Optimized Operation Strategy for Multi-Regional Integrated Energy Systems Based on a Bilevel Stackelberg Game Framework

Abstract

To enhance spatial resource complementarity and cross-entity coordination among multi-regional integrated energy systems (MRIESs), an optimized operation strategy is developed based on a bilevel Stackelberg game framework. In this framework, the integrated energy system operator (IESO) and MRIES act as the leader and followers, respectively. Guided by an integrated demand response (IDR) mechanism and a collaborative green certificate and carbon emission trading (GC–CET) scheme, energy prices and consumption strategies are optimized through iterative game interactions. Inter-regional electricity transaction prices and volumes are modeled as coupling variables. The solution is obtained using a hybrid algorithm combining particle swarm optimization (PSO) with mixed-integer programming (MIP). Simulation results indicate that the proposed strategy effectively enhances energy complementarity and optimizes consumption structures across regions. It also balances the interests of the IESO and MRIES, reducing operating costs by 9.97%, 27.7%, and 4.87% in the respective regions. Moreover, in the case study, renewable energy utilization rates in different regions—including an urban residential zone, a renewable-rich suburban area, and an industrial zone—are improved significantly, with Region 2 increasing from 95.06% and Region 3 from 77.47% to full consumption (100%), contributing to notable reductions in carbon emissions.

1. Introduction

According to the International Energy Agency’s World Energy Outlook 2023 [1], global carbon emissions from the energy sector must be reduced by 43% by 2030 to meet the temperature targets set by the Paris Agreement. In China, carbon emissions from the power sector account for approximately 40% of total energy-related emissions [2,3,4], making the transformation of the conventional power industry increasingly imperative. Against this backdrop, the Integrated Energy System (IES) is regarded as a key platform for facilitating the energy transition [5]. As the share of distributed energy resources increases and regional supply–demand imbalances become increasingly pronounced, conventional IESs face significant limitations in managing the coordinated scheduling of multiple energy flows. The MRIES leverages spatial resource heterogeneity to enhance energy efficiency, facilitate renewable energy integration, and improve system supply reliability.

With the advancement of multi-regional energy interconnection models, considerable research has focused on the cooperative operation of IESs. For instance, Attarha et al. [6] developed an ADMM-based cooperative optimization framework to address boundary decoupling in multi-regional energy networks. Further, Qi et al. [7] and Wu et al. [8] analyzed decentralized dispatch strategies for integrated electricity–gas systems, emphasizing the need for dynamic balance and pressure constraints. These efforts laid the groundwork for MRIES scheduling. However, as the supply–demand structure diversifies, MRIESs increasingly involve stakeholders with multiple interests, making fair benefit distribution and cross-regional coordination a major challenge. To this end, studies such as Xu et al. [9] proposed distributed operation models based on energy hubs to improve regional coordination. Additionally, Huang et al. [10] introduced a ladder-type carbon pricing model with flexible demand and hybrid energy storage, which provides valuable insights into emission cost coupling for MRIES operations.

In parallel, carbon emission trading (CET) and green certificate trading (GCT) have emerged as two essential policy tools under the “dual-carbon” strategy [11,12,13]. Several models have incorporated CET into IES scheduling [14], while others examined the impact of GCT on cost reduction and renewable integration [15,16]. For instance, Li et al. [17] considered both CET and GCT in hydrogen-integrated energy systems with demand response, demonstrating synergy between market incentives and decarbonization. Moreover, Cui et al. [18] and Zhang et al. [16] constructed joint scheduling models integrating carbon–green certificate equivalence, yet they often focused on single-region systems. Recent work by Li et al. [19] further explored flexible load management under CET–GCT integration, highlighting its potential for multi-agent coordination. Beyond Chinese studies, recent international contributions have explored related directions. X. Yang and L. Li [20] analyzed global coordination mechanisms for carbon and energy markets, while X. Li et al. [21] proposed incentive-compatible green certificate pricing with emission penalties. Y. Li et al. [22] introduced a demand response framework with emission constraints in multi-agent energy systems. However, most of these efforts remain limited to partial policy mechanisms and lack a unified game-based architecture for coordination under complex inter-regional scenarios.

To manage the hierarchical interactions between central coordinators and regional energy agents, game theory—particularly the Stackelberg model—has proven to be a powerful analytical tool. Previous work has introduced Stackelberg game approaches for multi-party energy systems. For example, Li et al. [23] formulated a Stackelberg model for multi-microgrid scheduling, integrating PV generation and demand response to align operator pricing and microgrid behavior. Yang et al. [24] developed a multi-level game strategy involving shared energy storage across regional IESs, demonstrating enhanced economic benefits and system flexibility. Additionally, Zhou et al. [25] and others have combined cooperative frameworks such as ADMM with bargaining mechanisms to address multi-agent scheduling under low-carbon constraints, offering complementary strategies for distributed coordination and privacy preservation. The European power balancing market has been undergoing structural reforms to enhance regional integration and flexibility [26]. Meanwhile, Germscheid et al. [27] analyzed local renewable energy generation combined with demand-side response in industrial processes, illustrating the adaptability of the multi-energy scheduling paradigm beyond the power system.

Despite recent advancements, three critical research gaps remain in the coordinated operation of MRIES:

• The lack of comprehensive models that simultaneously integrate CET, GCT, and IDR;
• The limited application of game-theoretic approaches that can reflect inter-entity strategic interactions under policy constraints;
• Insufficient consideration of inter-regional heterogeneity and cross-regional electricity pricing in existing centralized or single-agent frameworks.

Considering the above, the purpose of this study is to develop an operational optimization strategy for MRIES based on a bilevel Stackelberg game. The proposed approach addresses three key challenges:

• Spatial resource complementarity—leveraging the geographical diversity of renewable energy resources (e.g., solar-rich suburban areas and wind-intensive industrial zones) and flexible demand patterns across regions to improve energy balance and system efficiency;
• Collaboration among multiple stakeholders—including IESOs, regional electricity consumers, distributed renewable generators, and policy regulators—whose objectives and constraints must be harmonized through coordinated trading strategies;
• Carbon emission mitigation—achieving system-wide emission reductions through the joint optimization of CET quotas, green certificate transactions, and demand-side flexibility.

To achieve these goals, an interactive framework is constructed that combines IDR with a coordinated CET–GCT mechanism. The bilevel optimization model treats inter-entity electricity prices and transaction volumes as coupling variables and solves them iteratively until convergence. The effectiveness of the proposed strategy is evaluated across five key performance dimensions: transaction pricing efficiency, operational cost reduction, renewable energy integration rate, carbon mitigation performance, and energy mix optimization.

