Enhancing Regional Integrated Energy Systems Through Seasonal Hydrogen Storage: Insights from a Stackelberg Game Model

Abstract

This study enhances regional integrated energy systems by proposing a Stackelberg planning–operation model with seasonal hydrogen storage, addressing source–network separation. An equilibrium algorithm is developed that integrates a competitive search routine with mixed-integer optimization. In the price–energy game framework, the hydrogen storage operator is designated as the leader, while energy producers, load aggregators, and storage providers act as followers, facilitating a distributed collaborative optimization strategy within the Stackelberg game. Using an industrial park in northern China as a case study, the findings reveal that the operator’s initiative results in a revenue increase of 38.60%, while producer profits rise by 6.10%, and storage-provider profits surge by 108.75%. Additionally, renewable accommodation reaches 93.86%, reflecting an absolute improvement of 20.60 percentage points. Total net energy imbalance decreases by 55.70%, and heat-loss load is reduced by 31.74%. Overall, the proposed approach effectively achieves cross-seasonal energy balancing and multi-party gains, providing an engineering-oriented reference for addressing energy imbalances in regional integrated energy systems.

1. Introduction

1.1. Motivations and Literature Review

The accelerating global energy crisis and climate warming have made low-carbon and efficient renewable energy utilization a key direction for future energy development [1,2,3,4]. High-penetration renewable integration is a defining feature of future power systems [5,6,7,8,9]; however, the stochasticity, intermittency, and pronounced seasonal mismatches of renewable generation pose significant challenges to maintaining power balance [10,11].

Integrated energy systems (IESs) exploit complementarities among multiple energy carriers to enable cascade utilization and enhance renewable accommodation [12]. A substantial body of work has examined optimal IES operation. Lisa et al. developed a low-carbon economic operation model that coordinates cogeneration, carbon capture, and power-to-gas (P2G) units [13]. A multi-time-scale coordinated control strategy for IES with hybrid energy storage was proposed in [14]. Liu Y. et al. introduced a bidding strategy based on spectral risk measures for IES participation in short-term markets [15]. Kang L. et al. constructed an IES coupled with an organic Rankine cycle and P2G technology, formulating a multi-objective optimization framework spanning economic, energy, and environmental criteria [16]. Notably, these studies often treat the energy producer (EP), energy system operator (ESO), load aggregator (LA), and energy storage provider (ESP) as a single centralized decision entity. In practice, these stakeholders belong to different sectors and have access only to local information. Their interactions are therefore strategic, involving distributed decision making and limited information exchange.

Game-theoretic approaches have consequently become central to modeling multi-party interactions in IES. References [17,18,19] use Stackelberg games to analyze leader–follower dynamics between electricity retailers and users. Liu N. et al. designed operator pricing and producer–consumer selection models for community energy networks with combined cooling, heating, and power (CCHP), proposing a distributed management method grounded in Stackelberg games [20]. Wang H. et al. modeled the ESO as leader and EP/LA as followers, formulating a single-leader–multiple-follower (SLMF) distributed collaborative optimization framework [21]. Sheikhi A. et al. incorporated energy supply network participation to leverage inter-microgrid complementarities and reduce operating costs [22]. Li G. et al. further proposed an SLMF-based differential pricing model in which the system operator leads and heterogeneous producers/consumers follow, thereby guiding consumption behavior [23]. Summarizing this line of work, network-side system operators typically act as price-setting leaders, while source, load, and storage entities adjust their strategies in response—yet the operator’s profit model remains largely price spread driven and thus economically fragile in the face of structural imbalances.

In practice, weak connections to external heating networks restrict large-scale heat trading [21]. Without greater flexibility on the network side, seasonal mismatches between renewable generation and energy demand intensify multi-energy imbalances [24,25]. Few existing studies have explored long-duration, inter-seasonal storage as a structural solution. Seasonal storage technologies—mainly seasonal heat and hydrogen storage—provide a promising way to address these issues. As a clean and efficient energy carrier, hydrogen shows great potential in sustainable energy transitions [26]. Incorporating hydrogen storage into regional IESs helps mitigate seasonal mismatches and increases renewable utilization [27]. Furthermore, implementing hydrogen production, storage, and utilization on the network side improves energy balancing and diversifies the operator’s revenue sources across electricity, hydrogen, and heat streams.

However, most game-theoretic IES studies [20,21,22], including those using Stackelberg models, focus only on short-term operations or price-based interactions. They rarely capture long-term storage effects or the joint optimization of capacity and pricing under limited information. These gaps motivate the development of a more comprehensive Stackelberg framework that reflects multiscale and multiparty dynamics—an approach pursued in this study.

A central modeling challenge for seasonal hydrogen storage is to handle cross-scale optimization in a coherent way that links inter-day and inter-season dynamics with intra-day operations. Li L. et al. investigated inter-seasonal complementarity enabled by hydrogen transmission networks and achieved effective reductions in storage costs [28]. Renaldi R. et al. introduced seasonal storage devices within multi-time-grid full-chronology formulations [29]. Pan G. et al. modeled seasonal hydrogen storage with a stochastic–robust approach to address load uncertainty [30] and later optimized operating costs and the levelized cost of hydrogen in an electricity–hydrogen integrated energy system with seasonal storage [31]. However, full-year chronologies greatly increase the computational scale [29], and representative-day methods that ignore the carryover from the last typical day fail to capture essential seasonal storage characteristics [30,31].

1.2. Contributions

Building on existing research, this paper investigates the multi-participant game mechanism in an IES and the optimal configuration of a seasonal hydrogen storage system (SHSS) for the energy system operator (ESO) under a Stackelberg framework, with the goal of enhancing seasonal supply–demand balance and operator profits. The main contributions are as follows:

(1)

Methodological contribution: ESO-side SHSS with representative-season modeling and annual consistency. We develop an ESO-side configuration paradigm that supports long-duration energy storage across multiple carriers. The method integrates seasonal SHSS with both short-term and long-term hydrogen storage. Long-term storage is represented using representative seasonal days combined with cross-day state recursion and a season-end consistency constraint. This formulation preserves the annual energy balance without requiring full-year chronological data, while daily single-mode operation maintains distinct seasonal characteristics. Compared with traditional full-chronology approaches, it improves computational scalability and resolves the carryover problem found in representative-day models. As a result, the proposed method provides a tractable framework for enhancing economic performance, renewable integration, and system-level energy balance.

(2)

Mechanism and market design: SLMF Stackelberg coupling of price, capacity, and operation. We formulate a single-leader–multiple-follower Stackelberg game, where the ESO acts as the leader while the EP, LA, and ESP are the followers. The model endogenizes the interdependence among price signals, capacity planning, and operational strategies under limited information exchange. The resulting equilibrium provides the optimal pricing scheme for the operator, the sizing and operation of the SHSS, as well as the followers’ demand responses and device set-points. This mechanism captures multiple concurrent benefits, including higher operator revenue, improved renewable energy utilization, and reduced system imbalance and heat loss, while also explaining the absence of fuel cell adoption under current cost and efficiency conditions.

(3)

Algorithmic contribution: distributed CSA–MIP/QP equilibrium solution with convergence and practicality. We construct a Stackelberg equilibrium algorithm that combines a competitive search algorithm with mixed-integer and quadratic programming subproblems. The procedure converges within acceptable wall-clock time while preserving data privacy through minimal signal exchange. Case studies demonstrate improved multi-energy balance, increased revenues for market participants, and high renewable accommodation.

Unlike most previous game-theoretic IES studies [20,21,22], which mainly capture price–load or price–generation interactions in a single operational layer, the proposed framework establishes an SLMF game that integrates price, capacity, and operation decisions under seasonal hydrogen storage coupling. The ESO simultaneously determines pricing and sizing strategies, while the EP, ESP, and LA respond with operational adjustments. This hierarchical formulation thus bridges planning and operational timescales, incorporates long-duration storage dynamics, and preserves data privacy through a distributed CSA–MIP/QP algorithm. In summary, this hierarchical and distributed game-theoretic framework advances existing IES studies by capturing multi-agent interactions, long-duration storage coupling, and seasonal energy dynamics within a computationally tractable setting.

1.3. Paper Organization

The remainder of this paper is structured as follows: Section 2 provides an overview of the multi-subject energy system architecture with SHSS. Section 3 introduces the operation model of each subject. Section 4 constructs a multi-subject Stackelberg game framework and its solution method. Section 5 conducts numerical simulations to demonstrate the applicability and effectiveness of the proposed model. Finally, Section 6 concludes the paper, summarizing the key findings.

2. Multi-Subject Energy System Architecture with SHSS

In this paper, a novel IES including an SHSS integrated with an ESO, EP, ESP, and LA is proposed, termed IES-SHSS. The IES-SHSS uses the ESO as a link and the EP system as a foundation to interconnect external energy networks, achieving economic and efficient energy supply, and scientific and rational energy use. The specific energy interaction architecture is shown in Figure 1.

The ESO is adopted as the network side to facilitate energy transmission between each entity and the external energy network. It promotes the energy supply and demand balance of the system by formulating energy prices and obtains benefits from energy price differentials. In the process of energy trading, the energy price formulated by the ESO should be within a reasonable range to ensure the interests of each entity. In previous studies, the ESO could only use energy prices as a strategy to guide each entity’s response, but it could not directly dispatch the output of each piece of equipment. Additionally, the ESO must bear the risks of price fluctuations and the seasonal imbalance between the supply and demand of renewable energy output and load. When the output power of renewable energy and load cannot meet the demand, the ESO needs to purchase energy at high prices from the external energy network. Therefore, relying solely on price strategies to regulate the system’s energy will lead to energy supply and demand imbalances and pose a risk of economic loss.

Thus, in this study, the configuration of an SHSS, power-to-hydrogen (P2H) equipment, and fuel cell (FC) in the ESO are considered. The SHSS includes short-term hydrogen storage (STHS) and long-term hydrogen storage (LTHS) devices. The STHS is used for intra-day hydrogen energy complementation, while the LTHS is utilized for inter-day hydrogen energy complementation. Electric energy can be converted into hydrogen and heat energy through P2H to supply hydrogen and heat loads. Surplus hydrogen energy can be stored using STHS for short-term storage or LTHS for seasonal storage. Additionally, when power demand is high, FC can convert hydrogen energy in LTHS into electric and heat energy, achieving long-term storage and optimal utilization of cross-energy forms. This indirectly improves the seasonal characteristics of the power load.

It can be seen that the configuration of SHSS in the ESO can indirectly enhance the seasonal characteristics of the power load, giving the ESO stronger multi-energy supply capacity and comprehensive regulation ability. The ESO can promote the balance of energy supply and demand in the system, relieve the pressure of energy consumption, and reduce the risk of economic loss while meeting the diversified needs of users. Therefore, it is more reasonable to configure SHSS in the ESO compared with ESP.

