Two-Stage, Three-Layer Stochastic Robust Model and Solution for Multi-Energy Access System Based on Hybrid Game Theory

Abstract

This paper proposes a two-stage, three-layer stochastic robust model and its solution method for a multi-energy access system (MEAS) considering different weather scenarios which are described through scenario probabilities and output uncertainties. In the first stage, based on the principle of the master–slave game, the master–slave relationship between the grid dispatch department (GDD) and the MEAS is constructed and the master–slave game transaction mechanism is analyzed. The GDD establishes a stochastic pricing model that takes into account the uncertainty of wind power scenario probabilities. In the second stage, considering the impacts of wind power and photovoltaic scenario probability uncertainties and output uncertainties, a max–max–min three-layer structured stochastic robust model for the MEAS is established and its cooperation model is constructed based on the Nash bargaining principle. A variable alternating iteration algorithm combining Karush–Kuhn–Tucker conditions (KKT) is proposed to solve the stochastic robust model of the MEAS. The alternating direction method of multipliers (ADMM) is used to solve the cooperation model of the MEAS and a particle swarm algorithm (PSO) is employed to solve the non-convex two-stage model. Finally, the effectiveness of the proposed model and method is verified through case studies.

1. Introduction

With the proposal of carbon neutrality targets, the distribution grid will integrate a large amount of renewable energy in the future, thereby reducing the use of fossil fuels. In order to fully integrate renewable energy, multi-energy access systems (MEASs) have emerged [1]. MEASs are capable of optimizing their internal operational status under specified electricity price backgrounds, achieving a dynamic balance with the distribution grid’s energy interaction [2]. However, intermittent renewable energy poses challenges to the energy planning of MEASs [3]. Additionally, there will be more distributed MEASs connecting to the distribution grid in the future, which increases the difficulty of unified management for MEASs [4]. Therefore, addressing how to manage MEASs with uncertainty in energy and how to unify the management of multiple distributed MEASs are urgent challenges that need to be solved.

In recent years, scholars both domestically and internationally have conducted extensive research on the unified management of microgrids (MEASs). The general approach involves appointing a unified energy manager at the higher level of the MEAS, such as a grid dispatch department (GDD) [5]. The GDD is primarily responsible for coordinating energy interactions between the distribution grid and the MEAS, while also establishing optimal electricity prices for the MEAS to achieve overall optimal benefits for both the GDD and the MEAS. From the perspective of game theory, the relationship between the GDD and MEAS can be understood as a leader–follower game where the GDD acts as the upper-level leader, setting electricity prices for the MEAS, while the MEAS can be seen as the lower-level follower, planning energy usage based on these prices. The GDD–MEAS relationship can achieve game equilibrium solutions [6]. For example, [6] deals with energy trading among MEASs through leader–follower games and establishes a multi-agent optimization scheduling model with multiple microgrid systems as leaders and load aggregators as followers. Another study [7] builds upon leader–follower games to establish an optimization configuration model for multi-microgrid systems with multiple microgrid operators as leaders and distribution grids as followers.

With the advancement of market trading mechanisms, some MEASs that are geographically proximate or have similar energy forms will, under the guidance of decision-making information, form cooperative alliances through point-to-point transactions among MEASs. This is referred to as the cooperative operation mode. In this cooperative operation, the GDD permits MEASs to flexibly select transactions with other MEASs to enhance their own benefits. Meanwhile, compared to MEASs’ individual participation in market transactions, the formation of cooperative alliances also boosts the market competitiveness of MEASs [8]. One study [9] established a multi-MEAS cooperative optimization model by minimizing the electricity transaction costs between MEASs and market operators to optimize the centralized trading process and minimizing the electricity transaction costs among MEASs to optimize the decentralized trading process. Eventually, the Nash bargaining method was adopted to distribute the cooperative surplus value among multiple MEASs. From the perspective of game theory, this GDD–MEAS operation mode is actually a leader–follower–cooperative game model and the game model has expanded from a two-layer to a three-layer structure, with the intermediate layer being the follower cooperation layer. Another study [10] established a three-layer optimization model for the distribution network and multiple MEASs based on the leader–follower–cooperative game; studies [11,12] constructed a three-layer leader–follower–cooperative game model with the GDD as the leader and the collaborative operation of cold, heat, and electricity multi-energy coupled shared energy storage–prosumer.

At present, the main solution idea for the leader–follower–cooperative game model is the two-stage solution method; that is, the leader–follower problem is solved in the first stage and the cooperative problem in the second stage. Currently, methods for solving leader–follower game equilibrium solutions mainly include numerical methods based on Karush–Kuhn–Tucker (KKT) conditions [13,14,15], column constraint generation (C&CG) methods [16], heuristic intelligent algorithms [17,18,19], and the alternating direction method of multipliers (ADMM) [20]. Due to its decoupling characteristics and stable convergence, the ADMM commonly uses the second-stage distributed solving cooperative model [21]. The KKT numerical method is applicable to the scene where the second-stage problem is a two-level model, and can transform a two-level model into a single-level model, but this method is only applicable to the scene where the underlying model is a convex problem, so the underlying non-convex model needs to be simulated/convex in advance [22]. C&CG combined with KKT can solve the scenario where the second-stage problem is a three-level model. Generally, KKT is first used to convert the lowest two levels into a single level, so that the original problem can be simplified to a classical two-level problem, provided that the two-level model is convex [23]. Heuristic intelligent algorithms can overlook the non-convexity of the lower-level problem and are thus widely applied in solving leader–follower game models [24].

However, the above models of the master–slave game are mostly deterministic models, while there often exist uncertainty factors in MEASs, such as the uncertainty of output from renewable energy sources like wind turbines (WTs) and photovoltaics (PVs). The uncertainty factors in MEASs will affect the management of GDDs for MEASs [25]. For example, the paper [26] studies the cooperative Stackelberg game energy management scheme considering price discrimination and risk assessment. The upper layer uses a stochastic programming method with a conditional risk value to establish a retailer stochastic pricing model, while the lower layer is based on a Nash bargaining game follower–cooperation model. The experiment shows that the stochastic pricing information will affect the direct trading price and trading volume of the followers at the same time. To solve this problem, scholars have proposed uncertainty handling methods such as stochastic programming [27,28] and robust optimization [29,30,31]. Stochastic optimization uses historical experience to generate random probability scenarios and then seeks the scheduling scheme with the lowest cost expectation. For example, reference [27] used stochastic optimization to study the impact of WT output uncertainty on MEAS energy planning. Reference [28] modeled the randomness of WTs based on probability distribution functions to optimize the operation of distribution networks. Robust optimization describes uncertain variables using various uncertainty sets to find the optimal solution under the worst-case scenario. Reference [29] described WT output fluctuations using box uncertainty sets and constructed a two-stage robust optimization scheduling model. Reference [30] described the uncertainty of user participation using box uncertainty sets. However, both of these methods have a certain one-sidedness and limitations: scenario-based stochastic optimization is suitable for situations where probability distribution functions are relatively easy to obtain, but it may lead to a huge computational burden, while robust optimization is suitable for situations where the distribution of uncertain parameters is difficult to determine or there is little historical data, and it has good robustness, but it has conservatism issues [31,32].

To address the conservatism issue, researchers have proposed some improved models based on stochastic and robust methods. For example, to consider the probability uncertainty of multiple uncertain parameters, the idea of stochastic optimization is incorporated into robust optimization to propose a four-level stochastic robust model with predictive probabilities [33,34]. Reference [35] used stochastic optimization and robust handling of uncertainty to establish a max–max–max–min microgrid robust model, and the results showed that the conservatism of this model is moderate. Reference [36] studied the hybrid game optimization model of a micro-energy network and consumer, considering the uncertainty of WTs and consumer response, and adopted robust optimization, the chance constraint method, KKT condition, McCormick envelope method, and ADMM method to solve the model, which effectively coordinated the scheduling of the micro-energy network and consumer and ensured the fairness of the consumer cooperation alliance. However, these models also have some problems, as stochastic optimization still needs to simulate its probability function based on a large amount of historical data. In addition, the maximization in the second and third layers of the four-level model is coupled, making it more difficult to solve the model. Therefore, it is necessary to develop robust models adapted to uncertain factors and study corresponding solution algorithms.

In summary, this paper establishes a two-stage, three-level stochastic robust model considering the probability and output uncertainty of WT and PV scenarios. The model is for a two-stage problem based on mixed games, where the first stage is a stochastic pricing model for the GDD and the second stage is a three-level structured MEAS stochastic robust cooperative model. Considering the difficulty of decoupling in the three-level stochastic robust model, a variable alternating iteration algorithm (AIV–KKT) combined with KKT is proposed. Considering the nonlinearity and non-convexity of the two-stage model, a particle swarm algorithm is used for solving.

2. Transaction Mechanism Analysis

Figure 1 depicts the GDD–MEAS transaction framework. In this framework, the MEAS consists of electric chillers (EC), absorption chillers (AC), combined heat and power systems (CHP), gas boilers (GB), heat storage (HS), and electricity storage (ES), forming a multi-energy supply system for cooling, heating, and electricity. In the first stage, the GDD acts as the leader and personalized electricity pricing is formulated based on the electricity demand information from the MEASs, aiming to manage the MEASs effectively. The specific process is as follows: firstly, the GDD aggregates the electricity demand information from various MEASs, which is robust information considering both scenario and output uncertainties, thus resulting in a conservative estimate. Based on this information, the GDD sets a conservative pricing for electricity transactions. Subsequently, when an MEAS needs to purchase electricity, the GDD procures an equivalent amount from the electricity market at the grid price and resells it to the MEAS at the corresponding purchasing price; conversely, when an MEAS needs to sell electricity, the GDD purchases it from the MEAS at the corresponding selling price and resells it to the electricity market at the grid price. The GDD determines the purchasing and selling prices for each MEAS by maximizing the price difference.

