Optimal Research on the Optimal Operation of Integrated Energy Systems Based on Cooperative Game Theory

Abstract

This paper proposes a method based on interval linear robust optimization to address the potential impacts of multiple uncertainties on the operational security of Regional Integrated Energy Systems (RIESs). The model considers the uncertainty in user loads and renewable energy outputs and determines the value ranges of related parameters through statistical analysis to characterize the boundaries of these uncertainties. To transform the stochastic disturbances into a solvable problem, the model introduces energy balance constraints under the worst-case scenario, ensuring that the system remains feasible under extreme conditions. The research framework integrates Nash bargaining theory, demand response mechanisms, and tiered carbon trading policies, constructing a cooperative game model for RIESs to minimize the overall operation cost of the alliance while providing a reasonable revenue distribution scheme. This approach aims to achieve fairness and sustainability in regional cooperation. Simulation results show that the method can effectively reduce the collaborative operation cost and improve the fairness of revenue distribution. To address potential issues of information misreporting and dishonesty in real-world scenarios, the model introduces an adjustable fraud factor in the revenue distribution process to characterize the strategy deviations of participants. Even under potential fraud risks, the mechanism can maintain an optimal revenue structure and lead the participants toward a stable fraud equilibrium, thereby enhancing the robustness and reliability of the overall collaboration.

1. Introduction

The rapid development of the economy and the continuous growth in energy demand have made issues such as environmental pollution and global warming increasingly prominent. Low-carbon and clean development has become the dominant direction for restructuring modern energy systems. Against this background, regional integrated energy systems (RIESs), characterized by energy coupling and multi-energy complementarity, are widely regarded as an important pathway to promoting regional energy structure optimization and achieving green transition. However, single-park integrated energy systems (RIESs) are often operated independently, leading to limited utilization of renewable energy and insufficient exploitation of multi-energy synergy, which makes it difficult to meet the optimal scheduling requirements under high penetration of new energy. The establishment of a coordinated operation mechanism for multiple RIESs helps enhance local renewable energy consumption capacity and also offers significant energy-saving and emission-reduction benefits.

In [1], an electricity–heat interconnected system model is established based on an improved ADMM method, and a robust scheduling framework amenable to distributed solution is proposed, enabling a two-stage coordinated optimization between internal park operation and external systems. In [2], from the perspective of market operation, a market-oriented scheduling model for an integrated energy operator is developed, and a two-level coordinated trading scheme is proposed, providing a useful reference for the design of future energy market mechanisms. In [3], taking a community-level energy system as the research object, a Stackelberg leader–follower game model is constructed, in which users’ behavioral preferences and comfort levels are incorporated into the analysis to formulate energy-use strategies that better match user characteristics.

In [4], a leader–follower game is employed to coordinate the interests between the distribution network and different RIESs. In [5], a cooperative game approach is adopted to perform joint scheduling for a multi-RIES coalition, achieving a relatively balanced benefit allocation. In [6], a non-cooperative game model composed of CHP units, photovoltaics, and the power grid is proposed, demonstrating the feasibility of Nash equilibrium in multi-energy complementary scheduling. In [7], non-cooperative game theory is applied to the coordination of operator revenues and equipment reliability, enabling multiple entities in the source–grid–load–storage chain to reach acceptable operating outcomes. In [8], the capacity configuration of renewable resources such as wind and solar power is investigated from a game-theoretic perspective, balancing the investment returns of different stakeholders and enhancing their participation enthusiasm.

In the field of multi-microgrid research, Ref. [9] proposes a scheduling framework that combines integrated demand response with a leader–follower game, achieving bidirectional benefits between multiple microgrids and users. In [10], to address the problem of insufficient incentives for flexibility resources, a compensation mechanism is introduced, and a leader–follower game-based scheduling method is developed under a high wind-power-penetration scenario. Reference [11] designs energy pricing strategies through a bilevel game model, thereby reducing energy procurement costs and enhancing user participation. In [12], a multi-leader–multi-follower Stackelberg structure is proposed and solved in a decentralized manner, enabling integrated energy service providers and load aggregators to achieve coordinated behavior while maintaining their operational independence. Reference [13] constructs a non-cooperative game framework between generation operators and different prosumers and obtains the equilibrium solution via a distributed approach. In [14], models are formulated based on operator revenues and user utilities, and trading strategies are derived within a leader–follower game structure. In [15], the pricing behavior of integrated energy suppliers is analyzed using a Stackelberg game, which verifies the positive role of demand response in peak shaving, valley filling, and reducing operating costs.

In [16], a Wasserstein two-stage distributionally robust optimization method is applied to the planning of an electricity–heat–hydrogen–ammonia coupled microgrid. In [17], under a stepwise carbon trading mechanism, a Wasserstein two-stage distributionally robust optimal scheduling model is developed for a wind–PV–storage hybrid system. In [18], a distributionally robust optimization model based on the Wasserstein distance is established to achieve deep, coordinated peak-shaving between electrolytic aluminum loads and a coal-fired power–storage system, indicating that this class of methods has good applicability for uncertainty modeling in multi-energy systems; however, research extending them to three-layer energy trading scenarios remains relatively scarce.

Meanwhile, Ref. [19] characterizes conditional value-at-risk (CVaR) through the interaction cost between the RIES and the distribution network, thereby simplifying risk representation in integrated energy multi-microgrid scheduling. In [20], CVaR is introduced into the scheduling model of a virtual power plant with power-to-gas and carbon capture and storage (CCS) to describe the risks induced by carbon price uncertainty. Focusing on the energy optimization problem of prosumers in an integrated energy trading market, Ref. [21] employs CVaR to quantify the risk losses caused by source–load forecasting errors, providing a useful reference for the incorporation of risk metrics and risk management into integrated energy systems. In reference [22], motivated by China’s “dual-carbon” targets, a multi-agent collaborative optimization framework for a mining integrated energy system (MIES) is developed based on a bi-level game formulation. Specifically, a bi-level decision-making model is constructed between an aggregator service provider and energy suppliers/users. The aggregator is responsible for participating in electricity grid transactions and carbon market trading and for determining internal pricing signals. Given these prices, each agent optimizes its own supply–demand strategy as well as inter-zone energy exchanges within the mining area. Moreover, a tiered carbon trading mechanism is incorporated to more accurately capture carbon cost characteristics. To solve the proposed model, an adaptive differential evolution algorithm hybridized with CPLEX is employed.

Reference [23] addresses the curse of dimensionality and low training efficiency commonly encountered in multi-agent reinforcement learning (MARL) for coordinated scheduling of multiple microgrids. To this end, it proposes a joint scheduling strategy based on a Nash-competitive deep Q-network (DQN). An integrated electricity–heat–carbon trading dispatch model is established, where carbon capture and a tiered carbon trading mechanism are incorporated to enable coordinated optimization across electricity, heat, and carbon domains. Simulation results demonstrate that the proposed approach improves economic performance while reducing carbon emissions, achieving energy complementarity among microgrids, and balancing the distribution of operational benefits. Reference [24] focuses on the challenge of benefit allocation among multiple stakeholders in an integrated energy microgrid powered by hydrogen-blended natural gas. A low-carbon economic dispatch method based on a bi-level game framework is proposed. Specifically, models for electrolysis-based hydrogen production, hydrogen-blended gas turbines, and hydrogen-blended loads are developed. Carbon trading and demand response are further introduced, and a Stackelberg game is formulated between the microgrid operator (MGO) and load aggregators (LAs). The resulting bi-level optimization problem is solved using a black kite algorithm in combination with CPLEX. Case studies verify that the proposed method effectively reduces carbon emissions and enhances renewable energy utilization while ensuring a balanced benefit distribution among stakeholders. Reference [25] provides a unified taxonomy and bibliometric review of 162 metaheuristic algorithms, highlighting their generally low parameterization and strong global search capability, while noting issues such as premature convergence and redundancy in metaphor-driven variants.

Building on the above studies, this paper focuses on the coordinated operation of multi-regional integrated energy systems (RIESs) and introduces an interval linear robust optimization approach to mitigate the impact of load variations and renewable generation fluctuations on operational feasibility under multiple sources of uncertainty. The uncertain parameters are constructed by analyzing user demand patterns and the variability ranges of distributed generation, while energy balance constraints in the worst-case scenario are adopted to transform the stochastic operating environment into a tractable deterministic optimization model. On this basis, a game-theoretic model for cross-regional electric energy cooperation is developed by incorporating Nash bargaining, demand response strategies, and a tiered carbon trading mechanism, enabling the coalition to reduce its total operating cost while achieving a rational and acceptable benefit allocation scheme. To address potential dishonest behaviors among RIESs, such as cost misreporting, a dynamic fraud factor is designed, and its impact on the cooperative outcome is investigated so as to identify the conditions under which a stable benefit allocation can still be maintained in the presence of potential fraud. The proposed research provides theoretical support and methodological guidance for coordinated scheduling, benefit sharing, and stable game mechanisms in multi-regional integrated energy systems.

The proposed integrated framework of “robust optimization–demand response–tiered carbon trading–Nash bargaining” should not be interpreted as a simple superposition of independent modules. Robust optimization is introduced to guarantee dispatch feasibility under multiple sources of uncertainty, including renewable generation (wind/solar), load demand, and electricity prices. Demand response (DR) further provides controllable flexibility, which can hedge against adverse realizations in worst-case scenarios, thereby mitigating the excessive conservatism typically associated with robust scheduling. Meanwhile, the tiered carbon trading mechanism endogenizes emission costs and reshapes the marginal operating costs of energy conversion devices, enabling an adjustable trade-off between economic efficiency and low-carbon performance. Nash bargaining is employed to coordinate intra-coalition power mutual assistance and allocate cooperative surplus, improving overall system efficiency while ensuring individual rationality and fairness in benefit distribution. It is worth noting that potential trade-offs may arise among these mechanisms. For instance, increasing the robustness budget may reduce the feasible trading space and consequently shrink the cooperative surplus, whereas a tiered carbon price structure may alter the optimal incentive direction of DR. To address these interactions at the implementation level, key parameters—including the robustness budget (\Gamma), tiered carbon price settings, and DR compensation coefficients—are utilized to regulate the above trade-offs. Moreover, we recommend selecting appropriate parameter ranges via sensitivity analysis to balance risk protection, economic performance, decarbonization effectiveness, and fairness.

2. Configuration of the Regional Integrated Energy System

Figure 1 illustrates the structural configuration of a typical regionally integrated energy system (RIES). The system includes multiple types of energy supply units—such as the power grid, natural gas network, wind turbines, and photovoltaic (PV) generation—exhibiting the characteristic of multi-source coupling. To realize conversion and complementarity among different energy carriers, the system is equipped with various energy conversion devices, including gas-fired combined heat and power (CHP) units, heat pumps (HP), carbon capture and storage (CCS) facilities, and power-to-gas (P2G) units, enabling electricity, heat, and gas to flow flexibly within the system and thereby improving overall energy utilization efficiency. Energy storage units play a regulatory role by absorbing surplus power when energy is abundant and releasing energy when demand increases or when renewable output is insufficient. In this way, they mitigate load fluctuations, reduce the randomness of renewable generation, and enhance the local accommodation capability of clean energy. Through the coordinated dispatch of all these devices, the RIES can simultaneously satisfy electrical and thermal demands while achieving higher levels of energy utilization and operational efficiency.

3. Independent Optimization of RIES Under Multiple Uncertainties

3.1. Adjustable Interval-Based Robust Optimization Method for System Operation

In the dispatch optimization of multi-regional integrated energy systems, various uncertainty-handling approaches are commonly introduced to address the issues caused by renewable energy fluctuations and load variations. Typical methods include stochastic programming, fuzzy programming, interval programming, and robust optimization, all of which aim to enhance the stability and adaptability of scheduling schemes across multiple scenarios. In this paper, a robust optimization approach is adopted to deal with the random fluctuations of wind and PV generation, load demand, and electricity prices. Unlike traditional stochastic methods that rely on detailed probabilistic information, robust optimization does not require the exact probability distribution of uncertain parameters. Instead, it characterizes their possible values through discrete scenarios or interval ranges. The objective is to construct a solution that is immune to uncertainties within an acceptable level of optimality loss, such that all constraints are satisfied for any realization within the uncertainty set, thereby improving the robustness of the scheduling scheme.

This study adopts the interval-based adjustable robust optimization framework proposed by Bertsimas et al., in which a tuning factor is introduced to control the conservatism level of the model, and interval bounds are used to characterize the possible values of uncertain parameters, thereby enhancing the stability of the dispatch results. Under this mechanism, the originally complex interval uncertainty constraints can be reformulated as structurally clearer deterministic expressions or transformed into linear robust constraints that are more amenable to computation. On this basis, a corresponding scheduling model is established, together with tailored solution strategies, so as to effectively mitigate the impact of uncertainties and achieve the optimization objectives in a robust manner.