Building on its network-side responsibilities and superior system observability and controllability, the ESO is the only entity capable of internalizing the multi-energy coupling among electricity, heat, and hydrogen under the seasonal dynamics of SHSS. By co-optimizing internal pricing and the operation of cross-carrier assets (e.g., P2H and SHSS) while interfacing with external electricity and hydrogen markets, the ESO can guide EP, ESP, and LA toward optimal responses without full information sharing. Therefore, in this study, the ESO is modeled as the leader, whereas the EP, ESP, and LA act as followers in the subsequent SLMF formulation.

The EP mainly provides electric and heat energy in the system. Electric energy is primarily supplied by wind turbines (WT), photovoltaic (PV) systems, and combined heat and power (CHP) systems. Meanwhile, heat energy is mainly supplied by gas boilers (GB) and CHP systems. As the main provider of system energy, the EP adjusts the output of its units according to the price signals set by the ESO and sells electricity and heat to the ESO for maximum benefits.

As the energy storage side of the system, the ESP is composed of conventional electric energy storage (EES) and thermal storage (TT). Revenue is achieved in the ESP through the peak–valley arbitrage model of electricity and heat prices formulated by the ESO. The LA mainly has three kinds of load demand: electricity, heat, and hydrogen. As the load side of the system, the LA adjusts its energy demand according to the energy prices announced by the ESO, aiming to minimize the user’s energy purchase costs.

3. Operation Model of Each Subject

3.1. Operator Model of Energy System with Seasonal Hydrogen Storage

(1) Hydrogen Storage System Model: In this study, STHS is employed to achieve intraday hydrogen energy complementation, while LTHS is used to facilitate hydrogen energy interaction between different typical days. The operational model of the STHS device can be expressed as follows:

$$\left\{\begin{array}{c}\begin{array}{c}{S}_{w,t}^{\mathrm{STHS}}=(1-{\delta }_{\mathrm{loss}}^{\mathrm{STHS}})S_{w,t-1}^{\mathrm{STHS}}+\left(P_{w,t,\mathrm{abs}}^{\mathrm{STHS}}{\eta }_{\mathrm{abs}}^{\mathrm{STHS}}-P_{w,t,\mathrm{rel}}^{\mathrm{STHS}}/{\eta }_{\mathrm{rel}}^{\mathrm{STHS}}\right)\\ 0\le P_{w,t,\mathrm{abs}}^{\mathrm{STHS}}\le {\beta }^{\mathrm{STHS}}{C}^{\mathrm{STHS}}\\ 0\le P_{w,t,\mathrm{rel}}^{\mathrm{STHS}}\le {\beta }^{\mathrm{STHS}}{C}^{\mathrm{STHS}}\\ S_{w,1}^{\mathrm{STHS}}=S_{w,T}^{\mathrm{STHS}}\\ 0.1C^{\mathrm{STHS}}\le S_{w,t}^{\mathrm{STHS}}\le 0.9C^{\mathrm{STHS}}\end{array}\hfill \end{array}\right.$$

(1)

where $S_{w,t}^{\mathrm{STHS}}$ is the energy storage value of STHS at the time t of the $w^{\mathrm{th}}$ typical day; $P_{w,t,\mathrm{abs}}^{\mathrm{STHS}}$ and $P_{w,t,\mathrm{rel}}^{\mathrm{STHS}}$ are the hydrogen storage power and hydrogen release power of STHS; ${\eta }_{\mathrm{abs}}^{\mathrm{STHS}}$ and ${\eta }_{\mathrm{rel}}^{\mathrm{STHS}}$ are hydrogen storage efficiency and hydrogen desorption efficiency of STHS; ${\delta }_{\mathrm{loss}}^{\mathrm{STHS}}$ is the self-release rate of STHS; ${\beta }^{\mathrm{STHS}}$ is the power-capacity ratio of STHS; $C^{\mathrm{STHS}}$ is the capacity value for installing STHS; and T is the total scheduling period in each typical day.

The operational mechanism of LTHS is depicted in Figure 2. Assuming that LTHS can only charge hydrogen or only release hydrogen within a typical day, it can operate across different typical days to achieve inter-day hydrogen complementarity [31]. In practice, long-term hydrogen storage is commonly realized in underground salt caverns (and other underground formations), for which several studies report very low leakage/self-discharge losses under proper design and operation [32,33,34]. Accordingly, the energy self-dissipation rate of LTHS is taken to be nearly zero on an hourly basis in this work, and the hydrogen accumulated from the previous typical-day scenario is carried over to initialize the subsequent typical-day scenario. Consequently, the mathematical model of LTHS proposed in this study can be expressed as:

$$S_{1,t}^{\mathrm{LTHS}}=(1-{\delta }_{\mathrm{loss}}^{\mathrm{LTHS}})S_{1,t-1}^{\mathrm{LTHS}}+\left(P_{1,t,\mathrm{abs}}^{\mathrm{LTHS}}{\eta }_{\mathrm{abs}}^{\mathrm{LTHS}}-P_{1,t,\mathrm{rel}}^{\mathrm{LTHS}}/{\eta }_{\mathrm{rel}}^{\mathrm{LTHS}}\right)\mathrm{\Delta }t,2\le t\le 24$$

(2)

$$\left\{\begin{array}{c}0\le P_{w,t,\mathrm{abs}}^{\mathrm{LTHS}}\le {\mu }_{w,t,\mathrm{abs}}^{\mathrm{LTHS}}{\beta }^{\mathrm{LTHS}}{C}^{\mathrm{LTHS}}\hfill \\ 0\le P_{w,t,\mathrm{rel}}^{\mathrm{LTHS}}\le {\mu }_{w,t,\mathrm{rel}}^{\mathrm{LTHS}}{\beta }^{\mathrm{LTHS}}{C}^{\mathrm{LTHS}}\hfill \\ 0.1x^{\mathrm{LTHS}}{C}^{\mathrm{LTHS}}\le S_{w,t}^{\mathrm{LTHS}}\le 0.9x^{\mathrm{LTHS}}{C}^{\mathrm{LTHS}}\hfill \end{array}\right.;1\le w\le W$$

(3)

$$\left\{\begin{array}{c}{S}_{w,1}^{\mathrm{LTHS}}=\left(1-365{\delta }_{\mathrm{loss}}^{\mathrm{LTHS}}p(w-1)\mathrm{\Delta }t\right)·\left(S_{w-1,1}^{\mathrm{LTHS}}+365·p(w-1)\left(S_{w-1,24}^{\mathrm{LTHS}}-S_{w-1,1}^{\mathrm{LTHS}}\right)\right)\hfill \\ S_{w,t}^{\mathrm{LTHS}}=\left(1-365{\delta }_{\mathrm{loss}}^{\mathrm{LTHS}}\right)S_{w,t-1}^{\mathrm{LTHS}}+\left(P_{w,t,\mathrm{abs}}^{\mathrm{LTHS}}{\eta }_{\mathrm{abs}}^{\mathrm{LTHS}}-P_{w,t,\mathrm{rel}}^{\mathrm{LTHS}}/{\eta }_{\mathrm{rel}}^{\mathrm{LTHS}}\right)\mathrm{\Delta }t\hfill \end{array}\right.;\forall 2\le w\le W$$

(4)

$$S_{1,1}^{\mathrm{LTHS}}=\left((1-365{\delta }_{\mathrm{loss}}^{\mathrm{LTHS}}p\left(W\right)\mathrm{\Delta }t)S_{W,1}^{\mathrm{LTHS}}+365·p(w-1)\left(S_{W,24}^{\mathrm{LTHS}}-S_{W,1}^{\mathrm{LTHS}}\right)\right)$$

(5)

$${\mu }_{w,t,\mathrm{abs}}^{\mathrm{LTHS}}\le {\mu }_w^{\mathrm{abs}}\le \sum _{t=1}^T{\mu }_{w,t,\mathrm{abs}}^{\mathrm{LTHS}},{\mu }_{w,t,\mathrm{rel}}^{\mathrm{LTHS}}\le {\mu }_w^{\mathrm{rel}}\le \sum _{t=1}^T{\mu }_{w,t,\mathrm{rel}}^{\mathrm{LTHS}}$$

(6)

$${\mu }_{w,t,\mathrm{abs}}^{\mathrm{LTHS}}+{\mu }_{w,t,\mathrm{rel}}^{\mathrm{LTHS}}\le 1,{\mu }_w^{\mathrm{abs}}+{\mu }_w^{\mathrm{rel}}\le 1$$

(7)

where $P_{w,t,\mathrm{abs}}^{\mathrm{LTHS}}$, $P_{w,t,\mathrm{rel}}^{\mathrm{LTHS}}$, and $S_{w,t}^{\mathrm{LTHS}}$ are hydrogen storage power, hydrogen release, and storage value of LTHS at the time t of the $w^{\mathrm{th}}$ typical day; ${\eta }_{\mathrm{abs}}^{\mathrm{LTHS}}$, ${\eta }_{\mathrm{rel}}^{\mathrm{LTHS}}$, and ${\delta }_{\mathrm{loss}}^{\mathrm{LTHS}}$ are hydrogen storage and desorption efficiency and self-release rate of LTHS; ${\mu }_{w,t,\mathrm{abs}}^{\mathrm{LTHS}}$ and ${\mu }_{w,t,\mathrm{rel}}^{\mathrm{LTHS}}$ are hydrogen storage state variables and hydrogen release state variables of LTHS, which are 0–1 variables; $C^{\mathrm{LTHS}}$ and ${\beta }^{\mathrm{LTHS}}$ are the capacity value for installing LTHS and power-capacity ratio of LTHS; $p\left(w\right)$ is the probability of the wth typical day; and ${\mu }_w^{\mathrm{abs}}$ and ${\mu }_w^{\mathrm{rel}}$ are the hydrogen storage and release variables of LTHS, which are 0–1 variables representing the constraint that LTHS can only storage hydrogen or only release hydrogen on each typical day.

Equations (2)–(7) describe the operating conditions of LTHS. Equation (2) describes the capacity state of LTHS on the first typical day, and Equation (3) describes the maximum hydrogen charging and discharging constraints of LTHS on each typical day. Equation (4) describes the operating conditions of LTHS on other typical days, stating that the initial capacity of the next typical day is the hydrogen accumulation of the previous typical day. Equation (5) indicates that the initial capacity of LTHS on the first typical day is equal to the hydrogen accumulation of LTHS in the previous season.