In the second stage, each MEAS acts as a follower, formulating electricity purchase and sales plans based on the feedback from the GDD and arranging internal energy distribution reasonably. Typically, the selling price of electricity to the GDD is lower than the purchasing price from the GDD. Therefore, multiple MEASs can form cooperative alliances to reduce the overall electricity purchase from the GDD through internal electricity sharing, thereby saving electricity costs and improving economic benefits. Under the guidance of purchasing and selling prices, MEASs prioritize cooperation, which involves decentralized direct transactions. After completing direct transactions, MEASs report surplus or deficit electricity information to the GDD, which then determines the centralized trading electricity for each MEAS. After centralized trading, the GDD organizes Nash negotiations based on the basic information of each MEAS and allocates the cooperative surplus among multiple MEASs uniformly. In the proposed MEAS model, stochastic optimization and robust methods are utilized to handle uncertainties in WT and PV scenario probabilities and outputs. The uncertainty space of scenario probabilities is discretized based on a small amount of historical data to generate a typical scenario probability distribution, while the uncertainty space of WT output adopts a box-set form.

3. Two-Stage Stochastic Model

The objective of the two-stage stochastic model based on mixed games is to maximize operational profits.

$$U=U^{\mathrm{DSO}}+{\displaystyle \sum _{j=1}^N{U}_j^{\mathrm{m},\mathrm{hz}}}$$

(1)

where $U$ represents the objective function of the model; $U^{\mathrm{DSO}}$ represents the operational profit of GDD; and $U_j^{\mathrm{m},\mathrm{hz}}$ represents the operational profit under MEAS j. The maximizing model $U^{\mathrm{DSO}}$ is referred to as the stochastic pricing model for GDD, while the maximizing model $U_j^{\mathrm{m},\mathrm{hz}}$ is referred to as the MEAS cooperation model.

3.1. GDD’s Stochastic Pricing Model

3.1.1. Utility Function

As the leader, the GDD is obligated to manage the MEASs. The GDD determines the transaction prices between the GDD and each MEAS based on the spot electricity price and the grid electricity price. Generally, each MEAS’s selling price is lower than its buying price to encourage each MEAS to arrange internal energy, further reducing external dependence. Meanwhile, the GDD aims to maximize price difference benefits. Therefore, the GDD’s stochastic pricing model is

$$U^{\mathrm{DSO}}=\underset{x}{\max }{\displaystyle \sum _{s=1}^M{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{j=1}^N{\sigma }_{j,s}\left(\begin{array}{l}{\beta }_t^{\mathrm{f}}{P}_{j,t,s}^{\mathrm{m},\mathrm{S}}-{\beta }_t^{\mathrm{c}}{P}_{j,t,s}^{\mathrm{m},\mathrm{B}}+ \\ {\beta }_{j,t,s}^{\mathrm{m},\mathrm{B}}{P}_{j,t,s}^{\mathrm{m},\mathrm{B}}-{\beta }_{j,t,s}^{\mathrm{m},\mathrm{S}}{P}_{j,t,s}^{\mathrm{m},\mathrm{S}}\end{array}\right)}}}$$

(2)

where $x=\left({\beta }_{j,t,s}^{\mathrm{m},\mathrm{B}},{\beta }_{j,t,s}^{\mathrm{m},\mathrm{S}},{\sigma }_{j,s}\right)$ represents the set of decision variables. Among them, ${\beta }_{j,t,s}^{\mathrm{m},\mathrm{B}}$ and ${\beta }_{j,t,s}^{\mathrm{m},\mathrm{S}}$ denote the prices at which the j-th MEAS buys electricity from the GDD and sells electricity to the GDD at time t; namely, the purchase price and the selling price set by the GDD for the MEAS; $P_{j,t,s}^{\mathrm{m},\mathrm{B}}$ and $P_{j,t,s}^{\mathrm{m},\mathrm{S}}$ represent the amount of electricity purchased and sold by the j-th MEAS to the GDD at time t under scenario s, respectively; ${\sigma }_{j,s}$ represents the probability distribution of scenario s for the j-th MEAS; ${\beta }_t^{\mathrm{f}}$ and ${\beta }_t^{\mathrm{c}}$ represent the on-grid electricity price and off-grid electricity price at time t, respectively; M denotes the number of scenarios; N denotes the total number of MEASs; and T denotes the scheduling period.

3.1.2. Strategy Space

$${\beta }_t^{\mathrm{f}}\le {\beta }_{j,t,s}^{\mathrm{m},\mathrm{S}}\le {\beta }_{j,t,s}^{\mathrm{m},\mathrm{B}}\le {\beta }_t^{\mathrm{c}}$$

(3)

Equation (2) represents the strategy space ${\Omega }^{\mathrm{D}}$ of the GDD’s random pricing model. It constrains the pricing space of the GDD, where the purchasing price ${\beta }_{j,t,s}^{\mathrm{m},\mathrm{B}}$ set by the GDD is not greater than the off-grid electricity price ${\beta }_t^{\mathrm{c}}$, and the selling price ${\beta }_{j,t,s}^{\mathrm{m},\mathrm{S}}$ is not less than the on-grid electricity price ${\beta }_t^{\mathrm{f}}$. In order to maximize its own interests, the MEAS will choose to trade with the GDD.

3.2. Multi-MEAS Random-Robust Cooperation Model

3.2.1. Utility Function

The objective function of the MEAS is to minimize its costs, which is in the form of a max–max–min three-level random model. The first level involves finding the worst-case probability distribution of the WTs, the second level is to determine the worst-case output of the WTs, and the third level is to devise the optimal operating strategy for the MEAS to minimize its operational costs, including energy storage operational costs $C_j^{\mathrm{cn}}$, purchasing and selling electricity costs $C_j^{\mathrm{m}}$, and cooperation costs $C_{ji}^{\mathrm{hz}}$.

$$\left\{\begin{array}{l}{U}_j^{\mathrm{m},\mathrm{hz}}=\underset{{\sigma }_s}{\max } \underset{P_{j,t,s}^{\{.\}}}{\max } \underset{y_j}{\min }\left(C_j^{\mathrm{cn}}+C_j^{\mathrm{m}}+C_{ji}^{\mathrm{hz}}\right)\\ y_j=\left(\begin{array}{l}{P}_{ji,t,s}^{\mathrm{hz}},{\beta }_{ji,t,s}^{\mathrm{hz}},P_{j,t,s}^{\mathrm{m},\mathrm{S}},P_{j,t,s}^{\mathrm{m},\mathrm{B}},P_{j,t,s}^{\mathrm{esc}},P_{j,t,s}^{\mathrm{esdc}},\\ P_{j,t,s}^{\mathrm{hsc}},P_{j,t,s}^{\mathrm{hsdc}},g_{j,t,s}^{\mathrm{chp}},P_{j,t,s}^{\mathrm{ec}},g_{j,t,s}^{\mathrm{gb}},h_{j,t,s}^{\mathrm{ac}},\\ b_{j,t,s}^{\mathrm{esc}},b_{j,t,s}^{\mathrm{esdc}},b_{j,t,s}^{\mathrm{hsc}},b_{j,t,s}^{\mathrm{hsdc}}\end{array}\right)\end{array}\right.$$

(4)

where ${\sigma }_s$ represents the probability distribution of the s-th class; $P_{j,t,s}^{\{.\}}$ represents the WT and PV power of the j-th MEAS at time t under scenario s; $y_j=\left(\begin{array}{l}{P}_{ji,t,s}^{\mathrm{hz}},{\beta }_{ji,t,s}^{\mathrm{hz}},P_{j,t,s}^{\mathrm{m},\mathrm{S}},P_{j,t,s}^{\mathrm{m},\mathrm{B}},P_{j,t,s}^{\mathrm{esc}},P_{j,t,s}^{\mathrm{esdc}},P_{j,t,s}^{\mathrm{hsc}},P_{j,t,s}^{\mathrm{hsdc}},\\ g_{j,t,s}^{\mathrm{chp}},P_{j,t,s}^{\mathrm{ec}},g_{j,t,s}^{\mathrm{gb}},h_{j,t,s}^{\mathrm{ac}},b_{j,t,s}^{\mathrm{esc}},b_{j,t,s}^{\mathrm{esdc}},b_{j,t,s}^{\mathrm{hsc}},b_{j,t,s}^{\mathrm{hsdc}}\end{array}\right)$ is the set of operational decision variables for the j-th MEAS, where $P_{ji,t,s}^{\mathrm{hz}}$ represents the direct trading electric power between the j-th MEAS and the i-th MEAS at time t in scenario s; ${\beta }_{ji,t,s}^{\mathrm{hz}}$ is the direct trading price between the j-th and i-th MEASs at time t; $P_{j,t,s}^{\mathrm{esc}}$ and $P_{j,t,s}^{\mathrm{esdc}}$ respectively represent the charging and discharging power of the j-th MEAS’s ES in scenario s at time t; $b_{j,t,s}^{\mathrm{esc}}$ and $b_{j,t,s}^{\mathrm{esdc}}$ represent the state variables of the ES of the j-th MEAS at time t under scenario s, and $b_{j,t,s}^{\mathrm{esc}}$ = 1 indicates that the ES is charging, while $b_{j,t,s}^{\mathrm{esdc}}$ = 1 indicates that the ES is discharging. $P_{j,t,s}^{\mathrm{hsc}}$ and $P_{j,t,s}^{\mathrm{hsdc}}$ represent the charging and discharging power of the j-th MEAS’s HS at time t in scenario s; $b_{j,t,s}^{\mathrm{hsc}}$ and $b_{j,t,s}^{\mathrm{hsdc}}$ respectively represent the state variables of the j-th MEAS’s HS in scenario s at time t, and $b_{j,t,s}^{\mathrm{hsc}}$ = 1 indicates that the HS is currently charging, while $b_{j,t,s}^{\mathrm{hsdc}}$ = 1 indicates that the HS is discharging. $g_{j,t,s}^{\mathrm{chp}}$ represents the gas power of the CHP of the j-th MEAS at time t in scenario s; $P_{j,t,s}^{\mathrm{ec}}$ represents the EC of the j-th MEAS at time t in scenario s; $g_{j,t,s}^{\mathrm{gb}}$ represents the gas power generation of the j-th MEAS in scenario s at time t; and $h_{j,t,s}^{\mathrm{ac}}$ represents the heating power of the j-th MEAS’s AC in scenario s at time t.

