Hierarchical Distributed Low-Carbon Economic Dispatch Strategy for Regional Integrated Energy System Based on ADMM

Abstract

To further improve the economic benefits of operators and the low-carbon performance within the system, this paper proposes a hierarchical distributed low-carbon economic dispatch strategy for regional integrated energy systems (RIESs) based on the Alternating Direction Method of Multipliers (ADMM). First, the energy coupling relationships among conversion devices in RIESs are analyzed, and a structural model of RIES incorporating an energy generation operator (EGO) and multiple load aggregators (LAs) is established. Second, considering the stepwise carbon trading mechanism (SCTM) and the average thermal comfort of residents, economic optimization models for operators are developed. To ensure optimal energy trading strategies between conflicting stakeholders, the EGO and LAs are embedded into a master–slave game trading framework, and the existence of the game equilibrium solution is rigorously proven. Furthermore, considering the processing speed of the optimization problem by the operators and the operators’ data privacy requirement, the optimization problem is solved in a hierarchical distributed manner using ADMM. To ensure the convergence of the algorithm, the non-convex feasible domain of the subproblem bilinear term is transformed into a convex polyhedron defined by its convex envelope so that the problem can be solved by a convex optimization algorithm. Finally, an example analysis shows that the scheduling strategy proposed in this paper improves the economic efficiency of energy trading participants by 3% and 3.26%, respectively, and reduces the system carbon emissions by 10.5%.

1. Introduction

The increasingly prominent energy and environmental challenges have driven transformative changes in global energy consumption patterns. Enhancing energy utilization efficiency, mitigating environmental pollution, and achieving sustainable development have become critical concerns for policymakers and researchers. Integrated energy systems (IESs), particularly those centered on combined heat and power (CHP) units, coordinate the unified dispatch of electricity, thermal energy, natural gas, and renewable energy sources. By satisfying diversified load demands while improving economic and environmental performance, such systems represent a pivotal direction for future energy system development [1,2]. As a spatial implementation of the integrated energy paradigm, regional integrated energy systems (RIESs) have progressively emerged as a research hotspot [3].

Refs. [4,5] investigate multi-objective optimal dispatch in IESs with varying configurations, while refs. [6,7] focus on multi-time-scale optimal dispatch for IESs under diverse structural frameworks. However, these studies predominantly focused on centralized optimization frameworks. With the expansion of the system scale and the complexity of the operation environment, centralized algorithms face problems such as high model complexity, high computational burden, and high risk of data privacy [8]. Consequently, distributed optimization approaches have emerged as a dominant research trend. For instance, ref. [9] developed a cooperative game-theoretic model for multi-RIES coordination using data-driven two-stage distributed robust optimization, balancing economic efficiency and operational robustness while ensuring economic benefits for individual RIESs. Ref. [10] introduced a model-free deep reinforcement learning algorithm—distributed proximal policy optimization—to optimize intraday dispatch for microgrids. Ref. [11] proposes a consensus-based coordinated scheduling strategy for RIES. By exchanging information with neighbors, individual operators unify selected consensus variables to solve the real-time economic dispatch of high-penetration flexible loads. Furthermore, ref. [12] proposes distributed solutions based on Analytical Target Cascading for the Virtual Power Plant to coordinate the energy sharing coalition and iteratively compute with lower-level prosumers. The Alternating Direction Method of Multipliers (ADMM) provides a framework that strikes a better balance between decomposition capability, constraint handling, parallel efficiency, convergence robustness, and privacy preservation, making it a more common and reliable choice for complex IES distributed optimization problems. In addition, compared to dual decomposition, ADMM significantly enhances the convergence of the algorithm by adding a quadratic penalty term to the objective function, which has better applicability.

Game theory, which studies optimal decision-making processes among participants in competitive or adversarial scenarios based on available information and individual objectives, has emerged as a pivotal tool for addressing energy system optimization and energy management challenges [13]. Key branches of game theory include cooperative games [9], evolutionary games [14], and Stackelberg games [15]. In energy trading processes, the energy generation operator (EGO) first formulates pricing strategies based on the time-of-use (TOU) and feed-in tariffs (FIT), after which users adjust their consumption strategies according to the published energy prices. While EGOs act as leaders in energy transactions, users retain decision-making autonomy, resulting in sequential decision-making dynamics that are appropriately modeled using the Stackelberg game framework. Ref. [16] established a multi-operator model involving IESs, load aggregators (LAs), and end-users. By applying Karush–Kuhn–Tucker conditions and the big-M method, the bi-level game structure was reformulated as an IES-LA game model, subsequently solved via the Great White Shark Optimizer and GUROBI solver. Ref. [17] developed a hierarchical stochastic dispatch model for integrated demand response based on Stackelberg game theory, employing metaheuristic-based distributed iterative algorithms for solution derivation. Ref. [18] proposed a Nash–Stackelberg–Nash game framework to precisely characterize interactions between IESs and photovoltaic prosumers, formulating a three-stage robust optimization model. However, the heuristic optimization algorithm used in [17] is prone to local optimal solutions. In addition, unlike the refs. [16,17,18], ADMM decouples the user private variables from the public coordinator variables in solving the master–slave game, so that the coordinator only knows the weighted average of each user’s data, and cannot invert the individual private variables, which enhances the security of the user’s information.

Ref. [19] applies blockchain technology combined with a credit bidding mechanism to distributed power grids, enabling power companies to participate in smart dispatch in a fair and reasonable manner while ensuring the privacy of user data. Ref. [20] applies federated learning to decentralize the training of residential microgrid regulation strategies to protect user privacy while solving the problems of model complexity and the slow training speed of existing management methods. However, blockchain technology relies on asymmetric encryption and distributed ledgers, where user transaction data are transparent across the network but identity is anonymous. The data need to be broadcast globally, and there is a risk of indirect information leakage. Moreover, each transaction needs to be verified across the network by means of smart contract execution, etc., and the communication load increases dramatically when the number of nodes increases. Federated learning trains scheduling models locally at each operator and only uploads model parameters such as neural network weights. However, energy scheduling requires high-precision real-time data, and parameter inversion attacks may infer the original data distribution. Moreover, the parameter server needs to communicate with all subjects, and training is often interrupted in rural microgrids due to poor network coverage.

To achieve low-carbon objectives in IESs, supply-side strategies typically integrate carbon trading mechanisms (CTMs), where carbon emission costs are incorporated into the objective function or constraints of the EGO to limit emissions [21]. The “WindGas Falkenhagen 6 MW Wind Power-methane Demonstration Project” in Germany has injected more than 3.5 GWh of methane into the German gas pipeline network; the overall efficiency of electrolysis-methanation has been verified to be more than 80%, and the coupling of grid peaking and gas storage has been realized. Ref. [22] proposes an IES dispatch method integrating seasonal CTM and electricity carbon quota allocation, optimizing resource distribution through adaptive carbon pricing and quota rules. Ref. [23] quantifies carbon trading costs via lifecycle-based greenhouse gas emission coefficients and stepwise carbon trading prices (SCTPs). Currently, introducing power to gas (P2G) and carbon capture and storage (CCS) enables closed-loop carbon cycling by synthesizing methane from captured $\mathrm{C}{\mathrm{O}}_2$ and hydrogen produced via electrolysis, enhancing emission reduction [24]. The “Shanghai Lingang New District Integrated Energy System Demonstration Project” has embedded a stepwise carbon trading model in the integrated “source-grid-load-storage” platform. During the trial period of 2023–2024, the total operating costs of the park were reduced by 5.69%, carbon emissions were reduced by 17.06%, and the night-time wind abandonment rate was reduced from 9% to less than 3%. Ref. [25] further demonstrates that coupling CCS-P2G with carbon pricing models reduces emissions while increasing operating revenue. However, the above study only utilizes a single tool for EGO low carbon studies and does not use multiple methods for EGO low carbon retrofits. On the demand side, leveraging the thermal inertia of buildings and human thermal comfort ambiguity, flexible adjustment of heating loads within permissible ranges can lower energy costs without compromising comfort [26]. Ref. [27] establishes a heat–electricity coordinated dispatch model integrating the Predicted Mean Vote (PMV) index and thermal network dynamics, achieving full wind power utilization by optimizing thermal comfort flexibility. Additionally, ref. [28] highlights that deploying energy storage devices mitigates spatiotemporal energy mismatches, reduces peak-shaving pressure on generators, and further curtails carbon emissions.

In summary, with the transformation of energy system structures from unified dispatch to interactive competition, its distributed characteristics have become increasingly evident. Traditional centralized optimization methods are unable to reveal the energy interaction behaviors among operators within RIESs and neglect users’ initiative. Moreover, as the number of operators participating in the energy trading market continues to increase, addressing the processing speed of RIES optimization problems and operators’ data privacy requirements have become increasingly important. Additionally, current approaches dedicated to system energy-saving and emission reduction are overly simplistic, failing to fully exploit the multi-party, multi-mechanism potential for system energy-saving and emission reduction. To address the above issues, the main contributions and innovations of the method proposed in this paper are summarized as follows:

1.

A low-carbon operation framework for RIES incorporating EGO and LAs has been established, proposing a hierarchical distributed low-carbon economic dispatch strategy for RIESs. Conflicting EGO and LAs are embedded in a Stackelberg game framework to ensure optimal energy trading strategies for both parties. In the energy trading process, EGO acts as the upper-level leader while LAs serve as lower-level followers.

2.

To meet the low-carbon requirements of RIESs, P2G-CCS equipment has been introduced into the RIES operation framework to achieve energy recycling. During the establishment of optimal operation models for EGO and LAs, a stepwise carbon trading mechanism (SCTM) and the PMV index have been incorporated, respectively, further enhancing the system’s low-carbon economic operation capability.

3.

Based on the proven existence of the Stackelberg equilibrium solution in the multi-agent master–slave game transaction model, an ADMM-based approach is proposed to decouple the energy–power and energy–price balance coupling constraints between the leader and followers. Using the McCormick envelope relaxation method, operators’ optimization subproblems are transformed into convex optimization problems to solve the Stackelberg game transaction model, improving solution efficiency. Since only marginal power–price information needs to be exchanged between the leader and followers, the issue of information leakage among energy trading operators is avoided.

2. Bi-Level Optimization Scheduling Framework

2.1. Stackelberg Game Framework

In this framework, the LA, acting as an agent for community users in energy trading, interacts strategically with the EGO. Both the EGO and LA pursue optimal operating benefits under their respective operational constraints. As the leader, the EGO aims to maximize its operating revenue while complying with carbon emission limits. It adjusts the dispatch strategies of energy conversion equipment, coordinates interactions with external energy markets, and announces energy pricing to the LA. In response, the LA, as the follower, optimizes its thermal demand response and energy storage charging/discharging strategies based on the EGO’s pricing signals, with the objective of minimizing energy costs while satisfying end-user demand. The game reaches an equilibrium state when neither the EGO nor the LAs can further improve their benefits by altering their strategies.

2.2. RIES Architecture Incorporating EGO and LAs

The proposed RIES structure, incorporating electricity, thermal energy, and gas energy forms, is illustrated in Figure 1. The EGO utilizes a coal-fired power generator (CPG) and a CHP composed of a waste heat boiler (WHB) and a gas turbine (GT) as its primary supply-side facilities. To reduce system carbon emissions, P2G and CCS are deployed within the EGO. Additional infrastructure includes transformers (TRFs), an electrical bus (BUS), and a thermal energy bus (TEB). At the community level, energy storage systems (ESSs) and rooftop photovoltaic (PV) panels are installed. End-users purchase electricity and thermal energy, with the assumption that all thermal energy is consumed for indoor heating.