(2) Electric-Hydrogen Coupling Equipment: The electro-hydrogen coupling device is mainly composed of P2H and FC. Moreover, P2H considers the electrolyzer (EL). During the process of producing hydrogen energy by water electrolysis with EL and generating electric energy by burning hydrogen with FC, the generated heat energy can be supplied to the heat load using water as the working medium. The electric-hydrogen coupling devices and the energy flow situation are illustrated in Figure 1. Building on the model in [30] and accounting for the time-scale degradation of EL and FC, the following formulations are derived:

$${\gamma }^{\mathrm{EL}}{\mu }_{w,t}^{\mathrm{EL}}{\tilde{C}}_{w,t}^{\mathrm{EL}}\le P_{w,t}^{\mathrm{EL}}\le {\mu }_{w,t}^{\mathrm{EL}}{\tilde{C}}_{w,t}^{\mathrm{EL}},{\gamma }^{\mathrm{FC}}{\mu }_{w,t}^{\mathrm{FC}}{\tilde{C}}_{w,t}^{\mathrm{FC}}\le P_{w,t}^{\mathrm{FC}}\le {\mu }_{w,t}^{\mathrm{FC}}{\tilde{C}}_{w,t}^{\mathrm{FC}}$$

(8)

$${\mu }_{w,t}^{\mathrm{EL}}+{\mu }_{w,t}^{\mathrm{FC}}\le 1$$

(9)

$$\left\{\begin{array}{c}{P}_{w,t}^{\mathrm{EL},\mathrm{H}}={\tilde{\chi }}_{\mathrm{e},\mathrm{t}}^{\mathrm{EL}}{P}_{w,t}^{\mathrm{EL}},P_{w,t}^{\mathrm{FC}}={\tilde{\chi }}_{\mathrm{e},\mathrm{t}}^{\mathrm{FC}}{P}_{w,t}^{\mathrm{FC},\mathrm{H}}\hfill \\ P_{w,t}^{\mathrm{EL},\mathrm{h}}={\chi }_{\mathrm{h}}^{\mathrm{EL}}(1-{\tilde{\chi }}_{\mathrm{e},\mathrm{t}}^{\mathrm{EL}})P_{w,t}^{\mathrm{EL}},P_{w,t}^{\mathrm{FC},\mathrm{h}}={\chi }_{\mathrm{h}}^{\mathrm{FC}}(1-{\tilde{\chi }}_{\mathrm{e},\mathrm{t}}^{\mathrm{FC}})P_{w,t}^{\mathrm{FC},\mathrm{H}}\hfill \end{array}\right.$$

(10)

$$\left\{\begin{array}{c}{P}_{w,t}^{\mathrm{EL}}-P_{w,t-1}^{\mathrm{EL}}\le P_{w,t}^{\mathrm{EL},\mathrm{fluc}}\le M\hfill \\ -\left(P_{w,t}^{\mathrm{EL}}-P_{w,t-1}^{\mathrm{EL}}\right)\le P_{w,t}^{\mathrm{EL},\mathrm{fluc}}\le M\hfill \end{array}\right.$$

(11)

$$\left\{\begin{array}{c}\sum _{t=s}^{s+T_{\mathrm{ON},\min }^{\mathrm{EL}}-1}{\mu }_{w,t,\mathrm{ON}}^{\mathrm{EL}}\le 1\hfill \\ \sum _{t=s}^{s+T_{\mathrm{ON},\min }^{\mathrm{EL}}-1}\left({\mu }_{w,t,\mathrm{ON}}^{\mathrm{EL}}+{\mu }_{w,t,\mathrm{OFF}}^{\mathrm{EL}}\right)\le 1\hfill \\ \sum _{t=s}^{s+T_{\mathrm{OFF},\min }^{\mathrm{EL}}-1}{\mu }_{w,t,\mathrm{OFF}}^{\mathrm{EL}}\le 1\hfill \\ \sum _{t=s}^{s+T_{\mathrm{OFF},\min }^{\mathrm{EL}}-1}\left({\mu }_{w,t,\mathrm{ON}}^{\mathrm{EL}}+{\mu }_{w,t,\mathrm{OFF}}^{\mathrm{EL}}\right)\le 1\hfill \end{array}\right.$$

(12)

$$\left\{\begin{array}{c}{\tilde{C}}_{w,t}^{\mathrm{EL}}=C_0^{\mathrm{EL}}\left(1-d_C^{\mathrm{EL}}t\right)\hfill \\ {\tilde{C}}_{w,t}^{\mathrm{FC}}=C_0^{\mathrm{FC}}\left(1-d_C^{\mathrm{FC}}t\right)\hfill \\ {\tilde{\chi }}_{\mathrm{e},t}^{\mathrm{EL}}={\chi }_{\mathrm{e},0}^{\mathrm{EL}}\left(1-d_{\mathrm{e}}^{\mathrm{EL}}t\right)\hfill \\ {\tilde{\chi }}_{\mathrm{e},t}^{\mathrm{FC}}={\chi }_{\mathrm{e},0}^{\mathrm{FC}}\left(1-d_{\mathrm{e}}^{\mathrm{FC}}t\right)\hfill \end{array}\right.$$

(13)

$${\mu }_{w,t}^{\mathrm{EL}}-{\mu }_{w,t-1}^{\mathrm{EL}}={\mu }_{w,t,\mathrm{ON}}^{\mathrm{EL}}-{\mu }_{w,t,\mathrm{OFF}}^{\mathrm{EL}}$$

(14)

where $P_{w,t}^{\mathrm{EL}}$ and $P_{w,t}^{\mathrm{FC}}$ are the electric power of the EL and the FC at time t of the $w^{\mathrm{th}}$ typical day; ${\tilde{C}}_{w,t}^{\mathrm{EL}}$ and ${\tilde{C}}_{w,t}^{\mathrm{FC}}$ are the time-dependent effective capacities of the EL and the FC; ${\mu }_{w,t}^{\mathrm{EL}}$ and ${\mu }_{w,t}^{\mathrm{FC}}$ are the binary operation-state variables of the EL and the FC; ${\gamma }^{\mathrm{EL}}$ and ${\gamma }^{\mathrm{FC}}$ are the minimum load ratios of the EL and the FC; $P_{w,t}^{\mathrm{EL},\mathrm{h}}$ and $P_{w,t}^{\mathrm{FC},\mathrm{h}}$ are the heat power outputs of the EL and the FC; $P_{w,t}^{\mathrm{EL},\mathrm{H}}$ and $P_{w,t}^{\mathrm{FC},\mathrm{H}}$ are the hydrogen production of the EL and the hydrogen consumption of the FC; ${\chi }_{\mathrm{h}}^{\mathrm{EL}}$ and ${\chi }_{\mathrm{h}}^{\mathrm{FC}}$ are the waste-heat utilization efficiencies of the EL and the FC; ${\tilde{\chi }}_{\mathrm{e},t}^{\mathrm{EL}}$ and ${\tilde{\chi }}_{\mathrm{e},t}^{\mathrm{FC}}$ are the time-dependent conversion efficiencies of the EL and the FC; $P_{w,t}^{\mathrm{EL},\mathrm{fluc}}$ is the ramping variable of the EL; M is the upper limit of the EL ramping power; $T_{\mathrm{ON},\min }^{\mathrm{EL}}$ and $T_{\mathrm{OFF},\min }^{\mathrm{EL}}$ are the minimum on/off durations of the EL; ${\mu }_{w,t}^{\mathrm{EL},\mathrm{ON}}$ and ${\mu }_{w,t}^{\mathrm{EL},\mathrm{OFF}}$ are the binary start/stop state variables of the EL; $C_0^{\mathrm{EL}}$ and $C_0^{\mathrm{FC}}$ are the initial capacities of the EL and the FC; $d_C^{\mathrm{EL}}$ and $d_C^{\mathrm{FC}}$ are the capacity degradation rates of the EL and the FC; ${\chi }_{\mathrm{e},0}^{\mathrm{EL}}$ and ${\chi }_{\mathrm{e},0}^{\mathrm{FC}}$ are the initial conversion efficiencies of the EL and the FC; and $d_{\mathrm{e}}^{\mathrm{EL}}$ and $d_{\mathrm{e}}^{\mathrm{FC}}$ are the conversion-efficiency degradation rates of the EL and the FC.

Equation (8) constrains the output power range of EL and FC. Equation (9) constrains the working states of EL and FC, ensuring that EL and FC cannot operate simultaneously. Equation (10) describes the electrothermal power output relationship of FC. Equation (11) constrains the ramp power of EL, and Equation (12) constrains the minimum start/stop intervals of EL. Equation (13) specifies the degradation models for capacity and conversion efficiency, representing them as time-dependent linear functions used in Equations (8) and (10).

(3) Energy System Operator Revenue Model: The optimization objective of ESO is to maximize profit, as expressed by Equation (15).

$$U_{\mathrm{ESO}}=max365\sum _{w=1}^{W}p\left(w\right)(c_{w,\mathrm{ESO}}^{\mathrm{sell}}-c_{w,\mathrm{ESO}}^{\mathrm{buy}}-c_{w,\mathrm{ESO}}^{\mathrm{grid}}-c_{w,\mathrm{ESO}}^{\mathrm{ope}}-c_{w,\mathrm{ESO}}^{\mathrm{pean}})-c_{\mathrm{ESO}}^{\mathrm{cons}}$$

(15)

where $c_{w,\mathrm{ESO}}^{\mathrm{sell}}$ and $c_{w,\mathrm{ESO}}^{\mathrm{buy}}$ are the energy sales revenue and energy purchase cost of ESO to other subjects in the system; $c_{w,\mathrm{ESO}}^{\mathrm{grid}}$ is the energy purchase cost of ESO to the external distribution network; $c_{w,\mathrm{ESO}}^{\mathrm{ope}}$ is the operation and maintenance cost of ESO; $c_{w,\mathrm{ESO}}^{\mathrm{pena}}$ is the heat penalty cost of ESO; $c_{w,\mathrm{ESO}}^{\mathrm{cons}}$ is the annual investment cost of ESO configuration SHSS.

