A Stackelberg Game Approach for Collaborative Operation and Interest Balancing in Community-Based Integrated Energy Microgrids

Abstract

To address the limitation of traditional microgrid operator-led optimization models that compromise user-side benefits, this paper proposes a novel method for the collaborative optimal operation strategy of community-based integrated energy microgrids and diversified flexible resources. The method deeply integrates user-side flexibility resources into the decision-making process. Unlike existing research that only considers electro-heat coupling, our model integrates shared energy storage services into an integrated energy system, reflecting a more realistic future application. A Stackelberg game framework is then established with the microgrid operator (MGO) as the leader and the user aggregator as the follower. The user aggregator optimizes its energy strategy by coordinating user demand response, thereby increasing the profits of both itself and the shared energy storage operator. Meanwhile, this model guides the MGO’s pricing decisions for electricity and heat, balancing interests of all stakeholders. To solve the model, a hierarchical approach that merges the Harris Hawks Optimization algorithm with the CPLEX solver is employed. Finally, simulation results demonstrate that the proposed model and strategy significantly enhance user-side revenue and flexibility, achieve a win-win outcome for the user aggregator and MGO, and lay the foundation for future shared energy storage service providers to participate in market pricing as key game entities.

1. Introduction

Driven by the global energy transition and the low-carbon emission targets, the integrated energy microgrid (IEM), as a key technology for enhancing energy utilization efficiency and promoting the consumption of renewable energy, has become a research hotspot in both academia and industry [1,2,3,4,5]. Generally, IEM contains multiple energy sources, including electricity, natural gas, heat energy, etc. With the continuing development of energy conversion technologies, the coupling among energy sources in the IEM is becoming stronger. The optimal dispatch for IEM is to coordinate different energy systems and therefore it is different from the traditional power system dispatch that only focuses on electricity.

Currently, research on IEM has primarily evolved along the following two directions: multi-energy coordination and the integration of energy storage system. In the realm of multi-energy coordination, early studies established Stackelberg game models between the microgrid operator (MGO) and prosumer groups, typically focusing solely on electricity pricing strategies [6,7]. Subsequent research recognized the importance of heat energy, beginning to incorporate heat pricing into the operator’s regulatory strategy and considering the flexible adjustment of user-side heat loads [8,9,10,11]. A critical shortcoming of these models, however, is the neglect of energy storage’s role in mitigating fluctuations and enabling peak-shaving arbitrage, which restricts the system’s overall flexibility.

To address this issue, scholars have incorporated energy storage systems into the optimization framework. Reference [12] developed a collaborative optimization model for distribution network demand-side electricity pricing, energy storage operation strategies, and capacity configuration, aiming to maximize the profit of the distribution network. Driven by the sharing economy, which reduces economic costs and improves resource utilization [13,14], the shared energy storage model has emerged, addressing drawbacks of independent energy storage, such as high investment costs and the inability to achieve multi-party electricity complementarity, through resource integration [15]. Reference [16] pioneered a Stackelberg game model between a shared energy storage provider and multiple photovoltaic (PV) prosumers, to achieve win-win outcomes. Reference [17] verified that shared energy storage enhances revenues of residential/industrial users while creating profit opportunities for operators. Reference [18] further formulated the energy storage capacity configuration as an optimization problem, accounting for factors such as energy storage costs, electricity pricing systems, and load forecasting errors.

To achieve the efficient utilization of distributed energy, peer-to-peer (P2P) energy trading has emerged as an innovative trading paradigm [19,20]. Unlike the IEM, it enables prosumers to trade electricity directly, boosting their economic returns. Reference [21] proposed a community-level prosumers game-theoretic model, where buyers can adjust their energy consumption based on sellers’ price and quantity, with sellers’ price competition modeled as a non-cooperative game. Reference [22] optimized the energy storage systems-equipped trading strategies via three game-theoretic approaches. Recently, Reference [23] introduced a carbon-coupled network charge-guided bi-level optimization method for distribution system operators and prosumers to enhance economic efficiency. However, there is still a lack of comprehensive consideration of user-side load characteristics, specifically shiftable, curtailable, and transferable loads, as well as the integrated electric-heat demand response [24,25].

To address the limitation of traditional microgrid operator-led optimization models that compromise user-side benefits, this work proposes a community integrated energy system collaborative optimization approach based on Stackelberg game framework. The main contributions are as follows:

(1) A Stackelberg game-based collaborative optimization model is established to resolve interest conflicts and sequential decision-making between MGOs and user aggregators. As the leader, the MGO sets internal electricity and heat prices to maximize profits; as followers, user aggregators minimize costs by optimizing consumption strategies via integrating demand response (shiftable electrical loads, curtailable heat loads) and shared storage services.

(2) A shared energy storage operator is incorporated as an independent third-party provider. By treating the shared storage service as a “time-shifting” mechanism available to the user aggregator, the proposed method enhances the flexibility of the user side, enabling better management of renewable generation (PV and wind) intermittency and participation in peak-shaving arbitrage, thereby improving overall system efficiency beyond what individual user-side storage could achieve.

(3) A novel hybrid solution method is proposed for the complex bi-level optimization problem. The upper-level pricing problem (non-linear and non-convex) is tackled using the Harris Hawks Optimization (HHO) algorithm, selected for its superior global search capability. The lower-level scheduling problem is formulated as a Mixed-Integer Linear Programming (MILP) model and solved exactly by CPLEX. This combination ensures both computational efficiency for strategic pricing layer and optimality for operational scheduling.

The remainder of this paper is organized as follows: Section 2 presents the framework for the community-based integrated energy microgrid. The model for the MGO, shared energy storage operator and user aggregator is presented in Section 3 and Section 4, respectively. A Stackelberg game approach for collaborative operation and interest balancing is described in Section 5. In Section 6, a case study is conducted and the results are discussed. Finally, the conclusion is provided in Section 7.

2. Community-Based Integrated Energy Microgrid Framework

The community-based integrated energy microgrid (IEM) system architecture studied in this work is primarily composed of the MGO, the user aggregator, and the shared energy storage operator. The physical connections and information interactions among them are illustrated in Figure 1.