$$\begin{array}{c}{C}_j^{\mathrm{cn}}={\beta }_j^{\mathrm{e}}{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{s=1}^S{\sigma }_s\left(P_{j,t,s}^{\mathrm{esc}}+P_{j,t,s}^{\mathrm{esdc}}\right)}}+\\ {\beta }_j^{\mathrm{h}}{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{s=1}^S{\sigma }_s\left(P_{j,t,s}^{\mathrm{hsc}}+P_{j,t,s}^{\mathrm{hsdc}}\right)}}\end{array}$$

(5)

$$C_j^{\mathrm{m}}={\displaystyle \sum _{t=1}^T{\displaystyle \sum _{s=1}^S{\sigma }_s\left({\beta }_{j,t,s}^{\mathrm{m},\mathrm{B}}{P}_{j,t,s}^{\mathrm{m},\mathrm{B}}-{\beta }_{j,t,s}^{\mathrm{m},\mathrm{S}}{P}_{j,t,s}^{\mathrm{m},\mathrm{S}}\right)}}$$

(6)

$$C_{ji}^{\mathrm{hz}}={\displaystyle \sum _{i=1,j\ne i}^N{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{s=1}^S{\sigma }_s{\beta }_{ji,t,s}^{\mathrm{hz}}{P}_{ji,t,s}^{\mathrm{hz}}}}}$$

(7)

where ${\sigma }_s$ represents the probability distribution of scenario s; ${\beta }_j^{\mathrm{e}}$ represents the unit price of charging and discharging power for the j-th ES of the MEAS.

3.2.2. Strategy Space

(1) Uncertainty space of WT and PV scenario probabilities

The probability values of scenarios, denoted as ${\sigma }_s$, theoretically can be within any range. However, to ensure the economic and feasible nature of scheduling schemes, it is necessary to constrain the probabilities of the most adverse WT and PV output scenarios within reasonable bounds to ensure their fluctuations are within acceptable limits. In the group $M_{\mathrm{C}}$ of WT or PV output scenarios, the scenario reduction method is employed to select a subset S as the basis scenario, and the probability values of this subset, denoted as M, are taken as the initial scenario probabilities ${\sigma }_s^0$. The probability values in actual scenarios cannot be consistent with the initial scenario probabilities, and uncertainty still exists. In order to solve for the worst-case scenario and probability distribution, a comprehensive norm constraint is applied to the probabilities of each scenario, with the initial scenario probability values as the center:

$${\Omega }_{\sigma }=\left\{{\sigma }_s\left|{\sigma }_s\ge 0,{\displaystyle \sum _{s=1}^M{\sigma }_s}=1,{\displaystyle \sum _{s=1}^M\left|{\sigma }_s-{\sigma }_s^0\right|}\le {\theta }_1,\underset{1\le s\le M}{{\displaystyle \max }}\left|{\sigma }_s-{\sigma }_s^0\right|\le {\theta }_{\infty }\right.\right\}$$

(8)

where ${\Omega }_{\sigma }$ represents the probability uncertainty space for WT and PV classes; ${\theta }_1$ and ${\theta }_{\infty }$ are the allowable deviation values for actual scenario probabilities under the 1-norm condition and ∞-norm condition, respectively. According to reference [31], the following can be derived:

$$\left\{\begin{array}{l}{\theta }_1={\displaystyle \frac{M}{2M_{\mathrm{C}}}}\ln {\displaystyle \frac{2M}{1-{\alpha }_1}}\\ {\theta }_{\infty }={\displaystyle \frac{1}{2M_{\mathrm{C}}}}\ln {\displaystyle \frac{2M}{1-{\alpha }_{\infty }}}\end{array}\right.$$

(9)

where ${\alpha }_1$ and ${\alpha }_{\infty }$ represent the confidence levels of the 1-norm condition and the ∞-norm condition, respectively. Equation (8) indicates that as the historical data volume $S_C$ increases, the allowable deviation values ${\theta }_1$ and ${\theta }_{\infty }$ of the actual scenario probabilities under the 1-norm condition and the ∞-norm condition become smaller. The probability distributions of each scenario also tend to converge towards the true situation. Consequently, the uncertainty of WT and PV output considered in the scheduling scheme decreases, and its conservatism is also lower.

(2) Uncertainty sets of WT and PV output

The uncertainty sets of WT and PV output are represented as

$$D^j=\left[\begin{array}{l}{P}_j^{\{.\}}\in {\mathrm{R}}^{T\times S}:(P_{j,t,s}^{\{.\}}-P_{j,t,s}^{\{.\},\mathrm{f}})/{\Delta }_{j,t,s}^{\{.\}}={\xi }_{j,t,s}^{\{.\}},\\ {\displaystyle \sum _{t=1}^T{\xi }_{j,t,s}^{\{.\}}}\le {\Gamma }_{j,s}^{\{.\}},0\le {\xi }_{j,t,s}^{\{.\}}\le 1,\\ P_{j,t,s}^{\{.\},\mathrm{f}}-{\Delta }_{j,t,s}^{\{.\}}\le P_{j,t,s}^{\{.\}}\le P_{j,t,s}^{\{.\},\mathrm{f}}+{\Delta }_{j,t,s}^{\{.\}},\\ \{.\}=\left\{\mathrm{ES},\mathrm{HS}\right\}\end{array}\right]$$

(10)

where $D^j$ represents the uncertain space of WT and PV; $P_j^{\{.\}}$ is in the vector form of variable $P_{j,t,s}^{\{.\}}$; $P_{j,t,s}^{\{.\},\mathrm{f}}$ represents the predicted power of WT and PV of the j-th MEAS at time t; ${\Delta }_{j,t,s}^{\{.\}}$ represents the maximum fluctuation amplitude of the WT and PV of the j-th MEAS at time t; ${\xi }_{j,t,s}^{\{.\}}$ represents the fluctuation ratio of WT and PV of the j-th microgrid at time t; and ${\Gamma }_{j,s}^{\{.\}}$ represents the uncertainty margin of WT and PV of the j-th microgrid, used to adjust the conservatism of the uncertainty set.

(3) Electricity trading constraints

$$0\le P_{j,t,s}^{\mathrm{m},\mathrm{S}}\le P_{j,\max }^{\mathrm{m}}$$

(11)

$$0\le P_{j,t,s}^{\mathrm{m},\mathrm{B}}\le P_{j,\max }^{\mathrm{m}}$$

(12)

$$P_{j,t,s}^{\mathrm{m}}=P_{j,t,s}^{\mathrm{m},\mathrm{S}}-P_{j,t,s}^{\mathrm{m},\mathrm{B}}$$

(13)

$$0\le P_{ji,t,s}^{\mathrm{hz}}\le P_{j,\max }^{\mathrm{hz}}$$

(14)

where $P_{j,\max }^{\mathrm{m}}$ represents the maximum electricity trading volume between the j-th MEAS and GDD; $P_{j,t,s}^{\mathrm{m}}$ represents the total electricity trading volume between the j-th MEAS and GDD at time t under scenarios; and $P_{j,\max }^{\mathrm{hz}}$ represents the maximum electricity trading volume between the j-th MEAS and other MEASs.