2. Game-Theoretic Interaction Framework for Multi-Regional Integrated Energy Systems

Given the significant heterogeneity in geographic resource endowments and equipment configurations across individual IESs, multiple IESs are interconnected to form an MRIES, as schematically illustrated in Figure 1.

The MRIES is administered by an IESO, which supervises three regional IESs. Subject to electricity and gas price bounds specified by upstream transmission and distribution network operators, the IESO supplies electricity and natural gas to member systems. Simultaneously, a green GCT–CET synergistic mechanism is employed to optimize equipment dispatch, reduce carbon emissions, enhance renewable energy integration, and improve economic performance.

Within this Stackelberg-game-based interaction framework, the IESO functions as the leader, aiming to maximize its revenue. It first announces day-ahead energy purchase and sale price schedules for each time period. Subsequently, each regional IES (follower) optimizes its local operation by achieving the following:

• Minimizing energy procurement and IDR costs for electricity, heating, and cooling loads;
• Maximizing customer satisfaction metrics.

The resulting energy schedules are communicated to the IESO, which updates its pricing strategies through an iterative bidding process. Upon convergence of bidding rounds, the IESO determines the following:

• Locational marginal prices (LMPs) for electricity transactions;
• Electricity procurement volumes from the upper-level grid;
• Natural gas procurement volumes.

Concurrently, each IES finalizes the following:

• Its energy mix composition;
• Dispatch schedules for local generation assets;
• Carbon allowance allocations and green certificate trading volumes.

3. Green Certificate–Carbon Emission Trading Synergistic Mechanism

3.1. Synergistic Interaction Between GCT and CET

To achieve China’s dual-carbon targets, GCT and CET have emerged as key market-based instruments supporting the national energy transition [28].

CET Mechanism: Entities subject to CET regulation receive free carbon allowances based on verified emission reports. Those exceeding their quotas must purchase additional emission rights from the carbon market, while surplus allowances may be sold.

GCT Mechanism: Regulatory authorities impose mandatory Renewable Portfolio Standards (RPS) on IESs. By reporting verified renewable energy integration, IESs obtain green certificates to fulfill compliance obligations and monetize surplus certificates through market-based trading mechanisms.

CCER-Mediated Synergy: The Chinese Certified Emission Reduction (CCER) scheme serves as a bridge between CET and GCT markets [29]. Green certificates can be converted into CCER credits based on renewable energy displacement effects, and CCERs maintain a 1:1 conversion ratio with carbon allowances (1 CCER = 1 tonne CO₂). This cross-market linkage incentivizes energy portfolio optimization and promotes carbon emission reduction, as illustrated in Figure 2.

The GCT–CET synergy mechanism described in this model is designed based on China’s current renewable energy and carbon market policies. Specifically, parameters such as the green certificate quota coefficient (0.001), the conversion factor between green certificates and CCERs (1:1), and the stepwise carbon price escalation mechanism are directly derived from Chinese regulatory practices [13,15,16]. These values reflect localized compliance rules under China’s dual-carbon framework.

However, the proposed model structure itself is not limited to the Chinese policy environment. The hierarchical bilevel Stackelberg framework, IDR structure, and carbon–certificate interaction logic can be generalized to other national or regional energy markets. In countries where renewable energy targets are enforced via feed-in tariffs or RPS, the GCT component can be replaced by market-based incentive schemes or carbon tax equivalents. Similarly, the carbon offset ratio and certificate-to-carbon equivalence can be adapted to reflect the relevant cap-and-trade or emission performance standard regulations in different regions.

3.2. GCT Trading Mechanisms

The green certificate trading model is:

$$\left\{\begin{array}{l}{N}_{\mathrm{IES}}^{\mathrm{GCT}}=\epsilon {\displaystyle \sum _{t=1}^T\left(N_t^{\mathrm{ELOAD}}+N_t^{\mathrm{ECCHP}}\right)}\hfill \\ N_{\mathrm{NEW}}^{\mathrm{GCT}}=\beta {\displaystyle \sum _{t=1}^T\left(N_t^{\mathrm{WT}}+N_t^{\mathrm{PV}}\right)}\hfill \\ W_{\mathrm{GCT}}={\epsilon }_{\mathrm{GCT}}\left(N_{\mathrm{NEW}}^{\mathrm{GCT}}-N_{\mathrm{IES}}^{\mathrm{GCT}}\right)\hfill \end{array}\right.$$

(1)

where $W_{\mathrm{GCT}}$ denotes the cost of green certificate trading for the system; $N_{\mathrm{IES}}^{\mathrm{GCT}}$ and $N_{\mathrm{NEW}}^{\mathrm{GCT}}$ represent the number of green certificates issued by the system to consume new energy and the quota of the management department, respectively; $N_t^{\mathrm{ELOAD}}$ and $N_t^{\mathrm{ECCHP}}$ denote the power consumption of the user and the overall system, respectively; $N_t^{\mathrm{WT}}$ and $N_t^{\mathrm{PV}}$ denote the amount of renewable energy consumed; and $\epsilon$ and $\beta$ represent the quota allocation coefficient and green certificate-to-carbon conversion factor, respectively.

When allocating carbon emission quotas to an IES, the equivalent carbon reduction attributed to green certificates is calculated through the CCER mechanism. This reduction can partially offset the system’s carbon emissions, thereby enabling synergistic integration between the GCT and CET mechanisms. The corresponding calculation is expressed as follows:

$$Q_{\mathrm{CCER}}={\displaystyle \sum _{t=1}^T\left({\lambda }_{\mathrm{OM}}{y}_{\mathrm{e}1}+{\lambda }_{\mathrm{BM}}{y}_{\mathrm{e}2}\right)}{E}_{\mathrm{IES}-\mathrm{CCER}}$$

(2)

where $Q_{\mathrm{CCER}}$ denotes the amount of carbon offset achieved through CCER credits; ${\lambda }_{\mathrm{OM}}$ and ${\lambda }_{\mathrm{BM}}$ represent the marginal emission factors for electricity generation and installed capacity, respectively; $y_{\mathrm{e}1}$ and $y_{\mathrm{e}2}$ are the corresponding weighting coefficients for the electricity-based and capacity-based emission factors; and $E_{\mathrm{IES}-\mathrm{CCER}}$ denotes the residual green electricity corresponding to the portion of renewable energy exceeding the mandatory quota in the IES.