2.1. Microgrid Operator (MGO)

The MGO serves as a critical link between the microgrid and users, with dual roles in energy trading. From a market perspective, the MGO operates determines day-ahead internal electricity and heat prices for the user aggregator based on forecasted time-of-use (TOU) grid prices and its own operating costs, earning revenue through arbitrage margins. From a physical perspective, the MGO owns and operates distributed energy resources, such as micro gas turbines (MTs) and gas boiler (GB), to directly supply power and heat, ensuring local regional energy stability.

2.2. User Aggregator

The user aggregator represents the collective interests of the user side, responsible for integrating and optimizing the energy resources of its affiliated users. Within a microgrid, distributed user groups are typically modeled as a single user aggregator entity, with core functions and operational mechanisms structured as follows:

(1) Generation and Load: Each user is equipped with a photovoltaic (PV) and wind power system, forming a distributed generation unit. Simultaneously, the aggregator manages two types of flexible loads: electrical and heat.

(2) Operational Strategy: The aggregator prioritizes self-consumption of locally generated PV and wind power. In case of a deficit, it purchases electricity from the MGO or draws power from the shared energy storage system. Conversely, when PV and wind generation exceeds local demand, surplus generation can be sold back to the MGO or stored in the shared energy storage system.

(3) Heat management: For heat demand, the aggregator adopts a dual-supply mode. Part of the heat demand is met by the MGO, while the remaining portion is fulfilled by adjusting the shared energy storage system. This structure significantly enhances the flexibility and operational autonomy of the user side.

2.3. Shared Energy Storage Operator

The shared energy storage operator, acting as an independent third-party provider, offers capacity leasing services to user aggregators. Its primary revenue stream comes from is a service charged for the energy capacity users store or withdraw. By providing an effective energy “time-shifting” mechanism for energy, shared storage enables aggregators to manage the intermittency of PV and Wind generation, participate in demand response programs, and exploit peak-shaving arbitrage opportunities.

3. MGO Model

The MGO, acting as the leader, formulates its strategy by setting internal energy prices. To guide user-side energy behavior and maximize its own revenue, the MGO partitions the day into T discrete time slots and determines the electricity and heat prices for each time interval t (t = 1, 2, …, T).

3.1. Pricing Constraints

To ensure trading activity between the MGO and the grid, the MGO’s pricing strategy needs to satisfy

$${\lambda }_t^{EG,b}<{\lambda }_t^{MGO,s},{\lambda }_t^{MGO,b}<{\lambda }_t^{EG,s}$$

(1)

$${\gamma }_t^{MGO,\min }<{\gamma }_t^{MGO,s},{\gamma }_t^{MGO,b}<{\gamma }_t^{MGO,\max }$$

(2)

where ${\lambda }_t^{EG,b}$ and ${\lambda }_t^{EG,s}$ denote the grid’s electricity purchasing and selling prices in period t, respectively; ${\lambda }_t^{MGO,b}$ and ${\lambda }_t^{MGO,s}$ denote the MGO’s electricity purchasing and selling prices in period t, respectively; ${\gamma }_t^{MGO,b}$ and ${\gamma }_t^{MGO,s}$ denote the MGO’s heat purchasing and selling prices in period t, respectively; ${\gamma }_t^{MGO,\min }$ and ${\gamma }_t^{MGO,\max }$ are the minimum and maximum heat prices in period t, respectively.

Additionally, the electricity and heat selling prices must satisfy the following conditions:

$${\displaystyle \sum _{t=1}^T{\lambda }_t^{MGO,s}}\le T\times {\lambda }_{avg}^e$$

(3)

$${\displaystyle \sum _{t=1}^T{\gamma }_t^{MGO,s}}\le T\times {\gamma }_{avg}^h$$

(4)

where ${\lambda }_{avg}^e$ and ${\gamma }_{avg}^h$ represent the average selling prices of electricity and heat, respectively. As indicated by Equations (3) and (4), if the average price at which the MGO sells to users is higher than the average selling price of the grid, users may opt to purchase electricity directly from the grid.

3.2. MGO Equipment Modeling

The MGO utilizes micro gas turbine (MT) and gas boiler (GB), fueled by natural gas. Considering the variable-load efficiency characteristics of the GB and MT, the relationship between their output power and fuel cost is formulated as a quadratic function [26]:

$$C_t^{MT}=a_e{\left(P_t^{MT,e}\right)}^2+b_e{P}_t^{MT,e}+c_e+a_h{\left(P_t^{GB,h}\right)}^2+b_h{P}_t^{GB,h}+c_h$$

(5)

where $P_t^{MT,e}$ is the electrical power output of the MT in period t; $P_t^{GB,h}$ is the heat power output of the GB in period t; a_e, b_e, c_e (a_h, b_h, c_h) are the cost coefficients of the MT (GB), respectively.

In the integrated energy system, some parts of heat load supply are provided through waste heat exchangers. Ignoring the heat loss rate, the relationship between the recovered waste heat and the MT’s electrical power output in period t can be formulated as

$$P_t^{MT,h}={\displaystyle \frac{1-{\eta }_e^{MT}}{{\eta }_e^{MT}}}{\eta }_B{P}_t^{MT,e}$$

(6)

where $P_t^{MT,h}$ is the heat output of the MT in period t; ${\eta }_e^{MT}$ is the power generation efficiency; ${\eta }_B$ is the coefficient of performance for heating.

During the economic dispatch of the IEM, the MT is required to regulate its output in real time in response to dynamic load fluctuations. However, due to inherent output power limits and regulation characteristics dictated by its physical attributes, the MT’s output and ramp rate are typically restricted to a specific range, as expressed by the following equations:

$$0\le P_t^{MT,e}\le P_{\max }^{MT,e}$$

(7)

$$P_{down}^{MT,e}\le P_t^{MT,e}-P_{t-1}^{MT,e}\le P_{up}^{MT,e}$$

(8)

where $P_{\max }^{MT,e}$ represents the maximum output power; $P_{down}^{MT,e}$ and $P_{up}^{MT,e}$ denote the lower and upper limits of its ramp rate, respectively.

Since the MT cannot meet the total heat load, the GB supplies the shortfall. Similarly, the following conditions must be constrained:

$$0\le P_t^{GB,h}\le P_{\max }^{GB,h}$$

(9)

$$P_{down}^{GB,h}\le P_t^{GB,h}-P_{t-1}^{GB,h}\le P_{up}^{GB,h}$$

(10)

where $P_{\max }^{GB,h}$ denotes the maximum heat output; $P_{down}^{GB,h}$ and $P_{up}^{GB,h}$ denote the lower and upper limits of its ramp rate, respectively.

