitigates excessive electricity procurement costs during peak periods.

6. Conclusions

Based on the Stackelberg game theory, this paper proposes a capacity configuration method for SES considering wind and solar power integration. The method integrates optimization problems related to energy storage capacity configuration and operation. It treats a third-party SES investor as the primary player and the ADN as well as MGs as subordinate players. The proposed method employs a genetic algorithm nested within an interior point method for solutions. Through iterative optimization to reach a Nash equilibrium, the following main conclusions are drawn:

(1)

A SESO, as a third-party investment institution, constructs energy storage capacity to meet the energy storage demand of MGs and the ADN. It invests in energy storage power stations and charges corresponding capacity leasing fees to MGs and the ADN. This assists MGs and the ADN in reducing wind and solar curtailment, thereby lowering the operational costs of MGs and the ADN, thus generating profit.

(2)

By leasing SES, the annual cost for MGs is significantly reduced by 29.63%. The curtailment cost is reduced by 60.77%, significantly increasing the absorption rate of photovoltaic new energy within MGs. By optimizing internal flexible resources based on the ADN’s time-of-use electricity price information, the timing and amount of power interacting with ADN are optimized, resulting in a significant increase in the revenue of MGs from selling electricity to the ADN.

(3)

After leasing SES, the annual operational cost of the ADN decreases by 11.25%. There is a significant reduction in wind curtailment costs. The ADN decreased their purchased electricity from the upper-level power grid, significantly reducing the cost of purchased electricity from the upper-level power grid. By reducing network losses, the cost of network losses is reduced, and the annual operational cost of the ADN is still significantly reduced.

This paper does not consider the effect of wind–landscape randomness on the allocation of shared energy storage capacity and only verifies the effectiveness of the proposed method under the time-of-use model of a typical day. Subsequent research work will consider using a real-time price model and the random process to explore a more perfect effective optimization configuration method so as to promote the microgrid and coordinated sustainable development of the ADN.