(4) Power balance constraints

Power balance constraints primarily involve the balance of electrical power exchange between MEASs and the balance of supply and demand within each MEAS:

$$P_{ij,t,s}^{\mathrm{hz}}=P_{ji,t,s}^{\mathrm{hz}}$$

(15)

$$P_{j,t,s}^{\mathrm{m},\mathrm{B}}+P_{ji,t,s}^{\mathrm{hz}}+P_{j,t,s}^{\mathrm{w}}+P_{j,t,s}^{\mathrm{pv}}+{\alpha }_{\mathrm{e}}^{\mathrm{chp}}{g}_{j,t,s}^{\mathrm{chp}}+P_{j,t,s}^{\mathrm{esdc}}=P_{j,t}^{\mathrm{L}}+P_{j,t,s}^{\mathrm{esc}}+P_{j,t,s}^{\mathrm{m},\mathrm{S}}+P_{j,t,s}^{\mathrm{ec}}$$

(16)

$${\alpha }_{\mathrm{h}}^{\mathrm{chp}}{g}_{j,t,s}^{\mathrm{chp}}+{\alpha }_{\mathrm{h}}^{\mathrm{gb}}{g}_{j,t,s}^{\mathrm{gb}}+h_{j,t,s}^{\mathrm{hsdc}}=h_{j,t}^{\mathrm{L}}+h_{j,t,s}^{\mathrm{ac}}+h_{j,t,s}^{\mathrm{hsc}}$$

(17)

$${\alpha }_{\mathrm{c}}^{\mathrm{ec}}{P}_{j,t,s}^{\mathrm{ec}}+{\alpha }_{\mathrm{c}}^{\mathrm{ac}}{h}_{j,t,s}^{\mathrm{ac}}=c_{j,t}^{\mathrm{L}}$$

(18)

$$g_{j,t,s}^{\mathrm{chp}}+g_{j,t,s}^{\mathrm{gb}}=g_{j,t,s}^{\mathrm{buy}}$$

(19)

where, $P_{ij,t,s}^{\mathrm{hz}}$ represents the direct trading electric power between the i-th and j-th MEAS at time t under scenario s; ${\alpha }_{\mathrm{e}}^{\mathrm{chp}}$, ${\alpha }_{\mathrm{h}}^{\mathrm{chp}}$, ${\alpha }_{\mathrm{h}}^{\mathrm{gb}}$, ${\alpha }_{\mathrm{c}}^{\mathrm{ec}}$, ${\alpha }_{\mathrm{c}}^{\mathrm{ac}}$ represent the energy conversion efficiency of CHP, GB, EC, AC; $P_{j,t}^{\mathrm{L}}$, $h_{j,t}^{\mathrm{L}}$, $c_{j,t}^{\mathrm{L}}$ represent the fixed loads of the j-th MEAS at time t; $g_{j,t,s}^{\mathrm{buy}}$ represents the gas source power of the j-th MEAS at time t under scenario s.

(5) Price constraints

Due to the purchase price of electricity set by GDD for MEASs being greater than the selling price, assuming multiple MEASs form cooperation at the lower level, the participating MEASs will engage in electricity power exchange at a lower price among themselves. This is to facilitate energy sharing among the MEASs.

$${\beta }_{j,t}^{\mathrm{m},\mathrm{S}}\le {\beta }_{ji,t,s}^{\mathrm{hz}}\le {\beta }_{j,t}^{\mathrm{m},\mathrm{B}}$$

(20)

$${\beta }_{ji,t}^{\mathrm{hz}}={\beta }_{ij,t}^{\mathrm{hz}}$$

(21)

(6) Equipment models and constraints.

Constraints for GB, CHP, ES, HS, EC, AC, and GS are detailed in Appendix A (A1)–(A7).

The above Equations (8)–(21) and Equations (A1)–(A7) collectively form the strategy space ${\Omega }^{\mathrm{m},\mathrm{hz}}$ A of the MEAS stochastic model.

3.3. Dual-Layer Model

Building upon Section 3.1 and Section 3.2, we establish a dual-layer model for GDD and multiple MEASs based on the master–slave game, as follows:

$$\begin{array}{c}\left({\tilde{\beta }}_{i,t,s}^{\mathrm{m},\mathrm{S}},{\tilde{\beta }}_{i,t,s}^{\mathrm{m},\mathrm{B}},{\tilde{\beta }}_{j,t,s}^{\mathrm{m},\mathrm{S}},{\tilde{\beta }}_{j,t,s}^{\mathrm{m},\mathrm{B}}\right)=\arg \left\{U^{\mathrm{DSO}}\left(\begin{array}{l}{\tilde{P}}_{i,t,s}^{\mathrm{m},\mathrm{S}},{\tilde{P}}_{i,t,s}^{\mathrm{m},\mathrm{B}},{\tilde{\sigma }}_{i,s},{\tilde{P}}_{j,t,s}^{\mathrm{m},\mathrm{S}},\\ {\tilde{P}}_{j,t,s}^{\mathrm{m},\mathrm{B}},{\tilde{\sigma }}_{j,s}\end{array}\right)\right.\\ \mathrm{s}.\mathrm{t}. \left({\tilde{\beta }}_{i,t,s}^{\mathrm{m},\mathrm{S}},{\tilde{\beta }}_{i,t,s}^{\mathrm{m},\mathrm{B}},{\tilde{\beta }}_{j,t,s}^{\mathrm{m},\mathrm{S}},{\tilde{\beta }}_{j,t,s}^{\mathrm{m},\mathrm{B}}\right)\in {\Omega }^{\mathrm{D}}\\ \left({\tilde{P}}_{i,t,s}^{\mathrm{m},\mathrm{S}},{\tilde{P}}_{i,t,s}^{\mathrm{m},\mathrm{B}},{\tilde{\sigma }}_{i,s},{\tilde{P}}_{ij,t,s}^{\mathrm{hz}},{\tilde{\beta }}_{ij,t,s}^{\mathrm{hz}}\right)=\arg \left\{U_i^{\mathrm{m},\mathrm{hz}}\left(\begin{array}{l}{\tilde{\beta }}_{i,t,s}^{\mathrm{m},\mathrm{S}},{\tilde{\beta }}_{i,t,s}^{\mathrm{m},\mathrm{B}},\\ {\tilde{P}}_{ji,t,s}^{\mathrm{hz}},{\tilde{\beta }}_{ji,t,s}^{\mathrm{hz}}\end{array}\right)\right.\\ \mathrm{s}.\mathrm{t}. \left({\tilde{P}}_{i,t,s}^{\mathrm{m},\mathrm{S}},{\tilde{P}}_{i,t,s}^{\mathrm{m},\mathrm{B}},{\tilde{P}}_{ij,t,s}^{\mathrm{hz}},{\tilde{\beta }}_{ij,t,s}^{\mathrm{hz}}\right)\in {\Omega }_i^{\mathrm{m},\mathrm{hz}}\\ ,\dots ,\\ \left({\tilde{P}}_{j,t,s}^{\mathrm{m},\mathrm{S}},{\tilde{P}}_{j,t,s}^{\mathrm{m},\mathrm{B}},{\tilde{\sigma }}_{j,s},{\tilde{P}}_{ji,t,s}^{\mathrm{hz}},{\tilde{\beta }}_{ji,t,s}^{\mathrm{hz}}\right)=\arg \left\{U_j^{\mathrm{m},\mathrm{hz}}\left(\begin{array}{l}{\tilde{\beta }}_{j,t,s}^{\mathrm{m},\mathrm{S}},{\tilde{\beta }}_{j,t,s}^{\mathrm{m},\mathrm{B}},\\ {\tilde{P}}_{ij,t,s}^{\mathrm{hz}},{\tilde{\beta }}_{ij,t,s}^{\mathrm{hz}}\end{array}\right)\right.\\ \mathrm{s}.\mathrm{t}. \left({\tilde{P}}_{j,t,s}^{\mathrm{m},\mathrm{S}},{\tilde{P}}_{j,t,s}^{\mathrm{m},\mathrm{B}},{\tilde{P}}_{ji,t,s}^{\mathrm{hz}},{\tilde{\beta }}_{ji,t,s}^{\mathrm{hz}}\right)\in {\Omega }_j^{\mathrm{m},\mathrm{hz}}\end{array}$$

(22)

where ${\tilde{P}}_{i,t,s}^{\mathrm{m},\mathrm{S}},{\tilde{P}}_{i,t,s}^{\mathrm{m},\mathrm{B}},{\tilde{P}}_{j,t,s}^{\mathrm{m},\mathrm{S}},{\tilde{P}}_{j,t,s}^{\mathrm{m},\mathrm{B}}$ represents the power vectors transmitted by consumers i and j to the GDD, respectively, ${\tilde{\sigma }}_{i,s},{\tilde{\sigma }}_{j,s}$ represents the probability distribution values transmitted by consumers i and j to GDD, and ${\tilde{\beta }}_{i,t,s}^{\mathrm{m},\mathrm{S}},{\tilde{\beta }}_{i,t,s}^{\mathrm{m},\mathrm{B}},{\tilde{\beta }}_{j,t,s}^{\mathrm{m},\mathrm{S}},{\tilde{\beta }}_{j,t,s}^{\mathrm{m},\mathrm{B}}$ represents the price variables transmitted by the GDD to consumers i and j. These variables can be classified as master–slave variables; ${\tilde{P}}_{ij,t,s}^{\mathrm{hz}}$ and ${\tilde{P}}_{ji,t,s}^{\mathrm{hz}}$ is a pair of power coupling variable vectors between producer–consumer i and producer–consumer j, while ${\tilde{\beta }}_{ij,t,s}^{\mathrm{hz}}$ and ${\tilde{\beta }}_{ji,t,s}^{\mathrm{hz}}$ are a pair of price coupling variable vectors between producer–consumer i and producer–consumer j, classified as cooperative coupling vectors. The proposed model is a max–max–max–min two-stage, four-layer structure. The first stage is the first-layer max, which is the dynamic pricing model of the GDD, determining the purchase–sale electricity prices for the second stage. The second stage consists of the last three layers of max–max–min, representing a stochastic–robust model of the MEAS under producer–consumer cooperation. It relies on the purchase–sale electricity prices determined in the first stage to determine the optimal energy allocation and cooperative prices for producer–consumers. When ignoring the probability uncertainty of WTs, the model can be simplified into a max–max–min two-stage, three-layer robust model; when ignoring the output uncertainty of WTs, the model can be simplified into a max–max–min two-stage, three-layer stochastic model; and when simultaneously ignoring the probability-output uncertainty of the WT, the model can be simplified into a max–min two-stage two-layer deterministic model.