3.3. MGO Profit Function

In this work, the revenue of the MGO primarily derives from electricity and heat trading after deducting operational cost. The profit is formulated as follows:

$$C_{MGO}=C_{MGO}^{EG,e}+C_{MGO}^{f,h}-C_{MT}$$

(11)

where $C_{MGO}^{EG,e}$ represents the profit from electricity trading with the microgrid over the day; $C_{MGO}^{f,h}$ represents the daily revenue from heat sales to the user side; and $C_{MT}$ represents the operational cost.

3.3.1. Electricity Trading Revenue $C_{MGO}^{EG,e}$

For electricity trading, when the output of the MT is in surplus, the MGO gains revenue by selling electricity to the microgrid and supplying electricity to the user side, respectively. Conversely, the MGO purchases electricity from the microgrid when the MT output is insufficient. Considering that the electricity generated by distributed photovoltaics and wind power at the user side is traded directly with the MGO, the total revenue can be expressed as follows:

$$C_{MGO}^{EG,e}={\displaystyle \sum _{t=1}^T\left[\begin{array}{l}{\lambda }_t^{EG,s}\max \left(\left(L_t^{f,l}+P_t^{MT,e}\right)-L_t^{f,e},0\right)\\ +{\lambda }_t^{EG,b}\min \left(\left(L_t^{f,l}+P_t^{MT,e}\right)-L_t^{f,e},0\right)\\ +{\lambda }_t^{MGO,s}{L}_t^{f,e}-{\lambda }_t^{MGO,b}{L}_t^{f,l}\end{array}\right]}$$

(12)

where $L_t^{f,e}$ is the total electrical load of user aggregator; $L_t^{f,l}$ is the sum of the PV output power $P_t^{PV}$ and wind output power $P_t^{WT}$ from the user aggregator in period t, and is expressed as

$$L_t^{f,l}=P_t^{PV}+P_t^{WT}$$

(13)

3.3.2. Heat Trading Revenue $C_{MGO}^{f,h}$

Similarly, for heat trading, the MGO’s total revenue from supplying heat to internal loads via MT and GB is calculated as

$$C_{MGO}^{f,h}={\eta }_h{\displaystyle \sum _{t=1}^T{\gamma }_t^{MGO,s}\left(P_t^{GB,h}+P_t^{MT,h}\right)}$$

(14)

where ${\eta }_h$ represents the heat exchange efficiency.

3.3.3. Operational Cost $C_{MT}$

Regarding the operating costs of the MGO, fuel costs constitute the primary expenditure when the operation and maintenance costs of the MG, GB and other equipment are neglected. The total operational cost is calculated as follows:

$$C_{MT}={\displaystyle \sum _{t=1}^T{C}_t^{MT}}$$

(15)

4. Shared Energy Storage Operator Model

The shared energy storage system is an important component of the IEM. It achieves efficient energy utilization and optimized temporal allocation by storing and releasing energy. In this work, the shared energy storage system is managed as an independent asset providing an energy time-shifting service. Its operation adheres to the principle of energy conservation.

4.1. Operational Dynamics

For a community-level IEM with n users, batteries are widely utilized as the energy storage devices in the shared energy storage system. Generally, the State of Charge (SOC) of the shared electrical energy storage system at time t + 1 on the dispatch day is characterized by the following discrete-time state equation [7,22]:

$$E_{t+1}^{ESS}=E_t^{ESS}+\left({\eta }_c^{ESS}{P}_t^{f,c}-{\displaystyle \frac{P_t^{f,d}}{{\eta }_d^{ESS}}}\right)\mathsf{\Delta }t$$

(16)

where $E_{t+1}^{ESS}$ and $E_t^{ESS}$ represent the electrical energy storage capacity at times t + 1 and t; $P_t^{f,c}$ and $P_t^{f,d}$ are the charging and discharging power at period t; ${\eta }_c^{ESS}$ and ${\eta }_d^{ESS}$ are two parameters cover energy losses during the charging and discharging, respectively; $\mathsf{\Delta }t$ is time step duration (i.e., the time step between t and t + 1).

4.2. System Constraints

To ensure that the total charging or discharging power must not exceed the capacity limit of the shared energy storage system at any time during the period T, the system capacity $E_t^{ESS}$ at time t must satisfy the following constraint:

$$E_{\min }^{ESS}\le E_t^{ESS}\le E_{\max }^{ESS}$$

(17)

where $E_{\min }^{ESS}$ and $E_{\max }^{ESS}$ represent the minimum and maximum allowable SOC of the shared energy storage system, respectively.

To ensure sustainable and cyclic operation, the initial and final State of Charge (SOC) levels are set to be equal, i.e.,

$$E_0^{ESS}=E_T^{ESS}$$

(18)

Additionally, the charging and discharging power in any time period is limited by the allowable charging and discharging power of the shared energy storage system. Therefore, for all time t, the following conditions must be satisfied:

$$0\le P_t^{f,c}\le P_{\max }^{ESS,c}, 0\le P_t^{f,d}\le P_{\max }^{ESS,d}$$

(19)

where $P_{\max }^{ESS,c}$ and $P_{\max }^{ESS,d}$ are the maximum allowable charging and discharging power, respectively.

Similarly, for the heat storage system, the heat capacity $Q_t^{ESS}$ and the charging (discharging) heat power $Q_t^{f,c}$ ($Q_t^{f,d}$) must satisfy:

$$Q_{\min }^{ESS}\le Q_t^{ESS}\le Q_{\max }^{ESS}$$

(20)

$$0\le Q_t^{f,c}\le Q_{t,\max }^{f,c}, 0\le Q_t^{f,d}\le Q_{t,\max }^{f,d}$$

(21)

where $Q_{\min }^{ESS}$ and $Q_{\max }^{ESS}$ represent the minimum and maximum allowable heat capacity, respectively. $Q_{t,\max }^{f,c}$ and $Q_{t,\max }^{f,d}$ the maximum allowable charging and discharging power, respectively.