The optimal scheduling problem of regional integrated energy systems can typically be formulated as a mixed-integer linear programming (MILP) model, whose standard form is given in (1)–(4):

$$\min {\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{n}}{\mathrm{c}}_{\mathrm{j}}{\mathrm{x}}_{\mathrm{j}}$$

(1)

$$\mathrm{s}.\mathrm{t}.{\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{n}}{a}_{\mathrm{ij}}{\mathrm{x}}_{\mathrm{j}}\le {\mathrm{b}}_{\mathrm{j}},\forall \mathrm{i}=1,2,\cdots ,\mathrm{m}$$

(2)

$${\mathrm{x}}_{\mathrm{j}}\ge 0,\forall \mathrm{j}=1,2,\cdots ,\mathrm{n}$$

(3)

$${\mathrm{x}}_{\mathrm{j}}\in \left\{0,1\right\}Section\mathrm{j}=1,2,\cdots ,\mathrm{n}$$

(4)

In the proposed model, the parameters $a_{ij}$, ${\mathrm{c}}_{\mathrm{j}}$ and ${\mathrm{b}}_{\mathrm{j}}$ may be influenced by external factors and therefore exhibit a certain degree of uncertainty. When the above parameters fluctuate within given intervals, a robust optimization framework can be adopted to reformulate the scheduling problem as a mixed-integer linear programming (MILP) model, thereby mitigating the impact of parameter perturbations on the decision results. The uncertainty is mainly reflected in slight fluctuations of the variable coefficients in the constraints; specifically, $a_{\mathrm{ij}}$ represents the deviation in the coefficient associated with the $j$-th variable in the $i$-th constraint. These coefficients are usually treated as independent, symmetrically distributed, and bounded random variables, whose values lie in the interval $\left[{\overline{a}}_{ij}-{\widehat{a}}_{ij},{\overline{a}}_{ij}+{\widehat{a}}_{ij}\right]$, where ${\overline{a}}_{ij}$ denotes the nominal value of the coefficient and ${\widehat{a}}_{ij}$ represents its maximum deviation. If the prediction error with respect to the nominal value is denoted by ${\mathsf{\eta }}_{\mathrm{ij}}$, the coefficient $a_{ij}$ can then be further expressed as follows:

$$a_{ij}={\overline{a}}_{ij}\pm {\eta }_{ij}\cdot {\widehat{a}}_{ij}$$

(5)

The expression for ${\mathsf{\eta }}_{\mathrm{ij}}$ is given as follows:

$${\eta }_{ij}=\left({\overline{a}}_{ij}\mp {\widehat{a}}_{ij}\right)/{\overline{a}}_{ij}$$

(6)

Moreover, by introducing an integer control parameter ${\Gamma }_{\mathrm{i}}$, the conservatism of the robust optimization results can be flexibly adjusted. The value of this parameter is defined in the interval $\left[0,\left|{\mathrm{J}}_{\mathrm{i}}\right|\right]$, where ${\mathrm{J}}_{\mathrm{i}}$ denotes the index set of uncertain parameters. Considering that, in practical operation, it is impossible for all uncertain parameters to simultaneously reach both bounds of their fluctuation intervals, it is usually assumed in the $i$-th constraint that, for the uncertain coefficients $a_{ij}$ and their corresponding deviations ${\mathsf{\eta }}_{\mathrm{ij}}$, the total magnitude of these deviations does not exceed a prescribed threshold ${\Gamma }_{\mathrm{i}}$, so as to ensure the feasibility and rationality of the model. This constraint can be written as (7):

$${\displaystyle \sum }_{\mathrm{j}\in {\mathrm{J}}_{\mathrm{i}}}{\mathsf{\eta }}_{\mathrm{ij}}\le {\Gamma }_{\mathrm{i}}$$

(7)

To avoid an overly generic description of “interval linear robust optimization,” this study adopts the Bertsimas–Sim budgeted interval uncertainty set to characterize uncertain coefficients. Specifically, for the i-th constraint, let ${\mathrm{J}}_{\mathrm{i}}$ denote the index set of coefficients subject to perturbations. Each uncertain coefficient is represented by an interval in the form of “nominal value plus maximum deviation,” as given in (8).

$${\overline{a}}_{ij}=a_{ij}+{\widehat{a}}_{ij}\cdot {\gamma }_{ij},\left|{\gamma }_{ij}\right|\le 1, j\in J_i$$

(8)

In (8), ${\gamma }_{ij}$ denotes the proportional deviation coefficient, which quantifies the normalized perturbation intensity of the forecasting error relative to the nominal value. Furthermore, since uncertain parameters in practical operation are unlikely to simultaneously attain their respective interval bounds, a budget constraint is introduced in (7) to restrict the extent of “simultaneous worst-case” deviations. Accordingly, the resulting uncertainty set can be equivalently expressed as (9):

$$U_i\left(\Gamma \right)=\left\{{\gamma }_{ij}\left|\left|{\gamma }_{ij}\right|\le 1,{\displaystyle \sum _{j\in J_i}\left|{\gamma }_{ij}\right|}\le \Gamma \right.\right\}, \Gamma \in \left[0,\left|{\mathrm{J}}_{\mathrm{i}}\right|\right]$$

(9)

Case $\Gamma =0$ corresponds to the non-robust setting, i.e., the model degenerates into a deterministic formulation, whereas Case $\Gamma =\left|{\mathrm{J}}_{\mathrm{i}}\right|$ represents the most conservative configuration, in which all uncertain parameters simultaneously take their worst-case values over the full intervals.

The maximum deviation ${\overline{a}}_{ij}$ of the uncertainty interval (i.e., the deviation bounds for wind/PV output, load demand, and electricity price) is recommended to be determined based on statistical analysis of historical forecasting error samples. Taking wind power, photovoltaic generation, load demand, and electricity price forecasting as examples, the corresponding prediction error series is defined as in (10):

$$e_t^x={\displaystyle \frac{x_t^{act}-x_t^{pre}}{x_t^{pre}}},x\in \left\{WT,PV,L^e,L^h,{\pi }^e\right\}$$

(10)

Accordingly, the deviation bounds can be determined using quantile-based confidence limits. For example, given a confidence level of $1-\alpha$ (e.g., 90% or 95%), the corresponding bound can be obtained as in (11).

$$\Delta x_t=Quantil{e}_{1-\alpha }(\left|e_t^x\right|)\cdot x_t^{pre}$$

(11)

This approach enables a data-driven construction of a bounded uncertainty interval when the exact probability distribution is unavailable, while allowing the interval width to be explicitly aligned with the risk preference ($\alpha$).

To fully account for all potential uncertain constraint coefficients, the model introduces worst-case-based robust constraints, which transform the original optimization problem into a bilevel mixed-integer linear programming (MILP) problem. This reformulation not only guarantees the robustness of the solution but also provides a mathematically tractable way to handle the impact of uncertain parameters. The specific model formulation and constraint definitions are given in (12)–(17):

$$\min {\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{n}}{\mathrm{c}}_{\mathrm{j}}{\mathrm{x}}_{\mathrm{j}}$$

(12)

$$s.t.{\displaystyle \sum }_{i=1}^n{a}_{ij}{x}_j+\max {\displaystyle \sum }_{j\in J_i}{\widehat{a}}_{ij}{\eta }_{ij}{x}_j\le b_j,\forall i=1,2,\dots ,m$$

(13)

$${\mathrm{x}}_{\mathrm{j}}\ge 0,\forall \mathrm{j}=1,2,\cdots ,\mathrm{n}$$

(14)

$${\mathrm{x}}_{\mathrm{j}}\in \left\{0,1\right\}Section\hspace{0.33em}\hspace{0.33em}\mathrm{j}=1,2,\cdots ,\mathrm{n}$$

(15)

$${\displaystyle \sum }_{\mathrm{j}\in {\mathrm{J}}_{\mathrm{i}}}{\mathsf{\eta }}_{\mathrm{ij}}\le {\Gamma }_{\mathrm{i}}$$

(16)

$$0\le {\mathsf{\eta }}_{\mathrm{ij}}\le 1$$

(17)

The specific form of the inner mixed-integer programming model is given in (18)–(20):

$$\max {\displaystyle \sum }_{j\in J_i}{\widehat{a}}_{ij}{\eta }_{ij}\left|x_j\right|$$

(18)

$$\mathrm{s}.\mathrm{t}.{\displaystyle \sum }_{\mathrm{j}\in {\mathrm{J}}_{\mathrm{i}}}{\mathsf{\eta }}_{\mathrm{ij}}\le {\Gamma }_{\mathrm{i}}$$

(19)

$$0\le {\mathsf{\eta }}_{\mathrm{ij}}\le 1$$

(20)

According to duality theory, the inner model given in (18)–(20) can be transformed into its dual formulation as shown in (21)–(24):

$$\min \hspace{0.17em}{\mathrm{z}}_{\mathrm{i}}{\Gamma }_{\mathrm{i}}+{\displaystyle \sum }_{\mathrm{j}\in {\mathrm{J}}_{\mathrm{i}}}{\mathrm{p}}_{\mathrm{ij}}$$

(21)

$$s.t.z_i+p_{ij}\ge {\widehat{a}}_{ij}{y}_j,\forall i,j\in J_i$$

(22)

$$\left|{\mathrm{x}}_{\mathrm{j}}\right|\le {\mathrm{y}}_{\mathrm{j}}$$

(23)

$${\mathrm{z}}_{\mathrm{i}},{\mathrm{p}}_{\mathrm{ij}},{\mathrm{y}}_{\mathrm{j}}\ge 0$$

(24)

By introducing the auxiliary variables ${\mathrm{z}}_{\mathrm{i}}$, ${\mathrm{p}}_{\mathrm{ij}}$ and ${\mathrm{y}}_{\mathrm{j}}$ of the dual problem, constraints (21)–(24) are incorporated into (12)–(17), thereby constructing a solvable mixed-integer linear programming model for the worst-case scenario. After the robust optimization reformulation, this model can effectively cope with the impact of uncertainties, and its specific form is given in (25)–(31).

$$\min {\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{n}}{\mathrm{c}}_{\mathrm{j}}{\mathrm{x}}_{\mathrm{j}}$$

(25)

$$s.t.{\displaystyle \sum }_{j=1}^n{a}_{ij}{x}_j+z_i{\Gamma }_i+{\displaystyle \sum }_{j\in J_i}{p}_{ij}\le b_i$$

(26)

$${\mathbf{x}}_{\mathrm{j}}\ge 0,\forall \mathrm{j}=\mathrm{1,2},\cdots ,\mathrm{n}$$

(27)

$${\mathrm{x}}_{\mathrm{j}}\in \left\{0,1\right\}Section\hspace{0.33em}\mathrm{j}=1,2,\cdots ,\mathrm{n}$$

(28)

$$z_i+p_{ij}\ge {\widehat{a}}_{ij}{y}_j,\forall i,j\in J_i$$

(29)

$$\left|{\mathrm{x}}_{\mathrm{j}}\right|\le {\mathrm{y}}_{\mathrm{j}}$$

(30)

$${\mathrm{z}}_{\mathrm{i}},{\mathrm{p}}_{\mathrm{ij}},{\mathrm{y}}_{\mathrm{j}}\ge 0$$

(31)

3.2. Optimization of Integrated Energy Systems Considering Carbon Trading and Demand Response

Building on the mathematical models of the various energy devices in the system, this paper formulates an optimization model for a single integrated energy system with the objective of minimizing the total operating cost. Demand response strategies are incorporated in conjunction with a tiered carbon emission trading mechanism, thereby providing a directly applicable basic framework for the optimization and scheduling of multi-energy systems.

3.2.1. Objective Function

$${\mathrm{minC}}_{\mathrm{n}}^0={\displaystyle \sum }_{\mathrm{t}=1}^{24}({\mathrm{C}}_{\mathrm{n},\mathrm{t}}^{\mathrm{OPE}}+{\mathrm{C}}_{\mathrm{n},\mathrm{t}}^{\mathrm{DR}}+{\mathrm{C}}_{\mathrm{n},\mathrm{t}}^{\mathrm{Net}}+{\mathrm{C}}_{\mathrm{n},\mathrm{t}}^{\mathrm{CO}2})$$

(32)

Here, n denotes the index of different RIESs, $\mathrm{n}\in \left\{1,2,3\cdots \right\}$; ${\mathrm{C}}_{\mathrm{n}}^0$ represents the total operating cost of a RIES under independent operation; ${\mathrm{C}}_{\mathrm{n},\mathrm{t}}^{\mathrm{OPE}}$, ${\mathrm{C}}_{\mathrm{n},\mathrm{t}}^{\mathrm{DR}}$, ${\mathrm{C}}_{\mathrm{n},\mathrm{t}}^{\mathrm{CO}2}$ and ${\mathrm{C}}_{\mathrm{n},\mathrm{t}}^{\mathrm{Net}}$ denote, respectively, the equipment operation and maintenance (O&M) cost, demand response cost, carbon emission cost, and energy purchasing/selling cost at time period t.

(1) Equipment operation and maintenance (O&M) cost

$$\begin{array}{c}{\mathrm{C}}_{\mathrm{n},\mathrm{t}}^{\mathrm{OPE}}={\mathsf{\xi }}^{\mathrm{GT}}\times {\mathrm{G}}_{\mathrm{n},\mathrm{t}}^{\mathrm{Gas},\mathrm{GT}}+{\mathsf{\xi }}^{\mathrm{GB}}\times {\mathrm{G}}_{\mathrm{n},\mathrm{t}}^{\mathrm{Gas},\mathrm{GB}}+{\mathsf{\xi }}^{\mathrm{HP}}\times {\mathrm{H}}_{\mathrm{n},\mathrm{t}}^{\mathrm{h},\mathrm{HP}}+{\mathsf{\xi }}^{\mathrm{WT}}\times {\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{WT}}+{\mathsf{\xi }}^{\mathrm{PV}}\times {\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{PV}}\\ +{\mathsf{\xi }}^{\mathrm{ES}}\times \left({\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{ES},\mathrm{tro}}+{\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{ES},\mathrm{rel}}\right)+{\mathsf{\xi }}^{\mathrm{HS}}\times \left({\mathrm{H}}_{\mathrm{n},\mathrm{t}}^{\mathrm{HS},\mathrm{tro}}+{\mathrm{H}}_{\mathrm{n},\mathrm{t}}^{\mathrm{HS},\mathrm{rel}}\right)+{\mathsf{\xi }}^{\mathrm{CCS}}\times {\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{CCS}}+{\mathsf{\xi }}^{\mathrm{P}2\mathrm{G}}\times {\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{P}2\mathrm{G}}\end{array}$$

(33)

Here, ${\mathsf{\xi }}^{\mathrm{GT}}$, ${\mathsf{\xi }}^{\mathrm{GB}}$, ${\mathsf{\xi }}^{\mathrm{HP}}$, ${\mathsf{\xi }}^{\mathrm{WT}}$, ${\mathsf{\xi }}^{\mathrm{PV}}$, ${\mathsf{\xi }}^{\mathrm{ES}}$, ${\mathsf{\xi }}^{\mathrm{HS}}$, ${\mathsf{\xi }}^{\mathrm{CCS}}$ and ${\mathsf{\xi }}^{\mathrm{P}2\mathrm{G}}$ denote the O&M cost coefficients of the corresponding devices. ${\mathrm{G}}_{\mathrm{n},\mathrm{t}}^{\mathrm{Gas},\mathrm{GT}}$ and ${\mathrm{G}}_{\mathrm{n},\mathrm{t}}^{\mathrm{Gas},\mathrm{GB}}$ represent the natural gas consumption of the gas turbine and the gas boiler at time period t, respectively. ${\mathrm{H}}_{\mathrm{n},\mathrm{t}}^{\mathrm{h},\mathrm{HP}}$ denotes the heat output of the heat pump at time period t. ${\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{WT}}$ and ${\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{PV}}$ are the wind power and photovoltaic power generation at time period t, respectively. ${\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{ES},\mathrm{tro}}$ and ${\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{ES},\mathrm{rel}}$ denote the charging and discharging power of the electrical energy storage system at time period t, respectively. ${\mathrm{H}}_{\mathrm{n},\mathrm{t}}^{\mathrm{HS},\mathrm{tro}}$ and ${\mathrm{H}}_{\mathrm{n},\mathrm{t}}^{\mathrm{HS},\mathrm{rel}}$ $\mathrm{H}{\mathrm{S}}_{\mathrm{t}}^{\mathrm{R}\mathrm{E}\mathrm{L}}$ represent the charging (heat storage) power and discharging (heat release) power of the thermal energy storage system at time period t, respectively. ${\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{CCS}}$ and ${\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{P}2\mathrm{G}}$ denote the electric power of the CCS and P2G units at time period t, respectively.