In Equation (22), the GDD and MEAS formulate strategies with the objectives of maximizing revenue and minimizing operational costs, respectively. The GDD’s revenue is dependent on the set electricity prices and the trading volume of the MEAS: a larger price differential between buying and selling electricity results in a higher trading volume for the MEAS, thereby increasing the GDD’s revenue. However, the MEAS’s response to electricity prices also affects the GDD’s revenue: higher buying prices lead to a lower purchasing volume for the MEAS, while lower selling prices lead to a reduced selling volume, resulting in a decrease in the trading volume between the MEAS and GDD. Thus, there exists a game of interests between the GDD and MEAS. To maximize its own revenue, the GDD needs to consider the MEAS’s response to prices and seek a Nash equilibrium solution through game theory as the optimal pricing strategy.

Considering the equilibrium solution and its proof in principal–agent games, when there exists an equilibrium solution in principal–agent two-tier models, the following conditions must be satisfied [37]: (1) The strategy sets ${\Omega }^{\mathrm{D}}$, ${\Omega }_i^{\mathrm{m},\mathrm{hz}}$, and ${\Omega }_j^{\mathrm{m},\mathrm{hz}}$ are all non-empty bounded subsets of their respective Euclidean spaces; (2) $U_i^{\mathrm{m},\mathrm{hz}}$ and $U_j^{\mathrm{m},\mathrm{hz}}$ concern quasi-concave functions with respect to the variables ${\Omega }_i^{\mathrm{m},\mathrm{hz}}$ and ${\Omega }_j^{\mathrm{m},\mathrm{hz}}$; and (3) $U^{\mathrm{DSO}}$ is a continuous function concerning ${\Omega }^{\mathrm{D}}$, ${\Omega }_i^{\mathrm{m},\mathrm{hz}}$, and ${\Omega }_j^{\mathrm{m},\mathrm{hz}}$.

From Equation (2), it is clear that condition 3 is satisfied. Considering Equations (3), (8), (10)–(21), and (A1)–(A7), we can conclude that strategy sets ${\Omega }^{\mathrm{D}}$, ${\Omega }_i^{\mathrm{m},\mathrm{hz}}$, and ${\Omega }_j^{\mathrm{m},\mathrm{hz}}$ are all non-empty quasiconvex sets, thus satisfying condition 1. Furthermore, Equations (5)–(7) indicate that the cost indicators of the MEAS stochastic–robust model are all linear functions, implying that $U_i^{\mathrm{m},\mathrm{hz}}$ and $U_j^{\mathrm{m},\mathrm{hz}}$ are quasiconcave functions, fulfilling condition 2. Therefore, in Equation (22), assuming $A=\left({\tilde{\beta }}_{i,t,s}^{\mathrm{m},\mathrm{S}},{\tilde{\beta }}_{i,t,s}^{\mathrm{m},\mathrm{B}},{\tilde{\beta }}_{j,t,s}^{\mathrm{m},\mathrm{S}},{\tilde{\beta }}_{j,t,s}^{\mathrm{m},\mathrm{B}},{\tilde{P}}_{i,t,s}^{\mathrm{m},\mathrm{S}},{\tilde{P}}_{i,t,s}^{\mathrm{m},\mathrm{B}},{\tilde{P}}_{j,t,s}^{\mathrm{m},\mathrm{S}},{\tilde{P}}_{j,t,s}^{\mathrm{m},\mathrm{B}},{\tilde{\sigma }}_{i,s},{\tilde{\sigma }}_{j,s}\right)$ is a strategy set for the principal–agent game’s two-layer model, then according to the definition of the equilibrium solution in principal–agent games, it is considered an equilibrium solution when it satisfies the following conditions:

$$\left\{\begin{array}{l}{U}^{\mathrm{D}}\left(A\right)\ge U^{\mathrm{D}}\left(\begin{array}{l}{\beta }_{i,t,s}^{\mathrm{m},\mathrm{S}},{\beta }_{i,t,s}^{\mathrm{m},\mathrm{B}},{\beta }_{j,t,s}^{\mathrm{m},\mathrm{S}},{\beta }_{j,t,s}^{\mathrm{m},\mathrm{B}},\\ {\tilde{P}}_{i,t,s}^{\mathrm{m},\mathrm{S}},{\tilde{P}}_{i,t,s}^{\mathrm{m},\mathrm{B}},{\tilde{P}}_{j,t,s}^{\mathrm{m},\mathrm{S}},{\tilde{P}}_{j,t,s}^{\mathrm{m},\mathrm{B}},{\tilde{\sigma }}_{i,s},{\tilde{\sigma }}_{j,s}\end{array}\right)\\ U_i^{\mathrm{m}}\left({\tilde{\beta }}_{i,t,s}^{\mathrm{m},\mathrm{S}},{\tilde{\beta }}_{i,t,s}^{\mathrm{m},\mathrm{B}},{\tilde{P}}_{i,t,s}^{\mathrm{m},\mathrm{S}},{\tilde{P}}_{i,t,s}^{\mathrm{m},\mathrm{B}},{\tilde{P}}_{ij,t,s}^{\mathrm{hz}},{\tilde{\sigma }}_{i,s}\right)\le \\ U_i^{\mathrm{m}}\left({\tilde{\beta }}_{i,t,s}^{\mathrm{m},\mathrm{S}},{\tilde{\beta }}_{i,t,s}^{\mathrm{m},\mathrm{B}},P_{i,t,s}^{\mathrm{m},\mathrm{S}},P_{i,t,s}^{\mathrm{m},\mathrm{B}},{\tilde{P}}_{ji,t,s}^{\mathrm{hz}},{\sigma }_{i,s}\right)\\ U_j^{\mathrm{m}}\left({\tilde{\beta }}_{j,t,s}^{\mathrm{m},\mathrm{S}},{\tilde{\beta }}_{j,t,s}^{\mathrm{m},\mathrm{B}},{\tilde{P}}_{j,t,s}^{\mathrm{m},\mathrm{S}},{\tilde{P}}_{j,t,s}^{\mathrm{m},\mathrm{B}},{\tilde{P}}_{ji,t,s}^{\mathrm{hz}},{\tilde{\sigma }}_{j,s}\right)\le \\ U_j^{\mathrm{m}}\left({\tilde{\beta }}_{j,t,s}^{\mathrm{m},\mathrm{S}},{\tilde{\beta }}_{j,t,s}^{\mathrm{m},\mathrm{B}},P_{j,t,s}^{\mathrm{m},\mathrm{S}},P_{j,t,s}^{\mathrm{m},\mathrm{B}},{\tilde{P}}_{ij,t,s}^{\mathrm{hz}},{\sigma }_{j,s}\right)\end{array}\right.$$

(23)

4. Model Solving

4.1. Solution of Two-Layer Model

The two-layer model based on the principal–agent game is solved using the Particle Swarm Optimization (PSO) algorithm [38,39]. The schematic diagram of variable exchange in the model is shown in Figure 2. The specific steps are as follows:

(1)

Initialize system parameters, including the on-grid electricity price, off-grid electricity price, MEAS parameters, and relevant parameters of the PSO algorithm;

(2)

Generate initial sample points ${\beta }_{j,t,s,q}^{\mathrm{m},\mathrm{S}}$ and ${\beta }_{j,t,s,q}^{\mathrm{m},\mathrm{B}}$ based on Latin Hypercube Sampling (LHS), with a total of Q samples;

(3)

The MEAS calculates its collaborative model based on ${\beta }_{j,t,s,q}^{\mathrm{m},\mathrm{S}}(k)$ and ${\beta }_{j,t,s,q}^{\mathrm{m},\mathrm{B}}(k)$ (where k is the iteration count), obtaining the optimal purchase and sale electricity plans $P_{j,t,s,q}^{\mathrm{m},\mathrm{S}}(k)$ and $P_{j,t,s,q}^{\mathrm{m},\mathrm{B}}(k)$, and provides feedback to the GDD;

(4)

Solve the GDD stochastic pricing model based on Equation (1), and compute the objective function $U_q^{\mathrm{D}}(k)$;

(5)

Update individual historical best prices ${\beta }_{j,t,s,q}^{\mathrm{m},\mathrm{S}*}(k)$ and ${\beta }_{j,t,s,q}^{\mathrm{m},\mathrm{B}*}(k)$, as well as individual historical best profits $U^{\mathrm{D}*}$;

(6)

Perform a selection operation: if the condition $U_q^{\mathrm{D}}(k)$ > $U^{\mathrm{D}*}$ is met, designate ${\beta }_{j,t,s,q}^{\mathrm{m},\mathrm{S}}(k)$ and ${\beta }_{j,t,s,q}^{\mathrm{m},\mathrm{B}}(k)$ as the group’s historical best prices and select them as the internal electricity prices for the next iteration. Otherwise, designate ${\beta }_{j,t,s,q}^{\mathrm{m},\mathrm{S}*}(k)$ and ${\beta }_{j,t,s,q}^{\mathrm{m},\mathrm{B}*}(k)$ as the internal electricity prices for the next iteration and update the group’s historical best revenue to $U^{\mathrm{D}*}$;

(7)

Let k = k + 1. If k ≤ K (where K is the maximum number of iterations), return to step 2). Otherwise, output the optimal result.

4.2. Solving the Cooperation Model of Multiple MEAS

In the two-tier model, the second stage involves solving the cooperation prices and power exchange between the MEASs. The cooperation coupling vectors in Equation (22) cancel each other out, making direct solution difficult. Therefore, the MEAS cooperation model can be characterized as a Nash bargaining model and transformed into two sub-problems: maximizing cooperation benefits (P1) and distributing cooperation profits (P2), to avoid the non-convex problem of the Nash bargaining model. The detailed transformation process is described in Appendix B.

