A Multi-Objective Hybrid Game Pricing Strategy for Integrated Energy Operator-Load Aggregator Alliances Considering Integrated Demand Response

In recent years, with significant changes in electricity demand, load aggregators have begun to integrate customer demand-side resources and participate in market transactions as independent entities. This study aims to establish an alliance of integrated energy service providers and load integrators, which optimizes the operation of the integrated energy system and maximizes in order tothe interests of all participants through the collaborative operation mechanism of new Power to Gas (P2G) and Interruptible Demand Response (IDR). To this end, we first constructed a master-slave game model that considers the bargaining game between load integrators with a large number of photovoltaic users in the model, to solve the problem of competition and cooperation. We establish a multi-objective model to address the low-carbon economic equilibrium problem faced by comprehensive energy service providers in the game process, and use the Conditional Value at Risk (CVaR) theory to measure the uncertainty of renewable energy. Secondly, to achieve cooperation between load aggregators, we use Nash bargaining theory to decompose the cooperative game model into two subproblems: maximizing alliance benefits and allocating cooperation benefits. Finally, we use the Alternative Direction Method of Multipliers (ADMM) algorithm combined with dichotomy and compromise programming theory to solve the model, to maximize the interests of all parties in the continuous interaction process. The research results indicate that after considering the electricity trading and carbon emission targets between load aggregators, the interests of all parties are protected, fully promoting the exchange of interests between photovoltaic producers and sellers, and alleviating the energy supply pressure of IESO. At the same time, a comparison with Multi-objective Evolutionary Algorithm Based on Decomposition (MOEA/D) shows that the fuzzy compromise programming theory used in this paper has a faster solving speed and a smoother solving process.

SECTION I. Introduction

A. Motivation

Integrated energy systems (IES) are an effective means for addressing the volatility and uncertainty of renewable energy sources, as they enable the coordination and optimization of multiple heterogeneous energy systems [1]. As service providers, Integrated energy service providers (IESP) integrate diverse resources to ensure efficient energy usage and a stable and reliable energy supply on the demand side [2]. However, demand sides have undergone significant changes in recent years. The widespread adoption of photovoltaic technology has prompted many countries to issue policies that encourage the installation of photovoltaic energy systems on the consumer side [3]. This allows traditional electricity consumers to not only consume energy but also participate in electricity market transactions as producers. Nonetheless, many customers are unable to participate in market trading because of their dispersed geographic locations and insufficient power generation capacities. Load aggregators (LA) have emerged as a solution, because they combine the production of multiple photovoltaic electricity producers to create a market-competitive entity that can participate in electricity trading [4]. However, both the IESP and LAs are rational entities that seek to maximize their profit in market transactions, game theory supplies a suitable tool for modeling their interactions in a fresh operational framework underneath this decentralized paradigm, while accounting for the self-interest of individuals and the confidentiality of their related information [5].

With the establishment of a double-carbon target [6], carbon emissions have become a critical consideration in the design and optimization of integrated energy systems (IESs). However, optimizing carbon emissions often conflicts with economic objectives. Therefore, multi-objective optimization can help achieve a feasible and optimal design solution by incorporating carbon emissions as one of the optimization objectives [7]. In addition, coordinating the operating status of devices within IESs, promoting the consumption of renewable energy, reducing carbon emissions, and enhancing system flexibility are key issues for current IESs [8].

The integration of power-to-gas (P2G) technology and integrated demand response (IDR) can enhance clean energy utilization and enable flexible energy allocation, thus improving system efficiency [9]. IDR is a crucial approach for optimizing integrated energy systems (IES) as it helps balance the energy supply and demand [10]. To fully exploit the potential of both supply and demand while ensuring the active participation of all market participants, a reasonable IDR mechanism should be developed during the dispatching process. As a crucial link between electric and natural gas systems in an IES, P2G is considered a promising means of improving the system flexibility [11]. A typical P2G system comprises two processes: electrolysis (EL) and methanation (MR), in which hydrogen is first produced through the electrolysis of water via the EL process and then used to generate natural gas via a chemical reaction with CO₂ [12]. However, the use of hydrogen energy in the methanation process limits the regulation of P2G and restricts the flexibility of the system. Therefore, further exploration of the value of hydrogen energy in P2G conversion is necessary.

B. Literature Survey

With the rapid development of renewable energy and the deepening of electricity market reform, new energy service enterprises such as the Independent Electricity System Operator (IESP) and Load Aggregators (LAs) have emerged as important players in the energy industry. Several scholars have investigated this relationship. Reference [13] describes IESP, LA, and load as a hierarchical Stackelberg game and proposes an IES bilateral Interruptible Demand Response (IDR) model with LA at the core to achieve an equilibrium of interests between the parties. Reference [14] proposed a distributed and coordinated trading mechanism for community-integrated energy systems using a three-level game model, which was transformed into a two-level problem using the Karush-Kuhn-Tucker condition and convexification. Reference [15] accounts for market outcome uncertainties by modeling the interaction between LA and residential consumers as a stochastic Stackelberg game, and using LA as a mediator to participate in the day-ahead electricity market. Reference [16] proposes inter-community and intra-community energy-sharing strategies for day-ahead and real-time power management problems of sellers and community producers, respectively, and verified the framework efficiency through simulation calculations in a typical distribution network.

Although the above literature provides practical suggestions for the game relationship between IESP, LAs, and users, it ignores the fact that multiple LAs with production and marketing capabilities cannot obtain the overall optimal benefit when playing a master-slave game process with IESP [17]. Cooperative game theory can handle the electric energy interaction between LAs, taking into account individual and overall rationality, and improve the enthusiasm for inter-network electric energy trading. This paper proposes aggregating several alliances with photovoltaic production and marketing capabilities into the Load Aggregator Alliance (LAA) and nested cooperative games between alliances in the master-slave game process to deal with the competition and cooperation relationship in the model.

Secondly, as global warming continues to worsen, the Independent Electricity System Operator (IESP) must strive to minimize carbon emissions while maintaining system reliability, economy, and sustainability in the dispatch process. These goals are often conflicting, which necessitates the use of multi-objective optimization to balance them [18]. Reference [19] proposed a multi-objective planning model that considers both energy efficiency and economy. The original model was transformed into a convex problem using a convex relaxation method, which was then solved. Reference [20] proposes to solve the multi-objective optimal reactive power scheduling model using a weighted summation method to minimize voltage deviation and actual power loss in the load bus. Reference [21] established separate economic and environmental objective functions for a hybrid system, which were solved using the $\varepsilon$ -constraint method. Finally, the optimal solution was chosen using the fuzzy satisfaction technique. In addition to traditional multi-objective solution algorithms, proposals in the literature suggest the use of genetic algorithms [22], particle swarm optimization algorithms [23], MOEA/D algorithms [24], and other optimization techniques. However, each approach has its strengths and limitations. In this study, we first establish the economic objective function of IESP, then establish the carbon emission objective function, and introduce the stepped carbon trading mechanism. Finally, we adopt the compromise planning method proposed in the literature [25] to weigh the economy and low carbon emissions of the IESP in the game process.

In recent years, numerous studies have demonstrated that techniques such as interseasonal energy storage with an IDR and P2G can significantly enhance system flexibility. For instance, in [26], the authors proposed a cooperative scheduling strategy for P2G and pipeline storage capacity to enhance system flexibility. The economic and low-carbon performance of the proposed model was verified using arithmetic examples. However, the above literature neglects the internal energy of P2G - hydrogen. According to [27], P2G electric hydrogen production has an efficiency of 60%-80%, and the methanation efficiency is approximately 60%. If hydrogen energy is used solely for methanation, it does not contribute to the flexible regulation of P2G. Therefore, hydrogen energy must be exploited. Hydrogen fuel cells (HFC) can theoretically have an energy utilization rate of up to 80% under cogeneration [28], converting hydrogen produced by P2G into electrical and thermal energy to tap the value of P2G hydrogen energy. Additionally, the IDR can regulate customer electricity consumption behavior and respond to system changes [29]. Reference [30] introduces IDR in a multi-objective optimal scheduling model that considers low carbon and economic operation, solving the model using particle swarm algorithms. Hence, this paper considers HFC configuration in the P2G hydrogen link based on cooperative scheduling of P2G and IDR to achieve flexibility of the P2G hydrogen link. Table 1 lised the comparison between the study and the above reviewed major publications in detail.

