In recent years, small and micro enterprises have developed rapidly in Zhejiang Province, China. In order to facilitate the prosperous development of such enterprises, Zhejiang Province has standardized and renovated the existing small-micro industrial parks (SMIPs) based on the actual operation. However, the SMIPs do not dispatch enough power generation and energy storage (ES) devices, which results in a low capacity to withstand the operating risks. With the rapid development of SMIPs, the demand for electricity trading between SMIPs and distribution networks (DNs) is constantly increasing. On one hand, trading electricity with the DNs can help the SMIPs to withstand the operating risks. On the other hand, an optimal trading electricity strategy can help the SMIPs to save the operating costs. Then, how to optimize the electricity trading between the DNs and SMIPs is currently a highly important issue (He et al., 2021).

As the electricity market (EM) continues to expanding, the operators and agents in the DNs gradually participate in EM competition (Jin et al., 2022), and the economic operation of the DNs has been significantly improved (Li et al., 2023a). A market simulator for peer-to-peer electricity trading is proposed in Kuno et al. (2022), and a market agent is designed to perform power bidding and contract processing. Guo et al. (2022) showed that all agents can freely trade in an asynchronous mode without waiting for idle or inactive neighboring agents, and the sublinear regret upper bound is proved for the asynchronous mode, which can maximize the social welfare in the EM. A real-time congestion management strategy is applied by Klsch et al. (2022), and the strategy can enable grid-supportive operation of the operators without interventions. Considering the characteristic that natural gas can blend with hydrogen, Ding et al. (2023) proposed a multi-agent electricity–heat–hydrogen trading model by taking hydrogen produced on the load side. Tan et al. (2022) treated carbon as a direct trading object and proposed an internal multi-energy trading mechanism, which adopts an auction based on the demands for cooling, heating, electricity, and carbon. To further explore the multi-energy coupling capacity and carbon reduction potential of the integrated energy systems, Yang M. et al. (2023) proposed a cooling–heat–electricity–gas collaborative optimization model of integrated energy systems, given the ladder carbon trading mechanism and multi-energy demand response. Li Z. et al. (2023) proposed a medium-term, multi-stage, distributionally robust optimization scheduling approach for the price-taking of hydro–wind–solar complementary systems in the EM. A multi-agent deep reinforcement learning approach combining the multi-agent actor-critic algorithm with the twin-delayed deep deterministic policy gradient algorithm is proposed by Chen et al. (2022), and the proposed approach can handle the high-dimensional continuous action space and aligns with the nature of peer-to-peer energy trading. Yang et al. (2022) analyzed the impact of different bidding decisions on the distribution of wind farm revenues in a process where the interest of two markets is played against each other. Khaligh et al. (2022) introduced a stochastic agent-based model for the coordinated scheduling of multi-vector microgrids, considering interactions between electricity, hydrogen, and gas agents. Considering the power loss, flexible load demand, and other operating indicators, to maximize the user and supplier benefits, the real-time transaction electricity price model of the user side and the power supply side is constructed by Lyu et al. (2022). In order to meet the challenge of global low-carbon development, a multi-objective optimal scheduling model considering the participation of the park-level integrated energy system in the EM is proposed by Wang L. et al. (2022), which takes into account multiple uncertainties on the renewable energy and load. Li et al. (2023c) analyzed the impact of uncertainties for the multi-energy virtual power plant on the peak-regulation market, and the operation mechanism for the multi-energy virtual power plant in the peak-regulation market is proposed by considering the integrated demand response. As a user-side system, SMIPs can participate in electricity trading under the management of SMIP operators (Davoudi and Moeini-Aghtaie, 2022), which can develop the hierarchical structure of the EM (Pownall et al., 2021). Meanwhile, the agents in DNs can link the SMIP with EM (Anwar et al., 2022), which exerts a significant influence on the energy costs of such parks (Yuan et al., 2022). As SMIPs participate in electricity trading, how to balance the benefits between agents and operators, achieve the expected profit of agents, and reduce the operating costs of operators has become a key concern in current EM.