The above items can be expressed as Equations (16)–(21).

$$c_{w,\mathrm{ESO}}^{\mathrm{sell}}=\sum _{t=1}^T\left({\gamma }_{w,t}^{\mathrm{sell},\mathrm{e}}{p}_{w,t}^{\mathrm{sell},\mathrm{e}}+{\gamma }_{w,t}^{\mathrm{sell},\mathrm{h}}{p}_{w,t}^{\mathrm{sell},\mathrm{h}}+{\gamma }^{\mathrm{sell},\mathrm{H}}{p}_{w,t}^{\mathrm{sell},\mathrm{H}}\right)$$

(16)

$$c_{w,\mathrm{ESO}}^{\mathrm{buy}}=\sum _{t=1}^T\left({\gamma }_{w,t}^{\mathrm{buy},\mathrm{e}}{p}_{w,t}^{\mathrm{buy},\mathrm{e}}+{\gamma }_{w,t}^{\mathrm{buy},\mathrm{h}}{p}_{w,t}^{\mathrm{buy},\mathrm{h}}\right)$$

(17)

$$c_{w,\mathrm{ESO}}^{\mathrm{grid}}=\sum _{t=1}^T\left({\gamma }_t^{\mathrm{grid},\mathrm{in},\mathrm{e}}{p}_{w,t}^{\mathrm{grid},\mathrm{in},\mathrm{e}}+{\gamma }^{\mathrm{grid},\mathrm{in},\mathrm{H}}{p}_{w,t}^{\mathrm{grid},\mathrm{in},\mathrm{H}}-{\gamma }_t^{\mathrm{grid},\mathrm{out},\mathrm{e}}{p}_{w,t}^{\mathrm{grid},\mathrm{out},\mathrm{e}}\right)$$

(18)

$$c_{w,\mathrm{ESO}}^{\mathrm{pean}}=\sum _{t=1}^T{\zeta }^{\mathrm{loss},\mathrm{h}}{p}_{w,t}^{\mathrm{loss},\mathrm{h}}$$

(19)

$$\begin{array}{cc}\hfill c_{w,\mathrm{ESO}}^{\mathrm{ope}}& =\sum _{t=1}^T[P_{w,t}^{\mathrm{FC}}{\lambda }_{\mathrm{O}\&\mathrm{M}}^{\mathrm{FC}}+P_{w,t}^{\mathrm{EL}}{\lambda }_{\mathrm{O}\&\mathrm{M}}^{\mathrm{EL}}+\left(P_{w,t,\mathrm{abs}}^{\mathrm{STHS}}+P_{w,t,\mathrm{rel}}^{\mathrm{STHS}}\right){\lambda }_{\mathrm{O}\&\mathrm{M}}^{\mathrm{STHS}}+\hfill \\ \hfill & \left(P_{w,t,\mathrm{abs}}^{\mathrm{LTHS}}+P_{w,t,\mathrm{rel}}^{\mathrm{LTHS}}\right){\lambda }_{\mathrm{O}\&\mathrm{M}}^{\mathrm{LTHS}}+{\lambda }_{\mathrm{ON}}^{\mathrm{EL}}{\mu }_{w,t,\mathrm{ON}}^{\mathrm{EL}}+{\lambda }_{\mathrm{OFF}}^{\mathrm{EL}}{\mu }_{w,t,\mathrm{OFF}}^{\mathrm{EL}}+{\lambda }_{\mathrm{DEG}}^{\mathrm{EL}}{P}_{w,t}^{\mathrm{EL},\mathrm{fhc}}]\hfill \end{array}$$

(20)

$$c_{\mathrm{ESO}}^{\mathrm{cons}}={\displaystyle \frac{r{(1+r)}^m\left({\sum }_{\theta \in {\theta }_{\mathrm{ESO}}}{Q}^{\theta }{C}^{\theta }\right)}{{(1+r)}^m-1}}$$

(21)

where ${\gamma }_{w,t}^{\mathrm{sell},\mathrm{e}}$ and ${\gamma }_{w,t}^{\mathrm{sell},\mathrm{h}}$ are the unit prices of electricity and heat sold by ESO; ${\gamma }^{\mathrm{sell},\mathrm{H}}$ is the sold unit price of hydrogen; $p_{w,t}^{\mathrm{sell},\mathrm{e}}$, $p_{w,t}^{\mathrm{sell},\mathrm{h}}$, and $p_{w,t}^{\mathrm{sell},\mathrm{H}}$ are the electricity sales, heat sales, and hydrogen sales of ESO; ${\gamma }_{w,t}^{\mathrm{buy},\mathrm{e}}$ and ${\gamma }_{w,t}^{\mathrm{buy},\mathrm{h}}$ are the unit prices of electricity and heat purchased by ESO; $p_{w,t}^{\mathrm{buy},\mathrm{e}}$ and $p_{w,t}^{\mathrm{buy},\mathrm{h}}$ are the quantity of electricity and heat purchased by ESO; ${\gamma }_t^{\mathrm{grid},\mathrm{in},\mathrm{e}}$ and ${\gamma }_t^{\mathrm{grid},\mathrm{out},\mathrm{e}}$ are the unit prices of ESO purchasing and selling electricity from the external distribution network; ${\gamma }^{\mathrm{grid},\mathrm{in},\mathrm{H}}$ is the unit price of external hydrogen sales; $p_{w,t}^{\mathrm{grid},\mathrm{in},\mathrm{e}}$ and $p_{w,t}^{\mathrm{grid},\mathrm{out},\mathrm{e}}$ are the purchasing power and selling power of ESO from the external distribution network; $p_{w,t}^{\mathrm{grid},\mathrm{in},\mathrm{H}}$ is the external hydrogen purchasing power of ESO; ${\zeta }_{\mathrm{loss},h}$ is the cost of the heat loss penalty; $p_{w,t}^{\mathrm{loss},\mathrm{h}}$ is the heat loss power; ${\lambda }_{\mathrm{ON}}^{\mathrm{EL}}$, ${\lambda }_{\mathrm{OFF}}^{\mathrm{EL}}$, and ${\lambda }_{\mathrm{DEG}}^{\mathrm{EL}}$ are the start, stop, and degradation costs of EL, respectively, and the remaining $\lambda$ is the operation and maintenance cost of the corresponding equipment; r and m are the discount rate and equipment life; and $Q^{\theta }$ and $C^{\theta }$ are the configuration capacity and unit capacity cost of equipment $\theta$, ${\theta }_{\mathrm{ESO}}=\{\mathrm{STHS},L,\mathrm{LTHS},\mathrm{FC},\mathrm{EL}\}$.

The configuration of SHSS enables ESO to regulate the balance of electricity, heat, and hydrogen in the system by adjusting the output of EL and FC, as shown in Equations (22)–(24).

$$p_{w,t}^{\mathrm{sell},\mathrm{e}}+p_{w,t}^{\mathrm{grid},\mathrm{out},\mathrm{e}}+P_{w,t}^{\mathrm{EL}}=P_{w,t}^{\mathrm{FC}}+p_{w,t}^{\mathrm{buy},\mathrm{e}}+p_{w,t}^{\mathrm{grid},\mathrm{in},\mathrm{e}}$$

(22)

$$p_{w,t}^{\mathrm{buy},\mathrm{h}}+p_{w,t}^{\mathrm{loss},\mathrm{h}}+P_{w,t}^{\mathrm{EL},\mathrm{h}}+P_{w,t}^{\mathrm{FC},\mathrm{h}}=p_{w,t}^{\mathrm{sell},\mathrm{h}}+p_{w,t}^{\mathrm{abd},\mathrm{h}}$$

(23)

$$\begin{array}{cc}\hfill p_{w,t}^{\mathrm{sell},\mathrm{H}}=& p_{w,t}^{\mathrm{grid},\mathrm{in},\mathrm{H}}+P_{w,t}^{\mathrm{EL},\mathrm{H}}-P_{w,t}^{\mathrm{FC},\mathrm{H}}-P_{w,t,\mathrm{abs}}^{\mathrm{STHS}}{\eta }_{\mathrm{abs}}^{\mathrm{STHS}}+P_{w,t,\mathrm{rel}}^{\mathrm{STHS}}/{\eta }_{\mathrm{rel}}^{\mathrm{STHS}}\hfill \\ \hfill & -P_{w,t,\mathrm{abs}}^{\mathrm{LTHS}}{\eta }_{\mathrm{abs}}^{\mathrm{LTHS}}+P_{w,t,\mathrm{rel}}^{\mathrm{LTHS}}/{\eta }_{\mathrm{rel}}^{\mathrm{LTHS}}\hfill \end{array}$$

(24)

where $p_{w,t}^{\mathrm{abd},\mathrm{h}}$ is the abandoned heat.

In addition, it should be ensured that the buy/sell price of the ESO is slightly higher/lower than the market price to prevent the problem from degenerating and to prevent each entity from trading directly with external energy networks. Thus, the ESO also needs to satisfy the following constraints shown in Equation (25):

$$\left\{\begin{array}{c}{\gamma }_t^{\mathrm{e},\min }\le {\gamma }_{w,t}^{\mathrm{sell},\mathrm{e}}\le {\gamma }_t^{\mathrm{e},\max }\hfill \\ {\gamma }_t^{\mathrm{e},\min }\le {\gamma }_{w,t}^{\mathrm{buy},\mathrm{e}}\le {\gamma }_t^{\mathrm{e},\max }\hfill \end{array}\right.$$

(25)

$$\left\{\begin{array}{c}{\gamma }_t^{\mathrm{h},\min }\le {\gamma }_{w,t}^{\mathrm{sell},\mathrm{h}}\le {\gamma }_t^{\mathrm{h},\max }\hfill \\ {\gamma }_t^{\mathrm{h},\min }\le {\gamma }_{w,t}^{\mathrm{buy},\mathrm{h}}\le {\gamma }_t^{\mathrm{h},\max }\hfill \end{array}\right.$$

(26)

where ${\gamma }_t^{\mathrm{e},\max }/{\gamma }_t^{\mathrm{h},\max }$ and ${\gamma }_t^{\mathrm{e},\min }/{\gamma }_t^{\mathrm{h},\min }$ are the upper and lower limits of the electricity/heat price set by ESO.

3.2. Energy Producer Model

(1) Gas Units Model: The optimization objective of ESO is to maximize profit, as expressed by Equation (15).

The gas units in EP include CHP and GB. These gas units generate electric energy or heat energy by purchasing fuel for combustion. The CHP unit operates with the characteristic of “power determined by heat,” and the heat power output of CHP can be described as Equation (27).

$$max\left\{P_{min}^{\mathrm{CHP},\mathrm{e}}-h_1{P}_{w,t}^{\mathrm{CHP},\mathrm{h}},h_m(P_{w,t}^{\mathrm{CHP},\mathrm{h}}-P_{w,0}^{\mathrm{CHP},\mathrm{h}})\right\}\le P_{w,t}^{\mathrm{CHP},\mathrm{e}}\le P_{max}^{\mathrm{CHP},\mathrm{e}}-h_2{P}_{w,t}^{\mathrm{CHP},\mathrm{h}}$$

(27)

where $P_{\max }^{\mathrm{CHP},\mathrm{e}}$ and $P_{\min }^{\mathrm{CHP},\mathrm{e}}$ are the maximum and minimum power values that the CHP unit can output; $h_1$, $h_2$, and $h_m$ are electric heat conversion coefficients of the CHP units; $P_{w,t}^{\mathrm{CHP},\mathrm{e}}$ is the output power of the CHP unit; $P_{w,0}^{\mathrm{CHP},\mathrm{h}}$ is the corresponding output heat energy while the CHP unit outputs the minimum electric power; and $P_{w,t}^{\mathrm{CHP},\mathrm{h}}$ is the heat power output of the CHP.

In addition, CHP and GB should also meet the constraints of the upper and lower output power range shown in Equation (28).

$$\left\{\begin{array}{c}0\le P_{w,t}^{\mathrm{CHP},\mathrm{e}}\le Q^{\mathrm{CHP}}\hfill \\ 0\le P_{w,t}^{\mathrm{GB},\mathrm{h}}\le Q^{\mathrm{GB}}\hfill \end{array}\right.$$

(28)

where $Q^{\mathrm{CHP}}$ and $Q^{\mathrm{GB}}$ are the rated capacities of CHP and GB; and $P_{w,t}^{\mathrm{GB},\mathrm{h}}$ is the heat power output by GB.