TABLE 1 Comparison Between This Study and Other Major Publication

This paper proposes an IES game model that addresses issues related to the integration of IDR and P2G hydrogen energy refinement utilization. To facilitate the cooperative relationship between LAs and to account for the master-slave game relationship between the IESP and LAA, the model employs Nash bargaining theory. Moreover, to ensure low carbon emissions during system operation, a multi-objective function is formulated for the IESP, and compromise planning theory is employed to obtain a feasible solution that balances economic and low-carbon objectives. In addition, conditional value-at-risk [31] is utilized to address the inherent uncertainty in system operations and ensure system stability. Our main contributions are summarized as follows:

1. First, the cooperative game between LAs is nested within the master-slave game of the LAA and IESP to address the competitive and cooperative relationships that exist simultaneously within the model. Moreover, to reconcile the conflicting issues of carbon emissions and economic concerns in IES operations, two separate objective functions were established for the IESP: an economic objective function and a low-carbon objective function.

2. Second, to enhance the flexibility of the system operations, a novel mechanism of P2G and IDR cooperation was introduced. This mechanism enables efficient utilization of hydrogen energy by integrating HFC into the P2G hydrogen energy chain.

3. Finally, to address the multi-objective hybrid game model, we combine the dichotomous distributed optimization algorithm with the alternating direction multiplier method. In addition, we employ compromise programming theory in the dichotomous method to facilitate the model solution.

The remainder of this paper is organized as follows. Section II describes the IES operational framework. Section III describes the IESP and LAA hybrid game models. The LAA Nash Bargaining Model is described in Section IV. The solution process is presented in Section V and case studies are discussed in Section VI. Finally, conclusions are presented in Section VII.

SECTION II. Operation Framework of IES

A. Basic Framework of IES

This study investigates an IES consisting of an IESP, a consortium of multiple LAs equipped with photovoltaic (PV) power production functions, and extraneous energy equipments, as shown in Fig. 1. The optimization procedure is carried out through a supply phase at the upper level and a response phase at the lower level. The IESP supplies electricity and heat to LAs within the energy price range set by external energy facilities. It optimizes equipment output, reduces carbon emissions, and improves energy efficiency and economic benefits under a stepped carbon trading mechanism. LAs participate in an integrated demand response based on the energy price information provided by the IESP. Moreover, each load in the alliance is equipped with a PV power generation function, enabling it to purchase energy from other LAs when its power supply is insufficient and sell surplus power, thereby maximizing the interests of both the alliance and individual loads.

FIGURE 1.

The architecture diagram of IES.

B. Energy Flow of IES

The internal energy flow chart of the IES is presented in Fig. 2, and comprises four primary modules: supply side, energy conversion side, storage side, and load side. The supply side encompasses the power grid, solar power plant, wind power plant, and gas industry. The energy conversion equipment comprises Cogeneration Combined Heat and Power (CCHP), P2G, CCS, Gas Boilers (GB), and HFC. The CHP system is composed of a Gas Turbine (GT), Organic Ranking Cycle (ORC), and Waste Heat Boiler (WHB), which facilitate a flexible response of heat and electricity on the supply side by supplying waste heat from the GT to the ORC and WHB. Electric storage, thermal storage, and hydrogen storage are included on the energy storage side. The load side comprises electric and thermal loads.

FIGURE 2.

The energy flow of IES.

SECTION III. IESP and LAA Hybrid Game Model

Owing to policy and technical support, China has emerged as one of the leading nations in distributed photovoltaic (PV) installations worldwide [32]. This shift has transformed traditional energy consumers into productive consumers who can reap benefits from the Integrated Energy System (IES) and from each other. To address this paradigm shift, a Stackelberg model that considers the cooperative game relationship among multiple load aggregators is proposed. The model aims to balance the interests of both the IES Operator (IESP) and Local Alliances (LAs) while meeting the individual rational needs of alliance members. The model was divided into two stages:

Stage 1 of the proposed model is a master-slave game between the IESP and LAA. In this stage, the IESP, as the leader, sets the prices of electricity and heat based on the supply and demand market information. It also introduces a stepped carbon trading mechanism to minimize carbon emissions and operating costs while maximizing the revenue and carbon reduction capacity of the system. The established energy prices are then sent to Stage 2.

In Stage 2, each LA is tasked with devising a plan to determine the optimal volume and price of inter-LA power trading, based on the power price set by the higher-level leadership. The objective is to facilitate efficient energy usage and subsequently communicate the finalized energy purchase strategy back to Stage 1.

A. P2G-CCS-HFC Synergistic Operation Part

Initially, the EL employed electrical energy to generate hydrogen gas. Subsequently, a fraction of hydrogen serves as a feedstock for a chemical reaction with CO₂ obtained from CCS, leading to the formation of natural gas, whereas another portion can be directly converted into electricity and heat via HFC for efficient utilization of hydrogen energy. Finally, the residual hydrogen was stored in a dedicated hydrogen storage tank to enable electrical energy storage. In summary, the energy-conversion model is described as follows:

1) EL Equipment

$$\begin{align*} \begin{cases} \displaystyle {P_{EL,H_{2}} (t)=\eta _{EL} P_{e,EL} (t)} \\ \displaystyle {P_{e,EL}^{\min } \le P_{e,EL} (t)\le P_{e,EL}^{\max }} \\ \displaystyle {\Delta P_{e,EL}^{\min } \le P_{e,EL} (t+1)-P_{e,EL} (t)\le \Delta P_{e,EL}^{\max }} \end{cases} \tag {1}\end{align*}$$

where $P_{e,EL} (t),P_{EL,H_{2}} (t)$ are the electric energy input to EL and the hydrogen energy output from EL at time t. $\eta _{EL}$ is the energy conversion efficiency of EL. $P_{e,EL}^{\max },P_{e,EL}^{\min }$ are the upper and lower electric energy limits of input EL respectively. $\Delta P_{e,EL}^{\max },\Delta P_{e,EL}^{\min }$ are the upper and lower climbing limits of EL respectively.

2) MR Equipment

$$\begin{align*} \begin{cases} \displaystyle {P_{MR,g} (t)=\eta _{MR} P_{H_{2},MR} (t)} \\ \displaystyle {P_{H_{2},MR}^{\min } \le P_{H_{2},MR} (t)\le P_{H_{2},MR}^{\max }} \\ \displaystyle {\Delta P_{H_{2},MR}^{\min } \le P_{H_{2},MR} (t+1)-P_{H_{2},MR} (t)\le \Delta P_{H_{2},MR}^{\max }} \\ \displaystyle {C_{MR,CCS} (t)=\eta _{CCS} P_{MR,g} (t)} \\ \displaystyle {P_{e,CCS} (t)=\lambda _{CCS} C_{MR,CCS} (t)} \\ \displaystyle {0\le P_{e,CCS} (t)\le P_{e,CCS}^{\max }} \end{cases} \tag {2}\end{align*}$$

where $P_{MR,g} (t),P_{H_{2},MR} (t),C_{MR,CCS} (t)$ are the natural gas power output from the MR, hydrogen energy input to the MR, and mass of CO₂ captured by the CCS respectively. $\eta _{MR}$ is the power conversion efficiency of MR. $P_{H_{2},MR}^{\max },P_{H_{2},MR}^{\min }$ are the lower and upper hydrogen power boundaries for the input MR. $\Delta P_{H_{2},MR}^{\max },\Delta P_{H_{2},MR}^{\min }$ are the upper and lower limits for MR. $\eta _{CCS}$ is the carbon-to-gas conversion factor of MR. $\lambda _{CCS}$ is the energy consumption coefficient of the carbon capture devices. $P_{e,CCS}^{\max }$ is the maximum amount of CO₂ captured by the CCS.