3.3. Carbon Trading Mechanism with GCT–CET Synergy Consideration

In an IES, the primary carbon emission sources include the following:

• Electricity purchased from the main grid (primarily generated from coal-fired units),
• Micro gas turbines (MT),
• Gas boilers (GB).

Based on the baseline method, carbon emission allowances are freely allocated to these emission sources. Additionally, the carbon offset potential achieved through CCER credits is considered.

Accordingly, the total carbon emission quota for the IES can be formulated as follows:

$$\left\{\begin{array}{l}{Q}_{\mathrm{E},\mathrm{buy}}={\chi }_{\mathrm{E}}{\displaystyle \sum _{t=1}^T{E}_t^{\mathrm{buy}}}\hfill \\ Q_{\mathrm{MT}}={\chi }_{\mathrm{H}}{\displaystyle \sum _{t=1}^T\left({\phi }_{\mathrm{E}-\mathrm{H}}{E}_t^{\mathrm{MT}}+H_t^{\mathrm{MT}}\right)}\hfill \\ Q_{\mathrm{GB}}={\chi }_{\mathrm{H}}{\displaystyle \sum _{t=1}^T{H}_t^{\mathrm{GB}}}\hfill \\ Q_{{\mathrm{CO}}_2,\mathrm{IES}}=Q_{\mathrm{E},\mathrm{buy}}+Q_{\mathrm{MT}}+Q_{\mathrm{GB}}+Q_{\mathrm{CCER}}\hfill \end{array}\right.$$

(3)

where $Q_{{\mathrm{CO}}_2,\mathrm{IES}}$ denotes the total carbon emission allowance allocated to the system; $Q_{\mathrm{E},\mathrm{buy}}$, $Q_{\mathrm{MT}}$, and $Q_{\mathrm{GB}}$ represent the carbon quotas corresponding to electricity purchased from the main grid, MT, and GB, respectively; $E_t^{\mathrm{MT}}$ and $H_t^{\mathrm{MT}}$ denote the electric and thermal output power of MT units, respectively; ${\chi }_{\mathrm{E}}$ and ${\chi }_{\mathrm{H}}$ are the carbon allocation coefficients per unit of electricity and heat, set as 0.728 t/(MWh) and 0.102 t/(GJ), respectively; and ${\phi }_{\mathrm{E}-\mathrm{H}}$ denotes the electricity-to-heat conversion coefficient, set as 5.

The actual carbon emissions are calculated as follows:

$$\left\{\begin{array}{l}{m}_{\mathrm{E},\mathrm{buy}}={\mu }_{\mathrm{E},\mathrm{buy}}{\displaystyle \sum _{t=1}^T{E}_t^{\mathrm{buy}}}\hfill \\ m_{\mathrm{MT}}={\mu }_{\mathrm{MT}}{\displaystyle \sum _{t=1}^T\left({\phi }_{\mathrm{E}-\mathrm{H}}{E}_t^{\mathrm{MT}}+H_t^{\mathrm{MT}}\right)}\hfill \\ m_{\mathrm{GB}}={\mu }_{\mathrm{GB}}{\displaystyle \sum _{t=1}^T{H}_t^{\mathrm{GB}}}\hfill \\ m_{{\mathrm{CO}}_2,\mathrm{IES}}=m_{\mathrm{E},\mathrm{buy}}+m_{\mathrm{MT}}+m_{\mathrm{GB}}\hfill \end{array}\right.$$

(4)

where $m_{{\mathrm{CO}}_2,\mathrm{IES}}$ denotes the total actual carbon emissions of the system over one day; $m_{\mathrm{E},\mathrm{buy}}$, $m_{\mathrm{MT}}$, and $m_{\mathrm{GB}}$ represent the actual carbon emissions from electricity purchased from the grid, MT, and GB, respectively; and ${\mu }_{\mathrm{E},\mathrm{buy}}$, ${\mu }_{\mathrm{MT}}$, and ${\mu }_{\mathrm{GB}}$ are the corresponding actual carbon emission factors.

The amount of carbon emissions subject to CET is calculated as:

$$L_{{\mathrm{CO}}_2,\mathrm{IES}}=m_{{\mathrm{CO}}_2,\mathrm{IES}}-Q_{{\mathrm{CO}}_2,\mathrm{IES}}$$

(5)

The piecewise carbon trading cost model is provided in Appendix A.

3.4. Incentive Effects and Interactive Role of IDR and GC–CET Within the Game Framework

Within the proposed Stackelberg game, the IDR and GC–CET mechanisms are embedded as interactive economic incentives that directly influence participants’ decisions.

For MRIES units, IDR provides time-sensitive compensation for curtailment and load shifting, reducing operational costs and encouraging flexible energy use. GC–CET offers dual market incentives: surplus renewable energy can be converted into green certificates for revenue or carbon offset, while carbon-intensive regions can avoid penalties by acquiring carbon credits or trading certificates.

The IESO incorporates these responses into its pricing strategy to achieve better system coordination. Through iterative interactions, regions are economically motivated to cooperate: renewable-rich areas export energy and certificates, while deficit regions benefit from cleaner imports and reduced emission costs. This integrated mechanism strengthens cross-regional complementarity and ensures that local profit-seeking behaviors align with global optimization objectives.

4. Leader–Follower Stackelberg-Game-Based Bilevel Optimization Model

4.1. Objective Model of the IESO

To model the economic objectives of the IESO in the upper-level optimization, the objective function is designed to maximize the operator’s net profit derived from electricity trading activities. These activities involve purchasing electricity from the upper-level power grid and selling it to the lower-level MRIES, as well as buying surplus electricity from the MRIES when available.

Specifically, the objective function accounts for two types of bilateral transactions:

• Electricity purchased by the MRIES from the IESO at a price determined by the IESO;
• Electricity sold by the MRIES to the IESO, also at IESO-defined prices.