(2) Distributed Generation Output Model: The distributed generation in EP includes WT and PV, and their output model is shown in Equation (29).

$$\left\{\begin{array}{c}0\le P_{w,t}^{\mathrm{PV}}\le P_{w,t,\max }^{\mathrm{PV}}\hfill \\ 0\le P_{w,t}^{\mathrm{WT}}\le P_{w,t,\max }^{\mathrm{WT}}\hfill \end{array}\right.$$

(29)

where $P_{w,t}^{\mathrm{PV}}$ and $P_{w,t}^{\mathrm{WT}}$ are the output power of PV and WT; and $P_{w,t,max}^{\mathrm{PV}}$ and $P_{w,t,max}^{\mathrm{WT}}$ are the predicted outputs of PV and WT.

(3) Energy Producer Revenue Model: In the game process, EP optimizes the output of the equipment according to the energy prices set by ESO. The goal of EP is to maximize revenue, which can be expressed as follows:

$$U_{\mathrm{EP}}=max365\sum _{w=1}^{W}p\left(w\right)(c_{w,\mathrm{EP}}^{\mathrm{sell}}-c_{w,\mathrm{EP}}^{\mathrm{fuel}}-c_{w,\mathrm{EP}}^{\mathrm{ope}})$$

(30)

where $c_{w,\mathrm{EP}}^{\mathrm{sell}}$, $c_{w,\mathrm{EP}}^{\mathrm{gas}}$, and $c_{w,\mathrm{EP}}^{\mathrm{ope}}$ are the energy sales revenue, fuel cost, and operating cost of EP on the $w^{\mathrm{th}}$ typical day. The above can be expressed as Equations (31)–(33).

$$c_{w,\mathrm{EP}}^{\mathrm{sell}}=\sum _{t=1}^T\left({\gamma }_{w,t}^{\mathrm{buy},\mathrm{e}}{P}_{w,t}^{\mathrm{P}2\mathrm{E},\mathrm{e}}+{\gamma }_{w,t}^{\mathrm{buy},\mathrm{h}}{P}_{w,t}^{\mathrm{P}2\mathrm{E},\mathrm{h}}\right)$$

(31)

$$c_{w,\mathrm{EP}}^{\mathrm{fuel}}=\sum _{t=1}^T\left(a_{\mathrm{e}}{(P_{w,t}^{\mathrm{CHP},\mathrm{e}})}^2+b_{\mathrm{e}}{P}_{w,t}^{\mathrm{CHP},\mathrm{e}}+c_{\mathrm{e}}+a_{\mathrm{h}}{(P_{w,t}^{\mathrm{GB},\mathrm{h}})}^2+b_{\mathrm{h}}{P}_{w,t}^{\mathrm{GB},\mathrm{h}}+c_{\mathrm{h}}\right)$$

(32)

$$c_{w,\mathrm{EP}}^{\mathrm{ope}}=\sum _{t=1}^T\left(P_{w,t}^{\mathrm{PV}}{\lambda }_{\mathrm{O}\&\mathrm{M}}^{\mathrm{PV}}+P_{w,t}^{\mathrm{WT}}{\lambda }_{\mathrm{O}\&\mathrm{M}}^{\mathrm{WT}}+{\lambda }_{\mathrm{O}\&\mathrm{M}}^{\mathrm{GB}}{P}_{w,t}^{\mathrm{GB},\mathrm{h}}+{\lambda }_{\mathrm{O}\&\mathrm{M}}^{\mathrm{CHP}}{P}_{w,t}^{\mathrm{CHP},\mathrm{e}}\right)$$

(33)

where $P_{w,t}^{\mathrm{P}2\mathrm{E},\mathrm{e}}$ and $P_{w,t}^{\mathrm{P}2\mathrm{E},\mathrm{h}}$ are the electric and heat power of EP sold to ESO; ($a_{\mathrm{e}},b_{\mathrm{e}},c_{\mathrm{e}})/(a_{\mathrm{h}},b_{\mathrm{h}},c_{\mathrm{h}})$ is the gas cost coefficient of CHP/GB; and ${\lambda }_{\mathrm{O}\&\mathrm{M}}^{\mathrm{PV}}$, ${\lambda }_{\mathrm{O}\&\mathrm{M}}^{\mathrm{WT}}$, ${\lambda }_{\mathrm{O}\&\mathrm{M}}^{\mathrm{GB}}$, and ${\lambda }_{\mathrm{O}}\&{\mathrm{M}}^{\mathrm{CHP}}$ are the operation and maintenance costs for PV, WT, GB, and CHP.

In addition, EP needs to meet the corresponding power balance constraints, as shown in Equations (34) and (35).

$$P_{w,t}^{\mathrm{P}2\mathrm{E},\mathrm{e}}=P_{w,t}^{\mathrm{CHP},\mathrm{e}}+P_{w,t}^{\mathrm{PV}}+P_{w,t}^{\mathrm{WT}}$$

(34)

$$P_{w,t}^{\mathrm{P}2\mathrm{E},\mathrm{h}}=P_{w,t}^{\mathrm{GB},\mathrm{h}}+P_{w,t}^{\mathrm{CHP},\mathrm{h}}$$

(35)

where $P_{w,t}^{\mathrm{P}2\mathrm{E},\mathrm{e}}$ and $P_{w,t}^{\mathrm{P}2\mathrm{E},\mathrm{h}}$ are the total electric and heat power of EP sold to ESO.

3.3. Energy Storage Provider Model

(1) Energy Storage Equipment Model: The operation model of electric and heat energy storage devices in ESP is shown in Equations (36)–(39).

$$\left\{\begin{array}{c}{S}_{w,t}^{\mathrm{EES}}=\left(1-{\delta }_{\mathrm{loss}}^{\mathrm{EES}}\right)S_{w,t-1}^{\mathrm{EES}}+\left(P_{w,t}^{\mathrm{E}2\mathrm{S},\mathrm{e}}{\eta }_{\mathrm{abs}}^{\mathrm{EES}}-P_{w,t}^{\mathrm{S}2\mathrm{E},\mathrm{e}}/{\eta }_{\mathrm{rel}}^{\mathrm{EES}}\right)\hfill \\ S_{w,t}^{\mathrm{TT}}=\left(1-{\delta }_{\mathrm{loss}}^{\mathrm{TT}}\right)S_{w,t-1}^{\mathrm{TT}}+\left(P_{w,t}^{\mathrm{E}2\mathrm{S},\mathrm{h}}{\eta }_{\mathrm{abs}}^{\mathrm{TT}}-P_{w,t}^{\mathrm{S}2\mathrm{E},\mathrm{h}}/{\eta }_{\mathrm{rel}}^{\mathrm{TT}}\right)\hfill \end{array}\right.$$

(36)

$$\left\{\begin{array}{c}0\le P_{w,t}^{\mathrm{S}2\mathrm{E},\mathrm{e}}\le {\beta }^{\mathrm{EES}}{Q}^{\mathrm{EES}},0\le P_{w,t}^{\mathrm{E}2\mathrm{S},\mathrm{e}}\le {\beta }^{\mathrm{EES}}{Q}^{\mathrm{EES}}\hfill \\ 0\le P_{w,t}^{\mathrm{S}2\mathrm{E},\mathrm{h}}\le {\beta }^{\mathrm{TT}}{Q}^{\mathrm{TT}},0\le P_{w,t}^{\mathrm{E}2\mathrm{S},\mathrm{h}}\le {\beta }^{\mathrm{TT}}{Q}^{\mathrm{TT}}\hfill \end{array}\right.$$

(37)

$$S_{w,1}^{\mathrm{EES}}=S_{w,T}^{\mathrm{EES}},S_{w,1}^{\mathrm{TT}}=S_{w,T}^{\mathrm{TT}}$$

(38)

$$\left\{\begin{array}{c}0.1Q^{\mathrm{EES}}\le S_{w,t}^{\mathrm{EES}}\le 0.9Q^{\mathrm{EES}}\hfill \\ 0.1Q^{\mathrm{TT}}\le S_{w,t}^{\mathrm{TT}}\le 0.9Q^{\mathrm{TT}}\hfill \end{array}\right.$$

(39)

where $S_{w,t}^{\mathrm{EES}}$ and $S_{w,t}^{\mathrm{TT}}$ are the energy storage values of EES and TT; $P_{w,t}^{\mathrm{S}2\mathrm{E},\mathrm{e}}/P_{w,t}^{\mathrm{E}2\mathrm{S},\mathrm{e}}$ and $P_{w,t}^{\mathrm{S}2\mathrm{E},\mathrm{h}}/P_{w,t}^{\mathrm{E}2\mathrm{S},\mathrm{h}}$ are the electric and heat power of ESP sold/purchased from ESO; ${\eta }_{\mathrm{abs}}^{\mathrm{EES}}$ and ${\eta }_{\mathrm{rel}}^{\mathrm{EES}}$ are the storage efficiency and release efficiency of EES; ${\eta }_{\mathrm{abs}}^{\mathrm{TT}}$ and ${\eta }_{\mathrm{rel}}^{\mathrm{TT}}$ are the heat storage efficiency and heat release efficiency of TT; ${\delta }_{\mathrm{loss}}^{\mathrm{EES}}$ and ${\delta }_{\mathrm{loss}}^{\mathrm{TT}}$ are the self-release rates of EES and TT; ${\beta }^{\mathrm{EES}}$ and ${\beta }^{\mathrm{TT}}$ are the power-capacity ratio of EES and TT; and $Q^{\mathrm{EES}}$ and $Q^{\mathrm{TT}}$ are the maximum storage capacities of EES and TT.