3) HFC Equipment

$$\begin{align*} \begin{cases} \displaystyle {P_{HFC,e} (t)=\eta _{HFC}^{e} P_{H_{2},HFC} (t)} \\ \displaystyle {P_{HFC,h} (t)=\eta _{HFC}^{h} P_{H_{2},HFC} (t)} \\ \displaystyle {P_{H_{2},HFC}^{\min } \le P_{H_{2},HFC} (t)\le P_{H_{2},HFC}^{\max }} \\ \displaystyle {\Delta P_{H_{2},HFC}^{\min } \le P_{H_{2},HFC} (t+1)-P_{H_{2},HFC} (t)\le \Delta P_{H_{2},HFC}^{\max }} \\ \displaystyle {{\kappa _{HFC}^{\min } \le P_{HFC,h} (t)} \mathord {\left /{{\vphantom {{\kappa _{HFC}^{\min } \le P_{HFC,h} (t)} {P_{HFC,e} (t)\le \kappa _{HFC}^{\max }}}}}\right. \hspace {-1.2pt} } {P_{HFC,e} (t)\le \kappa _{HFC}^{\max }}} \end{cases} \tag {3}\end{align*}$$

where $P_{HFC,e} (t),P_{HFC,h} (t),P_{H_{2},HFC} (t)$ are the electricity output from the HFC, heat energy from the HFC, hydrogen power input to the HFC respectively. $\eta _{HFC}^{e},\eta _{HFC}^{h}$ are the efficiencies of HFC conversion to electricity and heat energy respectively. $P_{H_{2},HFC}^{\max },P_{H_{2},HFC}^{\min }$ are the upper and lower limits of hydrogen energy input to the HFC. $\kappa _{HFC}^{\max },\kappa _{HFC}^{\min }$ are the upper and lower limits of the thermoelectric ratio of the HFC respectively.

B. Upper-Level Scheduling Model

The IESP, as the leader in the master-slave game, sets electricity and heat prices based on the LAA’s energy purchase strategy. To ensure low carbon emissions and optimal system operation, the IESP proposes a multiobjective optimization model that establishes economic and carbon emission objective functions. This model improves upon the previous single strategy, which relies on weighting a multi-objective problem into a single objective problem.

1) Economic Objective Function Considering Cvar

To account for the uncertainty of the scenery output, the Monte Carlo method [33] was employed to generate a sample set of prediction errors for the scenery output at each dispatching moment. The clustering method was used to reduce the scenarios to obtain the set of scenarios $N_{\omega }$ , which can be expressed as Eq. (4). The probability of occurrence for each scenario was $p_{\omega }$ .

$$\begin{equation*} N_{\omega } =\left \{{{P_{WT\omega } (t),P_{PV\omega } (t),\omega =1,2\cdots,n}}\right \} \tag {4}\end{equation*}$$ where $P_{WT\omega } (t),P_{PV\omega } (t)$ are the scenery outside of t time of the day scenes. Minimizing operating costs is the economic goal of the IESP, which consists of energy sales revenue, energy purchase costs, and energy storage operating costs.The objective function in scenario $\omega$ can be expressed as Eq. (5). $$\begin{equation*} \psi _{1}^{\omega } =M_{buy} -M_{sell} +M_{bat} +f_{CO_{2}}^{price} \tag {5}\end{equation*}$$ where $\psi _{1}^{\omega },M_{buy},M_{sell},M_{bat},f_{CO_{2} }^{price}$ are the energy purchase cost, energy sales revenue, energy storage operation cost, and carbon trading cost of the IESP respectively. The specific of $M_{buy},M_{sell},M_{bat},f_{CO_{2}}^{price}$ can be expressed as Eq. (6), (7), (8), (9). $$\begin{align*} M_{sell} & =\sum \limits _{t=1}^{24} {\left ({{\begin{array}{l} (price_{e} (t)\ast P_{e,load} (t)+ \\ price_{h} (t)\ast P_{h,load} (t)) \\ \end{array}}}\right)} \tag {6}\\ M_{buy}& =\sum \limits _{t=1}^{24} {\left ({{\begin{array}{l} (price_{grid} (t)\ast P_{e,buy} (t))- \\ (price_{e} (t)\ast P_{e,sell} (t))+ \\ (price_{g,buy} (t)\ast P_{g,buy} (t)) \\ \end{array}}}\right)} \tag {7}\\ M_{bat}& =\xi _{bat} \ast \sum \limits _{t=1}^{24} {\left ({{\begin{array}{l} (P_{ES,1}^{cha} (t)-P_{ES,1}^{dis} (t))+ \\[3pt] (P_{ES,2}^{cha} (t)-P_{ES,2}^{dis} (t))+ \\[3pt] (P_{ES,3}^{cha} (t)-P_{ES,3}^{dis} (t)) \\[3pt] \end{array}}}\right)} \tag {8}\\ f_{CO_{2}}^{price} & =\begin{cases} \displaystyle {\lambda E_{IES,t}} \qquad {E_{IES,t} \le l} \\ \displaystyle {\lambda (1+\alpha)(E_{IES,t} -l)+\lambda l} \\ \displaystyle {l\le E_{IES,t} \le 2l} \\ \displaystyle {\lambda (1+2\alpha)(E_{IES,t} -2l)+\lambda (2+\alpha)l} \\ \displaystyle {2l\le E_{IES,t} \le 3l} \\ \displaystyle {\lambda (1+3\alpha)(E_{IES,t} -3l)+\lambda (3+3\alpha)l} \\ \displaystyle {3l\le E_{IES,t} \le 4l} \\ \displaystyle {\lambda (1+4\alpha)(E_{IES,t} -4l)+\lambda (4+6\alpha)l} \\ \displaystyle {E_{IES,t} \ge 4l} \end{cases} \tag {9}\end{align*}$$ $P_{e,load} (t),P_{h,load} (t)$ are the powers required for electrical and thermal loads. $P_{ES,i}^{cha} (t),P_{ES,i}^{dis} (t)$ are charge power for energy storage i. ($i=1$ ,2,3, Electricity is 1, heat is 2, hydrogen is 3). $price_{e} (t),price_{h} (t),price_{grid} (t),price_{g,buy} (t)$ are the price of electricity for the IESP, price of heat for the IESP, electricity purchased from the grid, and price of gas purchased from the gas grid. $f_{CO_{2} }^{price}$ is a carbon trading method that sets the price of carbon credit limits at different levels or steps based on certain standards or thresholds. $\lambda,\alpha,l$ are the base price of carbon trading, length of carbon emission interval and price growth rate respectively.

Considering the adverse impact of uncertainty on the operating cost of the Independent Electricity System Operator (IESP), the objective function incorporates the Conditional Value-at-Risk (CVaR) metric to account for the operator’s risk aversion. CVaR reflects the expected loss level when the risk exceeds a specific threshold, whereas risk aversion refers to the tendency of individuals or institutions to take conservative measures or make corresponding decisions when the risk exceeds the threshold. The model expressions for this approach are as follows:

$$\begin{align*} \mathop {\mathrm {CVaR}}\limits _{\tau }& =\min _{\delta }\left \{{{\varepsilon -\frac {1}{1-\tau } \sum _{\omega =1}^{n}\left [{{\psi _{1}^{\omega }-\varepsilon }}\right ]^{+} p_{\omega }}}\right \} \tag {10}\\ \sum _{\omega =1}^{n} p_{\omega }& =1 \tag {11}\end{align*}$$ where $\left [{{\psi _{1}^{\omega } -\varepsilon }}\right]^{+}=\max (\psi _{1}^{\omega },0),\tau$ is the confidence level, $\varepsilon$ is the Value at Risk(VaR). Combining with Eq. (4–​11), objective function1 accounting for CVaR can be expressed as Eq. (12). $$\begin{equation*} \min (1-\rho)\sum \limits _{\omega =1}^{n} {\psi _{1}^{\omega }p_{\omega }} +\rho \left [{{\varepsilon -\frac {1}{1-\tau }\sum \limits _{\omega =1}^{n} {p_{\omega }} \left [{{\psi _{1}^{\omega }-\varepsilon }}\right ]^{+}}}\right ] \tag {12}\end{equation*}$$ where $\rho$ is the risk weighting used to weigh the relationship between economic efficiency and the level of risk.