4.3. Cost and Revenue Model

Generally, users pay a service fee for using the storage. The daily electricity revenue associated with electrical storage services is

$$F_e^{ESS}={\displaystyle \sum _{t=1}^T\left(-{\lambda }_t^{ESS,c}{P}_t^{f,c}+{\lambda }_t^{ESS,d}{\eta }_d^{ESS}{P}_t^{f,d}\right)}\mathsf{\Delta }t$$

(22)

For simplicity, ${\lambda }_t^{ESS,c}$ and ${\lambda }_t^{ESS,d}$ are assumed to be equal the MGO’s internal electricity purchasing and selling prices in period t.

Similarly, for the shared heat energy storage system, the operator’s daily heat revenue can be expressed as

$$F_h^{ESS}={\displaystyle \sum _{t=1}^T\left(-{\gamma }_t^{ESS,c}{Q}_t^{f,c}+{\gamma }_t^{ESS,d}{\eta }_d^Q{Q}_t^{f,d}\right)}\mathsf{\Delta }t$$

(23)

where ${\gamma }_t^{ESS,c}$ and ${\gamma }_t^{ESS,d}$ are assumed to be equal the MGO’s heat purchasing and selling prices in the same period; ${\eta }_d^Q$ is the loss coefficient during the discharging heat.

5. User Aggregator Model

In this work, the user aggregator plays a vital role in the electricity market. Its core function lies in acting as a bridge between users and the MGO. Since the selling prices are given by the MGO, user aggregator can conduct demand response by utilizing load characteristics, in order to maximize the consumer surplus by optimizing the flexible loads.

5.1. Load Characteristic

In this work, the total electrical load $L_t^{f,e}$ is categorized into inflexible load $L_t^{e,rigid}$ and flexible load $L_t^{e,flex}$, and is expressed as

$$L_t^{f,e}=L_t^{e,rigid}+L_t^{e,flex}·$$

(24)

where flexible load $L_t^{e,flex}$ possesses high flexibility and does not require fixed power supply times, users can adjust their power consumption timing according to electricity prices, thereby enabling demand response. Therefore, it is subject to demand response constraints:

$$0\le L_t^{e,flex}\le L_{t,\max }^{e,flex}$$

(25)

$${\displaystyle \sum _{t=1}^T{L}_t^{e,flex}}=L^{flex}$$

(26)

where $L_{t,\max }^{e,flex}$ represents the upper limit of the shiftable load at time t; $L^{flex}$ denotes the total amount of shiftable load over the T time periods, ensuring the load is shifted rather reduced.

Similarly, the heat load consists of rigid heat load $L_t^{h,rigid}$ and curtailable heat load $L_t^{h,cut}$, and is defined as

$$L_t^{f,h}=L_t^{h,rigid}+L_t^{h,cut}$$

(27)

where $L_t^{h,cut}$ denotes the curtailable heat load that users can adjust in response to heat prices and demand signals. Generally, in accordance with user comfort levels and energy supply sufficiency, $L_t^{h,cut}$ should be subject to the following constraint:

$$0\le L_t^{h,cut}\le L_{t,\max }^{h,cut}$$

(28)

where $L_{t,\max }^{h,cut}$ is the upper limit of the curtailable heat load at time t.

By explicitly modeling the shifting potential of electricity and the curtailment capability of heat, user aggregator can adjust the flexible loads through demand response, thereby optimizing consumer surplus in accordance with the dynamic pricing mechanism.

5.2. Objective Function

In microeconomics, the utility function is commonly employed to measure consumer satisfaction resulting from the consumption of a particular good. To quantify the satisfaction consumers derive from purchasing electricity and heat, the utility function serves as an effective metric [27]. Therefore, in this work, the utility obtained by users from energy consumption can be expressed as

$$C_t^f=v_e{L}_t^{f,e}-{\displaystyle \frac{{\alpha }_e}{2}}{\left(L_t^{f,e}\right)}^2+v_h{L}_t^{f,h}-{\displaystyle \frac{{\alpha }_h}{2}}{\left(L_t^{f,h}\right)}^2$$

(29)

where v_e, α_e, v_h, α_h are the preference coefficients reflecting the user’s demand elasticity for electricity and heat, respectively.

During energy trading, the user aggregator determines its optimal demand for electrical and heat energy based on unit prices to maximize consumer surplus, as formulated as follows:

$$F_{user}={\displaystyle \sum _{t=1}^T\left[C_t^f-\left({\lambda }_t^{MGO,s}{L}_t^{f,e}+{\gamma }_t^{MGO,s}{L}_t^{f,h}\right)+{\lambda }_t^{MGO,b}{L}_t^{f,l}-\left(F_e^{ESS}+F_h^{ESS}\right)\right]}$$

(30)

It can be observed that Equation (30) consists of four components: the utility from energy consumption, the purchasing cost of electricity and heat, the revenue from selling PV and wind power, and the purchasing cost of energy storage.

6. A Stackelberg Game-Based Collaborative Optimization and Solution for Integrated Energy Systems

6.1. Stackelberg Game-Based Collaborative Optimization Model

In a community-based integrated energy system, the decision-making interaction between the MGO and the user aggregator is a typical sequential process. First, the MGO, acting as the market leader and price setter, determines the electricity and heat purchase and selling prices based on the supply-demand relationship and market information to maximize its own profit. It should be noted that the purchase and selling prices of the shared energy storage provider are also determined by the MGO in this work. Consequently, the storage operator acts as a passive facility subject to fixed constraints and is, therefore, not included as a player in the game model.

Subsequently, the user aggregators, acting as rational market participants, solve their own cost minimization based on the received price signals (including electricity, heat prices and shared energy storage leasing cost determined by the MGO), thereby determining their optimal electricity and heat purchasing plan. This plan is then transmitted back to the MGO as feedback information.

Since both parties aim to maximize their own benefits, their objectives are inherently in conflict. The MGO seeks to maximize its profit through pricing strategies (covering electricity, heat and shared energy storage), while the user aggregator endeavors to minimize its energy cost by adjusting its demand. This cyclical interaction of “one party sets the price, the other responds” continues until a state is reached where neither party can gain a higher profit by unilaterally changing its strategy. At this point, the game reaches an equilibrium.