The EGO formulates electricity and thermal energy pricing for LAs. To incentivize LA participation, the EGO sets electricity prices below the grid TOU and thermal energy prices within external upper/lower bounds, generating profits through energy transactions. Acting as an intermediary between LAs and external grids, the EGO purchases electricity when its output fails to meet LA demand or when generation costs exceed grid prices, ensuring reliable supply. Furthermore, the EGO assumes full responsibility for carbon trading costs within the RIES. End-users acquire electricity through three channels: purchasing from the EGO, discharging ESS, and utilizing PV generation. LAs store surplus electricity in the ESS during periods of low pricing or high PV output and discharge the stored energy during peak pricing intervals to minimize operating costs.

3. Optimization Models for Operators

3.1. Energy Generation Operator

3.1.1. Objective Function

The objective of EGO is to maximize operating revenue, where the operating revenue $F_g$ over a scheduling period is defined as follows.

$$F_{\mathrm{g}}=\sum _{t\in T}(F_{\mathrm{g},\mathrm{int},t}-F_{\mathrm{g},\mathrm{epg},t}-F_{\mathrm{g},\mathrm{gt},t}-F_{\mathrm{g},\mathrm{cpg},t})\Delta t-F_{\mathrm{g},\mathrm{c}}$$

(1)

$$F_{\mathrm{g},\mathrm{int},t}=\sum _{n\in N}(C_{\mathrm{g}-n,t}^{\mathrm{e}}{P}_{\mathrm{g}-n,t}^{\mathrm{e}}+C_{\mathrm{g}-n,t}^{\mathrm{h}}{P}_{\mathrm{g}-n,t}^{\mathrm{h}})$$

(2)

$$F_{\mathrm{g},\mathrm{epg},t}=C_t^{\mathrm{tou}}{P}_{\mathrm{g}-\mathrm{epg},t}^{\mathrm{b},\mathrm{e}}-C_t^{\mathrm{fit}}{P}_{\mathrm{g}-\mathrm{epg},t}^{\mathrm{s},\mathrm{e}}$$

(3)

$$F_{\mathrm{g},\mathrm{gt},t}=C_{\mathrm{gas}}(g_{\mathrm{gt},t}-g_{\mathrm{p}2\mathrm{g},t})$$

(4)

$$F_{\mathrm{g},\mathrm{cpg},t}=a_{\mathrm{cpg}}{\left(P_{\mathrm{cpg},t}^{\mathrm{e}}\right)}^2+b_{\mathrm{cpg}}{P}_{\mathrm{cpg},t}^{\mathrm{e}}+c_{\mathrm{cpg}}$$

(5)

where $F_{\mathrm{g},\mathrm{int},t}$ is the revenue from energy sales to LAs by the EGO; $F_{\mathrm{g},\mathrm{epg},t}$ is the cost of electricity transactions between the EGO and the external power grid (EPG); $F_{\mathrm{g},\mathrm{gt},t}$ is the operating cost of the GT; $F_{\mathrm{g},\mathrm{cpg},t}$ is the operating cost of the CPG; $F_{\mathrm{g},\mathrm{c}}$ is the carbon trading cost incurred by the EGO during a scheduling period (detailed in Section 3.2); t and T are the time index and duration of a scheduling period; $\Delta t$ is a time interval; n and N represent the identifiers and set of LA within the RIES; $C_{\mathrm{g}-n,t}^{\mathrm{e}}$ and $C_{\mathrm{g}-n,t}^{\mathrm{h}}$ are the electricity and thermal energy selling prices published by the EGO to the $\mathrm{L}{\mathrm{A}}_n$; $P_{\mathrm{g}-n,t}^{\mathrm{e}}$ and $P_{\mathrm{g}-n,t}^{\mathrm{h}}$ are the electricity and thermal power sold by the EGO to the $\mathrm{L}{\mathrm{A}}_n$; $C_t^{\mathrm{tou}}$ and $C_t^{\mathrm{fit}}$ are the TOU and FIT; $P_{\mathrm{g}-\mathrm{epg},t}^{\mathrm{b},\mathrm{e}}$ and $P_{\mathrm{g}-\mathrm{epg},t}^{\mathrm{s},\mathrm{e}}$ are the purchased and sold electricity power by the EGO from/to the EPG; $C_{\mathrm{gas}}$ is the natural gas price; $g_{\mathrm{gt},t}$ and $g_{\mathrm{p}2\mathrm{g},t}$ are the consumed natural gas volume by the GT and the synthesized natural gas volume by the P2G equipment, respectively; $P_{\mathrm{cpg},t}^{\mathrm{e}}$ is the electricity power output of the CPG; and $a_{\mathrm{cpg}}$, $b_{\mathrm{cpg}}$, and $c_{\mathrm{cpg}}$ are the operating cost coefficients of the CPG.

3.1.2. Constraints

The power balance constraints can be expressed as the following equations.

$$P_{\mathrm{gt},t}^{\mathrm{e}}+P_{\mathrm{cpg},t}^{\mathrm{e}}+P_{\mathrm{g}-\mathrm{epg},t}^{\mathrm{b},\mathrm{e}}=\sum _{n\in N}{P}_{\mathrm{g}-n,t}^{\mathrm{s},\mathrm{e}}+P_{\mathrm{ccs},t}^{\mathrm{e}}+P_{\mathrm{p}2\mathrm{g},t}^{\mathrm{e}}+P_{\mathrm{g}-\mathrm{epg},t}^{\mathrm{s},\mathrm{e}}$$

(6)

$$P_{\mathrm{whb},t}^{\mathrm{h}}=\sum _{n\in N}{P}_{\mathrm{g}-n,t}^{\mathrm{s},\mathrm{h}}$$

(7)

where $P_{\mathrm{gt},t}^{\mathrm{e}}$ is the electricity output of the GT; $P_{\mathrm{ccs},t}^{\mathrm{e}}$ is the electrical power consumption of the CCS; $P_{\mathrm{p}2\mathrm{g},t}^{\mathrm{e}}$ is the electrical power consumption of the P2G equipment; and $P_{\mathrm{whb},t}^{\mathrm{e}}$ is the thermal power output of the WHB.

The energy coupled device constraints can be expressed as the following equations.

$$P_{\mathrm{gt},t}^{\mathrm{e}}={\kappa }_{\mathrm{gt}}^{\mathrm{e}}{q}_{\mathrm{gas}}{g}_{\mathrm{gt},t}$$

(8)

$$P_{\mathrm{gt}}^{\mathrm{e},\min }\le P_{\mathrm{gt},t}^{\mathrm{e}}\le P_{\mathrm{gt}}^{\mathrm{e},\max }$$

(9)

$$P_{\mathrm{gt},t}^{\mathrm{h}}={\kappa }_{\mathrm{gt}}^{\mathrm{h}}{q}_{\mathrm{gas}}{g}_{\mathrm{gt},t}$$

(10)

$$\left|P_{\mathrm{gt},t+1}^{\mathrm{e}}-P_{\mathrm{gt},t}^{\mathrm{e}}\right|\le P_{\mathrm{gt}}^{\mathrm{e},\Delta \max }$$

(11)

$$\left|P_{\mathrm{cpg},t+1}^{\mathrm{e}}-P_{\mathrm{cpg},t}^{\mathrm{e}}\right|\le P_{\mathrm{cpg}}^{\mathrm{e},\Delta \max }$$

(12)

$$P_{\mathrm{who},t}^{\mathrm{h}}={\kappa }_{\mathrm{who}}^{\mathrm{h}}{P}_{\mathrm{gt},t}^{\mathrm{h}}$$

(13)

$$g_{\mathrm{p}2\mathrm{g},t}=P_{\mathrm{p}2\mathrm{g},t}^{\mathrm{e}}{\displaystyle \frac{3.6{\kappa }_{\mathrm{p}2\mathrm{g}}}{q_{\mathrm{gas}}}}$$

(14)

$$P_{\mathrm{p}2\mathrm{g}}^{\mathrm{e},\min }\le P_{\mathrm{p}2\mathrm{g},t}^{\mathrm{e}}\le P_{\mathrm{p}2\mathrm{g}}^{\mathrm{e},\max }$$

(15)

$$P_{\mathrm{ccs},t}^{\mathrm{e}}={\alpha }_{\mathrm{ccs}}{E}_{\mathrm{ccs},t}$$

(16)

$$P_{\mathrm{ccs}}^{\mathrm{e},\min }\le P_{\mathrm{ccs},t}^{\mathrm{e}}\le P_{\mathrm{ccs}}^{\mathrm{e},\max }$$

(17)

$$E_{\mathrm{ccs},t}={\alpha }_{\mathrm{p}2\mathrm{g}}{P}_{\mathrm{p}2\mathrm{g},t}^{\mathrm{e}}$$

(18)

where ${\kappa }_{\mathrm{gt}}^{\mathrm{e}}$ and ${\kappa }_{\mathrm{gt}}^{\mathrm{h}}$ are the electricity and thermal energy conversion efficiencies of the GT; $q_{\mathrm{gas}}$ is the calorific value of natural gas; $P_{\mathrm{gt}}^{\mathrm{e},\max }$ and $P_{\mathrm{gt}}^{\mathrm{e},\min }$ are the maximum and minimum power generation capacities of the GT, respectively; ${\kappa }_{\mathrm{whb}}^{\mathrm{h}}$ is the waste heat recovery efficiency of the WHB; ${\kappa }_{\mathrm{p}2\mathrm{g}}$ is the conversion efficiency of the P2G equipment; $E_{\mathrm{ccs},t}$ is the mass of carbon dioxide captured by the CCS; ${\alpha }_{\mathrm{p}2\mathrm{g}}$ is the carbon dioxide consumption coefficient in the P2G process; and ${\alpha }_{\mathrm{ccs}}$ is the energy consumption coefficient for $\mathrm{C}{\mathrm{O}}_2$ capture in the CCS.

The trading energy price constraints can be expressed as the following equations.

$$C_t^{\mathrm{fit}}\le C_{\mathrm{g}-n,t}^{\mathrm{e}}\le C_t^{\mathrm{tou}}$$

(19)

$$C^{\mathrm{h},\min }\le C_{\mathrm{g}-n,t}^{\mathrm{h}}\le C^{\mathrm{h},\max }$$

(20)

$$(\sum _{t\in T}{C}_{\mathrm{g}-n,t}^{\mathrm{e}})/T\ge C_{\mathrm{g}-n,\mathrm{ave}}^{\mathrm{e},\min }$$

(21)

$$(\sum _{t\in T}{C}_{\mathrm{g}-n,t}^{\mathrm{h}})/T\ge C_{\mathrm{g}-n,\mathrm{ave}}^{\mathrm{h},\min }$$

(22)

where $C^{\mathrm{h},\max }$ and $C^{\mathrm{h},\min }$ are the maximum and minimum thermal prices in the thermal energy market; $C_{\mathrm{g}-n,\mathrm{ave}}^{\mathrm{e},\min }$ and $C_{\mathrm{g}-n,\mathrm{ave}}^{\mathrm{h},\min }$ are the minimum average values of the electricity and thermal energy price, respectively.

3.2. Stepwise Carbon Trading Mechanism

The SCTM aims to encourage emission subjects to reduce carbon emissions by setting different levels of price standards for carbon emission quota. The CTM can be divided into three stages: allocating the carbon emission quota, calculating the actual carbon emissions of carbon emitting entities, and calculating the cost of carbon trading.