(2) Energy Storage Provider Revenue Model: The ESP determines optimal energy purchase and sale decisions for storage equipment in each period based on the prices set by the ESO. The objective accounts for sales revenue, purchase expenditure, and a simplified throughput-based O&M cost.

$$U_{\mathrm{ESP}}=max365\sum _{w=1}^{W}p\left(w\right)\left(c_{w,\mathrm{ESP}}^{\mathrm{sell}}-c_{w,\mathrm{ESP}}^{\mathrm{buy}}-c_{w,\mathrm{ESP}}^{\mathrm{O}\&\mathrm{M}}\right)$$

(40)

$$\left\{\begin{array}{c}{c}_{w,\mathrm{ESP}}^{\mathrm{sell}}=\sum _{t=1}^T\left({\gamma }_{w,t}^{\mathrm{sell},\mathrm{e}}{P}_{w,t}^{\mathrm{S}2\mathrm{E},\mathrm{e}}+{\gamma }_{w,t}^{\mathrm{sell},\mathrm{h}}{P}_{w,t}^{\mathrm{S}2\mathrm{E},\mathrm{h}}\right),\hfill \\ c_{w,\mathrm{ESP}}^{\mathrm{buy}}=\sum _{t=1}^T\left({\gamma }_{w,t}^{\mathrm{buy},\mathrm{e}}{P}_{w,t}^{\mathrm{E}2\mathrm{S},\mathrm{e}}+{\gamma }_{w,t}^{\mathrm{buy},\mathrm{h}}{P}_{w,t}^{\mathrm{E}2\mathrm{S},\mathrm{h}}\right)\hfill \end{array}\right.$$

(41)

$$c_{w,\mathrm{ESP}}^{\mathrm{O}\&\mathrm{M}}=\sum _{t=1}^T\left({\lambda }_{\mathrm{O}\&\mathrm{M}}^{\mathrm{ES},\mathrm{e}}\left(P_{w,t}^{\mathrm{E}2\mathrm{S},\mathrm{e}}+P_{w,t}^{\mathrm{S}2\mathrm{E},\mathrm{e}}\right)+{\lambda }_{\mathrm{O}\&\mathrm{M}}^{\mathrm{ES},\mathrm{h}}\left(P_{w,t}^{\mathrm{E}2\mathrm{S},\mathrm{h}}+P_{w,t}^{\mathrm{S}2\mathrm{E},\mathrm{h}}\right)\right)$$

(42)

where $U_{\mathrm{ESP}}$ is the expected annual profit of the ESP; $c_{w,\mathrm{ESP}}^{\mathrm{sell}}$ and $c_{w,\mathrm{ESP}}^{\mathrm{buy}}$ are the energy sales revenue and purchase expenditure of the ESP, respectively; $c_{w,\mathrm{ESP}}^{\mathrm{O}\&\mathrm{M}}$ is the throughput-based O&M cost of the ESP; $p\left(w\right)$ is the weight of representative day w; ${\gamma }_{w,t}^{\mathrm{sell},\mathrm{e}}$ and ${\gamma }_{w,t}^{\mathrm{sell},\mathrm{h}}$ are the unit selling prices for electricity and heat; ${\gamma }_{w,t}^{\mathrm{buy},\mathrm{e}}$ and ${\gamma }_{w,t}^{\mathrm{buy},\mathrm{h}}$ are the unit buying prices for electricity and heat; $P_{w,t}^{\mathrm{S}2\mathrm{E},\mathrm{e}}$ and $P_{w,t}^{\mathrm{S}2\mathrm{E},\mathrm{h}}$ are the storage-to-external power on the electricity and heat sides; $P_{w,t}^{\mathrm{E}2\mathrm{S},\mathrm{e}}$ and $P_{w,t}^{\mathrm{E}2\mathrm{S},\mathrm{h}}$ are the external-to-storage power on the electricity and heat sides; and ${\lambda }_{\mathrm{O}\&\mathrm{M}}^{\mathrm{ES},\mathrm{e}}$ and ${\lambda }_{\mathrm{O}\&\mathrm{M}}^{\mathrm{ES},\mathrm{h}}$ are the O&M cost coefficients for the electricity-side and heat-side storage.

3.4. Load Aggregator Model

(1) Load Model: Some small and medium-sized users are aggregated in this study with demand response capabilities through LA, representing them accepting supervision and participating in market transactions. On the basis of ensuring the fixed load for users’ normal lives, the transferable characteristics of electricity and heat loads are considered. Moreover, the curtailable characteristics of electricity, heat, and hydrogen loads are also considered. The electric-, heat-, and hydrogen-related load models are shown in Equations (43)–(45).

$$\left\{\begin{array}{c}{P}_{w,t}^{\mathrm{E}2\mathrm{L},\mathrm{e}}=P_{w,t}^{\mathrm{f},\mathrm{e}}+P_{w,t}^{\mathrm{tran},\mathrm{e}}+P_{w,t}^{\mathrm{cut},\mathrm{e}}\hfill \\ -P_{\max }^{\mathrm{tran},\mathrm{e}}\le P_{w,t}^{\mathrm{tran},\mathrm{e}}\le P_{\max }^{\mathrm{tran},\mathrm{e}}\hfill \\ \sum _{t=1}^T{P}_{w,t}^{\mathrm{tran},\mathrm{e}}=0\hfill \\ 0\le P_{w,t}^{\mathrm{cut},\mathrm{e}}\le k^{\mathrm{cut},\mathrm{e}}{P}_{w,t}^{\mathrm{f},\mathrm{e}}\hfill \end{array}\right.$$

(43)

$$\left\{\begin{array}{c}{P}_{w,t}^{\mathrm{E}2\mathrm{L},\mathrm{h}}=P_{w,t}^{\mathrm{f},\mathrm{h}}+P_{w,t}^{\mathrm{tran},\mathrm{h}}+P_{w,t}^{\mathrm{cut},\mathrm{h}}\hfill \\ -P_{\max }^{\mathrm{tran},\mathrm{h}}\le P_{w,t}^{\mathrm{tran},\mathrm{h}}\le P_{\max }^{\mathrm{tran},\mathrm{h}}\hfill \\ \sum _{t=1}^T{P}_{w,t}^{\mathrm{tran},\mathrm{h}}=0\hfill \\ 0\le P_{w,t}^{\mathrm{cut},\mathrm{h}}\le k^{\mathrm{cut},\mathrm{h}}{P}_{w,t}^{\mathrm{f},\mathrm{h}}\hfill \end{array}\right.$$

(44)

$$\left\{\begin{array}{c}{P}_{w,t}^{\mathrm{E}2\mathrm{L},\mathrm{H}}=P_{w,t}^{\mathrm{f},\mathrm{H}}+P_{w,t}^{\mathrm{cut},\mathrm{H}}\hfill \\ 0\le P_{w,t}^{\mathrm{cut},\mathrm{H}}\le k^{\mathrm{cut},\mathrm{H}}{P}_{w,t}^{\mathrm{f},\mathrm{H}}\hfill \end{array}\right.$$

(45)

where $P_{w,t}^{\mathrm{E}2\mathrm{L},\mathrm{e}}$, $P_{w,t}^{\mathrm{E}2\mathrm{L},\mathrm{h}}$, and $P_{w,t}^{\mathrm{E}2\mathrm{L},\mathrm{H}}$ are the electric, heat, and hydrogen loads required by LA; $P_{w,t}^{\mathrm{f},\mathrm{e}},P_{w,t}^{\mathrm{f},\mathrm{h}}$, and $P_{w,t}^{\mathrm{f},\mathrm{H}}$ are the basic electric, heat, and hydrogen loads; $P_{w,t}^{\mathrm{tran},\mathrm{e}}$ and $P_{w,t}^{\mathrm{tran},\mathrm{h}}$ are the transferred electric and heat loads; $P_{\max }^{\mathrm{tran},\mathrm{e}}$ and $P_{\max }^{\mathrm{tran},\mathrm{h}}$ are the upper limits of electric and heat load transfer; $k^{\mathrm{cut},\mathrm{e}},k^{\mathrm{cut},\mathrm{h}}$, and $k^{\mathrm{cut}},\mathrm{H}$ are the reduction coefficients of electric, heat, and hydrogen loads.

(2) Load Aggregator Objective Function: Since the demand response of the user load is considered, there is a baseline load in each period to represent the optimal energy use for that period. When the user energy deviates from this baseline load, a corresponding loss of satisfaction occurs, as shown in Equation (46).

$$\left\{\begin{array}{c}{U}_{w,\mathrm{SL}}=\sum _{t=1}^T\sum _{E\in E^{\prime }}\left({\displaystyle \frac{1}{2}}{\omega }_E{D}_{E,w,t}^2+{\varpi }_E{D}_{E,w,t}\right)\hfill \\ D_{E,w,t}=\left|P_{w,t}^{\mathrm{E}2\mathrm{L},E}-P_{w,t}^{E,\mathrm{B}}\right|\hfill \end{array}\right.$$

(46)

where ${\omega }_E$ and ${\varpi }_E$ are the satisfaction loss parameters of the load E; $E^{\prime }$ is the set of load types, $E^{\prime }=\left\{\mathrm{e},\mathrm{h},\mathrm{H}\right\}$; and $D_{E,w,t}$ is the adjustment of the load E at the time t, and its value is the difference between the actual load $P_{w,t}^{\mathrm{E}2\mathrm{L},\mathrm{E}}$ and the baseline load $P_{w,t}^{E,\mathrm{B}}.$

By introducing auxiliary variables $L_{w,t}^{\mathrm{E}2\mathrm{L},E}$, $L_{w,t}^{E,\mathrm{B}}$ and the constraints represented in Equations (48) and (49), the absolute values in Equation (46) can be transformed into the linear form shown in Equation (47).

$$D_{E,w,t}=L_{w,t}^{\mathrm{E}2\mathrm{L},E}+L_{w,t}^{E,\mathrm{B}}$$

(47)

$$P_{w,t}^{\mathrm{E}2\mathrm{L},E}-P_{w,t}^{E,\mathrm{B}}+L_{w,t}^{\mathrm{E}2\mathrm{L},E}-L_{w,t}^{E,\mathrm{B}}=0$$

(48)

$$L_{w,t}^{\mathrm{E}2\mathrm{L},E}\ge 0,L_{w,t}^{E,\mathrm{B}}\ge 0$$

(49)

Based on the energy prices set by ESO, LA adjusts its load with the goal of minimizing the sum of energy satisfaction loss and energy purchase cost. The objective function of LA can be described as Equation (50).

$$U_{\mathrm{LA}}=min365\sum _{w=1}^{W}p\left(w\right)(U_{w,\mathrm{SL}}+c_{w,\mathrm{LA}}^{\mathrm{buy}})$$

(50)

where $U_{\mathrm{LA}}$ is the user’s utility cost; and $c_{w,\mathrm{LA}}^{\mathrm{buy}}$ is the energy purchase cost on the $w^{\mathrm{th}}$ typical day, as shown in Equation (51).

$$c_{w,\mathrm{LA}}^{\mathrm{buy}}=\sum _{t=1}^T\left({\gamma }_{w,t}^{\mathrm{sell},\mathrm{e}}{P}_{w,t}^{\mathrm{E}2\mathrm{L},\mathrm{e}}+{\gamma }_{w,t}^{\mathrm{sell},\mathrm{h}}{P}_{w,t}^{\mathrm{E}2\mathrm{L},\mathrm{h}}+{\gamma }^{\mathrm{sell},\mathrm{H}}{P}_{w,t}^{\mathrm{E}2\mathrm{L},\mathrm{H}}\right)$$

(51)

4. Multi-Subject Stackelberg Game Framework

4.1. The Process of Stackelberg Game

From the above analysis, it can be seen that the EP, ESP, and LA optimize their respective objectives conditional on the internal energy prices announced by the ESO, and their responses in turn shape the ESO’s subsequent pricing and dispatch. As the only network-side agent with system-wide observability, cross-carrier control (P2H, SHSS), and authority to interface with external electricity and hydrogen markets, the ESO determines feasibility via price signals within regulatory bands (Equations (25) and (26)) and system balance constraints (Equations (1)–(14) and (22)–(24)), whereas the EP/ESP are carrier-specific and lack network-level coordination, and the LA has neither pricing nor dispatch authority. This leader–follower interdependence conforms to a dynamic Stackelberg structure; therefore, we formulate a single-leader–multiple-follower Stackelberg game with the ESO as the leader and the EP, ESP, and LA as the followers, consistent with industrial-park governance and established SLMF formulations [21,23]. The model framework is shown in Figure 3.