A Nash bargaining model is adopted to construct the producer–consumer cooperation model.

$$\begin{array}{l}{U}^{\mathrm{m},\mathrm{hz}}=\max {\displaystyle \prod _{j=1}^N\left(U_j^{\mathrm{m},\mathrm{hz}}-U_j^{\mathrm{m},0}\right)}\\ U_j^{\mathrm{m},\mathrm{hz}}=\underset{{\sigma }_s}{\max } \underset{P_{j,s,t}^{\{.\}}}{\max } \underset{y_j}{\min }\left(C_j^{\mathrm{cn}}+C_j^{\mathrm{m}}+C_{ji}^{\mathrm{hz}}\right)\\ \mathrm{s}.\mathrm{t}. U_j^{\mathrm{m},\mathrm{hz}}\ge U_j^{\mathrm{m},0},{\Omega }_j^{\mathrm{m},\mathrm{hz}}\end{array}$$

(24)

where $U_j^{\mathrm{m},0}$ represents the profit obtained by the j-th MEAS that did not participate in the cooperation.

P1:

$$\begin{array}{l}{U}_{\mathrm{p}1}=\max {\displaystyle \sum _{j=1}^N\underset{W_j^{\mathrm{m},\mathrm{hz}}}{\underbrace{\left[\underset{{\sigma }_s}{\max } \underset{P_{j,t,s}^{\{.\}}}{\max } \underset{{\beta }_{ji,t}^{\mathrm{hz}}\notin y_j,}{\min }\left(C_j^{\mathrm{en}}+C_j^{\mathrm{m}}\right)\right]}}}\\ \mathrm{s}.\mathrm{t}. {\Omega }_j^{\mathrm{m},\mathrm{hz}}\notin \left\{\left(7\right),\left(20\right)-\left(21\right)\right\}\end{array}$$

(25)

where $W_j^{\mathrm{m},\mathrm{hz}}$ represents the cost under cooperation, excluding P2P revenue. Since the transaction electricity price, total sold electricity power, and total purchased electricity power are all the same between producers and consumers, the transaction costs ${\displaystyle \sum _{i=1,j\ne i}^N{C}_{ji}^{\mathrm{hz}}}+{\displaystyle \sum _{j=1,i\ne j}^N{C}_{ij}^{\mathrm{hz}}}=0$ related to electricity power within producers and consumers can be omitted in Equations (7), (20), and (21). The transaction electricity price in Equations (20) and (21) will be solved in P2.

P2:

$$\begin{array}{l}{U}_{\mathrm{p}2}=\underset{{\beta }_{ji,t,s}^{\mathrm{hz}}}{\min }{\displaystyle \sum _{j=1}^N\ln \left(W_j^{\mathrm{m},\mathrm{hz}*}+C_{ji}^{\mathrm{hz}*}-U_j^{\mathrm{m},0}\right)}\\ C_{ji}^{\mathrm{hz}*}={\displaystyle \sum _{i=1,j\ne i}^N{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{s=1}^S{\sigma }_s{\beta }_{ji,t,s}^{\mathrm{hz}}{P}_{ji,t,s}^{\mathrm{hz}*}}}}\\ \mathrm{s}.\mathrm{t}. W_j^{\mathrm{m},\mathrm{hz}*}+C_{ji}^{\mathrm{hz}*}\ge U_j^{\mathrm{m},0}\\ \left(7\right),\left(20\right)-\left(21\right)\end{array}$$

(26)

where variables with superscript * denote the optimal solution for P1.

The distributed solving of P1 and P2 is achieved using ADMM, the solving principle of which is detailed in Appendix C. Consensus variables need to be introduced to decouple Equations (15) and (21), so that only information about cooperative plans needs to be provided without specific network topologies and operational modes, in order to protect their privacy, as shown in Equation (27):

$$\begin{array}{l}{\sigma }_{ij,t,s}^{\mathrm{p}1}=P_{ij,t,s}^{\mathrm{hz}},{\sigma }_{ji,t,s}^{\mathrm{p}1}=P_{ji,t,s}^{\mathrm{hz}}\\ {\sigma }_{ij,t,s}^{\mathrm{p}2}={\beta }_{ij,t,s}^{\mathrm{hz}},{\sigma }_{ji,t,s}^{\mathrm{p}2}={\beta }_{ji,t,s}^{\mathrm{hz}}\end{array}$$

(27)

4.3. Solving the MEAS Random–Robust Model

Equation (25) is solved through ADMM in a distributed manner, decomposed into j three-layered MEAS random–robust models, ensuring the optimality of the external electrical energy allocation for MEAS under the most adverse renewable energy scenarios and worst output conditions. Due to the problem’s nonlinear coupling nature, it cannot be directly solved. Therefore, the AIV algorithm is introduced to perform variable alternating iterative solving. Firstly, independent optimization variables are identified, then by fixing different independent optimization variables, the original optimization problem is decoupled into multiple simplified sub-problems, and finally, multiple sub-problems are alternately solved.

The random–robust model of the MEAS can be abbreviated as

$$\begin{array}{l}\underset{\sigma \in {\Omega }_{\sigma }}{\max }\underset{P_j^{\{.\}}\in D_j}{\max }\underset{Y_j^{\mathrm{m}}\in {\Omega }_j^{\mathrm{m},\mathrm{hz}}\left(x^{*},\sigma ,P_j^{\{.\}}\right)}{\min }{a}^{\mathrm{T}}{Y}_j^{\mathrm{m}}\\ \mathrm{s}.\mathrm{t}. A{Y}_j^{\mathrm{m}}\le B,C{Y}_j^{\mathrm{m}}\le f(P_j^{\{.\}}),\\ E\sigma \le h,I{P}_j^{\{.\}}\le J,\end{array}$$

(28)

where $\sigma$ represents the vector form of ${\sigma }^s$; $Y_j^{\mathrm{m}}$ represents the vector form of variables in ${\beta }_{ji,t}^{\mathrm{hz}}\notin y_j$; $x^{*}$ denotes the optimum value of the upper-level GDD; $f$ denotes a function related to uncertainty; and a, A, B, C, E, h, I, and J are matrices with constant coefficients. The first set of constraints includes Equations (11)–(15), (17)–(19), and (A1)–(A7). The second set of constraints is represented by Equation (16). The third set of constraints is represented by Equation (8). The fourth set of constraints is represented by Equation (10).

In Equation (28), $\sigma$ and $P_j^{\{.\}}$ are two sets of independent optimization variables, while $Y_j^{\mathrm{m}}$ belongs to ${\Omega }_j^{\mathrm{m},\mathrm{hz}}\left(\sigma ,P_j^{\{.\}}\right)$ and is a non-independent variable. Fixing $\sigma$ and $P_j^{\{.\}}$ separately decouples the original problem into two sub-problems: one with $\sigma$ fixed and the other with $P_j^{\{.\}}$ fixed.

(a) $\sigma$ Fixed Problem

$$\begin{array}{l}\underset{P_j^{\{.\}}\in D_j}{\max }\underset{Y_j^{\mathrm{m}}\in {\Omega }_j^{\mathrm{m},\mathrm{hz}}\left(x^{*},{\sigma }^{*},P_j^{\{.\}}\right)}{\min }{a}^{\mathrm{T}}{Y}_j^{\mathrm{m}}\\ \mathrm{s}.\mathrm{t}. A{Y}_j^{\mathrm{m}}\le B,C{Y}_j^{\mathrm{m}}\le f(P_j^{\{.\}}),\\ E{\sigma }^{*}\le h,I{P}_j^{\{.\}}\le J,\end{array}$$

(29)

where ${\sigma }^{*}$ is a constant obtained after fixing $\sigma$.

By utilizing KKT conditions [38], the fixed problem $\sigma$ is consolidated into a single-level model:

$$\begin{array}{l}\underset{P_j^{\{.\}},Y_j^{\mathrm{m}},\lambda }{\max }{a}^{\mathrm{T}}{Y}_j^{\mathrm{m}}\\ \mathrm{s}.\mathrm{t}. A{Y}_j^{\mathrm{m}}\le B,C{Y}_j^{\mathrm{m}}\le f(P_j^{\{.\}}),I{P}_j^{\{.\}}\le J,\\ {\lambda }_1^{\mathrm{T}}A+a^{\mathrm{T}}=0,{\lambda }_2^{\mathrm{T}}C+f^{\prime }(P_j^{\{.\}})+a^{\mathrm{T}}=0,{\lambda }_3^{\mathrm{T}}I=0,\\ {\lambda }_1^{\mathrm{T}}\left[A{Y}_j^{\mathrm{m}}-B\right]=0,{\lambda }_2^{\mathrm{T}}\left[C{Y}_j^{\mathrm{m}}-f(P_j^{\{.\}})\right]=0,\\ {\lambda }_3^{\mathrm{T}}\left[I{P}_j^{\{.\}}-J\right]=0,{\lambda }_1\ge 0,{\lambda }_2\ge 0,{\lambda }_3\ge 0\end{array}$$

(30)

Note that the complementary slackness constraints of Equation (30) can be linearized using the BIG-M method [39]:

$$\left\{\begin{array}{l}{\lambda }_1^{\mathrm{T}}\le M\left(1-\vartheta \right)\\ A{Y}_j^{\mathrm{m}}-B\ge -M\vartheta \end{array}\right.$$

(31)

where M is a large positive number and $\vartheta$ is a binary variable.

Then, the optimal solution $P_j^{\{.\}*}$ and the objective function value $U^1$ are obtained by solving Equation (30).