2) Low Carbon Objective Function

To realize low-carbon system operation, an objective function with a low-carbon focus is formulated, as shown in Eq. (13).

$$\begin{equation*} Min\psi _{2} =E_{total,a} =E_{IES,a} -E_{IES} -\sum \limits _{t=1}^{T} {C_{MR,CCS} (t)} \tag {13}\end{equation*}$$ where $E_{\textrm {IES,a}},E_{\textrm {IES}}$ are the actual carbon emissions and initial carbon emission allowances in the stepped carbon trading mechanism respectively.

C. The Constraints on the Upper Model

1) GT Operation Constraint

$$\begin{align*} \begin{cases} \displaystyle {P_{GT,e} (t)=\eta _{GT}^{e} P_{g,GT} (t)} \\ \displaystyle {P_{GT,h} (t)=\eta _{GT}^{h} P_{g,GT} (t)} \\ \displaystyle {P_{g,GT}^{\min } \le P_{g,GT} (t)\le P_{g,GT}^{\max }} \\ \displaystyle {\Delta P_{g,GT}^{\min } \le P_{g,GT} (t+1)-P_{g,GT} (t)\le \Delta P_{g,GT}^{\max }} \end{cases} \tag {14}\end{align*}$$

where $P_{GT,e} (t),P_{GT,h} (t),P_{g,GT} (t)$ are the electrical and thermal power outputs from the GT and natural gas power input to the GT respectively. $\eta _{GT}^{e},\eta _{GT}^{h}$ are the gas-to-electricity and gas-to-heat efficiencies of the GT respectively. $P_{g,GT}^{\max },P_{g,GT}^{\min }$ are the upper and lower limits of the GT input power respectively. $\Delta P_{g,GT}^{\max },\Delta P_{g,GT}^{\min }$ are the upper and lower climbing limits of GT respectively.

2) WHB and ORC Operating Constraint

$$\begin{align*} \begin{cases} \displaystyle {P_{GT,h} (t)=P_{h,WHB} (t)+P_{h,ORC} (t)} \\ \displaystyle {P_{ORC,e} (t)=\eta _{ORC} P_{h,ORC} (t)} \\ \displaystyle {P_{WHB,h} (t)=(1-\eta _{WHB})P_{h,WHB} (t)} \\ \displaystyle {P_{h,ORC}^{\min } \le P_{h,ORC} (t)\le P_{h,ORC}^{\max }} \\ \displaystyle {\Delta P_{h,ORC}^{\min } \le P_{h,ORC} (t+1)-P_{h,ORC} (t)\le \Delta P_{h,ORC}^{\max }} \end{cases} \tag {15}\end{align*}$$

where $P_{h,WHB} (t),P_{h,ORC} (t),P_{WHB,h} (t),P_{ORC,e} (t)$ are the thermal power input to the WHB, thermal power input to theORC, thermal power output from the WHB, and electrical power output from the ORC respectively. $\eta _{ORC},\eta _{WHB}$ are the conversion efficiency of the ORC and heat loss rate of the WHB respectively. $P_{h,ORC}^{\max },P_{h,ORC}^{\min }$ are the upper and lower limits of the ORC input thermal power. $\Delta P_{h,ORC}^{\max },\Delta P_{h,ORC}^{\min }$ are the upper and lower climbing limits of ORC respectively.

3) CHP Operation Constraint

$$\begin{align*} \begin{cases} \displaystyle {P_{CHP,e} (t)=P_{GT,e} (t)+P_{ORC,e} (t)} \\ \displaystyle {\begin{array}{cccccccccccccccccccc} {P_{CHP,h} (t)=P_{WHB,h} (t)} \\ {P_{CHP,h} (t)\ge 0} \\ \end{array}} \end{cases} \tag {16}\end{align*}$$

where $P_{CHP,e} (t),P_{CHP,h} (t)$ are the output electric and thermal power of CHP.

4) GB Operation Constraint

$$\begin{align*} \begin{cases} \displaystyle {P_{GB,h} (t)=\eta _{GB} P_{g,GB} (t)} \\ \displaystyle {P_{g,GB}^{\min } \le P_{g,GB} (t)\le P_{g,GB}^{\max }} \\ \displaystyle {\Delta P_{g,GB}^{\min } \le P_{g,GB} (t+1)-\le P_{g,GB} (t)\le \Delta P_{g,GB}^{\max }} \end{cases} \tag {17}\end{align*}$$

where $P_{GB,h} (t),P_{g,GB} (t)$ are the thermal power output of the GB and gas power consumed by the GB respectively. $\eta _{\textrm {GB}}$ is the energy conversion efficiency of the GB gas to heat. $P_{g,GB}^{\max },P_{g,GB}^{\min }$ are the upper and lower limits of GB gas consumption power respectively. $\Delta P_{g,GB}^{\max },\Delta P_{g,GB}^{\min }$ are climbing upper and lower limits of GB respectively.

5) Energy Storage Operation Constraint

To achieve flexible operation of the system, electric, thermal, and hydrogen energy storage systems have been introduced. A unified modeling approach was adopted for these three energy storage systems, and the model expression can be found in Eq. (18).

$$\begin{align*} \begin{cases} \displaystyle {0\le P_{ES,n}^{cha} (t)\le B_{ES,n}^{cha} (t)P_{ES,n}^{\max } (t)} \\ \displaystyle {0\le P_{ES,n}^{dis} (t)\le (1-B_{ES,n}^{cha} (t))P_{ES,n}^{\max } (t)} \\ \displaystyle {S_{n} (1)=S_{n} (T)} \\ \displaystyle {S_{n}^{\min } \le S_{n} (t)\le S_{n}^{\max }} \\ \displaystyle {S_{n} (t+1)=S_{n} (t)+P_{ES,n}^{cha} (t)\eta _{ES,n}^{cha} -P_{ES,n}^{dis} (t)/\eta _{ES,n}^{dis}} \end{cases} \tag {18}\end{align*}$$ where $P_{ES,n}^{cha} (t),P_{ES,n}^{dis} (t)$ are the nth energy storage device charging and discharging powers respectively. $B_{ES,n}^{cha} (t)$ are the nth energy storage device charging state parameter. $S_{n}^{\max },S_{n}^{\min }$ are the upper and lower limits of the capacity of the nth energy storage device respectively. ($n=1$ , 2, 3, where electricity is 1, heat is 2, and hydrogen is 3)

6) Electrical Power Balance Constraint

$$\begin{align*} & \hspace {-1pc}P_{e,buy} (t)-P_{e,sell} (t) \\ & =P_{e,load,1} (t)+P_{e,load,2} (t)+P_{e,load,3} (t)~ \\ & \quad +P_{e,EL} (t)+P_{ES,1}^{cha} (t)-P_{ES,1}^{dis} (t) \\ & \quad -P_{WT} (t)-P_{PV} (t) \\ & \quad +P_{CHP,e} (t)-P_{HFC,e} (t) \\ & \quad +P_{CCS,e} (t) \tag {19}\end{align*}$$

where $P_{e,load,1} (t),P_{e,load,2} (t),P_{e,load,3} (t)$ are the electrical power consumed by load aggregators 1, 2, and 3 respectively. $P_{WT} (t),P_{PV} (t)$ are the electric power provided to the IES by the wind and PV respectively. $P_{CCS,e} (t)$ is the electrical power consumed by the CCS unit. $P_{CHP,e} (t)$ is the electrical power generated by CHP.