(2) Demand response cost

$${\mathrm{C}}_{\mathrm{n},\mathrm{t}}^{\mathrm{DR}}={\mathsf{\xi }}_{\mathrm{e},\mathrm{tran}}\times {\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{tran}}+{\mathsf{\xi }}_{\mathrm{e},\mathrm{cut}}\times {\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{cut}}+{\mathsf{\xi }}_{\mathrm{h},\mathrm{tran}}\times {\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{h},\mathrm{tran}}+{\mathsf{\xi }}_{\mathrm{h},\mathrm{cut}}\times {\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{h},\mathrm{cut}}$$

(34)

Here, ${\mathsf{\xi }}_{\mathrm{e},\mathrm{tran}}$, ${\mathsf{\xi }}_{\mathrm{e},\mathrm{cut}}$, ${\mathsf{\xi }}_{\mathrm{h},\mathrm{tran}}$ and ${\mathsf{\xi }}_{\mathrm{h},\mathrm{cut}}$ denote the compensation coefficients for load shifting, electric load curtailment, and heat load curtailment in the RIES, respectively; and ${\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{tran}},{\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{cut}},{\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{h},\mathrm{tran}}$ and ${\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{h},\mathrm{cut}}$ represent, at time period t, the corresponding shifted load, electric load curtailment, and heat load curtailment amounts, respectively.

(3) Carbon trading cost

This study adopts a tiered (stepwise) carbon trading mechanism. Compared with conventional carbon trading schemes with a single carbon price, the tiered structure provides more flexible regulation of carbon emissions through segmented pricing. Given a predefined total carbon emission allowance, the available quota is divided into multiple tiers, each associated with a specific emission interval and a corresponding trading price. When actual emissions exceed the free allowance, the excess portion is charged according to the tier in which it falls, and higher tiers are assigned higher unit carbon prices. This increasing-price design raises the marginal cost of emissions and incentivizes enterprises to proactively reduce carbon emissions, thereby lowering the overall carbon trading expenditure while complying with emission reduction constraints. The carbon trading cost is calculated as follows:

$$C_{n,t}^{CO2}=\left\{\begin{array}{l}{C}_{CO2}\times G_{n,t}^{CO2} \left(G_{n,t}^{CO2}\le d\right)\hfill \\ C_{CO2}\times \left(1+\mathsf{\kappa }\right)\times \left(G_{n,t}^{CO2}-d\right)+C_{CO2}\times d \left(d<G_{n,\mathrm{t}}^{CO2}\le 2d\right)\hfill \\ C_{CO2}\times \left(1+2\mathsf{\kappa }\right)\times \left(G_{n,t}^{CO2}-2d\right)+\left(2+\mathsf{\kappa }\right)\times C_{CO2}\times d\left(2d<G_{n,t}^{CO2}\le 3d\right)\hfill \\ C_{CO2}\times \left(1+3\mathsf{\kappa }\right)\times \left(G_{n,t}^{CO2}-3d\right)+\left(2+3\mathsf{\kappa }\right)\times C_{CO2}\times d\left(3d<G_{n,t}^{CO2}\le 4d\right)\hfill \\ C_{CO2}\times \left(1+4\mathsf{\kappa }\right)\times \left(G_{n,t}^{CO2}-4d\right)+\left(4+6\mathsf{\kappa }\right)\times C_{CO2}\times d\left(4d<G_{n,t}^{CO2}\le 5d\right)\hfill \end{array}\right.$$

(35)

Here, ${\mathrm{c}}_{\mathrm{CO}2}$ denotes the baseline carbon trading price, $\mathsf{\kappa }$ is the incremental step of the unit carbon price, ${\mathrm{G}}_{\mathrm{n},\mathrm{t}}^{\mathrm{CO}2}$ represents the payable carbon trading volume, and $\mathrm{d}$ is the interval length of each carbon-emission tier.

(4) Energy purchasing/selling cost

$${\mathrm{C}}_{\mathrm{n},\mathrm{t}}^{\mathrm{Net}}={\mathrm{c}}_{\mathrm{e},\mathrm{buy}}\times {\mathrm{G}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{buy}}+{\mathrm{c}}_{\mathrm{e},\mathrm{sell}}\times {\mathrm{G}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{sell}}+{\mathrm{c}}_{\mathrm{gas},\mathrm{buy}}\times {\mathrm{G}}_{\mathrm{n},\mathrm{t}}^{\mathrm{gas},\mathrm{buy}}$$

(36)

Here, ${\mathrm{c}}_{\mathrm{e},\mathrm{buy}}$ and ${\mathrm{c}}_{\mathrm{e},\mathrm{sell}}$ denote the electricity purchase and selling prices of the RIES, respectively; ${\mathrm{c}}_{\mathrm{gas},\mathrm{buy}}$ is the natural gas purchase price; and ${\mathrm{G}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{buy}}$, ${\mathrm{G}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{sell}}$ and ${\mathrm{G}}_{\mathrm{n},\mathrm{t}}^{\mathrm{gas},\mathrm{buy}}$ represent, respectively, the electricity purchased from the energy retailer, the electricity sold to the retailer, and the natural gas purchased at time period t.

3.2.2. Constraints

(1) Device constraints

Constraints of the gas turbine (GT):

$$0\le {\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{GT}}\le {\mathrm{P}}_{\mathrm{n}}^{\mathrm{e},\mathrm{GT},\max }$$

(37)

Here, ${\mathrm{P}}_{\mathrm{n}}^{\mathrm{e},\mathrm{GT},\max }$ denotes the maximum electric power output of the gas turbine.

Constraints of the gas boiler (GB):

$$0\le {\mathrm{H}}_{\mathrm{n},\mathrm{t}}^{\mathrm{h},\mathrm{GB}}\le {\mathrm{H}}_{\mathrm{n}}^{\mathrm{h},\mathrm{GB},\max }$$

(38)

Here, ${\mathrm{H}}_{\mathrm{n}}^{\mathrm{h},\mathrm{GB},\max }$ denotes the maximum thermal output power of the gas boiler.

Constraints of the heat pump (HP):

$${\mathrm{H}}_{\mathrm{n},\mathrm{t}}^{\mathrm{h},\mathrm{HP}}={\mathsf{\eta }}_{\mathrm{e}}^{\mathrm{HP}}\times {\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{HP}}$$

(39)

Here, ${\mathrm{H}}_{\mathrm{n},\mathrm{t}}^{\mathrm{h},\mathrm{HP},\max }$ denotes the maximum heat output of the heat pump (HP).

Constraints of the electrical energy storage (EES):

$$0\le {\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{ES},\mathrm{tro}}\le {\mathsf{\zeta }}^{\mathrm{ES},\mathrm{tro}}\times {\mathrm{P}}_{\mathrm{n}}^{\mathrm{ES},\mathrm{tro},\max }$$

(40)

$$0\le {\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{ES},\mathrm{rel}}\le {\mathsf{\zeta }}^{\mathrm{ES},\mathrm{rel}}\times {\mathrm{P}}_{\mathrm{n}}^{\mathrm{ES},\mathrm{rel},\max }$$

(41)

$${\mathsf{\varsigma }}^{\mathrm{ES},\mathrm{tro}}+{\mathsf{\varsigma }}^{\mathrm{ES},\mathrm{rel}}\le 1$$

(42)

$${\mathrm{S}}_{\mathrm{n}}^{\mathrm{ES},\min }\le {\mathrm{S}}_{\mathrm{n},\mathrm{t}}^{\mathrm{ES}}\le {\mathrm{S}}_{\mathrm{n}}^{\mathrm{ES},\max }$$

(43)

$${\mathrm{S}}_{\mathrm{n},0}^{\mathrm{ES}}={\mathrm{S}}_{\mathrm{n},\mathrm{T}}^{\mathrm{ES}}$$

(44)

Here, ${\mathsf{\varsigma }}^{\mathrm{ES},\mathrm{tro}}$ and ${\mathsf{\varsigma }}^{\mathrm{HS},\mathrm{rel}}$ are binary indicators denoting the charging and discharging states of the electrical energy storage system, respectively; and ${\mathrm{S}}_{\mathrm{n}}^{\mathrm{ES},\min }$ and ${\mathrm{S}}_{\mathrm{n}}^{\mathrm{ES},\max }$ represent the minimum and maximum state of charge (SOC) limits of the EES, respectively.

Constraints of the thermal energy storage (TES):

$$0\le {\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{HS},\mathrm{tro}}\le {\mathsf{\varsigma }}^{\mathrm{HS},\mathrm{tro}}\times {\mathrm{P}}_{\mathrm{n}}^{\mathrm{HS},\mathrm{tro},\max }$$

(45)

$$0\le {\mathrm{P}}_{\mathrm{n},t}^{\mathrm{HS},\mathrm{rel}}\le {\mathsf{\varsigma }}^{\mathrm{HS},\mathrm{rel}}\times {\mathrm{P}}_{\mathrm{n}}^{\mathrm{HS},\mathrm{rel},\max }$$

(46)

$${\mathsf{\varsigma }}^{\mathrm{HS},\mathrm{tro}}+{\mathsf{\varsigma }}^{\mathrm{HS},\mathrm{rel}}\le 1$$

(47)

$${\mathrm{S}}_{\mathrm{n}}^{\mathrm{HS},\min }\le {\mathrm{S}}_{\mathrm{n},\mathrm{t}}^{\mathrm{HS}}\le {\mathrm{S}}_{\mathrm{n}}^{\mathrm{HS},\max }$$

(48)

$${\mathrm{S}}_{\mathrm{n},0}^{\mathrm{HS}}={\mathrm{S}}_{\mathrm{n},\mathrm{T}}^{\mathrm{HS}}$$

(49)

Here, ${\mathsf{\varsigma }}^{\mathrm{HS},\mathrm{tro}}$ and ${\mathsf{\varsigma }}^{\mathrm{HS},\mathrm{rel}}$ are binary indicators denoting the charging (heat storage) and discharging (heat release) states of the thermal energy storage system, respectively; and ${\mathrm{S}}_{\mathrm{n}}^{\mathrm{HS},\min }$ and ${\mathrm{S}}_{\mathrm{n}}^{\mathrm{HS},\max }$ represent the minimum and maximum stored heat limits of the TES, respectively.

(2) Demand response constraints

$${\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e}}={\mathrm{L}}_{\mathrm{e}}^0+{\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{tran}}+{\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{cut}}$$

(50)

$${\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{h}}={\mathrm{L}}_{\mathrm{h}}^0+{\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{h},\mathrm{tran}}+{\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{h},\mathrm{cut}}$$

(51)

$${\displaystyle \sum }_{\mathrm{t}=1}^{24}{\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{tran}}=0$$

(52)

$${\mathrm{L}}_{\mathrm{n}}^{\mathrm{e},\mathrm{cut},\min }\le {\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{cut}}\le 0$$

(53)

$${\mathrm{L}}_{\mathrm{n}}^{\mathrm{e},\mathrm{tran},\min }\le {\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{tran}}\le {\mathrm{L}}_{\mathrm{n}}^{\mathrm{e},\mathrm{tran},\max }$$

(54)

$${\displaystyle \sum }_{\mathrm{t}=1}^{24}{\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{h},\mathrm{tran}}=0$$

(55)

$${\mathrm{L}}_{\mathrm{n}}^{\mathrm{h},\mathrm{cut},\min }\le {\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{h},\mathrm{cut}}\le 0$$

(56)

$${\mathrm{L}}_{\mathrm{n}}^{\mathrm{h},\mathrm{tran},\min }\le {\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{h},\mathrm{tran}}\le {\mathrm{L}}_{\mathrm{n}}^{\mathrm{h},\mathrm{tran},\max }$$

(57)

Here, ${\mathrm{L}}_{\mathrm{e}}^0$ and ${\mathrm{L}}_{\mathrm{h}}^0$ denote the initial electric load and thermal load, respectively. ${\mathrm{L}}_{\mathrm{n}}^{\mathrm{e},\mathrm{cut},\min },{\mathrm{L}}_{\mathrm{n}}^{\mathrm{e},\mathrm{tran},\min }$ and ${\mathrm{L}}_{\mathrm{n}}^{\mathrm{e},\mathrm{tran},\max }$ represent, at time period t, the minimum electric load curtailment and the minimum/maximum transferable electric load, respectively; and ${\mathrm{L}}_{\mathrm{n}}^{\mathrm{h},\mathrm{cut},},{\mathrm{L}}_{\mathrm{n}}^{\mathrm{h},\mathrm{tran},\min }$, and ${\mathrm{L}}_{\mathrm{n}}^{\mathrm{h},\mathrm{tran},\max }$ denote, at time period t, the minimum thermal load curtailment and the minimum/maximum transferable thermal load, respectively.