According to Figure 3, the Stackelberg game model can be described as Equation (52).

$$\mathrm{\Omega }=\left\{\begin{array}{c}\mathrm{ESO}\cup \mathrm{EP}\cup \mathrm{ESP}\cup \mathrm{LA}\\ \left\{{\rho }_{\mathrm{ESO}}\right\},\left\{{\rho }_{\mathrm{EP}}\right\},\left\{{\rho }_{\mathrm{ESP}}\right\},\left\{{\rho }_{\mathrm{LA}}\right\}\\ \left\{U_{\mathrm{ESO}}\right\},\left\{U_{\mathrm{EP}}\right\},\left\{U_{\mathrm{ESP}}\right\},\left\{U_{\mathrm{LA}}\right\}\end{array}\right\}$$

(52)

where $\rho$ and U are the strategy set and optimization objective set of each subject.

If set $\left(\begin{array}{c}{\rho }_{\mathrm{ESO}}^{*},{\rho }_{\mathrm{EP}}^{*},{\rho }_{\mathrm{ESP}}^{*},{\rho }_{\mathrm{LA}}^{*}\end{array}\right)$ is the equilibrium solution of the IES game, and the game reaches the Stackelberg equilibrium when the participants meet the conditions of Equation (53), then no participant can obtain greater benefits by unilaterally changing their strategy.

$$\left\{\begin{array}{c}{U}_{\mathrm{ESO}}\left({\rho }_{\mathrm{ESO}}^{*},{\rho }_{\mathrm{EP}}^{*},{\rho }_{\mathrm{ESP}}^{*},{\rho }_{\mathrm{LA}}^{*}\right)\ge U_{\mathrm{ESO}}\left({\rho }_{\mathrm{ESO}},{\rho }_{\mathrm{EP}}^{*},{\rho }_{\mathrm{ESP}}^{*},{\rho }_{\mathrm{LA}}^{*}\right)\hfill \\ U_{\mathrm{EP}}\left({\rho }_{\mathrm{ESO}}^{*},{\rho }_{\mathrm{EP}}^{*},{\rho }_{\mathrm{ESP}}^{*},{\rho }_{\mathrm{LA}}^{*}\right)\ge U_{\mathrm{ESO}}\left({\rho }_{\mathrm{ESO}}^{*},{\rho }_{\mathrm{EP}},{\rho }_{\mathrm{ESP}}^{*},{\rho }_{\mathrm{LA}}^{*}\right)\hfill \\ U_{\mathrm{ESO}}\left({\rho }_{\mathrm{ESO}}^{*},{\rho }_{\mathrm{EP}}^{*},{\rho }_{\mathrm{ESP}}^{*},{\rho }_{\mathrm{LA}}^{*}\right)\ge U_{\mathrm{ESO}}\left({\rho }_{\mathrm{ESO}}^{*},{\rho }_{\mathrm{EP}}^{*},{\rho }_{\mathrm{ESP}},{\rho }_{\mathrm{LA}}^{*}\right)\hfill \\ U_{\mathrm{LA}}\left({\rho }_{\mathrm{ESO}}^{*},{\rho }_{\mathrm{EP}}^{*},{\rho }_{\mathrm{ESP}}^{*},{\rho }_{\mathrm{LA}}^{*}\right)\ge U_{\mathrm{ESO}}\left({\rho }_{\mathrm{ESO}}^{*},{\rho }_{\mathrm{EP}}^{*},{\rho }_{\mathrm{ESP}},{\rho }_{\mathrm{LA}}^{*}\right)\hfill \end{array}\right.$$

(53)

By solving the Stackelberg game model, the optimal pricing strategy for ESO energy, the optimal energy purchase strategy for users, and the optimal configuration capacity and output of the SHSS can be obtained.

4.2. Rationale for Stackelberg vs. Nash/Cournot

In industrial parks, the ESO uniquely owns cross-carrier assets (P2H, SHSS), observes system-wide states, and interfaces with external electricity and hydrogen markets under time-varying regulatory price bands (Equations (25) and (26)). A simultaneous-move Nash/GNE setting would ignore this governance hierarchy and the ESO’s anticipative price–capacity–operation coupling, leading to weaker internalization of seasonal coordination benefits. A Cournot model would shift to quantity competition with endogenous prices, which is at odds with our regulated price-band design and tends to induce quantity withholding rather than multi-carrier balancing. In contrast, a single-leader–multiple-follower Stackelberg game captures limited information exchange, preserves data privacy, and aligns with industrial-park practice in which the network-side operator sets internal prices within bands and coordinates cross-carrier assets. This is why the ESO is modeled as the leader while EP/ESP/LA are followers.

4.3. Solution Method

The Stackelberg game model is usually solved by converting it into a mixed-integer linear programming problem using the Karush–Kuhn–Tucker (KKT) conditions or the duality theorem [10,35]. The traditional method of centralized optimization requires knowledge of the participation information of all subjects during the solution process. However, in the actual competitive electricity market, the information of each subject is often opaque to others, and each participant needs to be optimized separately. Additionally, the model transformed by the KKT conditions or the duality theorem becomes too complicated due to the many integer variables in this study’s game model, and the decision of ESO involves a large-scale nonlinear optimization problem. The use of intelligent optimization algorithms can effectively reduce the complexity of the model [36], and using mixed-integer programming (MIP)/quadratic programming (QP) to optimize the decision making of each agent can improve the speed and accuracy of the solution [24]. Moreover, different subjects can complete the optimization process during the iterative embedding of MIP/QP into the optimization algorithm by transmitting only price signals and energy demand signals, effectively avoiding information leakage. This process simulates the independent decision making of each subject in the actual competitive market based only on public information. Therefore, a competitive search algorithm (CSA) [37], combined with an MIP/QP distributed equilibrium solution method (CSA-MIP/QP) is proposed in this study to solve the Stackelberg game model. The specific solution process is shown in Figure 4.

The implementation is as follows: initialization and seeds set the leader’s internal buy/sell price vectors within the time-varying exogenous bands (Equations (25) and (26)), with fixed random seeds for the metaheuristic and deterministic settings for the embedded QP/MIP solvers to ensure reproducibility and comparability across cases; the CSA hyperparameter policy adopts a standard configuration for bounded continuous decision spaces, using a population size of 30 divided into six groups with a competition rate of 0.5 to balance exploration with computational cost under hourly resolution on representative seasonal days, with hyperparameters kept constant across cases to avoid case-specific tuning, and candidate price updates projected onto the prescribed bands at each iteration to preserve feasibility; the algorithm terminates when the relative improvement of the best-so-far leader objective falls below ${10}^{-4}$ over 20 iterations, when the maximum absolute change in the leader price vector is less than ${10}^{-5}$, or when the iteration cap of 1000 is reached, with the earliest iterate retained in case of ties to ensure determinism.

5. Analysis of Examples

This section uses the IES of an industrial park in northern China as an example to analyze the impact of SHSS in ESO on the results and revenue of the system game equilibrium. All PV, WT, and load demand time series were obtained from the industrial park’s operational metering and SCADA systems and were subjected to anonymization, standardized resampling to an hourly resolution, temporal alignment, and rigorous quality control. Representative seasonal days were determined from normalized daily profiles via an unsupervised, data-driven selection process. Model adequacy was assessed using standard clustering validity criteria, and the representative day for each cluster was chosen as the profile with the minimum within-cluster distance.

5.1. Initial Parameters and Data

The upper and lower limits of the electricity price set by the ESO are consistent with its purchase and sale prices to the distribution network, as shown in

Table 1. Electricity purchase and sale price of distribution network.

Parameters	Value	Time Period
${\gamma }^{\mathrm{grid},\mathrm{in},\mathrm{e}}$	0.4 (CNY/kWh)	00:00–08:00
	0.8 (CNY/kWh)	09:00–11:00, 16:00–19:00, 23:00–24:00
	1.25 (CNY/kWh)	12:00–15:00, 20:00–22:00
${\gamma }^{\mathrm{grid},\mathrm{out},\mathrm{e}}$	0.35 (CNY/kWh)	00:00–24:00

. The upper and lower limits of the heat price are set to CNY 0.5/kWh and CNY 0.15/kWh, respectively [20]. In the energy plant (EP), six CHP units of 500 kW each and five GB units of 800 kW each are installed. The maximum storage capacity of both the energy storage system (ESS) and the thermal tank (TT) in the energy storage platform (ESP) is 10 MW.

The parameters of the electrolyzer (EL) are given in

Table 2. Parameters of EL.

Equipment	Parameters	Value	Parameters	Value
EL	Investment	2210 (CNY/kW)	O&M Cost	0.014 (CNY/kWh)
	Lifetime	20 (year)	Startup Cost	0.95 (CNY)
	Electrical	0.6	Shutdown Cost	0.048 (CNY)
	Thermal	0.88	Degradation cost	0.005 (CNY/kWh)

[30], while those of the remaining equipment are summarized in

Table 3. Parameters of other equipment.

Equipment	Investment	Lifetime
FC	4550/(CNY/kW)	13.5 (year)
STHS	130 (CNY/kW)	20 (year)
LTHS	1.95 (CNY/kW)	20 (year)

. Other simulation-related parameters are provided in

Table 4. Parameters related to systems.

Parameters	Value	Parameters	Value
$k^{\mathrm{cut},\mathrm{e}/\mathrm{cut},\mathrm{e}/\mathrm{cut},\mathrm{H}}$	0.15/0.2/0.2	$w^{\mathrm{e}/\mathrm{h}/\mathrm{H}}$	0.001/0.003/0.002
$p_{\max }^{\mathrm{tran},\mathrm{e}/\mathrm{tran},\mathrm{h}}$ (kW)	800/500	${\varpi }_{\mathrm{e}/\mathrm{h}/\mathrm{H}}$	0.15/0.16/0.15
${\lambda }_{\mathrm{O}\&\mathrm{M}}^{\mathrm{LTHS}/\mathrm{FC}/\mathrm{STHS}}$ (CNY/kWh)	0.01	${\delta }_{\mathrm{loss}}^{\mathrm{STHS}/\mathrm{ESS}/\mathrm{TT}}$	0.001
${\chi }_{\mathrm{h}}^{\mathrm{FC}}$	0.88	${\delta }_{\mathrm{loss}}^{\mathrm{LTHS}}$	0.0001
${\chi }_{\mathrm{e}}^{\mathrm{FC}}$	0.6	${\eta }_{\mathrm{abs}/\mathrm{rel}}^{\mathrm{EES}/\mathrm{TT}/\mathrm{STHS}/\mathrm{LTHS}}$	0.95/0.95/0.9/0.95
${\lambda }_{\mathrm{O}\&\mathrm{M}}^{\mathrm{PV}/\mathrm{WT}/\mathrm{GB}/\mathrm{CHP}}$ (CNY/kWh)	0.01	$h_1/h_2/h_{\mathrm{m}}$	0.15/0.2/0.85
$a_{\mathrm{e}}/b_{\mathrm{e}}/c_{\mathrm{e}}$	0.0015/0.16/0	$a_{\mathrm{h}}/b_{\mathrm{h}}/c_{\mathrm{h}}$	0.0005/0.11/0

, and the discount rate r for all ESO equipment is set to 5%.