7) Thermal Power Balance Constraint

$$\begin{align*} & \hspace {-1pc}P_{CHP,h} (t)+P_{GB,h} (t)+P_{HFC,h} (t) \\ & =P_{ES,2}^{cha} (t)-P_{ES,2}^{dis} (t)+P_{h1} (t)+P_{h2} (t)+P_{h3} (t) \tag {20}\end{align*}$$

where $P_{h1} (t),P_{h2} (t),P_{h3} (t)$ are the thermal power purchased by LA1, 2, and 3.

8) Gas Power Balance Constraint

$$\begin{equation*} P_{g,buy} (t)=P_{CHP,g} (t)+P_{GB,g} (t)-P_{MR,g} (t) \tag {21}\end{equation*}$$

where $P_{g,buy} (t)$ is the gas power purchased from Gas industry for IESP.

9) Hydrogen Power Balance Constraint

$$\begin{align*} P_{EL,H_{2}} (t)& =P_{H_{2},MR} (t)+P_{H_{2},HFC} (t) \\ & \quad +P_{ES,3}^{cha} (t)-P_{ES,3}^{dis} (t) \tag {22}\end{align*}$$

D. Lower-Level Scheduling Model

Upon receiving energy price information from the IESP, the LAA optimizes its energy purchase strategy to determine the overall power purchase and the heat and power trading volume among the load aggregators in the alliance. The resulting energy purchase information is then transmitted to the IESP. The objective of each individual in the LAA is to minimize the operating costs, as represented by Eq. (23).

$$\begin{equation*} MinC_{i} =C_{buy,i} +C_{dr,i} -C_{trade,i} \tag {23}\end{equation*}$$ where $C_{buy,i},C_{dr,i},C_{trade,i}$ are the energy purchase expenses, demand response costs, and power trading revenues of LAi. The specific model expressions are as follows: $$\begin{align*} C_{buy,i}& =\sum \limits _{t=1}^{24} {\left ({{\begin{array}{l} (price_{e} (t)\ast P_{e,load,i} (t))+ \\ (price_{h} (t)\ast P_{h,load,i} (t)) \\ \end{array}}}\right)} \tag {24}\\ C_{dr,i} & =\xi _{e,tran} \sum \limits _{t=1}^{24} {\left |{{P_{e,tran,i} (t)}}\right |} +\xi _{e,cut} \sum \limits _{t=1}^{24} {P_{e,cut,i} (t)} \\ & \quad +\;\;\xi _{h,cut} \sum \limits _{t=1}^{24} {P_{h,cut,i}} (t)+\xi _{h,tran} \sum \limits _{t=1}^{24} {\left |{{P_{h,tran,i} (t)}}\right |} \tag {25}\end{align*}$$ where $\xi _{e,tran},\xi _{e,cut}$ are the value factors of the transferable and curtailable electric loads. $\xi _{h,cut},\xi _{h,tran}$ are the value coefficients of the transferable and curtailable heat loads. $P_{e,tran,i} (t),P_{e,cut,i} (t),P_{h,tran,i} (t),P_{h,cut,i} (t)$ are the powers of the ith load aggregator transferable electrical load, curtailable electrical load, transferable thermal load, and curtailable thermal load respectively. $$\begin{equation*} C_{trade,i} =\sum \limits _{t=1}^{24} {\sum \limits _{j=1,j\ne i}^{N} {\left ({{P_{i\to j,t}^{P2P} \ast C_{i\to j,t}^{P2P}}}\right)}} \tag {26}\end{equation*}$$ where $P_{i\to j,t}^{P2P}$ is the power transfer from LAi to LAj at time t (i, j is 1or2or3). $C_{i\to j,t}^{P2P}$ is the price of electricity traded between LAi and LAj at time t.

E. The Constraints on the Lower-Model

1) Electricity/Heat Load Demand Response Constraints

$$\begin{align*} \begin{cases} \displaystyle {L_{e,i} (t)=L_{e0,i} -P_{e,cut,i} (t)+P_{e,tran,i} (t)} \\ \displaystyle {L_{h,i} (t)=L_{h0,i} -P_{h,cut,i} (t)+P_{h,tran,i} (t)} \\ \displaystyle {0\le P_{e,cut,i} (t)\le \varpi _{e} L_{e0,i}} \\ \displaystyle {-\varpi _{e} L_{e0,i} \le P_{e,tran,i} (t)\le \varpi _{e} L_{e0,i}} \\ \displaystyle {0\le P_{h,cut,i} (t)\le \varpi _{h} L_{e0,i}} \\ \displaystyle {-\varpi _{h} L_{e0,i} \le P_{h,tran,i} (t)\le \varpi _{h} L_{e0,i}} \\ \displaystyle {\sum \limits _{t=1}^{24} {P_{e,tran,i} (t)=0}} \\ \displaystyle {\sum \limits _{t=1}^{24} {P_{h,tran,i} (t)=0}} \end{cases} \tag {27}\end{align*}$$

where $L_{e,i} (t),L_{h,i} (t)$ are the actual electrical and thermal loads of the LAi after the demand response. $L_{e0,i},L_{h0,i}$ are the initial electrical and thermal loads determined by LAi at the initial moment based on the pricing of the IESP. $\varpi _{e},\varpi _{h}$ are the electric and thermal load demand response factor.

2) Electrothermal Power Balance Constraint

$$\begin{align*} \begin{cases} \displaystyle P_{PV,i} (t)+P_{e,load,i} (t)=L_{e,i} (t)+P_{tran,ij} \\ \displaystyle \qquad +P_{tran,ik} (i\ne j\ne k,i,j,k=1or2or3) \\ \displaystyle {P_{h,load,i} (t)=L_{h,i} (t)} \\ \displaystyle {0\le P_{e,load,i,0} (t)\le P_{h,load} (t)} \end{cases} \tag {28}\end{align*}$$

where $P_{PV,i} (t)$ is the PV production for load aggregator i. In this study, it was assumed that all PV were fed into the grid.

3) Interaction Power Constraint

The exchange of electricity between load aggregators should be conducted within predefined limits to ensure an equitable volume of transactions.

$$\begin{align*} \begin{cases} \displaystyle {-P_{\max }^{P2P} \le P_{i\to j,t}^{P2P} \le P_{\max }^{P2P}} \\ \displaystyle {P_{i\to j,t}^{P2P} +P_{j\to i,t}^{P2P} =0} \end{cases} \tag {29}\end{align*}$$ where $P_{\max }^{P2P}$ is a limit to the maximum amount of interactions between LAs.

4) Electricity Price Constraint

$$\begin{equation*} 0.2\le C_{i\to j,t}^{P2P} \le price_{e} (\textrm {t}) \tag {30}\end{equation*}$$

SECTION IV. LAA Nash Bargaining Model

A bargaining game, an important theoretical basis in cooperative games, can effectively depict the internal mechanism of cooperation and achieve a fair and effective distribution of benefits while ensuring the minimum overall operating costs of the alliance and maximizing the benefits of each cooperative player [34]. To ensure fairness, an LAA Nash negotiation model is constructed in this study, with reference to the literature [35], using the solution of non-participation in cooperation as the negotiation rupture point, as shown in Eq. (31).

$$\begin{align*} \begin{cases} \displaystyle {Max\prod \limits _{\textrm {i=1}}^{N} {\left ({{C_{i} -C_{i}^{0}}}\right)}} \\ \displaystyle {s.t. C_{i} \ge C_{i}^{0} } \end{cases} \tag {31}\end{align*}$$ where $C_{i},C_{i}^{0}$ are the benefits obtained from LAi’s participation in negotiations and the point of breakdown of the negotiations respectively. Referring to the literature [36], the Nash negotiation model is transformed into solving two subproblems of coalition operation cost minimization and cooperation benefit distribution, and then solving the two subproblems separately.