(3) Carbon trading constraints

$${\mathrm{G}}_{\mathrm{n},\mathrm{t}}^{\mathrm{CO}2}={\mathrm{G}}_{\mathrm{n},\mathrm{t}}^{\mathrm{A},\mathrm{CO}2}-{\mathrm{G}}_{\mathrm{n},\mathrm{t}}^{0,\mathrm{CO}2}-{\mathrm{G}}_{\mathrm{n},\mathrm{t}}^{\mathrm{CCS},\mathrm{CO}2}$$

(58)

$${\mathrm{G}}_{\mathrm{n},\mathrm{t}}^{\mathrm{A},\mathrm{CO}2}={\mathsf{\xi }}^{\mathrm{CO}2}\left({\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{GT}}+{\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{GB}}+{\mathrm{H}}_{\mathrm{n},\mathrm{t}}^{\mathrm{h},\mathrm{GT}}+{\mathrm{H}}_{\mathrm{n},\mathrm{t}}^{\mathrm{h},\mathrm{HP}}+{\mathrm{H}}_{\mathrm{n},\mathrm{t}}^{\mathrm{h},\mathrm{GB}}\right)$$

(59)

$${\mathrm{G}}_{\mathrm{n},\mathrm{t}}^{0,\mathrm{CO}2}={\mathsf{\xi }}^0\left({\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{GT}}+{\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{GB}}+{\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{WT}}+{\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{PV}}+{\mathrm{H}}_{\mathrm{n},\mathrm{t}}^{\mathrm{h},\mathrm{GT}}+{\mathrm{H}}_{\mathrm{n},\mathrm{t}}^{\mathrm{h},\mathrm{HP}}+{\mathrm{H}}_{\mathrm{n},\mathrm{t}}^{\mathrm{h},\mathrm{GB}}\right)$$

(60)

Here, ${\mathrm{G}}_{\mathrm{n},\mathrm{t}}^{\mathrm{A},\mathrm{CO}2}$ and ${\mathrm{G}}_{\mathrm{n},\mathrm{t}}^{0,\mathrm{CO}2}$ denote the actual carbon emissions of the microgrid and its allocated carbon emission allowance at time period t, respectively; and ${\mathsf{\xi }}^{\mathrm{CO}2}$ and ${\mathsf{\xi }}^0$ represent the carbon emission factor and the allowance (quota) coefficient, respectively.

(4) Energy purchasing/selling constraints

$$0\le G_{n,t}^{gas,buy}\le {\varsigma }^{B\mathrm{uy},\mathrm{e}}\times G_n^{gas,buy,\max }$$

(61)

$$0\le G_{n,t}^{gas,sell}\le {\varsigma }^{sell,e}\times G_n^{gas,sell,\max }$$

(62)

$${\varsigma }^{\mathrm{buy},e}+{\varsigma }^{sell,e}\le 1$$

(63)

$$0\le {\mathrm{G}}_{\mathrm{n},\mathrm{t}}^{\mathrm{gas},\mathrm{sell}}\le {\mathrm{G}}_{\mathrm{n}}^{\mathrm{gas},\mathrm{sell},\max }$$

(64)

Here, ${\mathrm{G}}_{\mathrm{n}}^{\mathrm{e},\mathrm{buy},\max }$ and ${\mathrm{G}}_{\mathrm{n}}^{\mathrm{gas},\mathrm{buy},\max }$ denote the maximum amounts of electricity and natural gas that the RIES can purchase, respectively; and ${\varsigma }^{\mathrm{buy},\mathrm{e}}$ and ${\varsigma }^{\mathrm{sell},\mathrm{e}}$ are binary status indicators for electricity purchasing and electricity selling, respectively.

(5) Power balance constraints

Taking a renewable generation unit consisting only of wind turbines as an example, its electric power output can be expressed as Equation (65).

$${\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{GT}}+{\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{ES},\mathrm{stro}}+{\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{Buy}}-{\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{ES},\mathrm{rel}}={\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e}}-{\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{WT}}$$

(65)

Thermal power balance:

$${\mathrm{H}}_{\mathrm{n},\mathrm{t}}^{\mathrm{h},\mathrm{GT}}+{\mathrm{H}}_{\mathrm{n},\mathrm{t}}^{\mathrm{h},\mathrm{GB}}+{\mathrm{H}}_{\mathrm{n},\mathrm{t}}^{\mathrm{HS},\mathrm{stro}}+{\mathrm{H}}_{\mathrm{n},\mathrm{t}}^{\mathrm{h},\mathrm{HP}}-{\mathrm{H}}_{\mathrm{n},\mathrm{t}}^{\mathrm{HS},\mathrm{rel}}={\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{h}}$$

(66)

It should be emphasized that demand response (DR) and the tiered carbon trading mechanism are not independent add-on terms that are simply superimposed. Instead, they jointly reshape the system’s effective marginal cost structure and consequently influence the conservatism of the subsequent robust dispatch. Specifically, the DR compensation coefficient defined in (34) reflects the opportunity cost of load shifting/curtailment, thereby providing controllable flexibility on the demand side. When the system encounters supply scarcity or an elevated carbon cost during unfavorable periods, the optimization tends to utilize DR to substitute part of the purchased electricity or the marginal output of high-emission units, which reduces operating costs and improves feasibility margins. Meanwhile, the tiered carbon trading scheme in (35) increases the marginal carbon price in a stepwise manner for excess emissions, significantly raising the effective marginal costs of emission-intensive devices such as gas turbines and gas boilers. This cost signal steers the dispatch toward low-carbon pathways, including renewable generation, energy storage, and carbon-mitigation technologies (e.g., CCS/P2G). Furthermore, after incorporating worst-case robust constraints under multi-source uncertainties in Section 3.2.3, the DR decision variables continue to participate in the power balance and load-related constraints through (50)–(57). The available DR adjustment range serves as a “buffer” against forecasting deviations, reducing the required reserve capacity and external energy procurement under worst-case scenarios for a given robustness budget. As a result, the excessive conservatism induced by robust optimization can be alleviated while preserving sufficient trading and coordination potential among the participating entities.

3.2.3. Robust Optimization Analysis Under Multiple Uncertainties

Since demand response and carbon trading reshape the marginal cost structure and the feasible adjustment domain, the mitigation strategy under the worst-case robust scenario will be jointly reflected in both the source-side dispatch decisions and the demand-side load adjustments. Building on the interval-adjustable robust optimization framework introduced in Section 3.1, this study develops a robust formulation for wind and photovoltaic generation outputs as well as load demand uncertainties, thereby establishing a RIES independent-operation model with robust optimization at its core.

(1) Uncertainty modeling of renewable generation and load demand

$${\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{WT}}\in \left[{\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{WT}}-{\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{WT}\hspace{0.17em},\mathrm{ld}},\hspace{0.17em}{\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{WT}}+{\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{WT}\hspace{0.17em},\mathrm{ud}}\right]$$

(67)

$${\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{PV}}\in \left[{\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{PV}}-{\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{PV},\mathrm{ld}}, {\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{PV}}+{\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{PV},\mathrm{ud}}\right]$$

(68)

$${\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e}}\in \left[{\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e}}-{\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{ld}},\hspace{0.33em}{\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{WT}}+{\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{ud}}\right]$$

(69)

$${\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{h}}\in \left[{\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{h}}-{\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{h},\mathrm{ld}}, {\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{h}}+{\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{h},\mathrm{ud}}\right]$$

(70)

Here, ${\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{WT},\mathrm{ld}}$ and ${\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{WT},\mathrm{ud}}$ denote the maximum allowable deviation bounds of the wind turbine output at time period t relative to its actual value; ${\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{PV},\mathrm{ld}}$ and ${\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{PV},\mathrm{ud}}$ denote the maximum allowable deviation bounds of the photovoltaic output at time period $\mathrm{t}$; ${\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{ld}}$ and ${\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{ud}}$ denote the maximum allowable deviation bounds of the electric load at time period t relative to its actual value; and ${\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{h},\mathrm{ld}}$ and ${\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{h},\mathrm{ud}}$ denote the maximum allowable deviation bounds of the thermal load at time period t relative to its actual value.

It should be noted that the maximum deviation bounds in (67)–(70) are determined using the historical forecast error statistics described in Section 3.1. Together with the robustness budget parameter $\Gamma$, they jointly specify the coverage intensity of the “worst-case scenario.” As the deviation upper bounds or $\Gamma$ increases, the model provides stronger risk protection, but it may also lead to higher reserve requirements and a more conservative dispatch outcome.

The core idea of robust optimization is to design decisions against the worst-case operating conditions, i.e., ensuring that the system can still maintain electricity–heat supply–demand balance and operate safely and stably when renewable generation and load forecast errors reach their extreme values. Taking a renewable generation unit in a RIES that is equipped only with wind turbines as an example, under the worst-case scenario the power balance constraints no longer follow the conventional forms in Equations (65) and (66) but should be reformulated as robust constraints, as given in Equations (71)–(76).

$${\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{GT}}+{\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{ES},\mathrm{stro}}+{\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{Buy}}-{\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{ES},\mathrm{rel}}={\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e}}-{\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{WT}}+\max \left\{\begin{array}{l}{\mathsf{\chi }}_{\mathrm{e},\mathrm{n},\mathrm{t}}^{\mathrm{ld}}\times {\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{ld}}+{\mathsf{\chi }}_{\mathrm{e},\mathrm{n},\mathrm{t}}^{\mathrm{ud}}\hfill \\ \times {\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{ud}}-{\mathsf{\chi }}_{\mathrm{WT},\mathrm{n},\mathrm{t}}^{\mathrm{ld}}\times {\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{WT},\mathrm{ld}}\hfill \\ -{\mathsf{\chi }}_{\mathrm{WT},\mathrm{n},\mathrm{t}}^{\mathrm{ud}}\times {\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{WT},\mathrm{ud}}\hfill \end{array}\right\}$$

(71)

$${\mathsf{\chi }}_{\mathrm{e},\mathrm{n},\mathrm{t}}^{\mathrm{ld}}+{\mathsf{\chi }}_{\mathrm{e},\mathrm{n},\mathrm{t}}^{\mathrm{ud}}+{\mathsf{\chi }}_{\mathrm{WT},\mathrm{n},\mathrm{t}}^{\mathrm{ld}}+{\mathsf{\chi }}_{\mathrm{WT},\mathrm{n},\mathrm{t}}^{\mathrm{ud}}\le {\Gamma }_{\mathrm{n},\mathrm{t}}^{\mathrm{e}}$$

(72)

$$0\le {\mathsf{\chi }}_{\mathrm{L},\mathrm{n},\mathrm{t}}^{\mathrm{ld}},{\mathsf{\chi }}_{\mathrm{L},\mathrm{n},\mathrm{t}}^{\mathrm{ud}},{\mathsf{\chi }}_{\mathrm{WTtn},\mathrm{t}}^{\mathrm{ld}},{\mathsf{\chi }}_{\mathrm{WT},\mathrm{n},\mathrm{t}}^{\mathrm{ud}}\le 1$$

(73)

$$H_{n,t}^{h,GT}+H_{n,t}^{h,GB}+H_{n,t}^{HS,stro}+H_{n,t}^{h,HP}-H_{n,t}^{HS,rel}=L_{n,t}^h+\max \left\{\begin{array}{l}{\chi }_{h,n,t}^{ld}\times L_{n,t}^{h,ld}\hfill \\ +{\chi }_{h,n,t}^{ud}\times L_{n,t}^{h,ud}\hfill \end{array}\right\}$$

(74)

$${\mathsf{\chi }}_{\mathrm{h},\mathrm{n},\mathrm{t}}^{\mathrm{ld}}+{\mathsf{\chi }}_{\mathrm{h},\mathrm{n},\mathrm{t}}^{\mathrm{ud}}\le {\Gamma }_{\mathrm{n},\mathrm{t}}^{\mathrm{h}}$$

(75)

$$0\le {\mathsf{\chi }}_{\mathrm{h},\mathrm{n},\mathrm{t}}^{\mathrm{ld}},{\mathsf{\chi }}_{\mathrm{h},\mathrm{n},\mathrm{t}}^{\mathrm{ud}}\le 1$$

(76)

Here, ${\Gamma }_{\mathrm{n},\mathrm{t}}$ is the robustness parameter used to adjust the conservativeness of the robust optimization; its value is restricted to ${\Gamma }_{\mathrm{n},\mathrm{t}}\in \left[0,\left|{\mathrm{J}}_{\mathrm{i}}\right|\right]$, where ${\mathrm{J}}_{\mathrm{i}}$ denotes the set (cardinality) of coefficients affected by uncertainty. $\mathsf{\chi }$ is the proportional deviation coefficient, defined as the ratio of the forecast error to the expected value, and it is used to characterize the relative fluctuation level of the uncertain variables.

It is worth noting that, as the robustness budget parameter ${\Gamma }_{\mathrm{n},\mathrm{t}}$ increases, the model must satisfy feasibility constraints over a broader uncertainty set, which typically leads to higher reserve requirements and/or increased external energy procurement. Meanwhile, the tradable electricity volume among RIESs may be reduced, thereby narrowing the coalition’s trading space and diminishing the allocable cooperative surplus. Given that demand response provides additional load-side adjustment margins (see (34) and (50)–(57)) and that tiered carbon trading raises the effective marginal costs of high-emission units (see (35)), both mechanisms interact with ${\Gamma }_{\mathrm{n},\mathrm{t}}$ in shaping dispatch decisions under the worst-case scenario. Therefore, we recommend jointly selecting ${\Gamma }_{\mathrm{n},\mathrm{t}}$ and the related parameters via sensitivity analysis so as to achieve a balanced trade-off among risk protection, economic performance, and decarbonization objectives.