3.2.1. Carbon Emission Quota

At present, the industry mainly adopts the gratuitous allocation of quota, and common gratuitous carbon emission quota acquisition methods include the following: the historical emission method, the industry benchmark method, the production benchmark method, the combination of the auction and gratuitous allocation methods, etc. Carbon emissions from RIESs mainly come from GT and CPG in EGO, and carbon emissions from external coal power units are also borne by EGO, and the carbon emissions are directly related to the energy production, so it is preferable to choose the production benchmark method to determine the unremunerated carbon emission quota of the system. The initial carbon emission quota $E_g$ is defined as follows.

$$E_{\mathrm{g}}=E_{\mathrm{g},\mathrm{epg}}+E_{\mathrm{g},\mathrm{gt}}+E_{\mathrm{g},\mathrm{cpg}}$$

(23)

$$E_{\mathrm{g},\mathrm{epg}}={\chi }_{\mathrm{epg}}\sum _{t\in T}(P_{\mathrm{g}-\mathrm{epg},t}^{\mathrm{b},\mathrm{e}}-P_{\mathrm{g}-\mathrm{epg},t}^{\mathrm{s},\mathrm{e}})\Delta t$$

(24)

$$E_{\mathrm{g},\mathrm{gt}}={\chi }_{\mathrm{gt}}\sum _{t\in T}({\chi }_{\mathrm{e}-\mathrm{h}}{P}_{\mathrm{gt},t}^{\mathrm{e}}+P_{\mathrm{gt},t}^{\mathrm{h}})\Delta t$$

(25)

$$E_{\mathrm{g},\mathrm{cpg}}={\chi }_{\mathrm{cpg}}\sum _{t\in T}{P}_{\mathrm{cpg},t}^{\mathrm{e}}\Delta t$$

(26)

where $E_{\mathrm{g},\mathrm{epg}}$ is the carbon emission quota obtained by the EGO when purchasing electricity from the EPG; $E_{\mathrm{g},\mathrm{gt}}$ and $E_{\mathrm{g},\mathrm{cpg}}$ are the carbon emission quotas generated by the power output of the GT and CPG, respectively; ${\chi }_{\mathrm{epg}}$ is the carbon emission quota coefficient for power generation units outside the RIES; ${\chi }_{\mathrm{gt}}$ is the carbon emission quota coefficient of the GT; ${\chi }_{\mathrm{e}-\mathrm{h}}$ is the conversion coefficient for scaling electric power output to thermal power output; and ${\chi }_{\mathrm{cpg}}$ is the carbon emission quota coefficient of the CPG. The carbon emission quota parameters refers to ref. [29].

3.2.2. Actual Carbon Emissions

The actual carbon emissions are defined as follows.

$$E_{\mathrm{g}}^{\mathrm{a}}=E_{\mathrm{g},\mathrm{epg}}^{\mathrm{a}}+E_{\mathrm{g},\mathrm{gt}}^{\mathrm{a}}+E_{\mathrm{g},\mathrm{cpg}}^{\mathrm{a}}-\sum _{t\in T}{E}_{\mathrm{ccs},t}$$

(27)

$$E_{\mathrm{g},\mathrm{epg}}^{\mathrm{a}}={\chi }_{\mathrm{epg}}^{\mathrm{a}}\sum _{t\in T}(P_{\mathrm{g}-\mathrm{epg},t}^{\mathrm{b},\mathrm{e}}-P_{\mathrm{g}-\mathrm{epg},t}^{\mathrm{s},\mathrm{e}})\Delta t$$

(28)

$$E_{\mathrm{g},\mathrm{gt}}^{\mathrm{a}}={\chi }_{\mathrm{gt}}^{\mathrm{a}}\sum _{t\in T}({\chi }_{\mathrm{e}-\mathrm{h}}{P}_{\mathrm{gt},t}^{\mathrm{e}}+P_{\mathrm{gt},t}^{\mathrm{h}})\Delta t$$

(29)

$$E_{\mathrm{g},\mathrm{cpg}}^{\mathrm{a}}={\chi }_{\mathrm{cpg}}^{\mathrm{a}}\sum _{t\in T}{P}_{\mathrm{cpg},t}^{\mathrm{e}}\Delta t$$

(30)

where $E_{\mathrm{g}}^{\mathrm{a}}$ is the actual carbon emission of the EGO; $E_{\mathrm{g},\mathrm{epg}}^{\mathrm{a}}$ is the carbon emission that the EGO must bear from purchasing electricity from the EPG; $E_{\mathrm{g},\mathrm{gt}}^{\mathrm{a}}$ and $E_{\mathrm{g},\mathrm{cpg}}^{\mathrm{a}}$ are the carbon emissions associated with the power outputs of the GT and CPG, respectively; ${\chi }_{\mathrm{epg}}^{\mathrm{a}}$ is the carbon emission coefficient for power generation units outside the RIES; ${\chi }_{\mathrm{gt}}^{\mathrm{a}}$ is the carbon emission coefficient for the GT; and ${\chi }_{\mathrm{cpg}}^{\mathrm{a}}$ is the carbon emission coefficient for the CPG. The actual carbon emission parameters refers to ref. [29].

3.2.3. Carbon Trading Costs

The SCTM specifies multiple carbon emission trading intervals. When the actual carbon emissions $E_{\mathrm{g}}^{\mathrm{a}}$ of the EGO exceed the free-allocated carbon emission quota $E_{\mathrm{g}}$, the EGO must purchase the deficit carbon emission quota. The SCTM model is defined as follows.

$$F_{\mathrm{g},\mathrm{c}}=\left\{\begin{array}{cc}-\gamma l-\gamma (1+\sigma )(-l-E_{\mathrm{g}}^{\mathrm{a}}+E_{\mathrm{g}})\hfill & E_{\mathrm{g}}-2l<E_{\mathrm{g}}^{\mathrm{a}}<E_{\mathrm{g}}-l\hfill \\ -\gamma (-E_{\mathrm{g}}^{\mathrm{a}}+E_{\mathrm{g}})\hfill & E_{\mathrm{g}}-l<E_{\mathrm{g}}^{\mathrm{a}}<E_{\mathrm{g}}\hfill \\ \gamma (E_{\mathrm{g}}^{\mathrm{a}}-E_{\mathrm{g}})\hfill & E_{\mathrm{g}}<E_{\mathrm{g}}^{\mathrm{a}}\le E_{\mathrm{g}}+l\hfill \\ \gamma l+\gamma (1+\sigma )(E_{\mathrm{g}}^{\mathrm{a}}-E_{\mathrm{g}}-l)\hfill & E_{\mathrm{g}}+l<E_{\mathrm{g}}^{\mathrm{a}}\le E_{\mathrm{g}}+2l\hfill \\ \gamma (2+\sigma )l+\gamma (1+2\sigma )(E_{\mathrm{g}}^{\mathrm{a}}-E_{\mathrm{g}}-2l)\hfill & E_{\mathrm{g}}+2l<E_{\mathrm{g}}^{\mathrm{a}}\hfill \end{array}\right.$$

(31)

where $\gamma$ is the carbon trading base price (CBP); $\sigma$ is the carbon trading price growth rate (CGR) of the SCTM; and l is the length of the carbon emission interval. The SCTM parameters refer to ref. [29]. Figure 2 illustrates the schematic diagram of carbon emission trading (CET) prices and costs under the SCTM.

3.3. Load Aggregator

3.3.1. Objective Function

The $\mathrm{L}{\mathrm{A}}_n$ aims to minimize its energy consumption cost, where the operating cost $F_n$ over a scheduling period is defined as follows.

$$F_n=\sum _{t\in T}(F_{n,\mathrm{int},t}+F_{n,\mathrm{ess},t}+F_{n,\mathrm{com},t})\Delta t$$

(32)

$$F_{n,\mathrm{int},t}=C_{n-\mathrm{g},t}^{\mathrm{e}}{P}_{n-\mathrm{g},t}^{\mathrm{e}}+C_{n-\mathrm{g},t}^{\mathrm{h}}{P}_{n-\mathrm{g},t}^{\mathrm{h}}$$

(33)

$$F_{n,\mathrm{ess},t}={\eta }_{\mathrm{ess}}\sum _{t\in T}(P_{n,t}^{\mathrm{e},\mathrm{ch}}+P_{n,t}^{\mathrm{e},\mathrm{dis}})$$

(34)

$$F_{n,\mathrm{com},t}={\eta }_{\mathrm{com}}\left|{\tau }_t^{\mathrm{in}}-{\tau }^{\mathrm{best}}\right|$$

(35)

where $F_{n,\mathrm{int},t}$ is the energy procurement cost of the $\mathrm{L}{\mathrm{A}}_n$; $F_{n,\mathrm{ess},t}$ is the operating and maintenance cost of the ESS in the $\mathrm{L}{\mathrm{A}}_n$; $F_{n,\mathrm{com},t}$ is the user discomfort penalty cost of the $\mathrm{L}{\mathrm{A}}_n$; $C_{n-\mathrm{g},t}^{\mathrm{e}}$ and $C_{n-\mathrm{g},t}^{\mathrm{h}}$ are the electricity and thermal energy procurement prices from the EGO for the $\mathrm{L}{\mathrm{A}}_n$; $P_{n-\mathrm{g},t}^{\mathrm{e}}$ and $P_{n-\mathrm{g},t}^{\mathrm{h}}$ are the purchased electricity and thermal power from the EGO for the $\mathrm{L}{\mathrm{A}}_n$; ${\eta }_{\mathrm{ess}}$ is the operating and maintenance cost coefficient of the ESS in the $\mathrm{L}{\mathrm{A}}_n$; $P_{n,t}^{\mathrm{e},\mathrm{ch}}$ and $P_{n,t}^{\mathrm{e},\mathrm{dis}}$ are the charging and discharging power of the ESS in the $\mathrm{L}{\mathrm{A}}_n$; ${\eta }_{\mathrm{com}}$ is the user discomfort penalty coefficient; and ${\tau }_t^{\mathrm{in}}$ and ${\tau }^{\mathrm{best}}$ are the indoor temperature and optimal indoor temperature of the user.

3.3.2. Thermal Energy Purchase Strategy Based on Thermal Inertia and PMV

Since buildings have thermal inertia, i.e., the property of storing and releasing heat, which buffers indoor temperature fluctuations, current research widely uses first-order equivalent thermal parameter models to characterize the thermal dynamics of buildings [30].

$$k_1{\tau }_t^{\mathrm{in}}-{\tau }_{t-1}^{\mathrm{in}}=k_2{P}_{n,\mathrm{load},t}^{\mathrm{h}}+k_3{\tau }_t^{\mathrm{out}}$$

(36)

where $k_1$, $k_2$, and $k_3$ are the coefficients of the thermal inertia difference equation; ${\tau }_t^{\mathrm{out}}$ is the outdoor temperature of the user; and $P_{n,\mathrm{load},t}^{\mathrm{h}}$ is the indoor heating power of the user. Thermal inertia equation parameters are all referenced in [30].

Currently, the thermal energy load in the IES system mainly includes two parts: the industrial thermal load and the residential thermal load. The industrial thermal load is constrained by the production plan of enterprises and difficult to regulate, while residents are sensitive to changes in energy prices and have flexibility in their energy-use habits. In addition, residents’ perception of indoor temperature has ambiguity, making full use of the thermal inertia of buildings and the ambiguity of the residents’ perception of indoor temperature to sacrifice the appropriate residents’ comfort within the acceptable range, and improve the demand-side response capability. The empirical equation for the human thermal comfort value can be expressed as the following equations.

$${\lambda }_{\mathrm{PMV},t}=2.43-{\displaystyle \frac{3.76({\tau }_{\mathrm{b}}-{\tau }_t^{\mathrm{in}})}{M(I_{\mathrm{clo}}+0.1)}}$$

(37)

$${\lambda }_{\mathrm{PMV}}^{\min }\le {\lambda }_{\mathrm{PMV},t}\le {\lambda }_{\mathrm{PMV}}^{\max }$$

(38)

where ${\lambda }_{\mathrm{PMV},t}$ is the human thermal comfort value; ${\tau }_{\mathrm{b}}$ is the average skin temperature of a person in a comfortable state; M is the human metabolic rate; and $I_{\mathrm{clo}}$ is the clothing thermal resistance value. To avoid persistent low indoor temperatures, the average indoor temperature should equal the ideal value. PMV parameters are all referenced in [27].