The profit is calculated as the revenue from selling electricity to the MRIES minus the cost of purchasing electricity from both the MRIES and the upper-level grid. The optimization variables in this function include the transaction prices $P_t^{\mathrm{NET}-\mathrm{buy}}$ and $P_t^{\mathrm{NET}-\mathrm{sell}}$ and the corresponding electricity trading quantities $E_t^{\mathrm{IES}-\mathrm{buy}}$ and $E_t^{\mathrm{IES}-\mathrm{sell}}$, which are jointly determined through upper-lower model interactions. The objective function is thus formulated as:

$$\max W_{\mathrm{IESO}}={\displaystyle \sum _{t=1}^T\left[E_t^{\mathrm{IES}-\mathrm{buy}}\cdot \left(P_t^{\mathrm{NET}-\mathrm{buy}}-P_t^{\mathrm{IES}-\mathrm{buy}}\right)+E_t^{\mathrm{IES}-\mathrm{sell}}\cdot \left(P_t^{\mathrm{NET}-\mathrm{sell}}-P_t^{\mathrm{IES}-\mathrm{sell}}\right)\right]}$$

(6)

where $W_{\mathrm{IESO}}$ denotes the profit of the IESO; $E_t^{\mathrm{IES}-\mathrm{buy}}$ and $E_t^{\mathrm{IES}-\mathrm{sell}}$ represent the electricity purchased from and sold to the IESO by the MRIES, respectively; $P_t^{\mathrm{NET}-\mathrm{buy}}$ and $P_t^{\mathrm{NET}-\mathrm{sell}}$ denote the electricity purchase and sale prices announced by the IESO; and $P_t^{\mathrm{IES}-\mathrm{buy}}$ and $P_t^{\mathrm{IES}-\mathrm{sell}}$ denote the electricity purchase and sale prices between the IESO and the upper-level distribution grid.

To ensure the feasibility of power exchanges, interaction constraints must be introduced between the IESO and the upper-level grid, as expressed in the following equation.

$$\left\{\begin{array}{l}{E}_{\min }^{\mathrm{IES}-\mathrm{buy}}\le E_t^{\mathrm{IES}-\mathrm{buy}}\le E_{\max }^{\mathrm{IES}-\mathrm{buy}}\hfill \\ E_{\min }^{\mathrm{IES}-\mathrm{sell}}\le E_t^{\mathrm{IES}-\mathrm{sell}}\le E_{\max }^{\mathrm{IES}-\mathrm{sell}}\hfill \end{array}\right.$$

(7)

where $E_{\min }^{\mathrm{IES}-\mathrm{buy}}$, $E_{\max }^{\mathrm{IES}-\mathrm{buy}}$, $E_{\min }^{\mathrm{IES}-\mathrm{sell}}$, and $E_{\max }^{\mathrm{IES}-\mathrm{sell}}$ denote the lower and upper bounds of electricity purchased from and sold to the distribution grid, respectively.

4.2. MRIES Benefit Model

The MRIES benefit model comprises several components:

• Operation and maintenance costs of local energy equipment within each IES region $W_{i,\mathrm{OP}}$;
• Revenues from participation in the IDR $W_{i,\mathrm{IDR}}$;
• Carbon trading costs $W_{i,\mathrm{CET}}$;
• Green certificate trading costs $W_{i,\mathrm{GCT}}$.

In addition, it accounts for inter-regional energy exchange costs ${\displaystyle \sum _j{W}_{i,j,\mathrm{IES}}}$, green certificate exchange costs ${\displaystyle \sum _j{W}_{i,j,\mathrm{GCT}}}$, and energy purchase/sale costs involving the IESO $W_{i,\mathrm{GNET}}$. The detailed expressions are as follows:

$$\min W_{\mathrm{MRIES}}={\displaystyle \sum _i\left(\begin{array}{l}{W}_{i,\mathrm{ENET}}+W_{i,\mathrm{GNET}}+W_{i,\mathrm{OP}}+W_{i,\mathrm{Ess}}+\\ W_{i,\mathrm{IDR}}+W_{i,\mathrm{GCT}}+W_{i,\mathrm{CET}}+\\ {\displaystyle \sum _j{W}_{i,j,\mathrm{GCT}}}+{\displaystyle \sum _j{W}_{i,j,\mathrm{IES}}}\end{array}\right)}$$

(8)

The carbon trading cost $W_{i,\mathrm{GCT}}$ and green certificate trading cost are defined in Equation (1) and Appendix A, respectively, while the remaining terms are formulated as follows:

$$\left\{\begin{array}{l}{W}_{i,\mathrm{OP}}={\displaystyle \sum _{t=1}^T{\displaystyle \sum _i{\kappa }_i{P}_{i,t}}}\hfill \\ W_{i,\mathrm{ESS}}={\displaystyle \sum _{t=1}^T{\displaystyle \sum _i{\displaystyle \sum _j{\kappa }_j\left(X_{t,j}^{\mathrm{chr}}+X_{t,j}^{\mathrm{dis}}\right)}}}\hfill \\ W_{i,\mathrm{IDR}}={\displaystyle \sum _i\left({\zeta }_{\mathrm{PY}}{\displaystyle \sum _{t=1}^T\mathsf{\Delta }{E}_{t,i}^{\mathrm{PY}}}+{\displaystyle \sum _{t=1}^T{P}_t^{\mathrm{RL}}\cdot E_{t,i}^{\mathrm{RL}}\cdot \mathsf{\Delta }t}\right)}\hfill \\ W_{i,j,\mathrm{IES}}={\displaystyle \sum _{t=1}^T\left(\begin{array}{l}{E}_{i,j,t}^{\mathrm{IES}-\mathrm{sell}-\mathrm{IES}}\cdot P_{i,j,t}^{\mathrm{IES}-\mathrm{sell}-\mathrm{IES}}-\\ E_{i,j,t}^{\mathrm{IES}-\mathrm{buy}-\mathrm{IES}}\cdot P_{i,j,t}^{\mathrm{IES}-\mathrm{buy}-\mathrm{IES}}\end{array}\right)}\hfill \\ W_{i,j,\mathrm{GCT}}={\displaystyle \sum _{t=1}^T\left({\epsilon }_{\mathrm{GCT}}\cdot N_{i,j,t}^{\mathrm{GCT}}\right)}\hfill \end{array}\right.$$

(9)

where ${\kappa }_i$ and ${\kappa }_j$ represent the operation and maintenance costs of energy conversion equipment and energy storage equipment, respectively, with i and j denoting their respective types; ${\zeta }_{\mathrm{PY}}$ denotes the subsidy cost for participating in IDR programs with shiftable loads; $P_t^{\mathrm{RL}}$ denotes the subsidy cost for participating in incentive-based IDR programs with curtailable loads; $X_{t,j}^{\mathrm{chr}}$ and $X_{t,j}^{\mathrm{dis}}$ denote the charging and discharging power of the energy storage system, where $X\in \{E,H,L\}$ refers to electricity, heat, and cooling, respectively; $E_{i,j,t}^{\mathrm{IES}-\mathrm{buy}-\mathrm{IES}}$ and $E_{i,j,t}^{\mathrm{IES}-\mathrm{sell}-\mathrm{IES}}$ denote the inter-regional exchanged power and corresponding transaction price between region mmm and region $P_{i,j,t}^{\mathrm{IES}-\mathrm{buy}-\mathrm{IES}}$ and $X_{t,j}^{\mathrm{dis}}$$P_{i,j,t}^{\mathrm{IES}-\mathrm{sell}-\mathrm{IES}}$; and $N_{i,j,t}^{\mathrm{GCT}}$ represents the volume of green certificate trading between regions.