Different stakeholders have different optimization goals in EM trading (Xiao et al., 2021), and a coupling relationship exists among the EM trading models (Yang et al., 2021). Finding the balanced benefits in EM trading has become a key way to keep the stability of the alliances (Cao et al., 2021). To find the balanced benefits in EM trading, Stackelberg game theory has become an effective tool (Li et al., 2022). To solve the inherent conflict among the players, a Stackelberg game-based technique is proposed by Haghifam et al. (2020), and the distribution system operator attempts to minimize its operating costs as the leader, while the distributed energy resource aggregator tends to maximize its profit as the follower. Huang et al. (2022) proposed a Stackelberg game-based optimization model for energy service providers and integrated energy systems based on the collaborative optimization of integrated energy systems and carbon transaction cost, where the energy service provider acts as a leader while the integrated energy systems serve as followers. A Stackelberg game model between the load aggregator and distribution system operator was proposed by Xu et al. (2022); the distribution system operator issues subsidies to decrease the frequency of voltage violations, and the load aggregator schedules the demand to maximize profits. A trading model based on the Stackelberg game model was proposed by Wei et al. (2022) to balance the interests of the supply side and demand side and reduce carbon emissions. To solve the problems of environmental pollution and conflict of interests among multiple stakeholders in the integrated energy system, Wang R. et al. (2022) proposed a novel collaborative optimization strategy for a low-carbon economy in the integrated energy system based on the carbon trading mechanism and Stackelberg game theory. Envelope et al. (2022) proposed a Stackelberg game-based optimal scheduling model for electro-thermal integrated energy systems, which seeks to maximize the revenue of the integrated energy operator and minimize the cost of users. Pu et al. (2023) constructed a two-stage supply chain consisting of a manufacturer and a retailer based on the dual-credit policy, considered three different power structure models, including the vertical Nash game model, the manufacturer Stackelberg game model, and the retailer Stackelberg game model, and explored the operational strategy issues of new energy vehicle enterprises under the dual-credit policy. Zhang et al. (2022) took the integrated energy system operator as the leader and each integrated energy system as the follower to construct the Stackelberg operation model, and the proposed model is constructed and solved by the double mutation differential evolution algorithm. Hua et al. (2023) proposed a framework of local energy markets to manage this transactive energy and facilitate the flexibility provision; the decision-making and interactions between a DN operator and multiple microgrid traders are formulated as the Stackelberg game-theoretic problem. Fattaheian et al. (2022) applied the Stackelberg game to model incentivizing resource scheduling optimization in post-contingency conditions, and a strong duality condition is used to re-cast the preliminary bi-level model into a one-level mathematical problem. The pricing strategies in existing research studies mainly belong to fixed pricing mechanisms (Du et al., 2022). In the future, as more SMIPs participate in EM trading, the fixed pricing mechanisms will not be satisfied to the flexible EM model. In this context, studying the dynamic pricing strategies for DN trading with SMIPs to enhance the economic operation of DNs is highly necessary.

As described previously, a two-layer optimal electricity trading method for DNs with SMIPs is proposed based on the Stackelberg game. The main contributions of this paper are summarized as follows.

(1) A two-layer optimization trading model is established for DNs and SMIPs, and the competitive game relationship between different stakeholders in the trading process is characterized by the Stackelberg game. The paper shows that through the proposed scheme, the profit of DNs increases by 14.8% in a typical day, and the operating costs of SMIPs decreased by 2.5% and 3.8%, respectively.

The remainder of this paper is organized as follows. A two-layer game optimization trading framework for multi-stakeholders is proposed in Section 2. In addition, a two-layer game optimization trading model based on the Stackelberg game is proposed in Section 3. In Section 4, the solution process of the proposed model is introduced. In Section 5, the case study is analyzed. The discussion and conclusion are described in Section 6.

In order to improve the operating benefits of DN and reduce the energy costs of SMIPs, a two-layer optimization trading framework for multi-stakeholders is established, as shown in Figure 1.