As indicated by Equations (71)–(76), to ensure the tractability (solvability) of the robust constraints, it is necessary to further handle the two embedded maximization subproblems involved, namely those in Equations (77) and (78).

$$\begin{array}{r}\hfill \left\{\begin{array}{l}\max \left({\mathsf{\chi }}_{\mathrm{e},\mathrm{n},\mathrm{t}}^{\mathrm{ld}}\times {\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{ld}}+{\mathsf{\chi }}_{\mathrm{e},\mathrm{n},\mathrm{t}}^{\mathrm{ud}}\times {\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{ud}}-{\mathsf{\chi }}_{\mathrm{WT},\mathrm{n},\mathrm{t}}^{\mathrm{ld}}\times {\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{WT},\mathrm{ld}}-{\mathsf{\chi }}_{\mathrm{WT},\mathrm{n},\mathrm{t}}^{\mathrm{ud}}\times {\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{WT},\mathrm{ud}}\right)\hfill \\ \mathrm{s}.\mathrm{t}.{\mathsf{\chi }}_{\mathrm{e},\mathrm{n},\mathrm{t}}^{\mathrm{ld}}+{\mathsf{\chi }}_{\mathrm{e},\mathrm{n},\mathrm{t}}^{\mathrm{ud}}+{\mathsf{\chi }}_{\mathrm{WT},\mathrm{n},\mathrm{t}}^{\mathrm{ld}}+{\mathsf{\chi }}_{\mathrm{WT},\mathrm{n},\mathrm{t}}^{\mathrm{ud}}\le {\Gamma }_{\mathrm{n},\mathrm{t}}^{\mathrm{e}}\hfill \\ 0\le {\mathsf{\chi }}_{\mathrm{L},\mathrm{n},\mathrm{t}}^{\mathrm{ld}},{\mathsf{\chi }}_{\mathrm{L},\mathrm{n},\mathrm{t}}^{\mathrm{ud}},{\mathsf{\chi }}_{\mathrm{WTtn},\mathrm{t}}^{\mathrm{ld}},{\mathsf{\chi }}_{\mathrm{WT},\mathrm{n},\mathrm{t}}^{\mathrm{ud}}\le 1\hfill \end{array}\right.\end{array}$$

(77)

$$\left\{\begin{array}{l}\max \left({\mathsf{\chi }}_{\mathrm{h},\mathrm{n},\mathrm{t}}^{\mathrm{ld}}\times {\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{h},\mathrm{ld}}+{\mathsf{\chi }}_{\mathrm{h},\mathrm{n},\mathrm{t}}^{\mathrm{ud}}\times {\mathrm{L}}_{\mathrm{h},\mathrm{t}}^{\mathrm{h},\mathrm{ud}}\right)\hfill \\ \mathrm{s}.\mathrm{t}.{\mathsf{\chi }}_{\mathrm{h},\mathrm{n},\mathrm{t}}^{\mathrm{ld}}+{\mathsf{\chi }}_{\mathrm{h},\mathrm{n},\mathrm{t}}^{\mathrm{ud}}\le {\Gamma }_{\mathrm{n},\mathrm{t}}^{\mathrm{h}}\hfill \\ 0\le {\mathsf{\chi }}_{\mathrm{h},\mathrm{n},\mathrm{t}}^{\mathrm{ld}},{\mathsf{\chi }}_{\mathrm{h},\mathrm{n},\mathrm{t}}^{\mathrm{ud}}\le 1\hfill \end{array}\right.$$

(78)

Based on the strong duality principle, by introducing the dual variables ${\mathsf{\lambda }}_{\mathrm{e}}^{\mathrm{n},\mathrm{t}}$, ${\mathsf{\pi }}_{\mathrm{e}1}^{\mathrm{n},\mathrm{t}+}$, ${\mathsf{\pi }}_{\mathrm{e}1}^{\mathrm{n},\mathrm{t}-}$, ${\mathsf{\pi }}_{\mathrm{WT}2}^{\mathrm{n},\mathrm{t}+}$, ${\mathsf{\pi }}_{\mathrm{WT}2}^{\mathrm{n},\mathrm{t}-}$, ${\mathsf{\lambda }}_{\mathrm{h}}^{\mathrm{n},\mathrm{t}}$, ${\mathsf{\pi }}_{\mathrm{h}1}^{\mathrm{n},\mathrm{t}+}$ and ${\mathsf{\pi }}_{\mathrm{h}1}^{\mathrm{n},\mathrm{t}-}$, Equations (77) and (78) can be transformed into their corresponding dual problems, as given in Equations (79) and (80).

$$\begin{array}{r}\hfill \left\{\begin{array}{l}\min \left({\mathsf{\lambda }}_{\mathrm{e}}^{\mathrm{n},\mathrm{t}}\times {\Gamma }_{\mathrm{n},\mathrm{t}}^{\mathrm{e}}+{\mathsf{\pi }}_{\mathrm{el}}^{\mathrm{n},\mathrm{t}+}+{\mathsf{\pi }}_{\mathrm{el}}^{\mathrm{n},\mathrm{t}-}\right.\hfill \\ \left.+{\mathsf{\pi }}_{\mathrm{WT}2}^{\mathrm{n},\mathrm{t}+}+{\mathsf{\pi }}_{\mathrm{WT}2}^{\mathrm{n},\mathrm{t}-}\right)\hfill \\ \mathrm{s}.\mathrm{t}.\hspace{1em}{\mathsf{\lambda }}_{\mathrm{LWT}}^{\mathrm{n},\mathrm{t}}+{\mathsf{\pi }}_{\mathrm{el}}^{\mathrm{n},\mathrm{t}+}\ge {\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{ld}}\hfill \\ \hspace{1em}{\mathsf{\lambda }}_{\mathrm{LWT}}^{\mathrm{n},\mathrm{t}}{\mathsf{\pi }}_{\mathrm{el}}^{\mathrm{n},\mathrm{t}-}\ge {\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{ud}}\hfill \\ \hspace{1em}{\mathsf{\lambda }}_{\mathrm{eWT}}^{\mathrm{n},\mathrm{t}}+{\mathsf{\pi }}_{\mathrm{WT}2}^{\mathrm{n},\mathrm{t}+}\ge -{\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{WT},\mathrm{ld}}\hfill \\ \hspace{1em}{\mathsf{\lambda }}_{\mathrm{eWT}}^{\mathrm{n},\mathrm{t}}+{\mathsf{\pi }}_{\mathrm{WT}2}^{\mathrm{n},\mathrm{t}-}\ge -{\mathrm{P}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{WT},\mathrm{ud}}\hfill \end{array}\right.\end{array}$$

(79)

$$\left\{\begin{array}{l}\min \left({\lambda }_h^{n,t}\times {\Gamma }_{n,t}^h+{\pi }_{h1}^{n,t+}+{\pi }_{h1}^{n,t-}\right)\hfill \\ s.t.\hspace{1em}{\lambda }_h^{n,t}+{\pi }_{h1}^{n,t+}\ge L_{n,t}^{h,ld}\hfill \\ \hspace{1em}{\lambda }_h^{n,t}+{\pi }_{h1}^{n,t-}\ge L_{n,t}^{h,ud}\hfill \end{array}\right.$$

(80)

Therefore, the power balance constraints in Equations (65) and (66) can be linearized using the robust optimization approach, leading to the following form:

$$\begin{array}{r}\hfill P_{n,t}^{e,GT}+P_{n,t}^{ES,stro}+P_{n,t}^{e,Buy}-P_{n,t}^{ES,rel}=L_{n,t}^e-P_{n,t}^{e,WT}+{\lambda }_{eWT}^{n,t}\times {\Gamma }_{n,t}^{eWT}\\ \hfill +{\pi }_{e1}^{n,t+}+{\pi }_{e1}^{n,t-}+{\pi }_{WT2}^{n,t+}+{\pi }_{WT2}^{n,t-}\end{array}$$

(81)

$${\mathrm{H}}_{\mathrm{n},\mathrm{t}}^{\mathrm{h},\mathrm{GT}}+{\mathrm{H}}_{\mathrm{n},\mathrm{t}}^{\mathrm{h},\mathrm{GB}}+{\mathrm{H}}_{\mathrm{n},\mathrm{t}}^{\mathrm{HS},\mathrm{stro}}+{\mathrm{H}}_{\mathrm{n},\mathrm{t}}^{\mathrm{h},\mathrm{HP}}-{\mathrm{H}}_{\mathrm{n},\mathrm{t}}^{\mathrm{HS},\mathrm{rel}}={\mathrm{L}}_{\mathrm{n},\mathrm{t}}^{\mathrm{h}}+{\mathsf{\lambda }}_{\mathrm{h}}^{\mathrm{n},\mathrm{t}}\times {\Gamma }_{\mathrm{n},\mathrm{t}}^{\mathrm{h}}+{\mathsf{\pi }}_{\mathrm{h}1}^{\mathrm{n},\mathrm{t}+}+{\mathsf{\pi }}_{\mathrm{h}1}^{\mathrm{n},\mathrm{t}-}$$

(82)

(2) Handling electricity price uncertainty

Following Method (1), the purchasing price of electricity from the external grid is bounded within a prescribed range:

$${\mathrm{c}}_{\mathrm{e},\mathrm{buy}}\in \left[{\mathrm{c}}_{\mathrm{e},\mathrm{buy}}-{\mathrm{c}}_{\mathrm{e},\mathrm{buy}}^{\mathrm{ld}},\hspace{0.33em}{\mathrm{c}}_{\mathrm{e},\mathrm{buy}}+{\mathrm{c}}_{\mathrm{e},\mathrm{buy}}^{\mathrm{ud}}\right]$$

(83)

Here, ${\mathrm{c}}_{\mathrm{e},\mathrm{buy}}^{\mathrm{ld}}$ and ${\mathrm{c}}_{\mathrm{e},\mathrm{buy}}^{\mathrm{ud}}$ denote the maximum deviation bounds of the electricity purchase price relative to its actual value.

Under the worst-case condition where the external grid purchase price reaches the upper bound of its allowable interval, the price constraint can be reformulated as follows by introducing the robustness parameter and the proportional deviation coefficient:

$$\left\{\begin{array}{l}\max \left({\mathsf{\chi }}_{\mathrm{buy},\mathrm{t}}^{\mathrm{ld}}\times {\mathrm{c}}_{\mathrm{e},\mathrm{buy}}^{\mathrm{ld}}+{\mathsf{\chi }}_{\mathrm{buy},\mathrm{t}}^{\mathrm{ud}}\times {\mathrm{L}}_{\mathrm{e},\mathrm{buy}}^{\mathrm{ud}}\right)\hfill \\ \mathrm{s}.\mathrm{t}.{\mathsf{\chi }}_{\mathrm{buy},\mathrm{t}}^{\mathrm{ld}}+{\mathsf{\chi }}_{\mathrm{buy},\mathrm{t}}^{\mathrm{ud}}\le {\Gamma }_{\mathrm{n},\mathrm{t}}^{\mathrm{buy}}\hfill \\ 0\le {\mathsf{\chi }}_{\mathrm{buy},\mathrm{t}}^{\mathrm{ld}},{\mathsf{\chi }}_{\mathrm{buy},\mathrm{t}}^{\mathrm{ud}}\le 1\hfill \end{array}\right.$$

(84)

Based on the strong duality principle, by introducing the dual variables ${\mathsf{\lambda }}_{\mathrm{buy}}^{\mathrm{t}}$, ${\mathsf{\pi }}_{\mathrm{buy}1}^{\mathrm{t}+}$ and ${\mathsf{\pi }}_{\mathrm{buy}1}^{\mathrm{t}-}$, Equation (84) can be transformed into the corresponding dual problem as follows:

$$\begin{array}{r}\hfill \left\{\begin{array}{l}\min \left({\mathsf{\lambda }}_{\mathrm{buy}}^{\mathrm{t}}\times {\Gamma }_{\mathrm{n},\mathrm{t}}^{\mathrm{buy}}+{\mathsf{\pi }}_{\mathrm{buy}1}^{\mathrm{t}+}+{\mathsf{\pi }}_{\mathrm{buy}1}^{\mathrm{t}-}\right)\hfill \\ \mathrm{s}.\mathrm{t}.{\mathsf{\lambda }}_{\mathrm{buy}}^{\mathrm{t}}+{\mathsf{\pi }}_{\mathrm{buy}1}^{\mathrm{t}+}\ge {\mathrm{c}}_{\mathrm{buy},\mathrm{t}}^{\mathrm{ld}}\hfill \\ {\mathsf{\lambda }}_{\mathrm{buy}}^{\mathrm{t}}+{\mathsf{\pi }}_{\mathrm{buy}1}^{\mathrm{t}-}\ge {\mathrm{c}}_{\mathrm{buy},\mathrm{t}}^{\mathrm{h},\mathrm{ud}}\hfill \end{array}\right.\end{array}$$

(85)

Therefore, when electricity price uncertainty is considered, the electricity purchasing/selling cost can be expressed as follows:

$$\begin{array}{r}\hfill {\mathrm{C}}_{\mathrm{n},\mathrm{t}}^{\mathrm{Net}}={\mathrm{c}}_{\mathrm{e},\mathrm{sell}}\times {\mathrm{G}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{sell}}+{\mathrm{c}}_{\mathrm{gas},\mathrm{buy}}\times {\mathrm{G}}_{\mathrm{n},\mathrm{t}}^{\mathrm{gas},\mathrm{buy}}\\ \hfill +\left({\mathrm{c}}_{\mathrm{e},\mathrm{buy}}+{\lambda }_{\mathrm{buy}}^{\mathrm{t}}\times {\Gamma }_{\mathrm{n},\mathrm{t}}^{\mathrm{buy}}+{\mathsf{\pi }}_{\mathrm{buy}1}^{\mathrm{t}+}+{\mathsf{\pi }}_{\mathrm{buy}1}^{\mathrm{t}-}\right)\times {\mathrm{G}}_{\mathrm{n},\mathrm{t}}^{\mathrm{e},\mathrm{buy}}\end{array}$$

(86)

It should be clarified that the treatment of electricity price uncertainty in (83)–(86) does not necessarily require adopting the “upper-bound worst case over all time periods.” Rather, it represents a commonly used risk-protection modeling paradigm in robust optimization. To ensure a proper balance between robustness and economic efficiency, the electricity price uncertainty is also incorporated into the budgeted interval robust framework, and an electricity-price robustness budget parameter ${\Gamma }_p$ is introduced to tune the degree of conservatism. Specifically, the purchasing price at time period (t) is assumed to satisfy the interval uncertainty model given in (87):

$${\pi }_t={\overline{\pi }}_t+{\gamma }_t^{\pi }\Delta {\pi }_t,\left|{\gamma }_t^{\pi }\right|\le 1$$

(87)

and is further subject to the following budget constraint in (88):

$$\sum \left|{\gamma }_t^{\pi }\right|\le {\Gamma }_p,{\Gamma }_p\in \left[0,\left|J_P\right|\right]$$

(88)

This indicates that ${\Gamma }_p$ controls the extent to which electricity prices in multiple time periods simultaneously approach their worst-case deviations. When ${\Gamma }_p=0$, the model degenerates into an economically optimal schedule based on the nominal electricity price ${\overline{\pi }}_t$; when ${\Gamma }_p=\left|J_P\right|$, it corresponds to the upper-bound robust schedule under the worst-case scenario across all time periods. Therefore, Equations (83)–(86) do not contradict the modeling objective of balancing robustness and economic efficiency, since the level of conservativeness can be tuned by adjusting ${\Gamma }_p$.