Given this sequential decision-making and conflicting interests, the interaction between the MGO and the user aggregator can be modeled as a Stackelberg game G, expressed as

$$G=\left\{\left(MGO\cup f\right);{\lambda }^{MGO,b},{\lambda }^{MGO,s},{\gamma }^{MGO,b},{\gamma }^{MGO,s};F_{IEM},F_{MGO},F_{user}\right\}$$

(31)

where the MGO is the leader and the user aggregator f serves as the follower; $F_{user}$ represents the daily profit of the user aggregator, calculated by Equation (30); $F_{IEM}$ denotes the daily profit of the IEM to pursue the maximum value, and is defined by

$$\max F_{IEM}=C_{sell}+C_{ESS}-C_{EG}-C_{MT}-C_{penalty}$$

(32)

where $C_{sell}$ represents the daily revenue from the user aggregator, expressed as

$$C_{sell}={\displaystyle \sum _{t=1}^T\left({\lambda }_t^{MGO,s}{L}_t^{f,e}+{\gamma }_t^{MGO,s}{L}_t^{f,h}\right)};$$

(33)

$C_{ESS}$ is the revenue from the shared energy storage system, expressed as

$$C_{ESS}=F_e^{ESS}+F_h^{ESS};$$

(34)

$C_{EG}$ is the revenue from selling electricity to the grid, expressed as

$$C_{EG}={\displaystyle \sum _{t=1}^T\left(\begin{array}{l}{\lambda }_t^{EG,s}\max \left(L_t^{f,e}-\left(L_t^{f,l}+P_t^{MT,e}\right),0\right)+\\ {\lambda }_t^{EG,b}\min \left(L_t^{f,e}-\left(L_t^{f,l}+P_t^{MT,e}\right),0\right)\end{array}\right);}$$

(35)

and $C_{penalty}$ denotes the penalty cost for heat supply interruptions, given by

$$C_{penalty}=\zeta \times {\displaystyle \sum _{t=1}^T\left(\max \left(L_t^{f,h}-\left(P_t^{GB,h}+P_t^{MT,h}\right),0\right)\right),}$$

(36)

where $\zeta$ is the penalty coefficient for a heat supply interruption.

6.2. Constraints

In this work, in addition to satisfying the operational constraints of the microgrid operator, the shared energy storage provider, and the user aggregators, the electrical and heat power balances must be satisfied as follows:

(1) Electrical Power Balance

The electrical power balance requires that the power supplied by generation units, such as the MT, PV, wind and microgrid, matches the electrical load of the users. Maintaining this balance is fundamental to system stability. Therefore, the electrical power balance constraint is formulated as follows:

$$P_t^{PV}+P_t^{WT}+P_t^{MT,e}+P_t^{net}-P_t^{f,c}+{\eta }_d^{ESS}{P}_t^{f,d}=P_{Load}+L_t^{e,flex}$$

(37)

where $P_t^{net}$ represents the interactive power between the MGO and the microgrid (positive for selling, negative for buying), and P_load represents the electrical load demand.

(2) Heat Power Balance

Similarly, the heat power balance constraint ensures that the total heat generation matches the total heat demand at any given time. This equilibrium is essential for the reliable supply of heating services. Therefore, the heat power balance constraint is formulated as:

$${\eta }_h\left(P_t^{MT,h}+P_t^{GB,h}\right)-Q_t^{f,c}+Q_t^{f,d}=Q_{Load}+\mathsf{\Delta }{L}_t^{f,h}$$

(38)

where ${\eta }_h$ is the heat exchange coefficient, Q_load is the heat load demand, and $\mathsf{\Delta }{L}_t^{f,h}$ denotes the curtailable heat load, which satisfies the following constraints:

$${\displaystyle \sum _{t=1}^T\mathsf{\Delta }{L}_t^{f,h}=0}$$

(39)

$$0\le \left|\mathsf{\Delta }{L}_t^{f,h}\right|\le L_{t,\max }^{h,cut}$$

(40)

where $L_{t,\max }^{h,cut}$ is the maximum amount of heat load that can be curtailed at time t.

During the dynamic scheduling of the IEM, the electrical and heat power balance constraints mentioned above serve as essential conditions for ensuring system stability. These constraints mandate an equilibrium between the total energy supply and total energy demand within the system.

6.3. Solution Method and Procedure

In the proposed game model, it can be considered as the bi-level optimization problem. In the upper-level problem. The MGO (leader) aims to maximize its daily profit by determining the optimal electricity and heat prices. In the lower-level problem, the user aggregator (follower) aims to maximize its daily profit by optimizing its flexible electricity load schedule, heat load curtailment, and its participation in the shared energy storage system.

To tackle the non-linearity of the upper-level problem (MGO pricing), the Harris Hawks Optimization (HHO) algorithm is employed for its robust global search. In contrast, the lower-level strategy problem is formulated as a Mixed-Integer Linear Programming (MILP) model and solved exactly using the CPLEX solver.

6.3.1. Harris Hawks Optimization Method

The HHO algorithm is inspired by the cooperative behavior and surprise attacks of Harris hawks [28]. It operates in the following three phases: global exploration, a transition from exploration to exploitation, and local exploitation. Within the algorithm, each hawk’s position is a candidate solution, while the prey represents the best solution.

(1) Global exploration

In the exploration phase, the Harris hawk population perches randomly at different locations, using their sharp eyesight to track and detect prey across the search space. A global search for the prey is conducted with equal probability using one of two strategies. If q < 0.5, each hawk moves based on the positions of other members and the prey. If q ≥ 0.5, the hawks randomly perch on a random tree within the range of population. The corresponding equations are as follows:

$$X\left(t+1\right)=\left\{\begin{array}{ll}{X}_{rand}\left(t\right)-r_1\left|X_{rand,t}-2r_{2}X\left(t\right)\right|& \hfill q\ge 0.5\\ \left(X_{rabbit}\left(t\right)-X_m\left(t\right)\right)-r_3\left(lb+r_4\left(ub-lb\right)\right)& \hfill q<0.5\end{array}\right.$$

(41)

where X(t) and X(t + 1) are the positions of an individual in the current and the next iteration, respectively, and t is the current iteration number. X_rand(t) is the position of a randomly selected individual, and X_rabbit(t) is the position of the prey (i.e., the individual with the best fitness). r₁, r₂, r₃, r₄, and q are random numbers in the interval [0, 1]. The parameter q is used to randomly select the strategy. X_m(t) is the average position of the population, expressed as

$$X_m\left(t\right)={\displaystyle \frac{1}{M}}{\displaystyle \sum _{i=1}^M{X}_i\left(t\right)}$$

(42)

(2) Transition from exploration to exploitation

The HHO algorithm divides the hunting process into exploration and exploitation behaviors based on the predatory habits of Harris hawks. During its escape, the prey’s energy gradually decreases. Therefore, the prey’s escape energy is used to dynamically select between exploration and exploitation. The prey’s escape energy is defined as

$$E=2E_0\left(1-{\displaystyle \frac{t}{T}}\right)$$

(43)

where E₀ is the initial escape energy of the prey, a random number in the interval [−1, 1]; t is the current iteration number; and M is the maximum number of iterations. When |E| ≥ 1, the algorithm is in the exploration phase. When |E| < 1, it transitions to the exploitation phase.