In the trading framework, the stakeholders include the EM, DN agent, and SMIP operators. The trading framework is divided into two layers, with EM and DN agent located in the upper layer and SMIP operators located in the lower layer. The stakeholders in the upper and lower layers have different benefits goals, and they mutually influence each other. In the upper layer, the DN agent makes decisions on the volume of purchased electricity or sold electricity based on the needs of EM and SMIPs. If EM supply cannot meet the demand, electricity trading occurs between two adjacent DN agents. Additionally, the DN agent will use dynamic trading prices to encourage the trading of lower-layer SMIP operators. In the lower layer, SMIP operators will dynamically adjust the outputs of internal generation units based on the trading prices set by DN agents, as well as the electricity demand on the user side. Simultaneously, SMIP operators provide real-time feedback to the upper layer DN agents regarding the electricity usage strategies. DN agents, based on the energy consumption situation, dynamically adjust the trading prices. Through the iterative process between the upper and lower layers, a balance point that aligns with the interests of both stakeholders is ultimately determined.

In the constructed two-layer optimization model, the upper and lower optimization models are coupled through the electricity trading price. The upper-layer DN agent, acting as the leader in the Stackelberg game, optimizes the electricity trading with EM and SMIP operators based on their own internal demand. The lower-layer SMIP operators, as followers in the Stackelberg game, optimize their internal energy consumption based on the electricity trading price provided by the DN agent.

Due to coupling in the two-layer model in this paper, the KKT condition transformation (Yang X. et al., 2023) is employed to convert the original coupled two-layer optimization model into a single-layer linear programming model. The specific process is as follows: c t sell Δ t − λ t 2 , ⁡ min + λ t 2 , ⁡ max − λ t 4 = 0 , (21) − c t buy Δ t − λ t 3 , ⁡ min + λ t 3 , ⁡ max + λ t 4 = 0 , (22) c DEG − ρ m , t min + ρ m , t max − λ t 4 = 0 . (23)

The dual variables above need to satisfy the following conditions: ρ m , t min ⊥ P m , t DEG − P m DEG , ⁡ min , (24) ρ m , t max ⊥ P m DEG , ⁡ max − P m , t DEG , (25) λ t 2 , ⁡ min ⊥ P t sell − P t sell , ⁡ min , (26) λ t 2 , ⁡ max ⊥ P t sell , ⁡ max − P t sell , (27) λ t 3 , ⁡ min ⊥ P t buy − P t buy , ⁡ min , (28) λ t 3 , ⁡ max ⊥ P t buy , ⁡ max − P t buy , (29) λ t 4 ⊥ P t load + P t buy + F − 1 α σ PV 2 + σ WT 2 + σ load 2 − ∑ m P m , t DEG + R m , t DEG − P t PV − P t WT − P t sell ) . (30)

In the above equations, x⊥y represents that x and y have at most one non-zero value. To linearize the model, this paper relaxes the above equations by introducing Boolean variables: 0 ≤ ρ m , t min ≤ M θ m , t min , (31) 0 ≤ P m , t DEG − P m DEG , ⁡ min ≤ M 1 − θ m , t min , (32) 0 ≤ P m DEG , ⁡ max − P m , t DEG ≤ M 1 − θ m , t max , (33) 0 ≤ ρ m , t max ≤ M θ m , t max , (34) 0 ≤ λ t 2 , ⁡ min ≤ M θ t 2 , ⁡ min , (35) 0 ≤ P t sell − P t sell , ⁡ min ≤ M 1 − θ t 2 , ⁡ min , (36) 0 ≤ λ t 2 , ⁡ max ≤ M θ t 2 , ⁡ max , (37) 0 ≤ P t sell , ⁡ max − P t sell ≤ M 1 − θ t 2 , ⁡ max , (38) 0 ≤ λ t 3 , ⁡ min ≤ M θ t 3 , ⁡ min , (39) 0 ≤ P t buy − P t buy , ⁡ min ≤ M 1 − θ t 3 , ⁡ min , (40) 0 ≤ λ t 3 , ⁡ max ≤ M θ t 3 , ⁡ max , (41) 0 ≤ P t buy , ⁡ max − P t buy ≤ M 1 − θ t 3 , ⁡ max , (42) 0 ≤ λ t 4 ≤ M θ t 4 , (43) 0 ≤ P t load + P t buy + F − 1 α σ PV 2 + σ WT 2 + σ load 2 − ∑ m P m , t DEG + R m , t DEG − P t PV − P t WT − P t sell ) ≤ M 1 − θ t 4 . (44)