4.3. Constraints

To ensure the feasibility and practicality of the proposed optimization model, this subsection outlines the key operational and market constraints governing the behavior of both the IESO and regional MRIES units.

1. Integrated Demand Response Load Constraints

$$\left\{\begin{array}{l}0\le E_{t,i}^{\mathrm{PY}}\le E_{\max }^{\mathrm{PY}}\hfill \\ {\displaystyle \sum _{t=1}^T{\displaystyle \sum _i{E}_{t,i}^{\mathrm{PY}}}=E_{t,i,0}^{\mathrm{PY}}}\hfill \\ 0\le E_{t,i}^{\mathrm{RL}}\le E_{\max }^{\mathrm{RL}}\hfill \end{array}\right.$$

(10)

where $E_{t,i}^{\mathrm{PY}}$ and $E_{t,i,0}^{\mathrm{PY}}$ represent the load profiles before and after participating in shiftable demand response programs; $E_{\max }^{\mathrm{PY}}$ denotes the maximum shiftable load at time $t$; and $E_{t,i}^{\mathrm{RL}}$ and $E_{\max }^{\mathrm{RL}}$ denote the curtailed load and its maximum allowable value under curtailable demand response programs.

2. Wind and Solar Output Constraints

$$\left\{\begin{array}{l}0\le E_{t,i,\mathrm{wind}}\le E_{t,i,\mathrm{wind}}^{\max }\hfill \\ 0\le E_{t,i,\mathrm{pv}}\le E_{t,i,\mathrm{pv}}^{\max }\hfill \end{array}\right.$$

(11)

where $E_{t,i,\mathrm{wind}}$ and $E_{t,i,\mathrm{wind}}^{\max }$ and $E_{t,i,\mathrm{pv}}$ and $E_{t,i,\mathrm{pv}}^{\max }$ are the power and its maximum value of wind power and PV at time t, respectively.

3. Electricity Pricing Constraints Set by the IESO

$$\left\{\begin{array}{l}{P}_{\min }^{\mathrm{IES}-\mathrm{buy}}\le P_t^{\mathrm{IES}-\mathrm{buy}}\le P_{\max }^{\mathrm{IES}-\mathrm{buy}}\hfill \\ P_{\min }^{\mathrm{IES}-\mathrm{sell}}\le P_t^{\mathrm{IES}-\mathrm{sell}}\le P_{\max }^{\mathrm{IES}-\mathrm{sell}}\hfill \\ P_{\min }^{\mathrm{NET}-\mathrm{buy}}\le P_t^{\mathrm{NET}-\mathrm{buy}}\le P_{\max }^{\mathrm{NET}-\mathrm{buy}}\hfill \\ \begin{array}{l}{P}_{\min }^{\mathrm{NET}-\mathrm{sell}}\le P_t^{\mathrm{NET}-\mathrm{sell}}\le P_{\max }^{\mathrm{NET}-\mathrm{sell}}\\ {\displaystyle \sum _{t=1}^T{P}_t^{\mathrm{IES}-\mathrm{sell}}}\le T\cdot \overline{P^{\mathrm{IES}-\mathrm{sell}}}\end{array}\hfill \end{array}\right.$$

(12)

where $P_{\min }^{\mathrm{IES}-\mathrm{buy}}$ and $P_{\max }^{\mathrm{IES}-\mathrm{buy}}$ and $P_{\min }^{\mathrm{IES}-\mathrm{sell}}$ and $P_{\max }^{\mathrm{IES}-\mathrm{sell}}$ denote the lower and upper bounds of electricity purchase and sale prices set by the IESO, respectively; $P_{\min }^{\mathrm{NET}-\mathrm{buy}}$ and $P_{\max }^{\mathrm{NET}-\mathrm{buy}}$ and $P_{\min }^{\mathrm{NET}-\mathrm{sell}}$ and $P_{\max }^{\mathrm{NET}-\mathrm{sell}}$ represent the lower and upper bounds of electricity prices announced by the distribution grid operator; and $\overline{P^{\mathrm{IES}-\mathrm{sell}}}$ indicates the upper limit of the average electricity selling price.

The operational constraints of energy equipment refer to [19].

4.4. Stackelberg Game Model

In the game framework, the IESO acts as the leader and takes the initiative to announce strategic variables such as electricity prices. The MRIES, as the follower, responds accordingly. A Stackelberg game model is thus established, formulated as follows:

$$G=\left\{\begin{array}{l}\left(\mathrm{IESO}\cup \mathrm{MRIES}\right);P_t^{\mathrm{IES}-\mathrm{buy}},P_t^{\mathrm{IES}-\mathrm{sell}};\hfill \\ E_t^{\mathrm{IES}-\mathrm{buy}},E_t^{\mathrm{IES}-\mathrm{sell}};W_{\mathrm{IESO}},W_{\mathrm{MRIES}}\hfill \end{array}\right\}$$

(13)

where $\left(\mathrm{IESO}\cup \mathrm{MRIES}\right)$ constitute the two players in the Stackelberg game; the strategy sets of the game include the electricity purchase and sale price vectors and corresponding demand quantities, denoted as $P_t^{\mathrm{IES}-\mathrm{buy}}$, $P_t^{\mathrm{IES}-\mathrm{sell}}$, $E_t^{\mathrm{IES}-\mathrm{buy}}$, and $E_t^{\mathrm{IES}-\mathrm{sell}}$; and $W_{\mathrm{IESO}}$ and $W_{\mathrm{MRIES}}$ represent the objective (profit) functions of the IESO and MRIES, respectively.

To further clarify the interaction logic and theoretical formulation of the proposed framework, the hierarchical structure between the IESO and the MRIES is abstracted into a formal bilevel optimization model. The decision-making mechanism and the underlying behavioral assumptions of both players are detailed in the following subsection.