(b) $P_j^{\{.\}}$ Fixed problem

$$\begin{array}{l}\underset{\sigma \in {\Omega }_{\sigma }}{\max }\underset{Y_j^{\mathrm{m}}\in {\Omega }_j^{\mathrm{m},\mathrm{hz}}\left(x^{*},\sigma ,P_j^{\{.\}*}\right)}{\min }{a}^{\mathrm{T}}{Y}_j^{\mathrm{m}}\\ \mathrm{s}.\mathrm{t}. A{Y}_j^{\mathrm{m}}\le B,C{Y}_j^{\mathrm{m}}\le f(P_j^{\{.\}*}),E\sigma \le h\end{array}$$

(32)

Although the objective function includes nonlinear terms composed of $\sigma$ and $Y_j^{\mathrm{m}}$ (as in Equations (8) and (A1)), the fixed problem $P_j^{\{.\}}$ can be decomposed into two independent steps without the need for strong duality/KKT transformations like those used in fixed problem $\sigma$, because there are no direct coupling constraints between $\sigma$ and $Y_j^{\mathrm{m}}$. Therefore, the fixed problem $P_j^{\{.\}}$ can be decoupled into the following S + 1 linear programming models:

First, the S linear programming models are solved:

$$\begin{array}{l}{y}_s^{*}=\underset{Y_j^{\mathrm{m}}\in {\Omega }_j^{\mathrm{m},\mathrm{hz}}\left(x^{*},{\sigma }_s,P_j^{\{.\}*}\right)}{\min }{a}^{\mathrm{T}}{Y}_j^{\mathrm{m}},s=1,\dots ,S\\ \mathrm{s}.\mathrm{t}. A{Y}_j^{\mathrm{m}}\le B,C{Y}_j^{\mathrm{m}}\le f(P_j^{\{.\}*})\end{array}$$

(33)

where $y_s^{*}$ is the optimal value for each s. Then, $y_s^{*}$ is substituted into the following model to obtain the optimal value ${\sigma }_s$ and the objective function value $U^2$.

$$\underset{\sigma \in {\Omega }_{\sigma }}{\max }{y}_s^{*}$$

(34)

Convergence can be achieved when the objective function values of the two fixed problems satisfy a small error ${\xi }_{\mathrm{RO}}$:

$$\left|U^1-L^2\right|\le \xi$$

(35)

In conclusion, the solution process diagram of the proposed double-layer, four-stage stochastic–robust optimization model is shown in Appendix D Figure A1.

5. Case Study Simulation

5.1. Case Description

The double-layer model is tested with one GDD and three MEASs as subjects. The MEAS model is derived from a comprehensive energy service demonstration project in Dongguan, which currently has 6 PVs, 6 energy storage units, 2 charging stations (20 charging piles), 3 flexible loads, and 2 MEASs formed by any combination of the above elements, along with one intelligent distribution room. It also has management capabilities for energy routers and combined cooling, heating, and power units, achieving full coverage of the main elements on the demand side. By setting different renewable energy output scenarios, we can expand this project into more MEASs. In this paper, we expand this project into 3 MEASs of equal size, with MEAS1 adding WT, MEAS2 replacing PV with WT, and MEAS3 remaining unchanged. Their parameters are listed in Appendix D,

Table A1. System Parameters.

Parameters	Parameter Values	Parameters	Parameter Values
$e_{j,\max }^{\mathrm{buy}},g_{j,\max }^{\mathrm{buy}}$	2500,4000	${\alpha }_{\mathrm{e}}^{\mathrm{chp}},{\alpha }_{\mathrm{h}}^{\mathrm{chp}}$	0.33,0.47
$e_{j,\max }^{\mathrm{chp}},h_{j,\max }^{\mathrm{chp}}$	1000,1400	${\alpha }_{\mathrm{h}}^{\mathrm{gb}}$	0.9
$h_{j,\max }^{\mathrm{gb}}$	1000	${\alpha }_{\mathrm{c}}^{\mathrm{ec}}$	3.5
$c_{n,\max }^{\mathrm{ec}},c_{n,\max }^{\mathrm{ac}}$	1750,1000	${\alpha }_{\mathrm{c}}^{\mathrm{ac}}$	1.2
$e_{n,\max }^{\mathrm{esc}},e_{n,\max }^{\mathrm{esdc}}$	250	${\lambda }_{\mathrm{e}},{\alpha }_{\mathrm{e}}^{\mathrm{es}}$	0.2,0.98
$S_{n,\min }^{\mathrm{e}},S_{n,\max }^{\mathrm{e}}$	100,900	${\eta }_{\mathrm{e}}^{\mathrm{esc}},{\eta }_{\mathrm{e}}^{\mathrm{esdc}}$	0.96
$h_{n,\max }^{\mathrm{hsc}},h_{n,\max }^{\mathrm{hsdc}}$	200	${\lambda }_{\mathrm{h}},{\alpha }_{\mathrm{h}}^{\mathrm{hs}}$	0.6,0.96
$S_{n,\min }^{\mathrm{h}},S_{n,\max }^{\mathrm{h}}$	100,800	${\eta }_{\mathrm{h}}^{\mathrm{hsc}},{\eta }_{\mathrm{h}}^{\mathrm{hsdc}}$	0.95
$p_t^{\mathrm{g},\mathrm{buy}},a_e^{\mathrm{buy}}$	0.31,0.04	$a_{\mathrm{h}}^{\mathrm{chp}},a_{\mathrm{e}}^{\mathrm{chp}}$	0.013,0.02
$b_{\mathrm{e}}^{\mathrm{ec}},b_{\mathrm{h}}^{\mathrm{ac}}$	0.01,0.016	$b_{\mathrm{g}}^{\mathrm{chp}},b_{\mathrm{g}}^{\mathrm{gb}}$	0.025,0.012
$b_{\mathrm{e}}^{\mathrm{pv}},b_{\mathrm{e}}^{\mathrm{wt}}$	0.024,0.02	$b_{\mathrm{e}}^{\mathrm{es}},b_{\mathrm{h}}^{\mathrm{hs}}$	0.002
$e_{n,\min }^{\mathrm{ex}},e_{n,\max }^{\mathrm{ex}}$	800,800	$a_{\mathrm{h}}^{\mathrm{gb}},b_{\mathrm{buy}}$	0.013

. The grid electricity price and the grid tariff are listed in Appendix D, Figure A2. The load curve of MEASs is shown in Figure 3.

Using the Monte Carlo simulation method, 365 WT scenes are randomly generated. Considering the trade-off between computational efficiency and accuracy, scene reduction techniques are employed to obtain 10 representative initial scenes, as detailed in Appendix D, Figure A3.

The underlying model utilizes MATLAB 2017 (Natick, Massachusetts, USA) calling the Cplex solver, solving on a personal computer with a 3.7GHz 6-core CPU. The convergence accuracy of AIV is set to 0.01. The original residuals of P1 and P2 are both 0.1, with maximum iteration counts of 50 and 100, respectively, and a penalty factor of 0.01. The uncertainty margins for WT and PV are set to 24 and 12, respectively. The scheduling interval is 1 h, with a maximum scheduling period of 24 h. With the initial settings of ${\alpha }_1$ and ${\alpha }_{\infty }$ as 0.9, according to Equation (9), we can determine that ${\theta }_1$ and ${\theta }_{\infty }$ are 0.073 and 0.0073, respectively.

5.2. The Impact of Uncertainty in Scenario Probability

Using WT analysis, the uncertainty in the spatial parameters of scenario probability includes confidence levels ${\alpha }_1$ and ${\alpha }_{\infty }$ for the 1-norm and ∞-norm conditions, which jointly affect the permissible deviation values ${\theta }_1$ and ${\theta }_{\infty }$ of scenario probability, thus influencing the scheduling of the MEASs and the purchase and sale electricity prices of the GDD.

5.2.1. Analysis of the Impact of Confidence Levels on MEAS Scheduling Situation

Setting ${\alpha }_1$ and ${\alpha }_{\infty }$ to 0.8, 0.9, and 0.98, according to the calculation in Equation (9), the allowable deviation values ${\theta }_1$ and ${\theta }_{\infty }$ for the probabilities of actual scenarios under the 1-norm condition and ∞-norm condition are shown in

Table 1. Allowable deviation values for actual scenario probabilities.

Variable	${\mathit{\theta }}_1$	${\mathit{\theta }}_{\mathit{\infty }}$	${\mathit{P}}_{\mathit{j}}^{\mathbf{w}}$
Fixed scenario probability	-	-	19.47
${\alpha }_1={\alpha }_{\infty }=0.8$	0.063	0.0063	19.42
${\alpha }_1={\alpha }_{\infty }=0.9$	0.073	0.0073	19.4
${\alpha }_1={\alpha }_{\infty }=0.98$	0.094	0.0094	19.37

. It can be observed that as the confidence level increases, the probabilities of ${\theta }_1$ and ${\theta }_{\infty }$ increase from 0.063 and 0.0063 to 0.094 and 0.0094, respectively, indicating a growing tolerance for deviations from the actual scenario probabilities. Combining this with the decrease in $P_j^{\mathrm{w}}$ from 19.47 MW to 19.37 MW, it is evident that as the confidence level increases, the output of WT becomes increasingly conservative. The root cause is as follows: it can be inferred that as the confidence level increases, the allowable deviation of scenario probabilities also increases, leading to a more adverse and extreme scenario probability distribution for the WTs. Specifically, scenarios with high outputs receive lower probability values, while scenarios with low outputs receive even lower probability values, resulting in lower WT outputs and greater scheduling conservatism for Monte Carlo simulation. Therefore, considering the uncertainty of scenario probability distributions can provide a wider range of scheduling options for the Monte Carlo simulation, which aligns with the scheduling requirements of real-world engineering.