A. Minimizing Alliance Operating Costs

$$\begin{align*} \begin{cases} \displaystyle MinC_{LAA} =C_{buy,1} +C_{dr,1} +C_{buy,2} +C_{dr,2} \\ \displaystyle \qquad +C_{buy,3} +C_{dr,3} \\ \displaystyle {s.t(27\sim 30)} \end{cases} \tag {32}\end{align*}$$

where $C_{LAA}$ is the alliance operating cost. The gains from LA participation in the cooperative game are not included in Eq. (32) because the revenue from transactions between individuals will not affect the total cost of energy consumption of the entire coalition when the LAs reach consensus on cooperation.

B. Distribution of Cooperation Revenue

$$\begin{align*} \begin{cases} \displaystyle {Max\prod \limits _{i=1}^{N} {\left [{{\left ({{P_{i\to j,t}^{P2P^{\ast }} \ast C_{i\to j,t}^{P2P}}}\right)-C_{i}^{LA^{\ast }} +C_{i}^{0}}}\right ]}} \\ \displaystyle {s.t\left ({{P_{i\to j,t}^{P2P^{\ast }} \ast C_{i\to j,t}^{P2P}}}\right)-C_{i}^{LA^{\ast }} +C_{i}^{0} \ge 0} \\ \displaystyle {\begin{array}{l} C_{i}^{LA^{\ast }} =C_{buy,i}^{\ast } +C_{dr,i}^{\ast } \\ 0.2\le C_{i\to j,t}^{P2P} \le price_{e} (\textrm {t}) \\ \end{array}} \end{cases} \tag {33}\end{align*}$$

where $P_{i\to j,t}^{P2P^{\ast }},C_{i}^{LA^{\ast }},C_{buy,i}^{\ast },C_{dr,i}^{\ast }$ are optimal interactive power, optimal operating cost of each LA, and optimal energy purchase expenditure, and the optimal demand response cost obtained in the subproblem (P1). For the convenience of the solution, the Eq. (33) is taken logarithmically, and because the natural logarithm is a strictly monotonically increasing convex function, the maximum problem can be converted to a minimum problem to facilitate the solution. Thus Eq. (33) can be converted into Eq. (34).

$$\begin{align*} \begin{cases} \displaystyle {Max\sum \limits _{i=1}^{N} {\ln \left [{{\sum \limits _{t=1}^{24} {\left ({{P_{i\to j,t}^{P2P^{\ast }} \ast C_{i\to j,t}^{P2P}}}\right)} -C_{i}^{LA^{\ast }} +C_{i}^{0}}}\right ]}} \\ \displaystyle {s.t\sum \limits _{t=1}^{24} {\left ({{P_{i\to j,t}^{P2P^{\ast }} \ast C_{i\to j,t}^{P2P}}}\right)} -C_{i}^{LA^{\ast }} +C_{i}^{0} \ge 0} \\ \displaystyle {\begin{array}{l} C_{i}^{LA^{\ast }} =C_{buy,i}^{\ast } +C_{dr,i}^{\ast } \\ 0.2\le C_{i\to j,t}^{P2P} \le price_{e} (\textrm {t}) \\ \end{array}} \end{cases} \tag {34}\end{align*}$$

SECTION V. Solution Process

This study constructs a complex multi-objective mixed game model, that is solved in two stages. The dichotomous nested compromise planning theory is employed to solve the stage 1 multi-objective master-slave game model, iteratively obtaining the optimal compromise solution of IESP and the optimal solution of the LAA. In stage 2, based on the solution of stage 1, the electric energy transaction volume between the LAs obtained in stage 1 is input, and the final output result is obtained using the ADMM algorithm for the distributed solution. The flow chart outlining this process is shown in Fig. 3.

FIGURE 3.

Algorithm flow chart.

A. Multi-Objective Problem Solution

To address the issues of poor convergence and slow solution speed associated with traditional multi-objective optimization algorithms, a fuzzy compromise planning method was employed to balance the conflicting carbon emission and operation cost objectives in the IESP objective function. The algorithm is formulated as follows:

• Step 1

  First, we compute the Pareto frontier solution set for our multi-objective optimization model by measuring the distance between the optimal values of the economic objective function and the low-carbon objective function, concerning a standard value determined by Eq. (35).

  $$\begin{align*} \begin{cases} \displaystyle {Min\psi =\delta } \\ \displaystyle {w_{1} +w_{2} =1} \\ \displaystyle {\delta \ge (\psi _{1} -\psi _{1,\min })(w_{1} /\psi _{1,\min })} \\ \displaystyle {\delta \ge (\psi _{2} -\psi _{2,\min })(w_{2} /\psi _{2,\min })} \end{cases} \tag {35}\end{align*}$$ where $\psi _{1,\min },\psi _{2,\min }$ are the optimal values for IESP’s economic and low carbon objectives. $\delta$ is the Chebyshev Distance.

• Step 2

  Next, we standardize the dimensions of the two objective function values by applying Eq. (36), which maps these to the [0, 1] interval.

  $$\begin{align*} \mu _{k}^{n} =\begin{cases} \displaystyle {1,} & {\psi _{k}^{n} \le \psi _{k,\min }} \\ \displaystyle {\frac {\psi _{k,\max } -\psi _{k}^{n}}{\psi _{k,\max } -\psi _{k,\min } },} & {\psi _{k,\min } \le \psi _{k}^{n} \le \psi _{k,\max }} \\ \displaystyle {0,} & {\psi _{k}^{n} \ge \psi _{k,\min }} \end{cases} \tag {36}\end{align*}$$ where $\psi _{k,\min },\psi _{k,\max }$ are the lower and upper limits of the number of Pareto solutions for the kth objective function respectively. $\mu _{k}^{n}$ is the nth iterative solution to the kth objective function.

• Step 3

  Finally, the optimal compromise solution was determined using Eq. (37).

  $$\begin{align*} \begin{cases} \displaystyle {\mu ^{n}=\min (\mu _{1}^{n},\mu _{2}^{n});\forall n=1,\ldots,N} \\ \displaystyle {\mu ^{\max }=\max (\mu ^{1},\mu ^{2},\ldots,\mu ^{N-1},\mu ^{N})} \end{cases} \tag {37}\end{align*}$$

B. Master-Slave Game Problem Solution

The proposed master-slave game model consists of two levels. In the upper level, the IESP selects the optimal operation strategy based on the energy purchase strategy of the LAA, with the objective of achieving economic and low-carbon optimization. At the lower level, the LAA minimizes its operational cost and updates its own operational policy based on the upper-level policy, aiming to achieve Stackelberg equilibrium through iterative optimization.

The Stackelberg model is a two-layer optimization model that is commonly solved using KKT conditions or heuristic algorithms. However, applying the KKT condition requires access to all LA constraints, which can compromise the privacy of the individual agents. However, the heuristic algorithm tends to converge slowly, leading to prolonged solution times in this complex two-layer optimization model. To overcome these challenges, we employ a bisection-based heuristic method [37] to solve the master-slave game model. Fig. 4 shows the bisection-based heuristic solution process. $x_{k,t}$ is the energy price of the kth iteration, which in this study represents $price_{e} (t),price_{h} (t)$ .

FIGURE 4.

The bisection-based heuristic method solution process.

C. Cooperative Game Problem Solution

Because the electric energy trading prices of LAi and LAj in Eq. (33) are coupled. Therefore $C_{i\to j,t}^{P2P},C_{j\to i,t}^{P2P}$ must be decoupled as shown in Eq. (38).

$$\begin{equation*} C_{i\to j,t}^{P2P} -C_{j\to i,t}^{P2P} =0 \tag {38}\end{equation*}$$ where $C_{i\to j,t}^{P2P},C_{j\to i,t}^{P2P}$ are the expected trading tariffs for LAi and LAj, respectively after solving for coalition operation cost minimization.