(3) Local exploitation

In the exploitation phase, after spotting the prey, the hawks encircle it, awaiting a chance to strike. The hunt is complex, as the prey can still escape, forcing the hawks to adapt. To model this, HHO uses the following four strategies: soft besiege, hard besiege, soft besiege with progressive rapid dives, and hard besiege with progressive rapid dives.

Let Sp be the prey’s escape probability, a random number in the interval (0, 1). Sp < 0.5 indicates that the prey has an opportunity to escape. The hunting strategy is then determined by combining the prey’s escape energy, |E|, and its escape probability, Sp.

Case 1: Soft Besiege (0.5 ≤ |E| < 1 and Sp ≥ 0.5)

The prey still has enough energy to escape and attempts to break out of the encirclement through random jumps. In this scenario, the hawks employ a soft besiege to exhaust the prey, creating an opportunity for a sudden surprise attack. The position update formula is as follows:

$$X\left(t+1\right)=\mathsf{\Delta }X\left(t\right)-E\left|J{X}_{rabbit}\left(t\right)-X\left(t\right)\right|$$

(44)

where $\mathsf{\Delta }X\left(t\right)=X_{rabbit}\left(t\right)-X\left(t\right)$ represents the difference between the prey’s position and the individual’s current position, and J is a random number in the interval [0, 2].

Case 2: Hard Besiege (|E| < 0.5 and Sp ≥ 0.5)

The prey has no energy left to escape. The Harris hawks then employ a hard besiege to capture the prey for the final surprise attack. The position update formula is as follows:

$$X\left(t+1\right)=X_{rabbit}\left(t\right)-E\left|\mathsf{\Delta }X\left(t\right)\right|$$

(45)

Case 3: Soft Besiege with Progressive Rapid Dives (0.5 ≤ |E| < 1 and Sp < 0.5)

The prey has sufficient energy to evade the hawks. However, the hawks will employ a soft besiege with progressive rapid dives, gradually correcting their position and direction based on the prey’s deceptive maneuvers. This is implemented by comparing two potential moves and selecting the better one. The update formula is as follows:

$$X\left(t+1\right)=\left\{\begin{array}{ll}Y:X_{rabbit}\left(t\right)-E\left|J{X}_{rabbit}\left(t\right)-X\left(t\right)\right|, & \hfill if F\left(Y\right)<F\left(X_t\right)\\ Z:Y+S\times LF\left(D\right),& \hfill if F\left(Z\right)<F\left(X_t\right)\end{array}\right.$$

(46)

where D is the dimension; F(·) is the fitness function; S is a D-dimensional random vector with elements that are random numbers in [0, 1]; and LF(·) is the Lévy flight function.

Case 4: Hard Besiege with Progressive Rapid Dives (|E| < 0.5 and Sp < 0.5)

The prey is exhausted but still has a chance to escape. The hawks employ a hard besiege with progressive rapid dives. The position update formula for this strategy is similar to Case 3. In this scenario, the hawk swarm attempts to reduce the distance to the average position of the target prey. The update formula is as follows:

$$X\left(t+1\right)=\left\{\begin{array}{ll}Y:X_{rabbit}\left(t\right)-E\left|J{X}_{rabbit}\left(t\right)-X_m\left(t\right)\right|, & \hfill if F\left(Y\right)<F\left(X_t\right)\\ Z:Y+S\times LF\left(D\right),& \hfill if F\left(Z\right)<F\left(X_t\right)\end{array}\right.$$

(47)

6.3.2. The Solution Procedure

The detailed solution procedure is as follows:

(1) Initialization: Set up the parameters for the MGO, the shared energy storage system, and the user aggregator. Initialize the iteration counter k = 0, the population size m = 30, the maximum number of iterations to 10.

(2) Initial price generation: Use the HHO algorithm to randomly generate an initial population of m candidate solutions, where each solution is a set of electricity and heat prices. These price sets are transmitted to the user aggregator.

(3) Iteration update: Increment the iteration counter: k = k + 1.

(4) Follower Optimization: For each of the m price sets, the profit-maximization problem in Equation (30) was solved by the CPLEX solver. This determines the optimal distribution of flexible electricity load, heat load curtailment, and shared energy storage participation. The user aggregator then calculates and stores its current daily profit $C_l^k$, and returns the purchased electricity and heat quantities to the MGO.

(5) Leader profit evaluation: For each of the m scenarios, the MGO calculates its daily profit $C_{MGO}^k$ based on the purchased electricity and heat quantities reported by the user aggregator.

(6) HHO update and re-evaluation: The HHO algorithm updates the population based on its exploration and exploitation mechanism, where the fitness function is Equation (32). Steps (4)–(5) are repeated for the new population to calculate the corresponding MGO’s profit ${C_{MGO}^k}^{\prime }$ and the user aggregator’s profit ${C_l^k}^{\prime }$.

(7) Solution update: if ${C_{MGO}^k}^{\prime }>C_{MGO}^k$, update $C_{MGO}^{k+1}={C_{MGO}^k}^{\prime }$ and $C_f^{k+1}={C_f^k}^{\prime }$; otherwise, $C_{MGO}^{k+1}=C_{MGO}^k$ and $C_f^{k+1}=C_f^k$ remain unchanged.

(8) Convergence check: If k reaches the maximum number of iterations, the algorithm is deemed terminates and outputs the best solution. Otherwise, return to step (3).