5.2.2. Analysis of the Impact of Confidence Level on Purchase and Sale Electricity

Figure 4 compares the effects on purchase and sale electricity prices of deterministic WT scenarios with a probability of 0.1 and uncertain WT scenarios with confidence levels of 0.8, 0.9, and 0.98 respectively. Positive values represent sale electricity prices, while negative values indicate purchase electricity prices. In pursuit of its own interests, the GDD tends to offer higher purchase prices and lower sale prices to MEASs. Comparing the purchase and sale electricity prices of each MEAS, it can be observed that as the confidence level increases, the GDD tends to set higher purchase prices and lower sale prices. However, as the confidence level increases towards saturation, the differentiation in pricing becomes smaller, such as the purchase and sale electricity prices at confidence levels of 0.9 and 0.98 showing no significant difference. This is because, as the confidence level increases, more adverse WT output scenarios occur, leading to a more conservative dispatch strategy by the MEASs. With the dual drive of maximizing its own interests and increasing MEAS conservatism, the GDD therefore raises purchase prices and lowers sale prices.

Figure 5 presents a comparison of the purchased and sold electricity volumes at different confidence levels. Positive values indicate the electricity sold by the MEAS to the GDD, while negative values indicate the electricity purchased by the MEAS from the GDD. With

Table 2. Total electricity purchases and sales at different confidence levels.

Scenario	Electricity Sales (MW)	Electricity Purchased (MW)	Purchase–Sale Gap (MW)
Fixed scenario probability	7.37	16.85	9.48
${\alpha }_1={\alpha }_{\infty }=0.8$	6.83	16.42	9.59
${\alpha }_1={\alpha }_{\infty }=0.9$	6.77	16.41	9.64
${\alpha }_1={\alpha }_{\infty }=0.98$	6.70	16.36	9.66

, it can be observed that compared to uncertain scenarios, in deterministic scenarios, the MEAS sells more electricity to the GDD, at 7.37 MW, and buys more electricity from the GDD, at 16.85 MW. Moreover, as the confidence level increases, the amount of electricity bought and sold by the MEAS to the GDD decreases, while the difference between the bought and sold amounts increases. This is attributed to the following: (1) the GDD raising the purchase price and lowering the selling price, driving a continuous reduction in the purchased and sold electricity volumes; (2) the uncertainty in scenarios increasing the uncertainty that the MEAS needs to consider regarding WT, leading to the MEAS needing to buy more electricity and sell less electricity to address the worst-case scenario probability distribution, hence the continuous increase in the buy–sell difference. However, as the confidence level saturates, the differences in purchased and sold electricity volumes at different confidence levels become smaller.

5.3. Dispatch Results Analysis

Table 3. Detailed dispatch cost statistics.

Confidence Level	Cost Item/JPY	MEAS1	MEAS2	MEAS3	Total Cost of Each of the MEAS/JPY	GDD Costs/JPY	GDD’s Net Gain/JPY
Fixed scenario probability	O&M costs	71.36	61.72	73.29	206.36	-	215.90
	Cooperation costs	−395.98	−313.26	−709.24	0.00	-
	Purchased–sold power costs	−107.49	−1040.21	11,321.17	10,173.47	−10,173.47
	GDD–electric grid turnover	-	-	-	-	9957.57
0.80	O&M costs	67.50	69.43	73.29	210.22	-	243.43
	Cooperation costs	−393.30	−323.31	−716.61	0.00	-
	Purchased–sold power costs	−83.78	−1034.43	11,396.35	10,278.15	−10,278.15
	GDD–electric grid turnover	-	-	-	-	10,034.72
0.90	O&M costs	63.65	69.43	69.43	202.51	-	253.18
	Cooperation costs	−405.66	−319.11	−724.77	1859.45	-
	Purchased–sold power costs	−48.19	−1004.34	11,728.08	10,675.55	−10,675.55
	GDD–electric grid turnover	-	-	-	-	10,422.37
0.98	O&M costs	65.57	61.72	69.43	196.72	-	260.58
	Cooperation costs	−404.22	−322.88	−727.11	0.00	-
	Purchased–sold power costs	−55.95	−1042.24	11,883.36	10,785.17	−10,785.17
	GDD–electric grid turnover	-	-	-	-	10,524.59

provides a detailed breakdown of the dispatch results at different confidence levels, including various costs for the MEAS, costs and revenues for the GDD, and the total cost for the GDD–MEAS, where negative values indicate revenue. It can be observed that with increasing confidence levels, the MEAS’s revenue from selling electricity decreases, while the cost of purchasing electricity increases, as a result of the dual driving forces of purchase and selling prices and transaction volumes. Additionally, the operational costs of MEAS’s energy storage also increase continuously, indicating that the MEAS has increased the scheduling intensity of energy storage under continuously adverse WT scenarios and outputs. However, the operational costs of the MEAS decrease continuously, indicating that the number of transitions between the trading status between the GDD–MEAS and the charging and discharging states of energy storage is decreasing.

Comparing the various cost items of the GDD at different confidence levels, the cooperation cost is the product of the cooperation volume (Figure A4) and the cooperation price (Figure A5). From the perspective of the GDD, the amount of money exchanged in the MEAS–GDD transactions represents the revenue for the GDD, while the amount of money exchanged in the GDD–grid transactions represents the cost for the GDD. It can be observed that with increasing confidence levels, both the costs and revenues of the GDD increase continuously, indicating that the amount of electricity traded between the GDD and the grid is continuously increasing. Additionally, combining with Table 3, it can be inferred that the sum of the GDD’s costs and revenues continuously increases, indicating that the growth rate of the GDD’s revenue is higher than its cost growth rate. This further suggests that the GDD has set higher purchase prices and lower selling prices to expand its revenue.

5.4. Model and Algorithm Comparative Analysis

Table 4. Comparison of different uncertainty treatments.

Uncertainty Optimization	Solution Duration/s	Costs/JPY	Iterations/k	Iterations/r
stochastic optimization	64.91	11,628.18	23	12
Model 1	42.86	12,232.79	22	3
Model 2	43.62	13,038.99	22	3
Model 3	53.85	12,484.33	22	5

compares the effects of four models on the cost and solving time of the GDD–MEAS, namely the stochastic optimization model (with deterministic scenario probabilities, ignoring output uncertainty), the three-layer stochastic model considering scenario probability uncertainty (Model 1), the three-layer robust model considering output uncertainty (Model 2), and the three-layer stochastic robust model considering both scenario probability and output uncertainty (Model 3). Among these models, the cost of stochastic optimization is JPY 11,628.18, demonstrating good economic efficiency. However, the stochastic optimization algorithm requires precise probability distribution functions, and many scenarios lead to long solving times of 64.91 s, limiting its applicability. Compared to Model 3, the cost of this model increases by JPY 856.15. This is mainly because the stochastic robust model sacrifices some economic efficiency to achieve conservatism by overcoming scenario probability and output uncertainty.

Meanwhile, Model 3 balances economic efficiency and conservatism compared to Models 1 and 2. For instance, Model 3 exhibits better economic efficiency than Model 2. This is because the inclusion of scenario probability uncertainty in Model 3 directly influences the GDD, which formulates purchase and sale electricity prices and quantities under the worst-case scenario. Under dual stimuli, the GDD’s revenue growth rate exceeds its cost growth rate, resulting in a more economical total cost for the GDD–MEAS. However, Model 3 is more conservative than Model 1, as the “output uncertainty” in Model 3 does not directly influence the GDD but indirectly affects the impact of WT output uncertainty through the electricity trading volume between the GDD and MEAS. As a result, the GDD’s revenue growth rate cannot cover its cost growth rate, resulting in a larger total cost for the GDD–MEAS, reflecting the conservatism of this approach.

However, Model 3 has a longer solving time, but it is still suitable for solving practical engineering problems. The difference in iteration times among models mainly lies in the solving of stochastic optimization, with the sub-problem iteration times of stochastic optimization being the highest at 12 times, while the iteration times of each robust model are relatively lower. Ultimately, this is because the stochastic optimization algorithm needs to select the economically optimal scenarios from many deterministic scenarios, while robust models only need to optimize in the uncertain space.

6. Conclusions

This paper establishes a two-stage, three-layer stochastic–robust optimization model for the MEAS–GDD relationship considering the probabilities of WT and PV scenarios and output uncertainty. A stochastic pricing model for the GDD considering the uncertainties of WT and PV scenario probabilities is developed, along with a three-layer stochastic-robust model for the MEASs considering the probabilities of WT and PV scenarios and output uncertainty. Additionally, an AIV–KKT algorithm is proposed based on KKT to solve the three-layer MEAS stochastic–robust model. The Simulation results indicate the following:

The confidence level and historical data volume both affect the economic efficiency and conservatism of the optimization results. A higher confidence level leads to higher conservatism and poorer economic efficiency of the model. The confidence level affects the trading situation of the GDD–MEAS. With a higher confidence level, the MEAS purchases more electricity and sells less. Both the GDD’s revenue and MEAS’s costs increase. Therefore, in practical engineering, the MEAS should carefully select confidence level parameters to ensure both the MEAS and GDD’s economic efficiency while meeting the conservatism requirements of their actual needs. Compared to traditional stochastic optimization and three-layer robust models, the proposed two-stage, three-layer stochastic-robust model avoids overly conservative scenarios while also considering economic feasibility to a certain extent.