Upon completing the decoupling transformation, we utilized the Alternating Direction Method of Multipliers (ADMM) algorithm to solve the cooperative revenue sharing problem. The solution procedure comprises the following steps:

• Step 1

  Establish the extended Lagrangian function of Eq. (38)

  $$\begin{align*} {\begin{cases} MinC_{i}\! =\!-\ln \left [{{\! {\sum \limits _{t=1}^{24} {\sum \limits _{j=1,j\ne i}^{N} {\!\left ({{ {P_{i\to j,t}^{P2P^{\ast }} \ast C_{i\to j,t}^{P2P}} \!}}\right)-C_{i}^{LA^{\ast }}\!+\!C_{i}^{0}}}} \!}}\right ] \\ +\sum \limits _{t=1}^{24} {\left [{{\lambda _{i\to j} \left ({{C_{i\to j,t}^{P2P} -C_{j\to i,t}^{P2P}}}\right)\!+\!\frac {\rho }{2}\mathop {\parallel C_{i\to j,t}^{P2P} -C_{j\to i,t}^{P2P} \parallel }\nolimits _{2}^{2}}}\right ]} \\ s.t\;\;\sum \limits _{t=1}^{24} {\sum \limits _{j=1,j\ne i}^{N} {\left ({{P_{i\to j,t}^{P2P^{\ast }} \ast C_{i\to j,t}^{P2P}}}\right)}} -C_{i}^{LA^{\ast }} +C_{i}^{0} \ge 0 \\ 0.2\le C_{i\to j,t}^{P2P} \le price_{e} (\textrm {t}) \\ \end{cases}} \tag {39}\end{align*}$$ where $\rho$ is Punishment Parameters. $\lambda _{i\to j}$ is the Lagrangian multiplier between LAi and LAj.

• Step 2

  To preserve the privacy of individual LAs, each agent calculates its trading price strategy and shares only the price information with others. Here, we denote the number of iterations as k. Within each iteration, the following steps are executed:

  LAi updates its decision variables ($C_{i\to j,t}^{P2P} (k+1))$ according to Eq. (40). The other LAj updates its own decision variables $C_{j\to i,t}^{P2P} (k+1)$ based on the updated $C_{i\to j,t}^{P2P} (k+1)$ .

  $$\begin{align*} \begin{cases} \displaystyle {C_{i\to j,t}^{P2P} (k+1)=\arg \min \left ({{C_{i\to j,t}^{P2P} (k),C_{j\to i,t}^{P2P} (k),\lambda _{i\to j} (k)}}\right)} \\[.5pc] \displaystyle {C_{j\to i,t}^{\textrm {P2P}} (k+1)=\arg \min \left ({{C_{j\to i,t}^{\textrm {P2P}} (k),C_{i\to j,t}^{\textrm {P2P}} (k),\lambda _{j\to i} (k)}}\right)} \end{cases} \tag {40}\end{align*}$$

• Step 3

  After completing a round of iterations, the Lagrange multiplier is updated in accordance with Eq. (41), as prescribed by the principles of scientific computing.

  $$\begin{equation*} \lambda _{i\to j} (k+1)=\lambda _{i\to j} (k)+\rho \ast (C_{i\to j,t}^{P2P} -C_{j\to i,t}^{P2P}) \tag {41}\end{equation*}$$

• Step 4

  Update the number of iterations $k=k+1$ .

• Step 5

  Determine the convergence according to Eq. (42).

  $$\begin{align*} \begin{cases} \displaystyle {\sum \limits _{t=1}^{T} {\sum \limits _{i=1,j\ne i}^{N} {\mathop {\parallel C_{i\to j,t}^{P2P} (k+1)-C_{j\to i,t}^{P2P} (k+1)\parallel }\nolimits _{2}^{2} \le \delta }}} \\ \displaystyle {k\ge k_{\max }} \end{cases} \tag {42}\end{align*}$$ where $\delta$ is convergence accuracy.

• Step 6

  If Eq. (42) is satisfied, and the iteration process is terminated. However, if it is not satisfied, the algorithm returns to Step 2 and recalculates until either convergence is achieved or the maximum allowed number of iterations is reached, in accordance with standard scientific computational practices.

SECTION VI. Case Study

A. Case Description

a multi-objective hybrid game model was used to consider the utilization of both IDR and P2G hydrogen refinements and the rationality of the proposed model was verified through simulation and analysis. The algorithm was implemented in MATLAB using the CPLEX solver, employing the dichotomous method with embedded fuzzy compromise programming theory, and combined with the ADMM solution model. The scheduling time was one day, divided into 24 time periods. Fig. 5 shows the predicted maximum output curves of photovoltaic power and wind power.

FIGURE 5.

The predicted maximum output curves of wind power and photovoltaic power.

B. Comparative Analysis of Programs

In this paper, we study the synergistic optimization mechanism between the IESP and LAA by considering the electric energy trading between different LAs. To explore the effectiveness of the proposed multi-objective hybrid game model, the following three cases were set up for simulation:

• Case 1:

  The model compromise considers the two objective functions of carbon emissions and operating costs of the IESP, the cooperative game among LAA alliance members, and the master-slave game between the IESP and LAA.

• Case 2:

  The model compromises the two objective functions of carbon emissions and operating costs of IESP, does not consider the cooperative game among LAA coalition members, and considers the master-slave game between the IESP and LAA.

• Case 3:

  The model considers only the operating cost objective function of IESP, the cooperative game among LAA consortium members, and the master-slave game between the IESP and LAA.

When comparing Case 1 to Case 2 in Table 2, it is evident that Case 1 considers the cooperative game among Local Authorities (LAs). Consequently, the total operating cost of the LAA in Case 1 was 9.804% lower than that in Case 2, although the operating cost of the IESP increased by 15.32%. However, this increase is due to the introduction of Nash negotiation theory by each LA, which ultimately results in electricity trading and the realization of benefits from cooperation. Specifically, through electricity trading, LAs could reap 3441.1962, 1268.1483, and 1190.7273 respectively. By ensuring a reduction in the overall operating cost of the alliance, the interests of the cooperative participants were protected, and the distribution of cooperative benefits was fair.

TABLE 2 Optimization Results of Different Cases

According to the results presented in the table, a comparison between Cases 1 and 3 shows that compromising the two objective functions of carbon emissions and operating costs of the IESP led to a 14.87% increase in carbon emissions in Case 1, whereas the total operating cost of the LAAs decreased by 7.35%. The decrease in the operating cost of LAA was due to the IESP bearing more carbon trading costs, resulting in higher electricity prices during the peak consumption periods. However, cooperation among LAs reduced their dependence on IESP, leading to a decrease in the IESP’s energy sales revenue.

In summary, the comparative analysis highlights the importance of considering cooperative games among LAs to fully leverage their multi-energy complementary characteristics and maximize the benefits for all stakeholders. The introduction of a multi-objective function in the optimization process allows for a compromise optimization of low carbon and economy, without adversely impacting the interests of the source load. This study provides empirical evidence supporting the effectiveness of the proposed multi-objective hybrid game model.

C. Analysis of Scheduling Results

1) Analysis of Pricing Results

In the existing one-master-multi-slave model for the energy market, there is a need to establish pricing for electricity sold by the IESP to the LAA, pricing for heat sold, and pricing for electricity interaction among LAs. Consequently, the pricing process is divided into two stages: stage 1 determines the energy price when the IESP engages in the master-slave game with the LAA, and stage 2 determines the price for electricity interaction among LAs. This two-stage process is essential for establishing a fair and efficient pricing mechanism in the energy market.