3.3.3. Constraints

The power balance constraints can be expressed as the following equations.

$$P_{n,\mathrm{fore},t}^{\mathrm{e}}+P_{n,t}^{\mathrm{e},\mathrm{ch}}=P_{n-\mathrm{g},t}^{\mathrm{b},\mathrm{e}}+P_{n,t}^{\mathrm{e},\mathrm{dis}}+P_{n,\mathrm{pv},t}^{\mathrm{e}}$$

(39)

$$P_{n,\mathrm{load},t}^{\mathrm{h}}=P_{n-\mathrm{g},t}^{\mathrm{h}}$$

(40)

where $P_{n,\mathrm{fore},t}^{\mathrm{e}}$ is the forecast electrical load power of users; $P_{n,\mathrm{pv},t}^{\mathrm{e}}$ is the forecast PV generation power of the community.

The energy storage device constraints can be expressed as the following equations.

$$S_{\mathrm{soe},t+1}=S_{\mathrm{soe},t}+({\kappa }_{\mathrm{ch}}{P}_{n,t}^{\mathrm{e},\mathrm{ch}}-P_{n,t}^{\mathrm{e},\mathrm{dis}}/\phantom{P_{n,t}^{\mathrm{e},\mathrm{dis}}{\kappa }_{\mathrm{dis}}}{\kappa }_{\mathrm{dis}})\Delta t$$

(41)

$$\sum _{t\in T}{\kappa }_{\mathrm{ch}}{P}_{n,t}^{\mathrm{e},\mathrm{ch}}-\sum _{t\in T}{P}_{n,t}^{\mathrm{e},\mathrm{dis}}/\phantom{P_{n,t}^{\mathrm{e},\mathrm{dis}}{\kappa }_{\mathrm{dis}}}{\kappa }_{\mathrm{dis}}=0$$

(42)

$$S_{\mathrm{soc}}^{\min }\le S_{\mathrm{soc},t}\le S_{\mathrm{soc}}^{\max }$$

(43)

$$0\le P_{n,t}^{\mathrm{e},\mathrm{ch}}\le {\mu }^{\mathrm{ch}}{P}_{n,\mathrm{ess}}^{\mathrm{e},\max }$$

(44)

$$0\le P_{n,t}^{\mathrm{e},\mathrm{dis}}\le {\mu }^{\mathrm{dis}}{P}_{n,\mathrm{ess}}^{\mathrm{e},\max }$$

(45)

$${\mu }^{\mathrm{ch}}+{\mu }^{\mathrm{dis}}\le 1$$

(46)

where $S_{\mathrm{soe},t}$ is the energy state of the ESS; ${\kappa }_{\mathrm{ch}}$ and ${\kappa }_{\mathrm{dis}}$ are the charging and discharging efficiencies of the ESS; $S_{\mathrm{soc}}^{\max }$ and $S_{\mathrm{soc}}^{\min }$ are the maximum and minimum states of charge (SOC) of the ESS; $P_{n,\mathrm{ess}}^{\mathrm{e},\max }$ is the maximum power transfer capacity of the ESS; and ${\mu }^{\mathrm{ch}}$ and ${\mu }^{\mathrm{dis}}$ are boolean variables indicating the charging and discharging states of the ESS.

The trading energy price constraints can be expressed as the following equations.

$$(\sum _{t\in T}{C}_{\mathrm{g}-n,t}^{\mathrm{e}})/T\le C_{\mathrm{g}-n,\mathrm{ave}}^{\mathrm{e},\max }$$

(47)

$$(\sum _{t\in T}{C}_{\mathrm{g}-n,t}^{\mathrm{h}})/T\le C_{\mathrm{g}-n,\mathrm{ave}}^{\mathrm{h},\max }$$

(48)

where $C_{\mathrm{g}-n,\mathrm{ave}}^{\mathrm{e},\max }$ and $C_{\mathrm{g}-n,\mathrm{ave}}^{\mathrm{h},\max }$ are the maximum average values of the electricity and thermal energy price, respectively.

4. Distributed Optimization Problem Solving

4.1. Proof of the Stackelberg Game Model

4.1.1. Analysis of the Stackelberg Game Model

The energy trading interaction between LAs and the EGO involves the following variables: energy pricing, power dispatch, purchasing pricing, and power demand. As derived from the operational models of these stakeholders, users’ consumption schedules and energy storage charging or discharging behaviors are determined based on the energy pricing issued by the EGO. Conversely, the optimization outcomes of the LAs in turn influence the EGO’s pricing and power dispatch decisions. This bidirectional dependency reflects conflicting interests and sequential decision-making processes between the two parties. Consequently, the energy trading dispatch between the EGO and LAs can be formulated as a Stackelberg game. The Stackelberg game model G is defined as follows.

$$\left\{\left(\mathrm{EGO}\cup \mathrm{LA}\right);\left(F_{\mathrm{EGO}}\cup F_{\mathrm{LA}}\right);\left(I_{\mathrm{EGO}}\cup I_{\mathrm{LA}}\right)\right\}$$

(49)

where $I_{\mathrm{EGO}}$ is the game strategy of the EGO; $I_{\mathrm{LA}}$ is the set of game strategies for the N lower-level followers.

Among all strategy combinations, if there exists a strategy $\left(I_{\mathrm{MGO}}^{\ast }\cup I_{\mathrm{LA}}^{\ast }\right)$ such that the objective functions of the EGO and LA satisfy Equation (50), it indicates that the Stackelberg game has reached equilibrium.

$$\left\{\begin{array}{c}{F}_{\mathrm{EGO}}(I_{\mathrm{EGO}}^{\ast },I_{\mathrm{LA}}^{\ast })\ge F_{\mathrm{EGO}}(I_{\mathrm{EGO}},I_{\mathrm{LA}}^{\ast })\\ F_{\mathrm{LA}}(I_{\mathrm{EGO}}^{\ast },I_{\mathrm{LA}}^{\ast })\ge F_{\mathrm{LA}}(I_{\mathrm{EGO}}^{\ast },I_{\mathrm{LA}})\end{array}\right\}$$

(50)

where $(I_{\mathrm{EGO}}^{\ast },I_{\mathrm{LA}}^{\ast })$ is the Nash equilibrium strategy of the EGO and LAs.

4.1.2. Existence Proof of Game Equilibrium

To address the Stackelberg game problem, it is necessary to prove the existence of its equilibrium. According to ref. [31], when the Stackelberg game satisfies the following conditions, a Stackelberg equilibrium solution exists.

1.

The leader’s strategy set $I_{\mathrm{EGO}}$ is a non-empty, convex, and bounded subset of the Euclidean space.

2.

The leader’s objective function is a non-empty continuous function on its strategy set $I_{\mathrm{EGO}}$.

3.

The follower’s objective function is a non-empty continuous function on its strategy set $I_{\mathrm{LA}}$.

4.

The follower’s objective function $F_n$ is a quasi-convex function with respect to its own strategy.

Clearly, according to the trading energy price constraints, Equations (19)–(22), the strategy set satisfies condition 1. Since the operating decision variable intervals of both EGO and LA are bounded, non-empty, and closed convex sets, their objective functions are both non-empty continuous functions with respect to the strategy set I, satisfying conditions 2 and 3. It remains to prove condition 4 that the objective function of each follower LA is a proposed convex function of its own strategy.

$$\begin{array}{c}{F}_n={\displaystyle \sum _{t\in T}}(F_{n,\mathrm{int},t}+F_{n,\mathrm{ess},t}+F_{n,\mathrm{com},t})\Delta t\hfill \\ ={\displaystyle \sum _{t\in T}}\left\{\left[C_{n-\mathrm{g},t}^{\mathrm{e}},C_{n-\mathrm{g},t}^{\mathrm{h}}\right]\left[\begin{array}{c}{P}_{n-\mathrm{g},t}^{\mathrm{e}}\\ P_{n-\mathrm{g},t}^{\mathrm{h}}\end{array}\right]+{\eta }_{\mathrm{ess}}(P_{n,t}^{\mathrm{e},\mathrm{ch}}+P_{n,t}^{\mathrm{e},\mathrm{dis}})\right.\hfill \\ \left.+{\eta }_{\mathrm{com}}\left|{\displaystyle \frac{k_2{P}_{n-\mathrm{g},t}^{\mathrm{h}}+k_3{\tau }_t^{\mathrm{out}}-{\tau }_{t-1}^{\mathrm{in}}}{k_1}}-{\tau }^{\mathrm{best}}\right|\right\}\Delta t\hfill \end{array}$$

(51)

The leader EGO transmits $C_{\mathrm{g}-n,t}^{\mathrm{e}}$ and $C_{\mathrm{g}-n,t}^{\mathrm{h}}$ information to the follower $\mathrm{L}{\mathrm{A}}_n$, when $C_{\mathrm{g}-n,t}^{\mathrm{e}}$ and $C_{\mathrm{g}-n,t}^{\mathrm{h}}$ are known variables. From Equation (51), $\mathrm{L}{\mathrm{A}}_n$ responds to $C_{\mathrm{g}-n,t}^{\mathrm{e}}$ and $C_{\mathrm{g}-n,t}^{\mathrm{h}}$ by adjusting its own decision variables $P_{n-\mathrm{g},t}^{\mathrm{e}}$ and $P_{n-\mathrm{g},t}^{\mathrm{h}}$. The first and third terms of the objective function vary linearly about $P_{n-\mathrm{g},t}^{\mathrm{e}}$ and $P_{n-\mathrm{g},t}^{\mathrm{h}}$, and the second term of the objective function is also linearly related to $P_{n-\mathrm{g},t}^{\mathrm{e}}$ and $P_{n-\mathrm{g},t}^{\mathrm{h}}$. The second term of the objective function varies linearly about $P_{n,t}^{\mathrm{e},\mathrm{ch}}$ and $P_{n,t}^{\mathrm{e},\mathrm{dis}}$, and the first and third terms of the objective function are also linearly related to $P_{n,t}^{\mathrm{e},\mathrm{ch}}$ and $P_{n,t}^{\mathrm{e},\mathrm{dis}}$. That is, the objective function $F_n$ of $\mathrm{L}{\mathrm{A}}_n$ is a linear function about its decision variables $P_{n-\mathrm{g},t}^{\mathrm{e}}$ and $P_{n-\mathrm{g},t}^{\mathrm{h}}$, and $P_{n,t}^{\mathrm{e},\mathrm{ch}}$ and $P_{n,t}^{\mathrm{e},\mathrm{dis}}$. The linear function is both a proposed concave function and a proposed convex function, which satisfies Condition 4. The proof is complete.

In summary, the constructed Stackelberg game G in this paper has a Stackelberg equilibrium solution. The Stackelberg game framework proposed in this study is structured as illustrated in Figure 3.

4.2. Distributed Optimization Problem

The ADMM enhances computational efficiency by decomposing a single optimization problem into multiple subproblems and performing parallel computations across different nodes, making it a mainstream distributed algorithm [32]. Based on the mathematical models of entities within the RIES, a Stackelberg game model solved via ADMM is formulated. In this model, both parties (the leader and followers) simultaneously optimize transaction power and prices to maximize their own interests. Clearly, when a transaction is concluded, the purchased power and energy price declared by the $\mathrm{L}{\mathrm{A}}_n$ to the EGO must align with the sold power and energy price published by the EGO to the $\mathrm{L}{\mathrm{A}}_n$. Thus, the consistency constraints for the transaction power and price between the EGO and $\mathrm{L}{\mathrm{A}}_n$ can be expressed as Equation (52).

$$\begin{array}{c}{P}_{\mathrm{g}-n,t}^{\mathrm{e}}=X_{n,t}^{\mathrm{e}}=P_{n-\mathrm{g},t}^{\mathrm{e}}\hfill \\ C_{\mathrm{g}-n,t}^{\mathrm{e}}=Y_{n,t}^{\mathrm{e}}=C_{n-\mathrm{g},t}^{\mathrm{e}}\hfill \\ P_{\mathrm{g}-n,t}^{\mathrm{h}}=X_{n,t}^{\mathrm{h}}=P_{n-\mathrm{g},t}^{\mathrm{h}}\hfill \\ C_{\mathrm{g}-n,t}^{\mathrm{h}}=Y_{n,t}^{\mathrm{h}}=C_{n-\mathrm{g},t}^{\mathrm{h}}\hfill \end{array}$$

(52)

where $X_{n,t}^{\mathrm{e}}$ and $Y_{n,t}^{\mathrm{e}}$ are virtual variables related to electricity power and prices between the EGO and $\mathrm{L}{\mathrm{A}}_n$; $X_{n,t}^{\mathrm{h}}$ and $Y_{n,t}^{\mathrm{e}}$ are virtual variables related to thermal power and prices between the EGO and $\mathrm{L}{\mathrm{A}}_n$.