In the above equations, θ m , t min , θ m , t max , θ 2 , t min , θ 2 , t max , θ 3 , t min , θ 3 , t max , and θ 4 , t are the Boolean variables. M is the maximum value. After the transformation, the objective function still contains two non-linear terms: ∑ t c t sell P t sell Δ t and ∑ t c t buy P t buy Δ t . To facilitate the solution, the strong duality principle (Ouyang et al., 2023) is employed to process the objective function, resulting in the transformation of the objective function into a linear programming model. max ∑ t c t sell P t sell Δ t − ∑ t c t buy P t buy Δ t + ∑ t π t − R t − Δ t − ∑ t π t + R t + Δ t − ∑ t π t R t Δ t − ∑ t c t tra P t sop Δ t ⇒ max ∑ t ∑ m ρ m , t min P m DEG , min − ρ m , t max P m DEG , max + ∑ t λ t 2 , min P t sell , min − λ t 2 , max P t sell , max + ∑ t λ t 3 , min P t buy , min − λ t 3 , max P t buy , max + ∑ t λ t 4 F − 1 α σ PV 2 + σ WT 2 + σ load 2 + P t load − P t PV − P t WT − R m , t DEG − ∑ t ∑ m c DEG P m , t DEG Δ t + ∑ t π t − R t − Δ t − ∑ t π t + R t + Δ t − ∑ t π t R t Δ t − ∑ t c t tra P t sop Δ t . (45)

By employing the above method, the non-linear dual-layer game model can be transformed into a mixed-integer linear single-layer optimization model. The optimal solution can be obtained using the commercial solver CPLEX (Zhao et al., 2022). The overall solving process of the model established in this paper is illustrated as follows and is shown in Figure 2:

Step 1:

The outputs of WTs, PVs, and user load demand are forecasted within the SMIP.

Step 2:

The data obtained from the previous step are utilized to generate EM trading scenarios, including the electricity generation volume and trading prices in EM.

Step 3:

A two-layer optimization trading model is established, with the upper-layer DN agent as the price leaders and the lower-layer SMIP operators as the price followers, thus forming the Stackelberg game optimization model.

Step 4:

KKT conditions are applied to transform the dual-layer problem into a single-layer problem. Dual theory, linear relaxation, and other methods are combined to convert the optimization model into a more easily solvable mixed-integer linear programming model. In this step, the specific electricity trading volume of each stakeholder can be calculated through the nested solution of electricity price and trading volume.

Step 5:

Through the real-time dynamic optimization processes, the optimal electricity price and trading volume are obtained, and an optimal solution for all stakeholders in the Stackelberg game is achieved.

With the rapid development of SMIP, the demand for electricity trading between SMIPs and DNs is constantly increasing. An optimal trading electricity strategy with the DN can help the SMIPs to withstand the operating risks, as well as help the SMIPs to save the operating costs. For the electricity trading between the DNs and SMIPs, the existing methods have the following issues that need to be addressed:

• The electricity trading mode between the DN and SMIP is unclear. How to characterize the competitive game relationship between the DN and SMIP in the trading process needs to be solved.

To address the above issues, a Stackelberg game-based optimal electricity trading method for DNs with SMIPs is proposed. Our conclusions are as follows:

• The proposed two-layer optimization trading model can characterize the competitive game relationship between the DNs and SMIPs. Furthermore, under the proposed scheme, the profit of DNs increases by 14.8% in a typical day, and the operating costs of SMIPs decreased by 2.5% and 3.8%, respectively.

The trading strategy proposed in this paper can effectively increase the profits of the DN agent while reducing the operating costs of SMIP operators. In future research, a two-layer robust optimization trading method for the DN agent and SMIP operators will be developed with the consideration of the uncertainty of PVs and WTs to make the methods more comprehensive.