4.5. Hierarchical Decision-Making Structure and Assumptions

In the proposed bilevel Stackelberg game framework, a hierarchical decision-making mechanism is established between the IESO as the leader and the MRIES as followers. The IESO determines electricity purchase and sale prices, anticipating the optimal reactions of the MRIES units. Each MRIES independently adjusts its energy consumption, local generation, demand response participation, and carbon/green certificate trading to minimize its total cost.

To support this bilevel formulation, the following modeling assumptions are adopted: Rational decision-making: Both the IESO and MRIES are assumed to be fully rational entities. The IESO aims to maximize system-wide economic benefit, while each MRIES minimizes its operational cost. Complete and symmetric information: The game is constructed under the assumption of complete information. All parties have full knowledge of system parameters, cost structures, trading rules, and environmental policies. The IESO is assumed to understand the decision models of MRIES and vice versa. Independent strategic behavior: Each MRIES operates independently without forming coalitions or colluding with others. Responses are based solely on the announced strategies of the IESO. Day-ahead static interaction: The decision process is carried out in a day-ahead scheduling horizon, following a quasi-static setup. The Stackelberg equilibrium is reached through iterative interaction simulated by the hybrid PSO–Gurobi algorithm, which captures both global strategy evolution and local response precision. This hierarchical framework enables coordinated optimization across multiple regions while preserving the autonomy of regional systems. It also reflects realistic market interactions under carbon-neutral policies and electricity market reform.

5. Solution Approach for the Stackelberg Game Model

The overall solution process is illustrated in Figure 3. To effectively solve the proposed bilevel Stackelberg model, which involves nonlinear coupling, discrete decision variables, and multi-agent coordination, a hybrid optimization algorithm combining PSO and MIP is employed.

Figure 3 illustrates the overall solution framework for the bilevel Stackelberg-game-based optimization model. The upper-level problem, controlled by the IESO, determines the electricity purchase and sale prices using PSO, while the lower-level problem, representing the operational responses of MRIES units, is solved using an MIP solver. During each iteration, the IESO updates its price strategy based on the feedback from the MRIES units. This feedback includes adjusted energy schedules, carbon trading decisions, and green certificate exchanges. The objective value—i.e., the IESO’s profit—is recalculated based on the latest pricing strategy and MRIES responses. The PSO algorithm searches for a better pricing scheme in the next iteration, guided by a fitness function that reflects the total profit improvement.

This hybrid approach integrates the advantages of both methods:

PSO performs a global search over the upper-level strategy space, i.e., the electricity purchase and sale prices determined by the IESO. It helps avoid premature convergence and identifies promising solution regions. Meanwhile, the lower-level subproblems, representing the MRIES’s operational responses (including equipment scheduling, energy transactions, and certificate decisions), are solved using Gurobi 9.5.x, a high-efficiency MIP solver, to ensure precise and feasible solutions under complex constraints.

To ensure convergence and computational efficiency, the following enhancements are applied: adaptive inertia weight and learning coefficients are used in PSO to balance exploration and exploitation across iterations; feasibility filtering is adopted to eliminate infeasible leader strategies early, reducing unnecessary MIP evaluations; and a convergence threshold based on the stabilization of the Stackelberg profit function is defined to terminate iterations.

Simulation results confirm that the proposed hybrid algorithm can achieve convergence within 30 iterations across all test scenarios, with total computation time under 5 min on standard computing platforms. Compared to standalone PSO or MIP methods, this approach significantly improves convergence stability and overall solution quality in the context of multi-regional energy optimization.

In the proposed PSO–MIP hybrid approach, the upper-level variables (electricity prices) are optimized using PSO, while the lower-level mixed-integer problems (regional operations) are solved using the Gurobi solver. The key PSO parameters are configured as follows: swarm size: 30; maximum iterations: 50; inertia weight: 0.9 → 0.4 (linearly decreasing); cognitive and social coefficients: c1 = c2 = 1.5. To improve convergence and reduce infeasible evaluations, the algorithm incorporates the following: feasibility filtering for candidate leader strategies; early termination if profit improvement < 0.1% over five iterations; and a penalty mechanism for strategy violations in energy prices or volume constraints. In the proposed PSO–MIP hybrid approach, the upper-level variables (electricity prices) are optimized using PSO, while the lower-level mixed-integer problems (regional operations) are solved using the Gurobi solver. The key PSO parameters are configured as follows:

• Swarm size: 30;
• Maximum iterations: 50;
• Inertia weight: 0.9 → 0.4 (linearly decreasing);
• Cognitive and social coefficients: c1 = c2 = 1.5.

To improve convergence and reduce infeasible evaluations, the algorithm incorporates the following:

• Feasibility filtering for candidate leader strategies;
• Early termination if profit improvement < 0.1% over five iterations;
• Penalty mechanism for strategy violations in energy prices or volume constraints.

A dynamic profit evolution is observed throughout the iterations: initially, the IESO profit fluctuates due to suboptimal pricing; however, as the algorithm progresses, feasible solutions converge toward a Stackelberg equilibrium, where the profit stabilizes. This change in optimal profit is used as a convergence criterion—when the improvement in profit becomes less than 0.1% over five consecutive iterations, the algorithm is terminated. In practice, convergence is typically achieved within 30 iterations, demonstrating the robustness and efficiency of the hybrid PSO–MIP approach.

6. Case Study

6.1. Basic Data

To validate the effectiveness of the proposed optimization strategy, a simulation case involving three interconnected integrated energy systems (IES1–IES3) is constructed, each representing a typical regional scenario in terms of energy supply, load demand, and carbon profile:

IES1: Urban residential area with comprehensive electricity, heating, cooling, and gas loads. It features high electricity demand and limited local generation, making it a power-deficient and carbon-intensive region.

IES2: Renewable-rich suburban area equipped with photovoltaic (PV), wind power, and a 3 MWh backup energy storage system. It has surplus renewable generation and acts as a power-exporting region with low carbon emissions.

IES3: Industrial area dominated by electricity and thermal loads, with limited renewable resources and high process energy intensity. It functions as a medium-load, medium-emission region with moderate energy flexibility.

The regional load profiles are illustrated in Figure 4, while the predicted renewable generation curves are shown in Figure 5. Energy prices for electricity and gas are presented in

Table 1. Energy price parameter.

Type of Energy	0:00–6:00	6:00–11:00	11:00–17:00	17:00–22:00	22:00–24:00
Purchased electricity price (USD/kWh)	0.064	0.173	0.104	0.173	0.064
Sold electricity price (USD/kWh)	0.039	0.104	0.063	0.104	0.039
Gas price (USD/m3)	0.5

All monetary values in this table are converted from Chinese Yuan (CNY) to U.S. Dollars (USD) based on an average exchange rate of 1 USD = 7.0 CNY.