Energy trading consistency constraints, i.e., game leader–follower layer coupling constraints. The augmented Lagrangian function is shown in Equation (53).

$$\begin{array}{c}L=\{{\displaystyle \sum _{t\in T}}(-F_{\mathrm{g},t})+{\displaystyle \sum _{t\in T}}{\displaystyle \sum _{n\in N}}[\left(F_{n,t}\right)+{\lambda }_{n,t}^{\mathrm{s},\mathrm{e},\mathrm{p}}(P_{\mathrm{g}-n,t}^{\mathrm{e}}-X_{n,t}^{\mathrm{e}})+{\lambda }_{n,t}^{\mathrm{b},\mathrm{e},\mathrm{p}}(P_{n-\mathrm{g},t}^{\mathrm{e}}-X_{n,t}^{\mathrm{e}})\hfill \\ +{\lambda }_{n,t}^{\mathrm{s},\mathrm{h},\mathrm{p}}(P_{\mathrm{g}-n,t}^{\mathrm{h}}-X_{n,t}^{\mathrm{h}})+{\lambda }_{n,t}^{\mathrm{b},\mathrm{h},\mathrm{p}}(P_{n-\mathrm{g},t}^{\mathrm{h}}-X_{n,t}^{\mathrm{h}})+{\lambda }_{n,t}^{\mathrm{s},\mathrm{e},\mathrm{c}}(C_{\mathrm{g}-n,t}^{\mathrm{e}}-Y_{n,t}^{\mathrm{e}})\hfill \\ +{\lambda }_{n,t}^{\mathrm{b},\mathrm{e},\mathrm{c}}(C_{n-\mathrm{g},t}^{\mathrm{e}}-Y_{n,t}^{\mathrm{e}})+{\lambda }_{n,t}^{\mathrm{s},\mathrm{h},\mathrm{c}}(C_{\mathrm{g}-n,t}^{\mathrm{h}}-Y_{n,t}^{\mathrm{h}})+{\lambda }_{n,t}^{\mathrm{b},\mathrm{h},\mathrm{c}}(C_{n-\mathrm{g},t}^{\mathrm{h}}-Y_{n,t}^{\mathrm{h}})\left]\right\}\hfill \\ +0.5\rho ({∥P_{\mathrm{g}-n}^{\mathrm{e}}-X_n^{\mathrm{e}}∥}_2^2+{∥P_{n-\mathrm{g}}^{\mathrm{e}}-X_n^{\mathrm{e}}∥}_2^2+{∥P_{\mathrm{g}-n}^{\mathrm{h}}-X_n^{\mathrm{h}}∥}_2^2+{∥P_{n-\mathrm{g}}^{\mathrm{h}}-X_n^{\mathrm{h}}∥}_2^2\hfill \\ +{∥C_{\mathrm{g}-n}^{\mathrm{e}}-Y_n^{\mathrm{e}}∥}_2^2+{∥C_{n-\mathrm{g}}^{\mathrm{e}}-Y_n^{\mathrm{e}}∥}_2^2+{∥C_{\mathrm{g}-n}^{\mathrm{h}}-Y_n^{\mathrm{h}}∥}_2^2+{∥C_{n-\mathrm{g}}^{\mathrm{h}}-Y_n^{\mathrm{h}}∥}_2^2)\hfill \end{array}$$

(53)

where $\lambda$ is the Lagrange multiplier; $\rho$ is the penalty factor.

The EGO-side subproblem objective function is shown in Equation (54).

$$\begin{array}{c}\min {\displaystyle \sum _{t\in T}}{\displaystyle \sum _{n\in N}}[(-F_{\mathrm{g},t})+{\lambda }_{n,t}^{\mathrm{s},\mathrm{e},\mathrm{p}}(P_{\mathrm{g}-n,t}^{\mathrm{e}}-X_{n,t}^{\mathrm{e}})+{\lambda }_{n,t}^{\mathrm{s},\mathrm{h},\mathrm{p}}(P_{\mathrm{g}-n,t}^{\mathrm{h}}-X_{n,t}^{\mathrm{h}})\hfill \\ +{\lambda }_{n,t}^{\mathrm{s},\mathrm{e},\mathrm{c}}(C_{\mathrm{g}-n,t}^{\mathrm{e}}-Y_{n,t}^{\mathrm{e}})+{\lambda }_{n,t}^{\mathrm{s},\mathrm{h},\mathrm{c}}(C_{\mathrm{g}-n,t}^{\mathrm{h}}-Y_{n,t}^{\mathrm{h}})]+\hfill \\ 0.5\rho ({∥P_{\mathrm{g}-n}^{\mathrm{e}}-X_n^{\mathrm{e}}∥}_2^2+{∥P_{\mathrm{g}-n}^{\mathrm{h}}-X_n^{\mathrm{h}}∥}_2^2+{∥C_{\mathrm{g}-n}^{\mathrm{e}}-Y_n^{\mathrm{e}}∥}_2^2+{∥C_{\mathrm{g}-n}^{\mathrm{h}}-Y_n^{\mathrm{h}}∥}_2^2)\hfill \end{array}$$

(54)

The LA-side subproblem objective function is shown in Equation (55).

$$\begin{array}{c}\min {\displaystyle \sum _{t\in T}}[F_{n,t}+{\lambda }_{n,t}^{\mathrm{b},\mathrm{e},\mathrm{p}}(P_{n-\mathrm{g},t}^{\mathrm{e}}-X_{n,t}^{\mathrm{e}})+{\lambda }_{n,t}^{\mathrm{b},\mathrm{h},\mathrm{p}}(P_{n-\mathrm{g},t}^{\mathrm{h}}-X_{n,t}^{\mathrm{h}})\hfill \\ +{\lambda }_{n,t}^{\mathrm{b},\mathrm{e},\mathrm{c}}(C_{n-\mathrm{g},t}^{\mathrm{e}}-Y_{n,t}^{\mathrm{e}})+{\lambda }_{n,t}^{\mathrm{b},\mathrm{h},\mathrm{c}}(C_{n-\mathrm{g},t}^{\mathrm{h}}-Y_{n,t}^{\mathrm{h}})]+\hfill \\ 0.5\rho ({∥P_{n-\mathrm{g}}^{\mathrm{e}}-X_n^{\mathrm{e}}∥}_2^2+{∥P_{n-\mathrm{g}}^{\mathrm{e}}-X_n^{\mathrm{e}}∥}_2^2+{∥C_{n-\mathrm{g}}^{\mathrm{h}}-Y_n^{\mathrm{h}}∥}_2^2+{∥C_{n-\mathrm{g}}^{\mathrm{h}}-Y_n^{\mathrm{h}}∥}_2^2)\hfill \end{array}$$

(55)

4.3. Nonlinear Term Processing

In both the leader’s and follower’s objective functions, convex hulls formed by the product of power optimization variables and energy price optimization variables are present, which are unfavorable for algorithmic solution. Under such circumstances, the McCormick envelope method is typically employed to linearly relax nonlinear terms [33]. The relaxation process is shown in Equation (56).

$$\begin{array}{c}{\omega }_{\mathrm{g}-n,t}^{\mathrm{e}}=C_{\mathrm{g}-n,t}^{\mathrm{e}}{P}_{\mathrm{g}-n,t}^{\mathrm{e}}\hfill \\ {\omega }_{\mathrm{g}-n,t}^{\mathrm{h}}=C_{\mathrm{g}-n,t}^{\mathrm{h}}{P}_{\mathrm{g}-n,t}^{\mathrm{h}}\hfill \\ {\omega }_{n-\mathrm{g},t}^{\mathrm{e}}=C_{n-\mathrm{g},t}^{\mathrm{e}}{P}_{n-\mathrm{g},t}^{\mathrm{e}}\hfill \\ {\omega }_{n-\mathrm{g},t}^{\mathrm{h}}=C_{n-\mathrm{g},t}^{\mathrm{h}}{P}_{n-\mathrm{g},t}^{\mathrm{h}}\hfill \end{array}$$

(56)

where ${\omega }_{\mathrm{g}-n,t}^{\mathrm{e}}$ and ${\omega }_{\mathrm{g}-n,t}^{\mathrm{h}}$ are nonlinear term substitution variables on the EGO side regarding electricity and thermal energy trading; ${\omega }_{n-\mathrm{g},t}^{\mathrm{e}}$ and ${\omega }_{n-\mathrm{g},t}^{\mathrm{h}}$ are nonlinear term substitution variables on the LA side regarding electricity and thermal energy trading. The substitution variables should satisfy the following slack supplement constraints.

The EGO-side relaxation supplement constraints are shown in Equations (57) and (58).

$$\begin{array}{c}{\omega }_{\mathrm{g}-n}^{\mathrm{e}}\ge P_{\mathrm{g}-n,t}^{\mathrm{e}}{C}_t^{\mathrm{e},min}+P_{\mathrm{g}-n}^{\mathrm{e},\min }{C}_{\mathrm{g}-n,t}^{\mathrm{e}}-P_{\mathrm{g}-n}^{\mathrm{e},\min }{C}_t^{\mathrm{e},min}\hfill \\ {\omega }_{\mathrm{g}-n}^{\mathrm{e}}\ge P_{\mathrm{g}-n,t}^{\mathrm{e}}{C}_t^{\mathrm{e},max}+P_{\mathrm{g}-n}^{\mathrm{e},\max }{C}_{\mathrm{g}-n,t}^{\mathrm{e}}-P_{\mathrm{g}-n}^{\mathrm{e},\max }{C}_t^{\mathrm{e},max}\hfill \\ {\omega }_{\mathrm{g}-n}^{\mathrm{e}}\le P_{\mathrm{g}-n,t}^{\mathrm{e}}{C}_t^{\mathrm{e},max}+P_{\mathrm{g}-n}^{\mathrm{e},\min }{C}_{\mathrm{g}-n,t}^{\mathrm{e}}-P_{\mathrm{g}-n}^{\mathrm{e},\min }{C}_t^{\mathrm{e},max}\hfill \\ {\omega }_{\mathrm{g}-n}^{\mathrm{e}}\le P_{\mathrm{g}-n,t}^{\mathrm{e}}{C}_t^{\mathrm{e},min}+P_{\mathrm{g}-n}^{\mathrm{e},\max }{C}_{\mathrm{g}-n,t}^{\mathrm{e}}-P_{\mathrm{g}-n}^{\mathrm{e},\max }{C}_t^{\mathrm{e},min}\hfill \end{array}$$

(57)

$$\begin{array}{c}{\omega }_{\mathrm{g}-n}^{\mathrm{h}}\ge P_{\mathrm{g}-n,t}^{\mathrm{h}}{C}_t^{\mathrm{h},min}+P_{\mathrm{g}-n}^{\mathrm{h},\min }{C}_{\mathrm{g}-n,t}^{\mathrm{h}}-P_{\mathrm{g}-n}^{\mathrm{h},\min }{C}_t^{\mathrm{h},min}\hfill \\ {\omega }_{\mathrm{g}-n}^{\mathrm{h}}\ge P_{\mathrm{g}-n,t}^{\mathrm{h}}{C}_t^{\mathrm{h},max}+P_{\mathrm{g}-n}^{\mathrm{h},\max }{C}_{\mathrm{g}-n,t}^{\mathrm{h}}-P_{\mathrm{g}-n}^{\mathrm{h},\max }{C}_t^{\mathrm{h},max}\hfill \\ {\omega }_{\mathrm{g}-n}^{\mathrm{h}}\le P_{\mathrm{g}-n,t}^{\mathrm{h}}{C}_t^{\mathrm{h},max}+P_{\mathrm{g}-n}^{\mathrm{h},\min }{C}_{\mathrm{g}-n,t}^{\mathrm{h}}-P_{\mathrm{g}-n}^{\mathrm{h},\min }{C}_t^{\mathrm{h},max}\hfill \\ {\omega }_{\mathrm{g}-n}^{\mathrm{h}}\le P_{\mathrm{g}-n,t}^{\mathrm{h}}{C}_t^{\mathrm{h},min}+P_{\mathrm{g}-n}^{\mathrm{h},\max }{C}_{\mathrm{g}-n,t}^{\mathrm{h}}-P_{\mathrm{g}-n}^{\mathrm{h},\max }{C}_t^{\mathrm{h},min}\hfill \end{array}$$

(58)

where $P_{\mathrm{g}-n}^{\mathrm{e},\max }$ and $P_{\mathrm{g}-n}^{\mathrm{e},\min }$ are the maximum and minimum power of the EGO and $\mathrm{L}{\mathrm{A}}_n$ to transmit electric energy; $P_{\mathrm{g}-n}^{\mathrm{h},\max }$ and $P_{\mathrm{g}-n}^{\mathrm{h},\min }$ are the maximum and minimum power of the EGO and $\mathrm{L}{\mathrm{A}}_n$ to transfer thermal energy.