7. Case Study and Results

7.1. Case Configuration

To validate the effectiveness of the proposed method, a community-level integrated energy microgrid is designed in this work. It comprises an MGO, a shared energy storage service, and user aggregators. The MGO is equipped with an MT and a GB. The shared energy storage provider possesses both electrical and heat energy storage devices. Each user within the user aggregators is equipped with PV and wind power generation systems, as well as other household electrical appliances. The proposed method is implemented through programming in MATLAB 2021b. The specific parameters of the method are shown in

Table 1. The parameter of MGO.

Parameter	${\mathit{\eta }}_{\mathit{e}}^{\mathit{M}\mathit{T}}$	${\mathit{\eta }}_{\mathit{B}}$	${\mathit{P}}_{\mathbf{max}}^{\mathit{G}\mathit{B},\mathit{h}}$	${\mathit{P}}_{\mathbf{max}}^{\mathit{M}\mathit{T},\mathit{e}}$	${\mathit{\lambda }}_{\mathit{a}\mathit{v}\mathit{g}}^{\mathit{e}}$	${\mathit{\lambda }}_{\mathit{a}\mathit{v}\mathit{g}}^{\mathit{h}}$	${\mathit{\eta }}_{\mathit{h}}$	${\mathit{P}}_{\mathit{d}\mathit{o}\mathit{w}\mathit{n}}^{\mathit{M}\mathit{T},\mathit{e}}$	${\mathit{P}}_{\mathit{u}\mathit{p}}^{\mathit{M}\mathit{T},\mathit{e}}$	${\mathit{P}}_{\mathit{d}\mathit{o}\mathit{w}\mathit{n}}^{\mathit{G}\mathit{B},\mathit{h}}$	${\mathit{P}}_{\mathit{u}\mathit{p}}^{\mathit{G}\mathit{B},\mathit{h}}$
value	0.35	0.83	800	900	0.7 CNY/kWh	0.45 CNY/kWh	0.8	−220	220	−400	500

,

Table 2. The parameter of utility function.

Parameter	ae	be	ce	ah	bh	ch	${\mathit{v}}_{\mathit{e}}$	${\mathit{\alpha }}_{\mathit{e}}$	${\mathit{v}}_{\mathit{h}}$	${\mathit{\alpha }}_{\mathit{h}}$
value	0.0013	0.16	0	0.0005	0.11	0	1.5	0.0009	1.1	0.0011

,

Table 3. The parameter of the shared energy storage system.

Parameter	${\mathit{\eta }}_{\mathit{c}}^{\mathit{E}\mathit{S}\mathit{S}}$	${\mathit{\eta }}_{\mathit{d}}^{\mathit{E}\mathit{S}\mathit{S}}$	${\mathit{E}}_{\mathbf{min}}^{\mathit{E}\mathit{S}\mathit{S}}$	${\mathit{E}}_{\mathbf{max}}^{\mathit{E}\mathit{S}\mathit{S}}$	${\mathit{P}}_{\mathbf{max}}^{\mathit{E}\mathit{S}\mathit{S},\mathit{c}}$	${\mathit{P}}_{\mathbf{max}}^{\mathit{E}\mathit{S}\mathit{S},\mathit{d}}$	${\mathit{Q}}_{\mathbf{min}}^{\mathit{E}\mathit{S}\mathit{S}}$	${\mathit{Q}}_{\mathbf{max}}^{\mathit{E}\mathit{S}\mathit{S}}$	${\mathit{Q}}_{\mathit{t},\mathbf{max}}^{\mathit{f},\mathit{d}}$	${\mathit{Q}}_{\mathit{t},\mathbf{max}}^{\mathit{f},\mathit{c}}$	${\mathit{\eta }}_{\mathit{d}}^{\mathit{Q}}$
Value	0.95	0.95	400 kWh	600 kWh	200 kW	200 kW	200	1500	150	200	0.9

and

Table 4. The parameter of user aggregator.

Parameter	${\mathit{L}}_{\mathit{t},\mathbf{max}}^{\mathit{e},\mathit{f}\mathit{l}\mathit{e}\mathit{x}}$	${\mathit{L}}_{\mathit{t},\mathbf{max}}^{\mathit{h},\mathit{c}\mathit{u}\mathit{t}}$	$\mathit{\zeta }$
Value	250	80	2 CNY/kWh

, and the YALMIP and CPLEX optimization toolboxes are used to assist in the modeling and analysis.

Figure 2 illustrates the forecasted curves for a typical day’s PV power, Wind power, and electrical and heat load in the entire community. The electrical load peaks at 10:00–14:00 and 18:00–22:00, while the heat load peaks at 4:00–8:00. It is assumed that the shiftable electrical load and curtailable heat load account for 20% of the total load. Notably, to maximize the profits for the user aggregator and MGO, this case comprehensively considers the user-side demand response for electricity and heat, as well as the shared energy storage mechanism, all within the framework of the MGO’s customized pricing strategy for electricity and heat, thus validating the effectiveness of the proposed method in this work.

7.2. Analysis of Stackelberg Game Results in the Integrated Energy Microgrid

Figure 3 presents the profit curves for the community IEM. It can be seen that, with the MGO’s price adjustments, the MGO’s profit shows a gradually increasing trend, while the user aggregator’s profit shows a gradually decreasing trend. Both eventually stabilize, and the results converge by the 8th iteration. This evolution of trends exemplifies the gaming process.

Figure 4 illustrates the pricing strategies of the MGO. The fluctuations in the hourly electricity and heat selling prices reflect the game interaction between the MGO and the user aggregator. This mechanism encourages energy trading in the community IEM. Furthermore, the game ultimately establishes an equilibrium between the user aggregator and the MGO that meets user needs.

Figure 5 presents the electrical and heat load curves of the user aggregator before and after demand response. As shown in the left panel of Figure 5, incentivized by electricity prices, the electrical load curve exhibits the characteristics of “peak shaving and valley filling” to reduce total electricity costs. The original electrical load curve had two peaks occurring at 11:00–12:00 and 18:00–19:00, when electricity prices were high. After user-side optimization, these load peaks are significantly reduced and shifted to the off-peak valley periods of 00:00–08:00 and 23:00–24:00, when electricity prices are lower. Consequently, the fluctuation of the electrical load curve is significantly reduced. As seen in the right panel of Figure 5, the overall heat load is curtailed. Furthermore, under the MGO’s heat price adjustments, the heat load reduction occurs primarily during the periods of 01:00–08:00 and 22:00–24:00.