, where time-of-use electricity pricing is applied. Green certificate and carbon trading parameters are summarized below (based on Chinese national standards):

• Green Certificate Parameters:
  • Conversion coefficient: 0.4;
  • Quota coefficient: 0.001;
  • Certificate price: 80 CNY/book (buy), 100 CNY/book (sell).
• Carbon Emission Parameters:
  • Marginal emission factors: 0.2135 t/MWh (electricity), 0.8042 t/MWh (installed capacity);
  • Carbon trading base price: 250 CNY/t;
  • Step interval: 0.1 t; price escalation rate: 25%.
• Demand Response Compensation:
  • A value of 0.3 CNY/kWh for curtailment-based DR;
  • A value of 0.1 CNY/kWh for load-shifting DR.

The load profiles and renewable generation data used in the case study are synthetically generated based on typical operating patterns and statistical characteristics of urban, suburban, and industrial regions in China. While not derived from real-time measurements, these data are calibrated using validated models and publicly available benchmarks (see [30]), ensuring that they reflect realistic seasonal variation, diurnal patterns, and capacity constraints suitable for MRIES simulation.

The input data arrays for electricity, heating, cooling, and gas loads, as well as renewable energy generation (wind and photovoltaic), are synthetically generated based on real-world statistical patterns and sectoral load archetypes. Specifically, the electricity and thermal load profiles for the three IES regions are derived from typical urban residential, suburban renewable-rich, and industrial zone consumption data published in the China Energy Statistical Yearbook and regional energy usage reports. The renewable generation profiles are generated using historical meteorological data (e.g., solar irradiance and wind speed) obtained from the China Meteorological Administration and processed through established power conversion models.

To ensure that the simulated data reflects realistic regional characteristics, the following calibration measures are taken:

• Hourly load distributions are matched to seasonal residential, commercial, and industrial usage patterns observed in cities like Beijing, Guangzhou, and Tianjin;
• Photovoltaic output is scaled based on solar exposure in suburban Hebei Province, while wind generation patterns follow turbine density and wind resource maps from northern and coastal China;
• The peak-to-valley ratios, average daily load factors, and generation–demand ratios are aligned with observed values reported in regional planning documents and academic benchmarks.

Although the data are not derived from real-time measurements in specific substations, they preserve the temporal dynamics, sectoral differences, and spatial diversity found in real-world MRIESs. This enables the case study to maintain both generalizability and engineering realism for testing the proposed optimization strategy.

6.2. Simulation Results Under Different Scenarios

6.2.1. Comparative Results of Three Scenarios

To assess the marginal and combined effects of each coordination mechanism, the following scenario configurations are adopted:

Scenario 1: Baseline—No inter-IES interaction; no CET, GCT, or IDR mechanisms are applied. Each IES operates independently with local optimization.

Scenario 2: Price-based interconnection—IESs are interconnected under time-sharing tariffs. No CET, GCT, or IDR mechanisms are applied.

Scenario 3: Full model—The proposed Stackelberg-game-based strategy is employed, and CET, GCT, and IDR mechanisms are all enabled to realize coordinated carbon-trading-aware dispatch and market-based complementarity.

The comparison of the operating costs of different scenarios is shown in

Table 2. Comparison of operating costs in different scenarios.

Scenarios	Running Costs (USD)
	IES1	IES2	IES3	IESO	SUM
1	6268.73	969.96	4632.11	—	11,870.8
2	5705.06	840.67	4519.04	—	11,064.8
3	5643.53	701.73	4406.3	216	10,751.6

All monetary values in this table are converted from Chinese Yuan (CNY) to U.S. Dollars (USD) based on an average exchange rate of 1 USD = 7.0 CNY.

.

A comparison between Scenario 1 and Scenario 2 indicates that the operating costs of the three IESs are reduced by 8.99%, 13.3%, and 2.44%, respectively, with the overall MRIES operating cost decreasing by 6.79%.

This demonstrates that system-level interconnection enables effective utilization of the spatial–temporal diversity and complementarity of regional energy supply and demand, thereby significantly reducing the total energy procurement cost through coordinated energy-sharing mechanisms.

A further comparison between Scenario 1 and Scenario 3 reveals that, guided by the IESO’s price signals, all participating entities engage in energy time-shifting transactions through dynamic price-based interactions and adopt a “buy-low-sell-high” strategy, resulting in operating cost reductions of 9.97%, 27.7%, and 4.87%, respectively.

These results confirm the practical effectiveness of the proposed game-theoretic interaction strategy in enhancing overall system economic performance.

The game-adjusted electricity pricing curve of the IESO is illustrated in Figure 6. The pricing strategy complies with the boundaries defined by the grid’s time-of-use tariff and feed-in tariff constraints.

The temporal variation in the tariff is closely aligned with the grid’s time-of-use pricing mechanism, thereby maintaining the IESO’s economic viability while ensuring compatibility with broader market-based operations during the iterative game process.

6.2.2. Inter-Regional Electricity Trading and Renewable Utilization

Under the guidance of the proposed game-theoretic interaction strategy, the MRIES operates in a coordinated manner, addressing the issue of spatial resource complementarity across regions through electricity trading among individual IESs and between the IESs and the upper-level IESO. IES2 and IES3 serve as electricity-surplus regions. During periods of power shortage in IES1, surplus electricity from IES2 and IES3 is transferred to IES1, thereby ensuring its stable operation. The inter-regional electricity exchange results among the integrated energy systems are illustrated in Figure 7.

When the IESs operate independently, IES2 and IES3 experience wind and solar curtailment during periods of high renewable output and low demand. In this case, the renewable energy utilization rates of IES1, IES2, and IES3 are 100%, 95.06%, and 77.47%, respectively. After applying the proposed MRIES coordination strategy, the overall renewable energy utilization rate increases to 100%, demonstrating the effectiveness of the proposed approach in enhancing renewable energy absorption and minimizing curtailment.

6.2.3. Carbon and Green Certificate Market Outcomes

Due to the deployment of gas-fired generation units, IES1 emitted 6741.9 kg of carbon, resulting in a shortfall of 2725.8 kg beyond its allocated carbon quota. To minimize carbon trading costs, 13.21 green certificates were utilized to offset 1385.65 kg of emissions via the CCER mechanism. The remaining deficit was covered by procuring additional green certificates from IES2. Benefiting from a high level of renewable energy utilization, IES2 initially held 21.75 green certificates. After the transaction, it retained 7.93 certificates and earned RMB 634.4 through market-based trading. IES3, relying on a renewable-based power supply system, maintained a low carbon footprint and traded 6.35 surplus certificates, generating RMB 508 in revenue. Through inter-regional certificate exchange, all three IESs effectively achieved carbon quota balancing while significantly reducing emission reduction costs.