The LA-side relaxation supplement constraints are shown in Equations (59) and (60).

$$\begin{array}{c}{\omega }_{n-\mathrm{g}}^{\mathrm{e}}\ge P_{n-\mathrm{g},\mathrm{t}}^{\mathrm{e}}{C}_t^{\mathrm{e},min}+P_{n-\mathrm{g}}^{\mathrm{e},\min }{C}_{n-\mathrm{g},\mathrm{t}}^{\mathrm{e}}-P_{n-\mathrm{g}}^{\mathrm{e},\min }{C}_t^{\mathrm{e},min}\hfill \\ {\omega }_{n-\mathrm{g}}^{\mathrm{e}}\ge P_{n-\mathrm{g},\mathrm{t}}^{\mathrm{e}}{C}_t^{\mathrm{e},max}+P_{n-\mathrm{g}}^{\mathrm{e},\max }{C}_{n-\mathrm{g},\mathrm{t}}^{\mathrm{e}}-P_{n-\mathrm{g}}^{\mathrm{e},\max }{C}_t^{\mathrm{e},max}\hfill \\ {\omega }_{n-\mathrm{g}}^{\mathrm{e}}\le P_{n-\mathrm{g},\mathrm{t}}^{\mathrm{e}}{C}_t^{\mathrm{e},max}+P_{n-\mathrm{g}}^{\mathrm{e},\min }{C}_{n-\mathrm{g},\mathrm{t}}^{\mathrm{e}}-P_{n-\mathrm{g}}^{\mathrm{e},\min }{C}_t^{\mathrm{e},max}\hfill \\ {\omega }_{n-\mathrm{g}}^{\mathrm{e}}\le P_{n-\mathrm{g},\mathrm{t}}^{\mathrm{e}}{C}_t^{\mathrm{e},min}+P_{n-\mathrm{g}}^{\mathrm{e},\max }{C}_{n-\mathrm{g},\mathrm{t}}^{\mathrm{e}}-P_{n-\mathrm{g}}^{\mathrm{e},\max }{C}_t^{\mathrm{e},min}\hfill \end{array}$$

(59)

$$\begin{array}{c}{\omega }_{n-\mathrm{g}}^{\mathrm{h}}\ge P_{n-\mathrm{g},\mathrm{t}}^{\mathrm{h}}{C}_t^{\mathrm{h},min}+P_{n-\mathrm{g}}^{\mathrm{h},\min }{C}_{n-\mathrm{g},\mathrm{t}}^{\mathrm{h}}-P_{n-\mathrm{g}}^{\mathrm{h},\min }{C}_t^{\mathrm{h},min}\hfill \\ {\omega }_{n-\mathrm{g}}^{\mathrm{h}}\ge P_{n-\mathrm{g},\mathrm{t}}^{\mathrm{h}}{C}_t^{\mathrm{h},max}+P_{n-\mathrm{g}}^{\mathrm{h},\max }{C}_{n-\mathrm{g},\mathrm{t}}^{\mathrm{h}}-P_{n-\mathrm{g}}^{\mathrm{h},\max }{C}_t^{\mathrm{h},max}\hfill \\ {\omega }_{n-\mathrm{g}}^{\mathrm{h}}\le P_{n-\mathrm{g},\mathrm{t}}^{\mathrm{h}}{C}_t^{\mathrm{h},max}+P_{n-\mathrm{g}}^{\mathrm{h},\min }{C}_{n-\mathrm{g},\mathrm{t}}^{\mathrm{h}}-P_{n-\mathrm{g}}^{\mathrm{h},\min }{C}_t^{\mathrm{h},max}\hfill \\ {\omega }_{n-\mathrm{g}}^{\mathrm{h}}\le P_{n-\mathrm{g},\mathrm{t}}^{\mathrm{h}}{C}_t^{\mathrm{h},min}+P_{n-\mathrm{g}}^{\mathrm{h},\max }{C}_{n-\mathrm{g},\mathrm{t}}^{\mathrm{h}}-P_{n-\mathrm{g}}^{\mathrm{h},\max }{C}_t^{\mathrm{h},min}\hfill \end{array}$$

(60)

where $P_{\mathrm{g}-n}^{\mathrm{e},\max }$ and $P_{\mathrm{g}-n}^{\mathrm{e},\min }$ are the maximum and minimum electrical energy transfer power between EGO and $\mathrm{L}{\mathrm{A}}_n$; $P_{\mathrm{g}-n}^{\mathrm{h},\max }$ and $P_{\mathrm{g}-n}^{\mathrm{h},\min }$ are the maximum and minimum thermal energy transfer power between EGO and $\mathrm{L}{\mathrm{A}}_n$.

4.4. Optimization Problem Solving Procedure

(1)

Set the number of iterations and the convergence margin value, set the initial values of the global variables, the Lagrange multiplier, the penalty factor, the power residual, and the price residual, and set the number of iterations k to zero.

(2)

$k=k+1$.

(3)

Solving subproblems.

The upper and lower subproblems are solved separately and in parallel by the corresponding computational bodies.

(1)

Solving the Stackelberg game leader EGO-side subproblem.

Objective function: Equation (54).

Restrictive condition: Equations (12)–(20), Equations (6)–(31) and (56)–(58).

(2)

Solving the Stackelberg game follower LA-side subproblem.

Objective function: Equation (55).

Restrictive condition: Equations (36)–(48), (56), (59) and (60).

(4)

Updating parameters.

Update global variables, Lagrange multipliers, power residuals, and price residuals. Update the global variables according to Equation (61).

$$\begin{array}{c}\begin{array}{c}{X}_{n,t}^{\mathrm{e},k+1}=0.5(P_{\mathrm{g}-n,t}^{\mathrm{e},k+1}+P_{n-\mathrm{g},t}^{\mathrm{e},k+1})\hfill \\ X_{n,t}^{\mathrm{h},k+1}=0.5(P_{\mathrm{g}-n,t}^{\mathrm{h},k+1}+P_{n-\mathrm{g},t}^{\mathrm{h},k+1})\hfill \end{array}\hfill \\ \begin{array}{c}{Y}_{n,t}^{\mathrm{e},k+1}=0.5(C_{\mathrm{g}-n,t}^{\mathrm{e},k+1}+C_{n-\mathrm{g},t}^{\mathrm{e},k+1})\hfill \\ Y_{n,t}^{\mathrm{h},k+1}=0.5(C_{\mathrm{g}-n,t}^{\mathrm{h},k+1}+C_{n-\mathrm{g},t}^{\mathrm{h},k+1})\hfill \end{array}\hfill \end{array}$$

(61)

Update the Lagrange multipliers according to Equations (62) and (63).

$$\begin{array}{c}{\lambda }_{n,t}^{\mathrm{s},\mathrm{e},\mathrm{p},k+1}={\lambda }_{n,t}^{\mathrm{s},\mathrm{e},\mathrm{p},k}+\rho (P_{\mathrm{g}-n,t}^{\mathrm{e},k+1}-X_{n,t}^{\mathrm{e},k+1})\hfill \\ {\lambda }_{n,t}^{\mathrm{b},\mathrm{e},\mathrm{p},k+1}={\lambda }_{n,t}^{\mathrm{b},\mathrm{e},\mathrm{p},k}+\rho (P_{n-\mathrm{g},t}^{\mathrm{e},k+1}-X_{n,t}^{\mathrm{e},k+1})\hfill \\ {\lambda }_{n,t}^{\mathrm{s},\mathrm{e},\mathrm{c},k+1}={\lambda }_{n,t}^{\mathrm{s},\mathrm{e},\mathrm{c},k}+\rho (C_{\mathrm{g}-n,t}^{\mathrm{e},k+1}-Y_{n,t}^{\mathrm{e},k+1})\hfill \\ {\lambda }_{n,t}^{\mathrm{b},\mathrm{e},\mathrm{c},k+1}={\lambda }_{n,t}^{\mathrm{b},\mathrm{e},\mathrm{c},k}+\rho (C_{n-\mathrm{g},t}^{\mathrm{e},k+1}-Y_{n,t}^{\mathrm{e},k+1})\hfill \end{array}$$

(62)

$$\begin{array}{c}{\lambda }_{n,t}^{\mathrm{s},\mathrm{h},\mathrm{p},k+1}={\lambda }_{n,t}^{\mathrm{s},\mathrm{h},\mathrm{p},k}+\rho (P_{\mathrm{g}-n,t}^{\mathrm{h},k+1}-X_{n,t}^{\mathrm{h},k+1})\hfill \\ {\lambda }_{n,t}^{\mathrm{b},\mathrm{h},\mathrm{p},k+1}={\lambda }_{n,t}^{\mathrm{b},\mathrm{h},\mathrm{p},k}+\rho (P_{n-\mathrm{g},t}^{\mathrm{h},k+1}-X_{n,t}^{\mathrm{h},k+1})\hfill \\ {\lambda }_{n,t}^{\mathrm{s},\mathrm{h},\mathrm{c},k+1}={\lambda }_{n,t}^{\mathrm{s},\mathrm{h},\mathrm{c},k}+\rho (C_{\mathrm{g}-n,t}^{\mathrm{h},k+1}-Y_{n,t}^{\mathrm{h},k+1})\hfill \\ {\lambda }_{n,t}^{\mathrm{b},\mathrm{h},\mathrm{c},k+1}={\lambda }_{n,t}^{\mathrm{b},\mathrm{h},\mathrm{c},k}+\rho (C_{n-\mathrm{g},t}^{\mathrm{h},k+1}-Y_{n,t}^{\mathrm{h},k+1})\hfill \end{array}$$

(63)

Update power residuals and price residuals according to Equations (64) and (65).