To avoid overly conservative solutions induced by excessive price robustness, this paper recommends selecting ${\Gamma }_p$ based on sensitivity analysis. Specifically, given an upper bound of electricity price deviations (which can be obtained from the quantile statistics of historical price forecasting errors), the operating cost and worst-case feasibility under different values of ${\Gamma }_p$ are compared. The range of ${\Gamma }_p$ that satisfies the prescribed risk tolerance while yielding a relatively low operating cost is then selected, thereby enabling a tunable trade-off between economic performance and risk protection.

(3) Single-region RIES model incorporating robust optimization

Building on the interval robust optimization method in Section 3.2, this study provides a unified treatment of multiple uncertainty sources in the system. Accordingly, the original power balance constraints in Equations (65) and (66) are reformulated into their robust counterparts in Equations (81) and (82), and the energy purchasing cost model is revised from Equation (36) to Equation (86). By further incorporating the output characteristics and operational constraints of various devices, the objective function and constraint set of the single-region RIES scheduling model under the robust optimization framework can be established as follows:

$$\begin{array}{r}\hfill \left\{\begin{array}{l}{\mathrm{minC}}_{\mathrm{n}}^0={\displaystyle {\displaystyle \sum }_{\mathrm{t}=1}^{24}}({\mathrm{C}}_{\mathrm{n},\mathrm{t}}^{\mathrm{OPE}}+{\mathrm{C}}_{\mathrm{n},\mathrm{t}}^{\mathrm{DR}}+{\mathrm{C}}_{\mathrm{n},\mathrm{t}}^{\mathrm{CO}2}+{\mathrm{C}}_{\mathrm{n},\mathrm{t}}^{\mathrm{Net}})\hfill \\ \mathrm{s}.\mathrm{t}.\left(33\right)\text{–}\left(35\right)\hfill \\ \&\left(37\right)\text{–}\left(64\right)\&\left(81\right)\&\left(82\right)\&\left(86\right)\hfill \end{array}\right.\end{array}$$

(89)

4. Nash Bargaining-Based Optimal Operation of Multiple RIESs

4.1. Framework of the RIES Alliance Structure

The system investigated in this study comprises three mutually independent regional integrated energy systems (RIESs), an external power grid, an external natural gas network, and a data center. Specifically, the generation units in RIES-1 consist of a combined heat and power (CHP) unit and a wind turbine (WT); RIES-2 is equipped with a CHP unit and photovoltaic (PV) generation; and RIES-3 includes a CHP unit and a WT, while the remaining device configurations are identical to the system structure shown in Figure 1. The external energy networks can supply electricity and natural gas to each RIES according to system requirements, thereby ensuring the secure and stable operation of the multi-RIES cluster. The energy management system deployed in the data center aggregates the load levels and operating states of all RIESs in real time. When the load demand of a given RIES exceeds its local supply capability, electricity trading with other RIESs or the external power grid can be used to regulate the imbalance, thereby achieving system-wide energy balance and coordinated optimal operation.

4.2. Nash Bargaining-Based Optimal Operation Model for a Multi-RIES Alliance

4.2.1. Nash Bargaining Model

Within the cooperative game framework developed in this study, the alliance consists of n participating entities. The primary objective is to reduce the overall operating cost of the system through coordinated decision-making, while ensuring that each participant’s minimum benefit requirements are satisfied. Nash bargaining, a representative mechanism in cooperative game theory, can characterize the strategic behavior of participants during benefit allocation, thereby yielding a profit-sharing scheme that balances efficiency and fairness. In multi-agent interactions where participants’ objectives may conflict, Nash bargaining provides a principled way to reconcile competing interests and mitigate disagreements, offering theoretical support for both the stability and efficiency of alliance cooperation. It should be further clarified that the Nash bargaining scheme in this paper is not employed to directly determine the physical dispatch decisions. Instead, it serves as a settlement and coordination mechanism for inter-RIES power mutual assistance and benefit allocation within the coalition. Specifically, based on the robust dispatch model that incorporates demand response, tiered carbon trading (i.e., low-carbon constraints), and multiple sources of uncertainty, the optimal generation outputs of each RIES under cooperative operation, as well as the inter-regional power exchange quantities, are first determined, thereby achieving the minimization of the coalition-level total operating cost and the low-carbon operation objectives. On this basis, Nash bargaining allocates the overall system benefit generated by cooperation (i.e., the cooperative surplus) by determining the settlement prices and transfer payments associated with internal electricity transactions. This allocation guarantees individual rationality for all participants (i.e., each RIES is no worse off than its disagreement payoff under independent operation) while maintaining fairness in benefit distribution, thereby enhancing the stability and sustainability of coalition cooperation.

(1) Electricity trading among RIESs

Under the cooperative electricity-based optimal operation framework, the total cost of the regional integrated energy system (RIES) cluster includes not only the individual operating costs of each RIES but also the electricity trading among RIESs and the associated constraints. The corresponding mathematical formulation is given in Equations (90) and (91).

$$C_{n,t}^{Trade}=c_{n\to k,t}^e\times P_{n\to k,t}^e$$

(90)

$$P_{n\to k}^{e,\min }\le P_{n\to k,t}^e\le P_{n\to k}^{e,\max }$$

(91)

Here, $c_{n\to k,t}^e$ denotes the electricity trading price between RIES-n and RIES-k; $P_{n\to k,t}^e$ represents the amount of electricity exported from or imported by RIES-n to/from RIES-k; and $P_{n\to k}^{e,\min }$ and $P_{n\to k}^{e,\max }$ indicate the lower and upper bounds of the electricity trading volume, respectively.

(2) Standard form of the Nash bargaining model

When electricity trading payments are taken into account, the objective function and constraints can be expressed as follows:

$$\begin{array}{r}\hfill \left\{\begin{array}{l}\min C_n={\displaystyle {\displaystyle \sum }_{t=1}^{24}}(C_{n,t}^{OPE}+C_{n,t}^{DR}+C_{n,t}^{CO2}+C_{n,t}^{Net}-C_{n,t}^{Trade})\hfill \\ \mathrm{s}.\mathrm{t}.\left(33\right)\text{–}\left(35\right)\&\left(37\right)\text{–}\left(64\right)\hfill \\ \&\left(81\right)\&\left(82\right)\&\left(86\right)\&\left(90\right)\&\left(91\right)\hfill \end{array}\right.\end{array}$$

(92)

Building on Nash bargaining theory and incorporating the payment settlement results for the electricity sharing/trading among the RIESs described above, the corresponding standard Nash bargaining model can be formulated as follows:

$$\left\{\begin{array}{l}\max {\displaystyle {\displaystyle \prod }_{n=1}^3}\left(C_n^0-C_n\right)\hfill \\ s.t.\hspace{1em}{C}_n^0\ge C_n\hfill \end{array}\right.$$

(93)

Here, $C_n^0$ denotes the operating cost of RIES-$n$ under independent operation, which serves as its disagreement point in the Nash bargaining model. Since Equation (93) is a multivariable-coupled, nonconvex, and nonlinear optimization problem, it is difficult to solve directly. To improve computational efficiency, the model is decomposed into two relatively independent subproblems: (i) minimization of the total alliance cost and (ii) benefit allocation after cooperative operation. These two subproblems are solved sequentially in a “cost optimization first, benefit allocation second” manner. The rationale of this two-stage procedure lies in the fact that the internal electricity trading payments within the coalition are transfer payments. As they are budget-balanced at the coalition level, they do not alter the coalition’s total operating cost. Therefore, the optimal physical dispatch and inter-regional power exchange quantities can be determined first, and Nash bargaining can then be applied to settle and allocate the cooperative surplus. In this way, the proposed framework maintains both computational tractability and consistency in economic interpretation.

(3) Cost minimization subproblem

Under the multi-RIES alliance operation scenario, the total operating cost of the system can be expressed as the sum of the operating costs of the three RIESs under cooperative optimal dispatch. As shown in Figure 2, electricity trading is settled internally among the three RIESs, and the corresponding payment is calculated according to Equation (90), where ${\displaystyle {\displaystyle \sum }_{n=1}^3}{\displaystyle {\displaystyle \sum }_{t=1}^{24}}{C}_{n,t}^{Trade}=0$. To simplify the subsequent analysis and highlight the impact of cooperative scheduling on the operating cost itself, the electricity trading payments are not explicitly included in the objective function in the following formulation. Accordingly, the resulting cost minimization model can be expressed as follows:

$$\left\{\begin{array}{l}\min \mathrm{U}={\displaystyle {\displaystyle \sum }_{n=1}^3}{C}_n^{*}={\displaystyle {\displaystyle \sum }_{n=1}^3}{\displaystyle {\displaystyle \sum }_{t=1}^{24}}\left(C_{n,t}^{OPE}+C_{n,t}^{DR}+C_{n,t}^{CO2}+C_{n,t}^{Net}\right)\hfill \\ \mathrm{s}.\mathrm{t}.\left(33\right)\text{–}\left(35\right)\hfill \\ \&\left(37\right)\text{–}\left(64\right)\&\left(81\right)\&\left(82\right)\&\left(86\right)\hfill \end{array}\right.$$

(94)

Here, $C_n^{*}$ denotes the operating cost of RIES-n obtained after cooperative electricity-based optimization, excluding energy trading payments.

(4) Benefit allocation subproblem

By substituting the inter-RIES electricity exchange quantities $P_{n\to k,t}^e$ obtained from Equation (94) into Equation (95), the corresponding benefit allocation subproblem can be formulated as follows:

$$\left\{\begin{array}{l}\max {\displaystyle {\displaystyle \prod }_{n=1}^3}\left(C_n^0-C_n^{*}+{\displaystyle {\displaystyle \sum }_{t=1}^{24}}{C}_{n\to k,t}^{trade}\right)\hfill \\ s.t.C_n^0\ge C_n^{*}-{\displaystyle {\displaystyle \sum }_{t=1}^{24}}{C}_{n\to k,t}^{trade}\hfill \end{array}\right.$$

(95)

Since Equation (95) remains a nonconvex and nonlinear optimization problem, directly solving its maximization form is computationally challenging. By exploiting the strictly increasing property of the natural logarithm function, Equation (96) can be logarithmically transformed, whereby the original maximization problem is equivalently converted into a corresponding minimization problem. This transformation simplifies the model structure and improves the tractability and computational efficiency of the solution procedure.

$$\left\{\begin{array}{l}\min {\displaystyle {\displaystyle \prod }_{n=1}^3}-\ln \left(C_n^0-C_n^{*}+{\displaystyle {\displaystyle \sum }_{t=1}^{24}}{C}_{n\to k,t}^{trade}\right)\hfill \\ s.t.C_n^0\ge C_n^{*}-{\displaystyle {\displaystyle \sum }_{t=1}^{24}}{C}_{n\to k,t}^{trade}\hfill \end{array}\right.$$

(96)

4.2.2. Model Solution

(1) Distributed solution based on the ADMM algorithm

The alternating direction method of multipliers (ADMM) is an iterative algorithm that is widely used to solve decomposable convex optimization problems. The main idea is to decompose a large-scale optimization model into several relatively independent subproblems, solve them separately, and then coordinate information exchange across subproblems through iterative updates of Lagrange multipliers and coupling variables. In this way, the distributed solutions progressively converge to the optimum (or an acceptable solution) of the overall problem. The ADMM-based model formulation and the detailed solution procedure are presented as follows:

$$\left\{\begin{array}{l}\min f\left(x\right)+g\left(z\right)\hfill \\ \mathrm{s}.\mathrm{t}. Ax+Bz=c\hfill \end{array}\right.$$

(97)

Here, $f\left(x\right)$ and $g\left(x\right)$ are convex functions; $x$ and $z$ denote the decision variables; and A, B and c represent the given coefficient matrices and constant vector, respectively.

To solve the above optimization problem, the augmented Lagrangian function is employed. Compared with the standard Lagrangian, a quadratic penalty term is added to strengthen constraint satisfaction and to improve the convergence and numerical stability of the solution process.

$$L_p\left(x,y,z\right)=f\left(x\right)+g\left(z\right)+y^T\left(Ax+Bz-c\right)+{\displaystyle \frac{\rho }{2}}{\parallel Ax+Bz-c\parallel }_2^2$$

(98)

Here, $y$ denotes the Lagrange multiplier, and $\rho$ is the penalty parameter (with $\rho$ > 0) that controls the weight of the penalty term.