Fig. 6 illustrates the optimal pricing strategy obtained for stage 1, which involves the IESP setting energy prices based on feed-in tariffs and feed-in heat tariffs in the previous day phase, while the LAA adjusts its energy consumption strategy in real-time at each moment based on the IESP’s tariff plan. This allows the LAA to maximize its profits by scheduling multiple LAs. Furthermore, the IESP redefines the intraday pricing strategy based on energy purchased from multiple sources. The upper and lower levels of the pricing strategy make independent decisions, and their benefits iteratively optimize each other until an optimal pricing strategy is found. The Fig. 5 reveals that after the introduction of the IDR, the actual service prices set by the IESP are controlled within the thresholds of the upper tariffs for electricity and heat. In addition, the actual service prices are lower during periods when the upper energy prices are higher. This outcome can be attributed to the LAA’s implementation of aggressive peaking measures, which enable it to optimize profits.

FIGURE 6.

The Pricing diagram of master-slave game.

Fig. 7 displays the trading tariff between LAs after optimization, which highlights that each LA possesses independent and autonomous pricing power within the threshold of the electricity price set by scheme 1. The feed-in tariff is maintained within the peak range to foster collaboration among LAs through electricity trading, thereby ensuring individual rationality and fairer rights to trade with the IESP.

FIGURE 7.

The pricing diagram between LAs.

2) Analysis of Optimization Results

To evaluate the effectiveness of the new P2G and IDR cooperative operation mechanism for source-load double-layer scheduling optimization, we simulated the optimized scheduling results of the IESP, LA1, LA2, and LA3 in Case 1, as illustrated in Fig. 8, 9, 10, and 11. Fig. 8(a) reveals that during the peak hours of wind power output at night and the peak hours of photovoltaic output in the afternoon, the renewable energy sources generate hydrogen through EL equipment and consume all renewable energy. The hydrogen produced by electrolysis is preferentially transferred to the HFC for thermoelectric production because of its high energy efficiency and reduced intermediate energy conversion compared to the natural gas production process by the MR. The surplus hydrogen energy is utilized by MR to produce natural gas, which is then supplied to the GB and CHP for energy provision. In addition, excess heat energy is stored during periods of low customer heat load and high unit output. Furthermore, CCS as shown in Fig. 8(d) achieves an integrated energy system’s low-carbon operation by consuming electrical power to capture CO₂ for MR. Finally, hydrogen is stored in the hydrogen storage tank when the natural gas supply in the system is sufficient to enable electrical energy storage.

FIGURE 8.

The optimized scheduling results of IESP.

FIGURE 9.

Scheduling optimization results after LA1 participates in IDR.

FIGURE 10.

Scheduling optimization results after LA2 participates in IDR.

Fig. 9 illustrates the impact of LA1’s participation in carbon trading and IDR on the electricity and heat load curves. In graph (a), the optimization results of the electricity load scheduling are presented under the optimal price strategy of the IESP, which involves the IDR. Before optimization, the peak electricity load periods are mainly concentrated between 7-10 h and 18-22 h. However, after participating in the IDR and being subjected to time-sharing tariffs, electricity loads are shifted from peak to valley periods.In addition, a certain degree of reduction was achieved through the carbon trading mechanism. In graph (b), the peak heat load periods before optimization are mainly distributed between 1-5 h and 17-24 h. After participating in the IDR, the heat load was reduced through load reduction and vertical transfer, based on the high and low heat prices. Fig. 10 and Fig. 11 demonstrate that the introduction of the IDR can reduce the peak-to-valley load difference, lower carbon emissions, promote energy supply and demand balance, and enable the time shift of energy demand in both horizontal and vertical dimensions.

FIGURE 11.

Scheduling optimization results after LA3 participates in IDR.

D. Algorithm Solution Analysis

This study proposes a two-stage solution to address the optimization problem in the context of the IESP and LAA. In the first stage, a dichotomous approach is used to develop the two-layer optimization model, where the embedded tradeoff programming theory is utilized to identify the solution that optimizes the carbon emission and economic tradeoff of the IESP. The second stage involves solving the cooperative game problem between the LAs using the ADMM algorithm.

Fig. 12 illustrates the iterative convergence of the LAA running cost, which is obtained by the dichotomous method is employed to solve the master-slave game upper-lower maximization problem. Fig. 12 indicates that the dichotomous method exhibits excellent convergence characteristics and efficiently solves the upper and lower level problems, while also considering the co-optimization of each subject. This is facilitated by the dichotomous method’s ability to determine whether the results of two adjacent iterations are the same during the solution process and converge rapidly to the final iterative results.

FIGURE 12.

The iterative convergence of the LAA running cost.

Second, to validate the effectiveness of the proposed fuzzy compromise programming theory in solving the multi-objective mixed game model, two different approaches were employed: the fuzzy compromise programming theory and MOEA/D multi-objective algorithm. The results obtained using the two approaches are illustrated in Fig. 13, which shows the set of Pareto solutions. Fig. 13 demonstrates that the results obtained by using fuzzy compromise programming theory are characterized by a smoother distribution than those obtained by MOEA/D, and better achieve the trade-off between carbon emissions and operating costs. A comparison of the solving time of the two algorithms is shown in Table 3. It can be observed from Table 3 that the compromise planning method is faster to solve.

TABLE 3 Optimization Results of Different Algorithms

FIGURE 13.

Pareto solution sets of different algorithms.

Finally, Fig. 14 shows the iterations of each LA obtained by implementing the ADMM algorithm. The algorithm exhibited excellent convergence performance, as it converged to within $10^{-3}$ of the residuals after 15 iterations. Furthermore, the computation time was 21 s, indicating that the ADMM optimization algorithm was computationally efficient in solving the model proposed in this study. Notably, the algorithm also considers privacy protection and day-ahead optimal scheduling requirements of each LA.

FIGURE 14.

The iterations of each LA.

E. Analysis of Risk Results

To assess the impact of renewable energy uncertainty on operators’ risks, we conducted an analysis by gradually increasing the risk weights, and the results are shown in Fig. 15. Fig. 15 shows that as the risk weight increases, the operating costs of the IESP increase. This is because a higher risk weight indicates that the IESP is more cautious in avoiding potential risks, which leads to a compromise in overall economics. Additionally, as the risk weight increases, the decrease in CVaR leads to a greater increase in IESP operating costs. Therefore, to minimize the risk of revenue fluctuation arising from renewable energy uncertainty for the IESP, a trade-off must be established between risk weighting, IESP operating costs, and CVaR. This will allow for a reduction in IESP operating costs while avoiding excessive risk.

FIGURE 15.

Relation curve between IESP operating cost and CVaR under different risk factors.

SECTION VII. Conclusion

1. The proposed multi-objective hybrid game model can effectively balance the economic and low-carbon requirements of IES operation, and fully leverage the multi-energy complementary characteristics of different LAs to optimize the benefits of each subject.

2. During the dispatching process, the energy price of the IESP and the electricity trading price between LAs are established independently. By realizing the coordinated operation of the IESP and LAA, cooperation and revenue distribution among LAs are achieved, thereby ensuring that LAs possess more equitable trading rights with the IESP. Furthermore, it has been demonstrated that new P2G and IDR technologies can work together effectively, enhancing the efficient use of renewable energy and providing greater flexibility for IES dispatch and management. It is important to note that the uncertainty associated with the energy landscape brings revenue risks to the system’s operation; hence, it is necessary to carefully weigh the risk weight, IESP operation cost, and CVaR in order to achieve a balance.

3. The proposed model is solved using a dichotomous method that is embedded with compromise programming theory and combined with the ADMM algorithm. The results demonstrate that the proposed algorithm can efficiently solve the complex model, which verifies the effectiveness of the proposed algorithm. Furthermore, a comparison between compromise planning theory and the Pareto solution of MOEA/D shows that compromise planning theory can achieve a smoother and quicker compromise between carbon emissions and operating costs.

   The model proposed in this study considers only the cooperative and competitive relationships between LAs and IESP, as well as between load aggregators. The impact of user electricity consumption behavior on the interests of each party was not consiered. In future research, the proposed algorithm can be combined with simulated annealing optimizers, which have already been widely used in multi-objective models, to achieve better solution performance. References [38] and [39] provide more information on this topic.