$$\begin{array}{c}{∥r_{\mathrm{p},\mathrm{s}}^{\mathrm{e},k+1}∥}_2^2={∥P_{\mathrm{g}-n}^{\mathrm{e},k+1}-X_n^{\mathrm{e},k+1}∥}_2^2\hfill \\ {∥r_{\mathrm{p},\mathrm{b}}^{\mathrm{e},k+1}∥}_2^2={∥P_{n-\mathrm{g}}^{\mathrm{e},k+1}-X_n^{\mathrm{e},k+1}∥}_2^2\hfill \\ {∥r_{\mathrm{c},\mathrm{s}}^{\mathrm{e},k+1}∥}_2^2={∥C_{\mathrm{g}-n}^{\mathrm{e},k+1}-Y_n^{\mathrm{e},k+1}∥}_2^2\hfill \\ {∥r_{\mathrm{c},\mathrm{b}}^{\mathrm{e},k+1}∥}_2^2={∥C_{n-\mathrm{g}}^{\mathrm{e},k+1}-Y_n^{\mathrm{e},k+1}∥}_2^2\hfill \end{array}$$

(64)

$$\begin{array}{c}{∥r_{\mathrm{p},\mathrm{s}}^{\mathrm{h},k+1}∥}_2^2={∥P_{\mathrm{g}-n}^{\mathrm{h},k+1}-X_n^{\mathrm{h},k+1}∥}_2^2\hfill \\ {∥r_{\mathrm{p},\mathrm{b}}^{\mathrm{h},k+1}∥}_2^2={∥P_{n-\mathrm{g}}^{\mathrm{h},k+1}-X_n^{\mathrm{h},k+1}∥}_2^2\hfill \\ {∥r_{\mathrm{c},\mathrm{s}}^{\mathrm{h},k+1}∥}_2^2={∥C_{\mathrm{g}-n}^{\mathrm{h},k+1}-Y_n^{\mathrm{h},k+1}∥}_2^2\hfill \\ {∥r_{\mathrm{c},\mathrm{b}}^{\mathrm{h},k+1}∥}_2^2={∥C_{n-\mathrm{g}}^{\mathrm{h},k+1}-Y_n^{\mathrm{h},k+1}∥}_2^2\hfill \end{array}$$

(65)

where ${∥r_{p,s}^{e,k+1}∥}_2^2$ and ${∥r_{p,b}^{e,k+1}∥}_2^2$ are the residual sets for the sold and purchased electricity power; ${∥r_{p,s}^{h,k+1}∥}_2^2$ and ${∥r_{p,b}^{h,k+1}∥}_2^2$ are the residual sets for the sold and purchased thermal power; ${∥r_{c,s}^{e,k+1}∥}_2^2$ and ${∥r_{c,b}^{e,k+1}∥}_2^2$ are the residual sets for the sold and purchased electricity price; and ${∥r_{c,s}^{h,k+1}∥}_2^2$ and ${∥r_{c,b}^{h,k+1}∥}_2^2$ are the residual sets for the sold and purchased thermal price.

(5)

Determine whether the current residuals satisfy the ADMM convergence determination condition according to Equation (66).

$$\begin{array}{c}{∥r^{\mathrm{p},k+1}∥}^2\le r^{\mathrm{p}}\hfill \\ {∥r^{\mathrm{c},k+1}∥}^2\le r^{\mathrm{c}}\hfill \end{array}$$

(66)

where ${∥r^{\mathrm{p},k+1}∥}^2$ and ${∥r^{\mathrm{c},k+1}∥}^2$ are the sum of power residuals and price residuals; $r^{\mathrm{p}}$ and $r^{\mathrm{c}}$ are the convergence margins of power residuals and price residuals.

If Equation (66) is satisfied, the algorithm is judged to have converged, the algorithm stops iterating, and the optimal scheduling policy is output. Otherwise, go back to step (2). The framework of the distributed interactive scheduling is shown in Figure 4. The flow of the algorithm is shown in Figure 5.

5. Calculus Analysis

To deploy and validate the distributed optimal scheduling strategy of a RIES in a real scenario, we need to follow the four-stage validation process of “modeling-simulation-semi-physical-field’’:

• Modeling stage: Construct the optimal operation model of the EGO and LAs.
• Digital simulation: Verify the convergence of the algorithm and the speed of solving the optimization problem on the PSCAD/MATLAB platform.
• Hardware-in-the-loop testing: Connect the real controller with the real-time grid simulation equipment through the RT-LAB platform to test the communication delay and control response time.
• On-site validation: The strategy is implemented through a layered distributed architecture (physical layer–edge layer–cloud): the terminal equipment collects data, the edge controllers perform local fast response (demand management, energy storage control), and the virtual power plant platform in the cloud coordinates the global optimization to ensure that the regulation delay meets the requirements by combining the 5G/fiber optic hybrid communication.

The core indicators for deployment verification in real scenarios should include economy, reliability, regulation capability, etc., and the verification should cover typical scenarios such as peak and valley tariff hours.

5.1. Case Configuration

Typical winter daily load data from three communities in a region were selected for simulation experiments. The scheduling period is set to be 24 h and the time scale is 1 h. The electrical energy load and PV power prediction in the three communities are shown in Figure 6. The outdoor temperatures of the customers are shown in Figure 7. The TOU prices for EPG, FIT, as well as the leader’s minimum average energy selling price and the followers’ maximum average energy purchasing price are presented in

Table 1. Electricity prices of EPG.

Types of Electricity Price	Periods	Value (CNY/kWh)
TOU	Peak (11:00–14:00), (19:00–22:00)	1.20
	Flat (7:00–10:00), (15:00–18:00)	0.75
	Valley (1:00–6:00), (23:00–24:00)	0.40
FIT	Peak (11:00–14:00), (19:00–22:00)	0.65
	Flat (7:00–10:00), (15:00–18:00)	0.45
	Valley (1:00–6:00), (23:00–24:00)	0.25

. The model parameters of the operators in the RIES are shown in

Table 2. Model parameters of operators in the RIES.

Parameter	Value	Parameter	Value
$C_{\mathrm{gas}}$	3.2 CNY/m3	${\chi }_{\mathrm{gt}}^{\mathrm{a}}$	0.234
$a_{\mathrm{cpg}}$	0.001	${\chi }_{\mathrm{cpg}}^{\mathrm{a}}$	1.08
$b_{\mathrm{cpg}}$	0.43	$\gamma$	0.2 CNY/kg
$c_{\mathrm{cpg}}$	1000	$\sigma$	50%
${\kappa }_{\mathrm{gt}}^{\mathrm{e}}$	0.4	l	10,000 kg
${\kappa }_{\mathrm{gt}}^{\mathrm{h}}$	0.45	${\eta }_{\mathrm{ess}}$	0.1 CNY/kWh
$q_{\mathrm{gas}}$	9.7	${\eta }_{\mathrm{com}}$	100 CNY/°C
$P_{\mathrm{gt}}^{\mathrm{e},\max }$	20 MWh	${\tau }^{\mathrm{best}}$	20 °C
$P_{\mathrm{gt}}^{\mathrm{e},\min }$	2 MWh	$k_1$	0.66
$P_{\mathrm{cpg}}^{\mathrm{e},\max }$	20 MWh	$k_2$	0.03
$P_{\mathrm{cpg}}^{\mathrm{e},\min }$	2 MWh	$k_3$	0.34
${\kappa }_{\mathrm{whb}}^{\mathrm{h}}$	0.95	${\tau }^{\mathrm{b}}$	33.5 °C
${\kappa }_{\mathrm{p}2\mathrm{g}}$	0.7	M	80
${\alpha }_{\mathrm{p}2\mathrm{g}}$	0.55	$I_{\mathrm{clo}}$	0.11
${\alpha }_{\mathrm{ccs}}$	0.4	${\kappa }_{\mathrm{ch}}$	0.98
$C^{\mathrm{h},\max }$	0.8 CNY/kWh	${\kappa }_{\mathrm{dis}}$	0.98
$C^{\mathrm{h},\min }$	0.3 CNY/kWh	$S_{\mathrm{soc}}^{\max }$	0.1
${\chi }_{\mathrm{epg}}$	0.728	$S_{\mathrm{soc}}^{\min }$	0.9
${\chi }_{\mathrm{gt}}$	0.367	$P_{\mathrm{g}-n}^{\mathrm{e},\max }$	10 MWh
${\chi }_{\mathrm{cpg}}$	0.728	$P_{\mathrm{g}-n}^{\mathrm{e},\min }$	0
${\chi }_{\mathrm{e}-\mathrm{h}}$	1.67	$P_{\mathrm{g}-n}^{\mathrm{h},\max }$	10 MWh
${\chi }_{\mathrm{epg}}^{\mathrm{a}}$	0.756	$P_{\mathrm{g}-n}^{\mathrm{h},\min }$	0

. Herein, C1, C2, and C3 represent Community 1, Community 2, and Community 3, respectively.

5.2. Optimal Dispatch Results

In the hierarchical distributed RIES optimization scheduling model constructed in this paper, all subproblems, after processing, are linear models and can be solved using the GUROBI 11.04 called via YALMIP in the MATLAB 2023.

In the experiments, the maximum number of iterations for the ADMM algorithm was set to 45. The convergence tolerances for the power and energy price residuals were set to $1\times {10}^{-3}$ and 1, respectively. The ADMM algorithm converged at the 10th iteration. The results show that the revenue of the EGO is 296,639 CNY, while the energy costs for $\mathrm{L}{\mathrm{A}}_1$, $\mathrm{L}{\mathrm{A}}_2$, and $\mathrm{L}{\mathrm{A}}_3$ are 76,493 CNY, 85,950 CNY, and 81,326 CNY, respectively. The residual convergence process of the ADMM algorithm is shown in Figure 8. The iteration process of the objective functions for each stakeholder in the RIES is shown in Figure 9.

5.2.1. EGO Operating Optimization Analysis

The optimized scheduling results for the electricity and thermal energy of the EGO over one scheduling cycle are shown in Figure 10 and Figure 11.

From Figure 10 and Figure 11, it can be seen that during the 1:00–6:00 and 23:00–24:00 time periods, the EGO purchases a large amount of power from the grid to satisfy the power demand of the users in each LA during these time periods because the TOU tariffs are low, lower than the cost of the GT and CPG generation within the EGO. The outdoor temperature is lower during this time period, and the LA purchases more thermal power from the EGO to maintain the indoor temperature of the customers, which results in the higher thermal generation power of the CHP units and the lower CPG output power. As time passes from 7:00 to 13:00, the outdoor temperature gradually increases, the thermal energy purchased from the EGO by the LA gradually decreases, the thermal generation power of the CHP unit gradually decreases, and the output power of the CPG increases. As time passes from 13:00 to 22:00, the outdoor temperature gradually decreases, the thermal power purchased from the EGO by the LA gradually increases, the thermal power generated by the CHP unit gradually increases, and the power generated by the CPG decreases. Moreover, the TOU tariff in the period from 7:00 to 22:00 is higher than the internal GT and CPG generation cost of the EGO, so the EGO stops purchasing electricity from the EPG. During 6:00–8:00, 12:00–13:00, and 19:00–21:00, the power consumption of users peaks, and the EGO maintains the energy supply mainly by increasing the CPG generation power. The operation cost of P2G-CCS is mainly reflected in the consumption of electrical power, and the TOU reaches the maximum during 11:00–14:00, and P2G-CCS reduces the equipment power consumption to ensure the economic efficiency of the EGO.

5.2.2. LAs Operating Optimization Analysis

The optimized scheduling results for electricity of the LAs over one scheduling cycle, along with the electricity prices agreed upon with the EGO, are illustrated in Figure 12, Figure 13 and Figure 14. The optimized scheduling results for thermal energy of the LAs over one scheduling cycle, along with the thermal prices agreed upon with the EGO, are depicted in Figure 15, Figure 16 and Figure 17. The indoor temperature conditions of users are presented in Figure 18.