By iteratively performing Equations (99)–(101), at each step two variables are treated as fixed while updating the remaining one, i.e., $x$ and $z$ are updated in an alternating manner. This procedure is repeated until the convergence criterion in Equation (102) is satisfied, at which point the algorithm is deemed to have converged and an optimal solution to the optimization problem is obtained.

$$x^{k+1}=\arg \underset{x}{\min } L_p\left(x,y^k,z^k\right)$$

(99)

$$z^{k+1}=\arg \underset{y}{\min } L_p\left(x^{k+1},y^k,z\right)$$

(100)

$$y^{k+1}=y^k+\rho \left(A{x}^{k+1}+B{z}^{k+1}-c\right)$$

(101)

Convergence criterion:

$${\parallel x^{k+1}-z^{k+1}\parallel }_2^2\le \epsilon$$

(102)

It should be noted that the Nash bargaining solution in this paper is implemented via a two-stage framework. The first stage determines the coalition-level physical dispatch (i.e., operating cost minimization and power exchange quantities), while the second stage performs the negotiation-based settlement and allocation of the cooperative surplus. Since the two stages differ fundamentally in their mathematical properties, the convergence discussion of ADMM should be distinguished accordingly. For the second-stage benefit allocation problem, after applying a logarithmic transformation to the Nash product, the resulting objective function becomes concave, and the feasible set is defined by linear constraints. Hence, the problem can be reformulated as a standard convex optimization model, for which the use of ADMM is supported by classical convergence theory. In contrast, the first-stage physical dispatch involves mixed-integer decisions such as unit commitment, leading to a non-convex MILP formulation. Therefore, this paper does not claim a global convergence guarantee of ADMM for the first-stage problem. Instead, ADMM is adopted as a widely used practical decomposition-and-coordination approach to enforce consistency through iterative updates, where solution stability is ensured via residual-based stopping criteria and a maximum iteration limit. Moreover, the iteration performance observed in the case studies is reported to empirically demonstrate the applicability of the proposed algorithm.

(2) Solution of the cost minimization subproblem

Augmented Lagrangian formulation: by introducing the Lagrange multiplier ${\lambda }_{n\to k}^1$ and the penalty parameter ${\rho }_n^1$, the constraints of the optimization problem are incorporated into the augmented Lagrangian function, enabling efficient iterative solution.

$$L_1=\min \left[C_n^{*}+{\displaystyle \sum }_{t=1}^{24}{\lambda }_{n\to k,t}^{1e}\left(P_{n\to k,t}^e+P_{k\to n,t}^e\right)+\left({\rho }_n^{1e}/2\right)\times {\displaystyle \sum }_{t=1}^{24}{\parallel P_{n\to k,t}^e+P_{k\to n,t}^e\parallel }_2^2\right]$$

(103)

Figure 3 is flowchart of the proposed ADMM-based solution procedure.

(a) Initialization: Set the iteration index $x=1$ and the maximum number of iterations to 100. Initialize the power exchange quantities of all microgrids at the initial state as $P_{n\to k}^e=0$, and specify the convergence tolerance as ${\tau }_1=0.01$.

(b) Power trading update: Solve the model in Equation (103) and compute $P_{n\to k}^e$.

(c) Lagrange multiplier update: ${\lambda }_{n\to k,t}^{le,x+1}={\lambda }_{n\to k,t}^{le,x}+{\rho }_n^{1e}\left({\rho }_{k\to n,t}^{e,x+1}+{\rho }_{n\to k,t}^{e,x+1}\right)$.

(d) Iterative update: $x=x+1$.

(e) Convergence check: Examine the convergence of the ADMM algorithm and verify whether the termination condition in ${\displaystyle {\displaystyle \sum }_{n=1}^3}{\displaystyle {\displaystyle \sum }_{t=1}^{24}}{\parallel P_{n\to k,t}^{e,x+1}-P_{n\to k,t}^{e,x}\parallel }_2^2\le {\tau }_1$ is satisfied. If the condition holds, terminate the iteration; otherwise, return to Step (b) and continue until either ${\displaystyle {\displaystyle \sum }_{n=1}^3}{\displaystyle {\displaystyle \sum }_{t=1}^{24}}{\parallel P_{n\to k,t}^{e,x+1}-P_{n\to k,t}^{e,x}\parallel }_2^2\le {\tau }_1$ is satisfied or the iteration count exceeds the prescribed maximum limit.

(3) Solution of the benefit allocation subproblem

Augmented Lagrangian formulation: by introducing the Lagrange multiplier ${\lambda }_{n\to k}^2$ and the penalty parameter ${\rho }_n^2$, the augmented Lagrangian function is constructed as follows:

$$L_2=\min -\ln \left[\begin{array}{l}{C}_n^0-C_n^{*}+{\displaystyle \sum _{t=1}^{24}{C}_{n\to k,t}^{trade}}+{\displaystyle \sum _{t=1}^{24}{\lambda }_{n\to k,t}^{2e}}(P_{n\to k,t}^e+P_{k\to n,t}^e)+(P_n^{2e}/2)\\ \times {\displaystyle \sum _{t=1}^{24}{\parallel P_{n\to k,t}^e+P_{k\to n,t}^e\parallel }_2^2}\end{array}\right]$$

(104)

Figure 4 is flowchart of the proposed ADMM-based solution procedure.

(a) Initialization: Set the iteration index $x=1$ and the maximum number of iterations to 100. Initialize the electricity prices of all RIESs in the initial state as $c_{n\to k}^e=0$, and specify the convergence tolerance as ${\tau }_2=0.01$. Import the energy exchange quantities obtained by solving Equation (104).

(b) Electricity trading payment update: Solve the model in Equation (104) and compute $c_{n\to k}^e$.

(c) Lagrange multiplier update: $x=x+1$.

(d) Iterative update: $x=x+1$.

(e) Convergence check: Examine the convergence of the ADMM algorithm and verify whether the termination condition in ${\displaystyle {\displaystyle \sum }_{n=1}^3}{\displaystyle {\displaystyle \sum }_{t=1}^{24}}{‖c_{n\to k,t}^{e,x+1}-c_{n\to k,t}^{e,x}‖}_2^2\le {\tau }_2$ is satisfied. If the condition holds, terminate the iteration; otherwise, return to Step (b) and continue until either ${\displaystyle {\displaystyle \sum }_{n=1}^3}{\displaystyle {\displaystyle \sum }_{t=1}^{24}}{‖c_{n\to k,t}^{e,x+1}-c_{n\to k,t}^{e,x}‖}_2^2\le {\tau }_2$ is satisfied or the iteration count exceeds the prescribed maximum limit.

4.3. Analysis of Fraudulent Behavior Among RIESs

In the multi-RIES cooperative electricity-based optimization model, potential fraudulent behavior should be further considered in the benefit allocation stage. To maximize individual benefits, each RIES may strategically misreport its cooperation cost during allocation so as to obtain higher electricity trading revenue. Despite such strategic behavior, the system can still reach a stable state among the three RIESs, i.e., a fraudulent Nash equilibrium.

4.3.1. Definition of Fraud

The fraudulent cost is defined as follows:

$$C_{n,f}^{gap}=C_n^{gap}-{\gamma }_n\times \left|C_n^{gap}\right|$$

(105)

Here, $C_n^{gap}$ denotes the difference between the operating cost of a RIES under independent operation and its cost under cooperative electricity trading, i.e.,$C_n^{gap}=C_n^0-C_n^{*}=C_n^0-C_n-{\displaystyle {\displaystyle \sum }_{t=1}^{24}}{C}_{n\to k,t}^{trade}$. ${\gamma }_n$ is the fraud factor; when ${\gamma }_n=0$, the cost reported by the RIES corresponds to the true cost.

It should be emphasized that the tunable fraud factor ${\gamma }_n$ introduced in this paper is not an arbitrarily selected empirical parameter. Instead, it is a strategic parameter designed to characterize the potential for strategic cost reporting (i.e., misreporting) by RIES participants during the settlement stage. Specifically, in the coalition settlement process, each participant is allowed to report a cost that deviates from its true operating cost within a prescribed range, thereby influencing the subsequent benefit allocation outcome.

To clarify the game-theoretic basis of the proposed “tunable fraud factor,” this paper formalizes the cost reporting process as a non-cooperative reporting game.

$$G=〈N,{\left\{S_n\right\}}_{n\in N},{\left\{U_n\right\}}_{n\in N}〉$$

(106)

Here, $N=\left\{1,2,\dots ,N\right\}$ denotes the set of RIESs participating in the electricity cooperation. $S_n$ represents the strategy space of participant $n$, whose decision variable is the fraud factor ${\gamma }_n$, subject to the following constraint:

$${\gamma }_n\in S_n=\left[0, {\gamma }_n^{\lim it}\right]$$

(107)

Here, ${\gamma }_n=0$ corresponds to truthful reporting, whereas ${\gamma }_n>0$ indicates that the participant strategically reports a biased cost difference. The misreported (fraudulent) cost difference is given by Equation (105). Furthermore, $U_n$ denotes the utility function of participant $n$ after the Nash bargaining settlement, which can be characterized as the difference between the “disagreement payoff under independent operation” and the “settled payoff under cooperative operation,” i.e.,

$$U_n\left({\gamma }_n, {\gamma }_{-n}\right)=C_n^0-\left(C_n^{*}-{\displaystyle {\displaystyle \sum }_{t=1}^{24}}{C}_{n\to k,t}^{trade}\right)$$

(108)

In this equation, $C_n^0$ denotes the operating cost of RIES-n under independent operation, while $C_n^{*}$ represents its actual operating cost under cooperative operation. $C_{n\to k,t}^{trade}$ denotes the settlement payment term determined by the benefit allocation model (96), which is affected by the reported information of all participants ${\gamma }_n$. Based on the above definitions, when the strategy profile ${\gamma }_n^{\ast }=\left\{{\gamma }_1^{\ast },\dots ,{\gamma }_N^{\ast }\right\}$ satisfies

$$U_n\left({\gamma }_n^{\ast }, {\gamma }_{-n}^{\ast }\right)\ge U_n\left({\gamma }_n,{\gamma }_{-n}^{\ast }\right),{\gamma }_n\in S_n,\forall n\in N$$

(109)

then ${\gamma }_n^{\ast }$ is referred to as the fraud Nash equilibrium of the reporting game.

4.3.2. Fraudulent Nash Equilibrium

To obtain the subsequent fraudulent Nash equilibrium, the following constraints must be satisfied:

Based on Equations (105) and (110), the upper bound of the fraud factor can be derived as follows:

$${\displaystyle {\displaystyle \sum }_{n=1}^3}{C}_{n,f}^{gap}>0$$

(110)

$${\gamma }_n^{\lim it}=\psi \left(C_n^{gap}+{\displaystyle {\displaystyle \sum }_{n=1}^3}{C}_{k,f}^{gap}-C_{n,f}^{gap}\right)/\left|C_n^{gap}\right|$$

(111)

Here, $\psi$ denotes a tuning parameter. When $\psi \in \left(0,1\right)$, it can be used to adjust the magnitude of ${\gamma }_n^{\lim it}$, thereby controlling the bargaining strength of the major energy-sharing party in the fraudulent bargaining process and avoiding unreasonable bargaining outcomes.

According to the benefit allocation model in Equation (95) and the arithmetic–geometric mean (AM–GM) inequality, the objective in Equation (95) attains its maximum if and only if the multiplicative terms in the product are equal. Therefore, Equation (95) can reach its maximum when all terms take the same value, i.e., ${\displaystyle {\displaystyle \sum }_{n=1}^3}(C_n^0-C_n^{*}+{\displaystyle {\displaystyle \sum }_{t=1}^{24}}{C}_{n\to k,t}^{trade})/3$. Since ${\displaystyle {\displaystyle \sum }_{n=1}^3}{\displaystyle {\displaystyle \sum }_{t=1}^{24}}{C}_{n,t}^{Trade}=0$, Equation (95) achieves its maximum when all terms satisfy ${\displaystyle {\displaystyle \sum }_{n=1}^3}(C_n^0-C_n^{*})/3$.

Consequently, the following solution can be obtained:

$$C_n^{trade}=\left(C_n^0-C_n^{*}\right)-{\displaystyle {\displaystyle \sum }_{n=1}^3}(C_n^0-C_n^{*})/3=C_n^{gap}-{\displaystyle {\displaystyle \sum }_{n=1}^3}{C}_n^{gap}$$

(112)

From the perspective of a given RIES, assume that all other RIESs report their true costs $C_n^{gap}$, whereas this RIES reports a cost lower than $C_n^{gap}$ by $C_{n,f}^{gap}$. While keeping its actual cost $C_n^{gap}$ unchanged, the RIES can obtain a higher payoff $C_n^{trade}$. Taking RIES-1 as an example, the reported (fraudulent) cost can be expressed as follows:

$$\left\{\begin{array}{l}{C}_1^{trade}=C_1^{gap}-\left(C_{1,f}^{gap}+C_2^{gap}+C_3^{gap}\right)/3\hfill \\ C_{1,f}^{gap}<C_1^{gap}\hfill \end{array}\right.$$

(113)

5. Case Study

5.1. Case Study Data

This section conducts a case study based on the three-area RIES cooperative electricity trading-based optimal operation model shown in Figure 2. Figure 5 illustrates the renewable energy generation outputs, load profiles, and electricity purchasing/selling prices of each RIES, while

Table 1. Summary of key parameters.

Parameter	Definition	Unit
${\Gamma }_{\mathrm{i}}$	Robustness budget parameter
${\gamma }_{ij}$	Relative deviation coefficient
${\mathrm{c}}_{\mathrm{CO}2}$	Carbon trading benchmark price	CNY/(kgCO2)
$\mathsf{\kappa }$	Incremental carbon price per unit	CNY/(kgCO2)
$\mathrm{d}$	Length of the carbon emission tier interval	kgCO2
${\mathsf{\xi }}^{\mathrm{CO}2}$	Carbon emission factor	kgCO2/kWh
${\mathsf{\xi }}^0$	Carbon allowance coefficient	kgCO2/kWh
${\mathsf{\xi }}_{\mathrm{e},\mathrm{tran}}$, ${\mathsf{\xi }}_{\mathrm{e},\mathrm{cut}}$, ${\mathsf{\xi }}_{\mathrm{h},\mathrm{tran}}$, ${\mathsf{\xi }}_{\mathrm{h},\mathrm{cut}}$	Compensation coefficients (load shifting/electricity curtailment/heat curtailment)	CNY/kWh
${\gamma }_n$	Tunable fraud factor
$\psi$	Fraud game adjustment factor
$\rho$	ADMM penalty parameter

summarizes the key parameters used in the case study.