The carbon offset contributions and green certificate transactions among the three IES regions are illustrated in Figure 8, which shows the effectiveness of inter-regional certificate allocation in reducing emission costs and balancing carbon quotas.

The results illustrate the importance of integrating multiple market mechanisms for optimal operation. In additional comparative tests, removing the GCT mechanism led to a 6.2% increase in carbon trading cost in IES1 and reduced green electricity monetization in IES2 by over 30%. Similarly, deactivating IDR caused peak load penalties and increased regional operating costs by 3.5–4.1%. These observations underscore the complementary roles of GCT, CET, and IDR in enhancing system-level flexibility and economic efficiency.

6.2.4. Sensitivity Analysis on Policy and Resource Uncertainty

To evaluate the robustness of the proposed strategy under external uncertainties, a sensitivity analysis is conducted on three key dimensions:

• Carbon Price Variations: The base carbon trading price is varied within ±30% (from 175 CNY/t to 325 CNY/t). The results show that higher carbon prices lead to increased utilization of green certificates and a shift in dispatch strategy toward renewable-dominant regions. Total operating cost increases by 3.1% under high carbon pricing, indicating moderate sensitivity.
• Electricity Price Fluctuations: The upper and lower bounds of electricity purchase/sale prices are expanded by ±20%. The model adapts through price signal rebalancing and demand response scheduling. Operating costs vary within ±2.5%, demonstrating acceptable robustness under market volatility.
• Renewable Generation Uncertainty: Wind and solar profiles are perturbed with a ±15% deviation to simulate forecast error. The results indicate that the energy-sharing mechanism across regions helps compensate for local shortages, and the system still maintains a 95–100% renewable utilization rate. This confirms the model’s effectiveness in mitigating generation uncertainty through inter-regional coordination.

The above sensitivity experiments reveal that the proposed mechanism exhibits strong adaptability under fluctuating policy, price, and resource conditions. Compared with baseline coordination approaches such as centralized optimization, the game-theoretic strategy maintains greater autonomy for regional entities while achieving comparable or superior system-wide cost and emission performance. Additionally, the iterative interaction structure allows price signals and market incentives to self-balance across heterogeneous regions—an advantage over static, one-shot centralized strategies. These findings support the applicability of the method under diverse operational contexts.

6.3. Applicability to Large-Scale Power Systems

The proposed optimization framework, although demonstrated on a simplified MRIES, is designed with scalability in mind and can be extended to more complex power systems containing numerous buses, diverse load types, and geographically distributed resources.

Firstly, the bilevel Stackelberg game model inherently features a hierarchical structure. The upper-level decision variables (e.g., electricity prices set by the IESO) and lower-level operational responses (e.g., scheduling decisions of each regional IES) are modeled independently, allowing for the integration of detailed subsystems without altering the core interaction logic. In practical applications, each MRIES unit can represent a regional area, an industrial park, or a distribution substation connected to multiple buses.

Secondly, the model structure can be coupled with conventional power flow models. By embedding nodal constraints, line capacity limits, and voltage bounds into the lower-level optimization, the methodology can accommodate AC or DC optimal power flow (OPF) formulations. This extension enables coordination with detailed transmission or distribution network models.

Thirdly, from a computational perspective, the hybrid PSO–MIP algorithm supports parallel processing. As each MRIES responds independently to the upper-level pricing strategy, their optimization tasks can be executed simultaneously, which improves computational efficiency and facilitates application to large-scale systems.

Finally, the bilevel structure aligns with the decentralized and layered characteristics of real-world power systems. With appropriate model adjustments and incorporation of real-time data, the proposed method can be adapted to support rolling dispatch or online decision-making under market-based scheduling mechanisms.

These features indicate that the proposed strategy holds potential for deployment in actual power system operations, especially in scenarios involving regional coordination, cross-entity energy trading, and policy-driven carbon management.

7. Conclusions

In this paper, a strategy for the optimized operation of a multi-regional integrated energy system based on a Stackelberg game framework is proposed. IESO and MRIES are cited to represent the two subjects, and the inter-subject trading tariffs and electricity quantities are used as coupling variables for the game to realize the system optimization operation. The feasibility of the strategy is verified through example analysis, and the conclusions are as follows:

(1)

Under the guidance of the game tariff signal, the subjects’ operation costs are reduced by 9.97%, 27.7%, and 4.87%, respectively, through the buy-low-sell-high strategy;

(2)

Under the guidance of the gaming strategy, the MRIES operates cooperatively, and multi-regional spatial resource complementarity is solved among IESs and between them and their higher-level IESOs, which improves the stability of system operation;

(3)

The spatial resource complementarity among multi-regional integrated energy systems is fully utilized, and the new energy consumption rate of each IES is increased from 100%, 95.06%, and 77.47% all the way to 100% full consumption;

(4)

Through coordinated carbon rights allocation and green certificate trading among IESs, the final carbon emissions remain within quota limits, avoiding excessive carbon costs.

Although PSO is a relatively mature algorithm, its adoption in this study is aligned with the specific structure of the proposed bilevel optimization framework. PSO offers a favorable balance between global search capability and algorithmic simplicity, which facilitates its integration with exact solvers such as Gurobi in solving the lower-level mixed-integer subproblems. This hybrid structure enables effective handling of the nonlinear, multi-agent characteristics inherent in the multi-regional energy coordination problem. The authors recognize that more recent optimization techniques—such as differential evolution, ant colony optimization, and reinforcement learning—may offer potential advantages in terms of convergence speed, adaptability, or solution accuracy under different system conditions. Future research will aim to evaluate and incorporate such methods within the proposed framework, particularly for large-scale, real-time, or highly uncertain integrated energy systems.

Future research may explore several extensions: incorporating power flow constraints to reflect grid security and capacity bottlenecks; applying the method to larger-scale systems with more regions and multiple energy carriers (e.g., hydrogen, cooling); integrating stochastic programming or reinforcement learning to handle deep uncertainties in prices, loads, and renewable outputs; and testing the model under real-world regulatory structures in non-Chinese markets to verify its global applicability.

Overall, this work offers a transferable decision-making tool for integrated energy system coordination under evolving policy environments.