Combined with Figure 6 and Figure 12, it is jointly analyzed that in the 1:00–4:00 time period, the user’s energy power is relatively low, and because the TOU tariff is in the valley, the EGO releases a tariff at a relatively low level, and a large amount of power is purchased from the EGO and stored in the energy storage device. In the 5:00–7:00 time period, the user’s energy consumption power gradually increases. During the 5:00–6:00 time period, the TOU tariff is still at a low level, while the EGO has issued a relatively low price; so, power is directly purchased from the EGO to meet the energy demand. At 7:00, the TOU tariff increases and less power is purchased from the EGO, and instead power is drawn from the ESS. At 8:00, the consumer’s energy power further decreases, the PV panels of community 1 start to output power, and the tariff issued by the EGO after gaming decreases compared to the previous time period. In the 9:00 time slot, the community 1 PV power and load power increase simultaneously, but the increased PV power is not enough to make up for the power deficit caused by the increase in load power, resulting in an increase in the power purchased from the EGO, and the tariff increases. In the 11:00–14:00 time period, the user energy power and PV power generation reach their peaks at the same time, and the power purchased from the EGO does not increase significantly, resulting in the negotiated tariffs between the EGO and the consumers remaining in the middle, and the tariffs fluctuate along with the trend of the power purchased by the consumers. In the 16:00–17:00 time period process, the community 1 PV power and load power compared to the previous time period decreased at the same time, but the reduction in the PV power is less than the reduction in the load power, so in the 16:00–17:00 time period, power is transferred to the ESS for the storage of electricity and the EGO power purchase power and the purchase price of electricity remains stable. In the 19:00–21:00 time period, the developed power of the PV is zero, and the load power reaches the peak. In order to reduce the energy cost of the user, power is taken from the ESS, and the amount of power purchased from the EGO and the price of the traded power do not show significant fluctuations.

When analyzed together with Figure 7, Figure 15 and Figure 18, compared to the thermal energy required to maintain the optimal indoor temperature, the thermal energy purchase strategy considering thermal inertia and user comfort decreases the power of thermal energy purchase in the 2:00–6:00 and 17:00–18:00 time periods and increases the power of thermal purchase in the 11:00–16:00 and 22:00–22:00 time periods. In the 2:00–6:00 time period, the outdoor temperature of the user is around −6 °C, more thermal energy is needed to maintain the indoor temperature, and the energy price issued by the EGO is at a high level; so, in order to reduce the energy cost of the user, $\mathrm{L}{\mathrm{A}}_1$ reduces the power purchased from the EGO, and the indoor temperature of the user drops to below 19 °C. During 7:00–10:00, as the outdoor temperature increases, the thermal energy required to maintain the indoor temperature gradually decreases, $\mathrm{L}{\mathrm{A}}_1$ purchases almost the same amount of power as in the traditional thermal energy purchasing strategy, the indoor temperature gradually rises, and the thermal energy price issued by the EGO gradually decreases. During 11:00–14:00, the outdoor temperature reaches the highest and remains relatively stable. At this time, the user can purchase a small amount of thermal energy to maintain a high level of indoor temperature, and the user’s room temperature is maintained at about 21.6 °C. In the 15:00–20:00 time period, the outdoor temperature drops sharply, but due to the thermal inertia of the indoor temperature, in the 15:00–16:00 time period, the indoor temperature does not drop immediately, but is maintained for two hours under the premise of the increase in purchased thermal energy power. In the period 17:00–20:00, the indoor temperature decreases due to the outdoor temperature and the decrease in the $\mathrm{L}{\mathrm{A}}_1$ thermal energy purchase power compared to the conventional thermal energy purchase strategy. Due to the similarity of the energy use patterns in different communities, the energy purchase and trading prices in other communities are not repeated.

5.3. Sensitivity Analysis

Figure 19 and Figure 20 show the changes in the EGO’s operating revenue and carbon emission cost with SCTM parameters (CBP and CGR). According to Figure 19, carbon trading cost is positively correlated with the change in CBP, and the EGO operating revenue is negatively correlated with the change in CBP. With the increase in CBP, the proportion of carbon trading cost in EGO operating cost is increasing, which leads to the decrease in the EGO operating revenue. When the CBP exceeds 0.25 CNY/kg, the carbon emission level of the system tends to stabilize, and so does the EGO operating income. This indicates that SCTM can indirectly restrain the carbon emission behavior of the EGO by influencing the carbon trading cost.

According to Figure 20, the carbon trading cost is positively correlated with the change in CGR, and the EGO operating revenue is negatively correlated with the change in CGR. With the increase in CGR, the proportion of the carbon trading cost in the EGO operating cost is increasing, which leads to the decrease in the EGO operating revenue. When the CGR is greater than 60%, the carbon emission level of the system tends to stabilize, and so does the EGO operating revenue. This indicates that SCTM can indirectly restrain the carbon emission behavior of the EGO by influencing the carbon transaction cost.

5.4. Scalability Analysis

For larger-scale integrated energy systems, the scalability and robustness of the model can be improved from three aspects:

• Algorithmic level: Firstly, strengthen the distributed algorithm, adopt asynchronous ADMM algorithm to reduce the communication waiting time, and dynamically adjust the parameters to accelerate the convergence; secondly, implement the hierarchical optimization strategy, divide the LA group into multiple sub-systems according to the region, and optimize locally first and then realize the global synergy through the coordination layer.
• Model structure level: Modular design can be used to encapsulate the LA and EGO functional modules, and only the corresponding modules need to be added when adding new LAs. Secondly, the parameters can be dynamically adjusted to establish the mapping relationship between the parameters and the system scale and LA characteristics, such as optimizing the load forecast coefficients according to user types. Finally, robust optimization can be extended to incorporate load and new energy uncertainty into interval constraints to improve anti-interference capability.
• Communication architecture level: Optimize topology, adopt hierarchical distributed communication, use tree topology between the EGO and LA, and design redundant links; choose MQTT lightweight protocol, customize message format and compress data, and encrypt for security.

5.5. Comparative Case Analysis

To further analyze the rationality and effectiveness of the proposed method, numerical experiments are conducted on multiple test cases. The following five arithmetic scenarios are set as shown in

Table 3. Description of the 5 comparison scenarios.

Scenario	SCTM	P2G-CCS	ESS	PMV
1	√	√	√	√
2	×	√	√	√
3	√	×	√	√
4	√	√	×	√
5	√	√	√	×

“√” = mechanism considered; “×” = mechanism not considered.

.

The operating revenue of the EGO, the operating cost of each LA, and the carbon emissions of the RIES are shown in

Table 4. Operating revenue of EGO, operating cost of LAs, and carbon emissions of RIES.

Scenario	EGO Operating Revenue/CNY	$\mathbf{L}{\mathbf{A}}_1$ Operating Cost/CNY	$\mathbf{L}{\mathbf{A}}_2$ Operating Cost/CNY	$\mathbf{L}{\mathbf{A}}_3$ Operating Cost/CNY	RIES $\mathbf{C}{\mathbf{O}}_2$ Emissions /kg
1	296,639	76,493	85,950	81,326	388,835
2	305,350	76,411	85,799	81,219	434,447
3	243,382	76,040	85,418	80,787	328,298
4	305,622	78,419	89,630	83,815	397,736
5	297,568	77,640	88,098	83,033	388,936

.

Comparing scenarios 1 and 2, although the consideration of the SCTM leads to a reduction of 8711 CNY (3%) in the EGO’s operating benefits, this situation is within the acceptable range, and the energy costs of the three LAs remain almost unchanged, with the operating benefits and electricity costs of the trading parties remaining within the normal fluctuation range. On the contrary, after the introduction of the SCTM, the system carbon emissions are significantly reduced by 45,612 kg (10.5%), indicating that the SCTM introduced in this paper can make the carbon-emitting body pay the economic cost when emitting carbon by pricing the carbon emissions, so as to prompt the enterprises to pay attention to their own carbon emission behaviors, to reduce the carbon emissions, and to push the society as a whole to change to a low-carbon life style.

Comparing scenarios 1 and 3, it is found that the EGO’s actual carbon emissions from the generating unit increased by 60,537 kg (15.6%) after the introduction of the P2G-CCS equipment; however, most of the increased $\mathrm{C}{\mathrm{O}}_2$ was converted to natural gas after carbon capture. This reduces the cost of natural gas purchased by the EGO, resulting in an increase of 53,257 CNY (12.2%) in the EGO’s efficiency without any detriment to the customer’s interests. This demonstrates that the coupling device transforms $\mathrm{C}{\mathrm{O}}_2$ from a cost burden to a profitable resource through the synergistic model of “carbon resource utilization+energy conversion”.

Comparing scenarios 1 and 4, planning energy storage devices on the energy-use side of RIES will improve the three indicators of the EGO operating revenue, the energy cost of each LA, and the system $\mathrm{C}{\mathrm{O}}_2$ emission, and the EGO operating revenue is improved by 8983 CNY (3%), and the energy cost of each LA is reduced by 1926 CNY (2.5%), 3680 CNY (4.2%), and 2489 CNY (3.1%), respectively. And the system carbon emission is reduced by 1099 kg (1%). This shows that the planning of energy storage equipment in the RIES in this paper can achieve the optimal use of energy and cost saving, improve the efficiency of energy use, and enhance the reliability of the power supply.

Comparing scenarios 1 and 5, the thermal inertia demand response will keep the system $\mathrm{C}{\mathrm{O}}_2$ emissions almost unchanged, so that there is an improvement in the EGO’s revenue and the energy cost of each LA. The EGO’s revenue experiences a slight increase, and the energy cost of each LA decreases by 1147 CNY (1.5%), 2148 CNY (2.5%), and 1707 CNY (2.1%), respectively. This shows that the introduction of thermal inertia-based demand response in the RIES in this paper can adjust the demand for energy purchase according to the EGO published tariff, improve the flexibility of user energy use, and increase the economic benefits of users under the premise of ensuring that the indoor temperature meets the comfort of the users. At the same time, the introduction of demand response also indirectly reduces the dependence of the energy system on fossil fuels, which helps to reduce greenhouse gas emissions.

6. Conclusions

This study investigates the low-carbon economic optimal dispatch of an RIES from the perspective of economic benefits for various stakeholders. Incorporating factors such as the SCTM, P2G-CCS, and thermal energy demand response, an EGO-LA transaction process is embedded within a Stackelberg game framework. This establishes a multi-participant interactive energy trading mechanism with the EGO as the leader and the LAs as the followers. The following conclusions are drawn:

• Effectiveness of the RIES distributed optimal dispatch: The proposed hierarchical distributed low-carbon economic dispatch strategy for the RIES, based on the ADMM, effectively coordinates the interests of the energy trading parties, achieving efficient and economic system operation. Case studies demonstrate that this strategy exhibits significant advantages in enhancing the economic benefits for all stakeholders and improving user energy consumption flexibility.
• Positive impact on system emission reduction: The introduction of SCTM, P2G-CCS, and user thermal energy demand response leads to a significant reduction in the system’s actual carbon emissions. This indicates that the series of measures proposed in this study, including carbon emission pricing and carbon management via P2G-CCS, prompts enterprises to pay greater attention to their carbon emission behaviors. Consequently, these mechanisms effectively promote a broader societal shift towards a low-carbon lifestyle. This approach not only contributes to achieving the “Dual Carbon” goals but also provides economic incentives for enterprises, fostering a win–win outcome for both environmental protection and economic benefits.

In this paper, the types of flexible power equipment and working forms are not sufficiently considered, and the utilization of energy forms is relatively homogeneous. Subsequently, further consideration will be given to aggregating a variety of flexible loads for demand response analysis from an integrated energy system perspective.

The methodology is applied in a wider range of real-world cases and systematically analyzed against relevant proven optimization strategies to further validate its generalizability and competitive advantage.

Integration of demand and PV output uncertainty modeling. In the short term, scenario analysis will be used to generate multiple uncertainty scenarios of demand and PV output based on historical data, which will be integrated into the existing model through a stochastic optimization framework in order to quantify the impact of uncertainty on the system optimization results. In the medium and long term, it is planned to introduce robust optimization methods to consider the fluctuating range of uncertainty parameters and construct a decision model with anti-interference capability to ensure that the system can still maintain stable operation under extreme scenarios.