The figure illustrates the intraday variations in electrical load, thermal load, renewable generation output, and external energy prices for the three RIESs. As shown in panels (a) and (b), the electric and heat demands of each RIES exhibit a typical daily profile with morning and evening peaks and lower demand during nighttime; however, notable differences exist in both the load level and fluctuation magnitude. In particular, RIES-1 has the highest load, whereas RIES-3 has the lowest, leaving room for subsequent cross-regional electricity mutual support. Panel (c) indicates that the wind turbines deployed in RIES-1 and RIES-3 provide output throughout the day with relatively smooth variations, while the PV output in RIES-2 follows a single-peaked daytime pattern, reflecting the temporal complementarity between wind and solar resources. Panel (d) shows that the external electricity price follows a time-of-use (TOU) tariff with peak and off-peak periods, whereas the natural gas price remains nearly constant over the day. This price structure incentivizes each RIES to purchase electricity during low-price periods and rely on local natural gas and renewable generation during high-price periods, thereby laying the foundation for electricity cooperation and coordinated optimal dispatch in the subsequent analysis.

5.2. Convergence Analysis

(1) Convergence analysis of the cost minimization problem

The aforementioned load profiles, renewable generation outputs, and energy price parameters are substituted into both the independent-operation model of each RIES and the multi-RIES cooperative electricity-based operation model. Under independent operation, the operating cost of each RIES can be directly computed when no energy exchange is allowed. Under cooperative operation, the proposed model is solved in a distributed manner using the ADMM algorithm. The total operating cost of the alliance gradually converges to the optimum after 41 iterations, and the electricity trading quantities among the RIESs are obtained simultaneously. The convergence trajectories of the operating costs are shown in Figure 6, where the final converged cost of the RIES alliance (RIES-A) equals the sum of the converged costs of all individual RIESs.

(2) Convergence analysis of the benefit allocation problem

After the electricity exchange results for each time period are determined through cost optimization in the previous step, they are treated as known parameters and substituted into the Nash bargaining model. The model is then solved in a distributed manner using the ADMM algorithm to obtain the settlement prices and payment amounts for electricity trading among the RIESs, i.e., the bargaining outcomes under Nash bargaining. As shown in Figure 7, the algorithm essentially converges at around the 21st iteration, after which the electricity trading payments of each RIES exhibit only minor variations and gradually stabilize. Since electricity trading occurs only among the three RIESs and the budget-balance constraint is satisfied, ${\displaystyle {\displaystyle \sum }_{\mathrm{n}=1}^3}{\displaystyle {\displaystyle \sum }_{\mathrm{t}=1}^{24}}{\mathrm{C}}_{\mathrm{n},\mathrm{t}}^{{\mathrm{T}}_{\mathrm{Trade}}}=0$; consequently, the converged net payment of the overall alliance (RIES-A) is zero, which is consistent with the sum of the converged payments of the three individual RIESs (see Figure 7d).

(3) Convergence analysis of fraudulent behavior

The computations in stages (1) and (2) do not account for potential dishonest reporting behaviors by the RIESs. Based on the independent operating cost of each RIES and the optimal operating cost under alliance operation, the corresponding cost differences can be further derived, which are then used to define the fraudulent costs reported by each RIES to the data center. After introducing the fraud factor, an iterative algorithm is applied to model the dynamic evolution of the system, which converges to a fraudulent Nash equilibrium at approximately the 70th iteration. The convergence trajectories of the fraudulent costs for all RIESs are shown in Figure 8.

The iterative convergence of the fraud factors and their upper bounds is shown in Figure 9. After the system converges to the fraudulent equilibrium, the benefits within the alliance are reallocated, and the iterative convergence trajectories of the electricity trading settlement payments for each RIES are presented in Figure 10. The overall trend is essentially consistent with that in the case without fraud. When fraudulent behavior is considered, the converged net settlement payment of the RIES alliance (RIES-A) remains zero, which is consistent with the sum of the converged payments of all individual RIESs.

5.3. Analysis of Electricity Exchange Results

The optimized electricity trading results among the RIESs are presented in Figure 11, which depicts the electricity exchange power among the three RIESs. During 1–5 h, RIES-1 mainly acts as the electricity exporter, RIES-2 remains in a high-power net importing state, and the exchange power of RIES-3 is relatively small. During 9–13 h, when PV output is high, RIES-2 becomes the main supplier and transmits electricity to RIES-1 and RIES-3, both of which are mostly net importers. After 17 h, as PV generation decreases, RIES-1 again becomes the primary exporter, while RIES-2 shifts back to net importing. Overall, the three RIESs dynamically switch between supplier and consumer roles across different time periods, achieving peak shaving and valley filling at the inter-regional level. This indicates that the cooperative electricity trading mechanism is effective in mitigating renewable generation variability.

5.4. Power Balance Analysis

Under both the independent operation of each RIES and the alliance-based cooperative operation, the electric power balance results (Figure 12) and the thermal power balance results (Figure 13) can be obtained, respectively. The interval robust optimization method developed in Section 3 reformulates the power balance constraints of each RIES into a deterministic equivalent under the worst-case realization of uncertainties. Accordingly, the maximum deviation between the load demand and the wind/PV generation can be characterized. The corresponding results are illustrated in the electric power balance plots by the “net electric load” curve, as discussed below:

(1) Figure 12d–f illustrates the electric power balance of each region under the multi-RIES cooperative electricity-based optimal operation. It can be observed that the inter-RIES exchanged electric power profiles are consistent with the optimized trading results shown in Figure 11. This indicates that, in the cooperative dispatch model, the electricity trading decisions are well aligned with the power balance constraints, and the case study results exhibit good consistency and plausibility in terms of both electricity trading and power balance.

(2) By comparing the operating results of each RIES before and after cooperation in Figure 12 and Figure 13, it is evident that electricity cooperation leads to significant changes in the purchased/sold electricity with the external grid as well as the electricity exchanges among RIESs. The power inflow/outflow of the external grid and the charging/discharging profiles of electrical energy storage become smoother, and in some periods the system shifts from relying on large-scale external electricity purchases to prioritizing internal surplus power sharing within the alliance. Through cooperative operation, the RIESs coordinate the allocation of multiple energy resources (e.g., electricity and heat) at the alliance level. When determining the hour-ahead dispatch strategy, carbon emission costs, energy purchasing costs, and the temporal constraints induced by electricity–heat coupling are jointly considered, together with demand response management on the load side. Overall, this cooperative strategy reduces the system operating cost.

(3) Although a thermal energy cooperation mechanism is not incorporated in the RIES alliance’s optimal operation, cooperative electricity-based operation can still perturb the thermal power balance of each RIES due to the electricity–heat coupling relationship. The resulting variations are shown in Figure 13.

5.5. Negotiation Results of Electricity Trading Prices

Figure 14a,b presents the inter-RIES electricity trading prices in each time period under the scenarios without fraud and with fraud, respectively. As can be seen, the trading price is always constrained within the interval bounded by each RIES’s electricity purchase price and selling price. When a RIES has a power surplus in a given period, it tends to sell electricity to other RIESs at a price lower than that of the external grid; when its local power supply is insufficient, it compares the external grid price with the selling quotations from other RIESs and purchases electricity from the cheaper source. These results indicate that the alliance-internal electricity trading prices formed under the Nash bargaining mechanism facilitate the preferential utilization of inter-regional mutual support potential and reduce reliance on transactions with the external grid, thereby lowering the overall energy procurement cost of the alliance.

After fraudulent behavior is introduced, each RIES can manipulate the Nash bargaining equilibrium by reporting artificially adjusted cost information to the data center, thereby affecting the negotiated electricity trading prices and the resulting benefit allocation. It should be noted that such fraudulent behavior does not change the total operating cost of the alliance under cooperative operation; rather, it shifts the internal benefit distribution toward outcomes that better reflect each participant’s self-interested incentives under fraudulent motives.

5.6. Cost and Benefit Analysis

After introducing inter-RIES cooperative electricity trading, both the cost composition and the corresponding values of each RIES change to some extent. The electricity trading settlement payment represents the actual monetary amount settled for electricity exchanges during cooperative operation. Since electricity trading occurs only under cooperative operation, the electricity trading settlement payment of each RIES is zero under independent operation.

Table 2. Operating costs of the RIESs (unit: CNY).

Operating Mode	RIES-1	RIES-2	RIES-3	RIES-A
Independent operation	8315.02	20,716.68	11,046.54	40,077.87
Cooperative electricity trading (without fraud)	6953.71	18,663.94	9463.17	35,080.82
Cooperative electricity trading (with fraud)	7156.94	18,334.62	9586.15	35,077.71

reports the operating cost of each RIES and the total cost of the RIES alliance. Compared with the independent-operation case, the cooperative electricity trading scheme reduces the operating costs of RIES-1, RIES-2, and RIES-3 by CNY 1361.31, CNY 2052.74, and CNY 1583.37, respectively. Overall, the total operating cost decreases by CNY 4997.05 after introducing electricity cooperation.

After accounting for fraudulent behavior in cooperative electricity trading, the cost allocation results among the RIESs change to some extent. However, such fraud manifests only as dishonest reporting strategies adopted by individual participants during the benefit allocation stage to maximize their own interests; it does not alter the physical dispatch decisions of the alliance or the overall operating cost under the given operating conditions. Consequently, the total cost remains around CNY 35,000. As shown in Table 2, the operating cost of the RIES alliance is CNY 35,077.71 when fraud is considered, compared with CNY 35,080.82 in the no-fraud case. The difference is negligible and mainly arises from numerical errors introduced by the increased computational complexity of the optimization problem after incorporating fraud modeling, rather than any substantive change in the operating strategy.

To further quantify and validate the effectiveness of the proposed method, three baseline schemes are selected for comparison, namely: (i) “no cooperation (independent operation),” (ii) “cooperation (without fraud),” and (iii) “cooperation (with fraud).” Let the total operating cost of the coalition under the non-cooperative mode be $C^0$, and that under the cooperative mode be $C^{\ast }$. The absolute cost saving and the cost-saving ratio of the coalition are then defined as follows:

$$\Delta C=C^0-C^{\ast }$$

(114)

$$\eta ={\displaystyle \frac{C^0-C^{\ast }}{C^0}}\times 100\%$$

(115)

As shown in Table 2 and

Table 3. Comparison of cost savings and convergence speed under different baseline schemes.

Indicator	Cooperation (No Fraud)	Cooperation (with Fraud)
Absolute cost saving (CNY)	4997.05	5000.16
Cost-saving ratio (%)	12.47%	12.48%
Cost-minimization iterations	41	41
Benefit-allocation iterations	21	21
Fraud-equilibrium iterations		70

, compared with the non-cooperative baseline (i.e., $C^0=40077.87$ CNY), the “cooperation without fraud” scheme reduces the coalition-wide total operating cost to $C^{\ast }=35080.82$ CNY, corresponding to a cost saving of $\Delta C=4997.05$ CNY and a saving ratio of $\eta =12.47\%$. These results demonstrate that mutual power assistance and coordinated dispatch can significantly enhance the overall economic performance of the system. Furthermore, when fraudulent behavior is considered, the fraud factor only affects the reporting and settlement process in the benefit allocation stage and does not alter the physical dispatch decisions under the given operating conditions. Therefore, under “cooperation with fraud,” the coalition’s total operating cost remains at approximately 35,077.7 CNY, and the cost-saving performance is essentially consistent with that of the fraud-free case. In terms of convergence, the distributed ADMM converges within approximately 41 iterations in the cost-minimization stage and within about 21 iterations in the benefit allocation stage. When the fraud dynamics are incorporated, the system converges to the fraud Nash equilibrium within around 70 iterations, indicating that the proposed framework achieves cost-benefit improvement with an acceptable convergence speed and practical computational feasibility.

6. Conclusions

Based on the overall architecture of integrated energy systems (IESs) and the mathematical models of typical energy devices, this study proposes an integrated modeling and solution framework for regional integrated energy systems (RIESs) featuring “independent operation–alliance cooperation–fraud-aware reallocation”. First, an adjustable interval-robust optimization model is developed for independent RIES operation. Then, a Nash bargaining-based electricity cooperation model for multiple RIESs is formulated, and a fraud factor is introduced at the benefit allocation layer to characterize dishonest cost reporting. Finally, the alternating direction method of multipliers (ADMMs) is employed to solve, in a distributed iterative manner, the two coupled subproblems of alliance-wide total cost minimization and benefit allocation. Case studies validate the effectiveness of the proposed approach in reducing the total operating cost of the alliance and in revealing how fraud affects trading prices and the resulting benefit distribution.

(1) At the independent-operation level, this study formulates a RIES optimization model that minimizes the sum of equipment operation and maintenance costs, demand response expenditures, carbon trading costs, and energy purchasing/selling costs, subject to energy balance constraints as well as device output limits and operational logic. To address the uncertainties in key parameters such as distributed renewable generation, user demand, and electricity purchase/selling prices, an adjustable interval robust optimization approach is adopted for unified modeling. By introducing tunable parameters, the uncertainty set and the conservativeness of the solution can be flexibly controlled, and the original objective and constraints are equivalently transformed into a tractable deterministic counterpart under the worst-case realization. As a result, a robustified independent-operation model for the RIES is obtained.

(2) At the cooperative-operation level, an alliance-scale Nash bargaining model in the standard Nash form is established and decomposed into two coupled subproblems: minimization of the total operating cost of the alliance and intra-alliance benefit allocation. On this basis, potential fraudulent behavior among RIESs is further considered by introducing a fraud factor in the benefit allocation stage, thereby formulating a fraud-aware Nash equilibrium model to characterize benefit reallocation and its impact on electricity trading prices. The distributed ADMM-based solution results indicate that the total cost under cooperative alliance operation is significantly lower than the sum of the independent operating costs of all RIESs. Moreover, variations in the degree of fraud can substantially alter both the price formation mechanism of intra-alliance electricity trading and the resulting benefit allocation pattern.