Figure 6 illustrates the optimal scheduling results for electrical energy in the IEM. Considering the environmental benefits of renewable energy, the output from PV and wind power is prioritized, while gas turbines and energy storage serve as supplements to compensate for renewable energy deficits. During the electricity valley periods after demand response (10:00–13:00 and 17:00–21:00), gas generators operate at high output to maximize profits, and surplus power is stored in the shared energy storage system. Conversely, during the peak electricity load periods (10:00–15:00, 18:00–20:00), the insufficient energy is purchased from the microgrid.

Figure 7 presents the optimal scheduling results for heat energy in the IEM. Heat is supplied jointly by the waste heat from the gas generators and the gas boilers, where the quantity of waste heat is directly correlated with the power generation of the gas generators. To guarantee the heat supply and avoid penalty costs, the MGO adjusts the heat purchasing price to guide the boiler output, thereby achieving a balance between supply and demand.

7.3. Comparison Analysis

7.3.1. Searching Strategy Comparison

To demonstrate the superiority of the searching method in this work, the results from running the classic GA algorithm (crossover = 0.8, mutation = 0.05) and the HHO with same size of populations are shown in Figure 8.

Table 5. Average time consumption.

Method	GA	HHO
Value/s	46.42	60.35

lists the average time consumption. The experimental results indicate that the HHO method introduced into this work possesses superior optimization capability and faster convergence, although it costs more time than that of GA. This fact can be explained that each iteration undergoes optimization by CPLEX whose runtime is uncertain. Moreover, the time consumed by CPLEX optimization constitutes a much larger proportion than that of population evolution.

7.3.2. Scenario Comparison

To further demonstrate the effectiveness of the proposed model, two scenarios are established for analysis:

(1) Scenario 1: Without game-theoretic pricing, where calculations are performed using the time-of-use (TOU) electricity price and the average heat price at each time step.

(2) Scenario 2: The proposed method without the shared energy storage system.

Table 6. Cost comparison results.

Item	${\mathit{C}}_{\mathit{M}\mathit{G}\mathit{O}}$/CNY	${\mathit{F}}_{\mathit{u}\mathit{s}\mathit{e}\mathit{r}}$/CNY	${\mathit{C}}_{\mathit{E}\mathit{S}\mathit{S}}$/CNY	${\mathit{F}}_{\mathit{I}\mathit{E}\mathit{M}}$/CNY
Case 1	8642.6	10,305.2	−1128.1	−18,653.0
Case 2	3826.9	12,327.0	0	−16,183.2
Ours	5670.6	13,604.1	−500.3	−18,321.4

presents the results for the above two scenarios. Through comparison, it can be observed that the revenue obtained by the user aggregator in Scenario 1 is lower than that of the proposed method, whereas the revenue of the MGO is higher. It can be explained that the purchase and sale prices of energy storage in this work are based on those of the MGO, thus possibly leading to negative profits because of differing pricing strategies during charging and discharging periods. However, the revenue of the shared energy storage provider in the proposed method is higher than that of Scenario 1, further indicating that game pricing can effectively avoid electricity price peaks. Additionally, the revenues of both the user aggregator and the MGO in Scenario 2 are lower than those of the proposed method. This demonstrates that the shared energy storage system can increase the benefits for both the MGO and the user aggregator, which also indirectly illustrates that shared energy storage provides peak shaving and valley filling effects for the entire system.

7.3.3. Sensitivity Analysis

To effectively demonstrate the effectiveness of the dynamic pricing strategy in handling fluctuations in PV and wind power, an additional experiment was conducted by introducing a 10% random fluctuation into the output of PV and wind power. A comparative analysis was performed using the following two strategies:

Strategy 1: Utilizing pre-determined optimal prices to calculate economic benefits.

Strategy 2: Re-adjusting prices to calculate economic benefits.

Table 7. Comparison results for the above two strategies.

Item	${\mathit{C}}_{\mathit{M}\mathit{G}\mathit{O}}$/CNY	${\mathit{F}}_{\mathit{u}\mathit{s}\mathit{e}\mathit{r}}$/CNY	${\mathit{C}}_{\mathit{E}\mathit{S}\mathit{S}}$/CNY	${\mathit{F}}_{\mathit{I}\mathit{E}\mathit{M}}$
Strategy 1	7421.4	12,669.6	−990.6	−18,535.6
Strategy 2	4657.7	14,863.5	−652.7	−18,218.4

illustrates the results for the above two strategies. It can be seen that, the objective optimization value F_IEM of Strategy 1 is higher than that of Strategy 2, indicating that re-adjusting prices during the optimization process yields the optimal objective function value. However, due to fluctuations in PV and wind power, the MGO tends to compensate for these impacts to balance the electric and heat loads, consequently generating relatively higher revenue. Additionally, the benefits of user aggregator of Strategy 2 are relatively higher than that of Strategy 1, which indicates that dynamic pricing enables users to adjust their flexible loads to pursue profit maximization. Ultimately, the goal is to leverage the advantages of dynamic price adjustments to achieve a profit equilibrium between the MGO and the user aggregator.

8. Conclusions

This paper addresses the benefit imbalance between the MGO and user aggregators by proposing a collaborative operation approach integrated with shared energy storage. A Stackelberg game model is constructed for the electricity-heat coupling community integrated energy system, where user aggregators optimize their energy consumption through the coordinated use of demand response and shared storage services. For the complex bi-level optimization problem, a hierarchical solution method combining the Harris Hawks Optimization algorithm (for upper-level pricing) and the CPLEX solver (for lower-level scheduling) is adopted. Simulation results demonstrate that the proposed model significantly improves user-side profits and energy flexibility, achieving a win-win outcome for both user aggregators and the MGO. Furthermore, the model provides valuable decision-making support for users and MGO by considering electrical and heat demand response as well as shared storage utilization. In future work, the shared energy storage provider will be incorporated into the game model, with the formulation of its daily pricing schemes representing a key issue for further investigation. Additionally, the mechanism of peer-to-peer energy trading will be introduced to promote economic efficiency and environmental sustainability, further facilitating the establishment of a sustainable energy ecosystem.