The market price of hydrogen ranges from CNY 19.2 to 38.4/kg. Considering the high purity of hydrogen produced by electrolysis, CNY 35/kg is adopted. Given that the hydrogen density is 0.089 kg/m³ and its energy content is 37.87 kWh/kg, the corresponding unit energy price of hydrogen is CNY 0.924/kWh. The output power of PV and WT units and the load demand on four typical days in spring, summer, autumn, and winter are illustrated in Figure 5.

5.2. Analysis of Optimization Results

(1) Optimization Results of Difference Cases: Different cases are set to verify the economy and feasibility of the proposed model and strategy. The considerations for each case are shown in

Table 5. Difference considered in six cases.

Case	1	2	3	4	5	6
EL	✓	✓	✓	✓	✓	x
STHS	✓	x	✓	✓	x	x
LTHS	✓	✓	×	✓	x	x
FC	✓	✓	✓	x	x	x

. Case 1 is the model proposed in this study, where ESO configures EL, STHS, LTHS, and FC. Cases 2, 3, and 4 do not consider the configuration of LTHS, STHS, and FC, respectively. Case 5 only configures EL. Case 6 is the ESO model from previous studies, which only relies on the price strategy to regulate the system energy and does not configure any equipment.

The optimization iteration of ESO, EP, ESP, and LA under six cases is shown in Figure 6. In addition, post-convergence best-response checks confirmed that unilateral gains for all players satisfied $\mathrm{\Delta }\le {10}^{-4}$ in relative terms; thus, no profitable unilateral deviations were observed. It can be seen that the benefits of ESO in each case gradually increase, the benefits of ESP and EP gradually decrease, and the utility cost of LA gradually increases as the number of iterations increases. This reflects the game process between the leaders and the followers and also highlights the dominant position of ESO as the leader in the Stackelberg game. Additionally, it can be observed that the CSA-MIP/QP solution method used in this study has a better convergence effect. Furthermore, the results reach Stackelberg equilibrium with the iterations occurring 303 times, 528 times, 513 times, 441 times, 246 times, and 564 times for Cases 1–6, respectively. At this point, no participant can obtain more benefits by changing their strategy. The final convergence cost of each subject in each case and the optimal configuration capacity of each piece of equipment in ESO are shown in

Table 6. Economic optimization results of each case (unit: ${10}^4$)/CNY.

Subjects	Case 1	Case 2	Case 3	Case 4	Case 5	Case 6
ESO	1086.9	963.9	934.0	1086.9	784.2	784.2
LA	4327.2	4210.4	4269.2	4327.2	4135.0	4159.2
EP	866.0	825.9	855.2	866.0	860.5	816.2
ESP	50.1	28.5	51.4	50.1	37.4	24.0

and

Table 7. Capacity optimization results of each case.

Equipment	Case 1	Case 2	Case 3	Case 4	Case 5	Case 6
EL(MW)	3.56	3.69	1.39	3.56	0.77	0
LTHS(MWh)	165.31	0	300	165.31	0	0
STHS(MWh)	17.22	18.32	0	17.22	0	0
FC(MW)	0	0	0	0	0	0

, respectively. Table 7 shows that FC is not configured in Cases 1–6 because the configuration cost of FC is higher, and the energy loss in the process of electricity–hydrogen conversion is significant. Moreover, a large peak–valley price difference is needed to achieve profitability. Therefore, ESO’s configuration of FC is more conservative. Since FC is not configured in Case 1, the game equilibrium solutions of Case 1 and Case 4 are the same. Consequently, Case 4 is no longer analyzed separately in the following discussion.

The final pricing strategy of ESO in each case is shown in Figure 7. The heat energy sold by EP to ESO and the output power of CHP in EP in each scheme on a single typical day are shown in Figure 8. The charging and discharging power and SOC curves of ESS and TT in ESP are shown in Figure 9. It can be seen that the trend of EP selling heat energy closely follows the price fluctuation trend of ESO’s real-time heat energy purchases. Because the CHP unit has the working characteristic of “power is determined by heat,” the output electric power of the CHP unit is similar to the trend of its heat output. Therefore, the ESO unit can indirectly affect the output electric power of the CHP unit by setting the heat purchase price. The formulation of the electricity purchase price has a significant influence on EES in ESP. During periods when ESO sells electricity at a lower price, EES will choose to purchase electricity. Conversely, EES will choose to sell electricity to obtain income when the price is higher. The behavior of TT is similar to that of EES.

It can be observed from Table 6 and Table 7 that in Case 1, ESO has more storage capacity for the hydrogen energy generated by EL when configuring both LTHS and STHS simultaneously, compared to Cases 2, 3, and 5. To maximize hydrogen sales revenue, ESO sets a higher average purchase price for electricity to encourage ESP and EP to generate electricity, and establishes a higher average selling price to prompt LA to respond to electricity demand. Consequently, the utility cost of LA and the income of EP in Case 1 are the highest among Cases 2, 3, and 5, amounting to CNY 43.272 million and CNY 8.66 million, respectively. Additionally, ESO’s income increases by CNY 1.23 million, CNY 1.53 million, and CNY 3.027 million, respectively.

It can be seen from Figure 7 that the selling prices of peak-time electricity in Cases 1–5 are higher than those in Case 6, because EL is configured in Cases 1–5. The configuration of EL allows ESO to consume excess electric energy for hydrogen production. Additionally, since the CHP unit in EP operates under the characteristic of “power is determined by heat,” the average heat purchase prices in Cases 1–5 are higher than those in Case 6, thereby benefiting ESO.

The configuration of SHSS on the ESO side also benefits the promotion of system power balance and enhances renewable energy consumption.

Table 8. Optimization index of each case.

Index	Case 1	Case 2	Case 3	Case 4	Case 5	Case 6
HLP/kWh	86,631	94,562	113,271	86,631	97,446	126,921
HAP/kWh	34,951	34,646	27,723	34,951	23,100	10,219
TNI/kWh	51,679	59,916	85,547	51,679	74,346	116,703
REAR	93.86%	93.12%	81.63%	93.86%	80.83%	73.26%

shows the total heat loss power (HLP), heat abandonment power (HAP), total net imbalance (TNI) of ESO, and the renewable energy accommodation rate (REAR) for each case. It can be observed from Table 8 that there is still a phenomenon of heat loss load despite the configuration of EL in ESO, due to the system’s inability to interact with the external heating network. However, the system’s heat supply efficiency improves after EL configuration, with Case 1 showing the most significant improvement, reducing heat loss load by nearly 31.74%.

However, the abandoned heat in Case 1 remains the highest among the different schemes. The primary reason is that the simultaneous configuration of LTHS and STHS in Case 1 makes EL’s hydrogen production more active, thereby increasing ESO’s heat supply capacity initiative. Furthermore, the net unbalanced power in Case 1 is the smallest compared to other schemes, and the energy imbalance issue mainly manifests as temporal imbalances. ESO can incentivize ESP to charge and discharge more heat energy by subsidizing ESP’s energy storage service fees, thereby achieving greater energy balance. Additionally, Table 8 shows that the local accommodation rate of renewable energy in Case 1 reaches 93.86%, whereas in Case 6 it is only 73.26%, significantly enhancing local consumption of clean energy.

Moreover, achieving a renewable energy accommodation rate of 93.86% not only represents a significant technical accomplishment but also has profound implications on policy and environmental levels. The increased integration of renewable energy can provide policymakers with valuable insights when considering adjustments to related policies, potentially fostering the advancement of clean energy technologies and contributing to the achievement of sustainable development goals. Furthermore, the enhanced utilization of renewable energy is expected to effectively reduce greenhouse gas emissions, leading to improvements in air quality and public health. This positive impact on the ecological environment underscores the necessity for policy attention on the integration of renewable energy, facilitating more effective responses to climate change. Additionally, such integration may lead to an increase in green jobs, thereby injecting vitality into local economic growth.

(2) Result Analysis in Case 1: Because Case 1 demonstrates better economic performance and promotes system power balance, the operational results of each typical day in Case 1 are selected for further analysis in this section. Figure 10 illustrates the overall electric, thermal, and hydrogen power balance curves of IES. Taking a typical spring day as an example, we analyze the implementation of IES’s overall energy operation strategy. LA responds to demands based on price incentives, as depicted in Figure 10, thereby exhibiting characteristics of peak load shifting in electric load.

The peak electric load occurs from 10:00 to 12:00 and from 17:00 to 20:00 before the demand response. However, as shown in Figure 9a, ESO sets higher electric energy prices during these periods. Therefore, LA shifts a portion of the electric load to the period 00:00–08:00 to reduce utility costs. This action slows down the fluctuation of the electric load curve and brings numerous indirect benefits to IES and the external distribution network. Furthermore, LA adjusts the overall heat and hydrogen load to an appropriate range after the demand response, reducing energy purchase costs. This also alleviates the energy supply pressure on the ESO system, achieving a win-win situation.

It is shown in Figure 11 that the comparison of energy purchase cost and utility cost of LA before and after the demand response indicates a decrease in energy cost by CNY 6.955 million and a decrease in utility cost by CNY 3.462 million. Therefore, implementing demand response can reduce energy purchase costs while ensuring user comfort.

It can be seen from Figure 10 that the electricity, heat, and hydrogen loads all occur during the off-peak period from 00:00 to 08:00 on a typical spring day, coinciding with the low pricing period of the external distribution network. ESO aims to purchase electricity from the external distribution network during this time for electricity–hydrogen conversion and storage in SHSS, thereby enhancing ESO’s overall economic benefits. Additionally, ESO sets lower electricity sales prices during this period to incentivize ESP to store electricity and adjusts heat purchase prices to stimulate the CHP unit in EP to increase heat production, resulting in increased electricity generation. However, due to the adjustment in heat energy prices, EP generates excessive heat energy, leading to partial heat abandonment. From 09:00 to 11:00 is the normal electricity pricing period in the distribution network. During this time, ESO increases electricity purchase prices to encourage EP and ESP to sell electricity. Furthermore, higher energy sales prices are set to limit LA’s energy consumption, although it still cannot meet electricity and heat load demands. Therefore, ESO purchases electricity from the distribution network during this period and reduces some heat supply. From 12:00 to 15:00, which is the peak electricity pricing period in the external network, ESO sets higher purchase prices to encourage ESP to generate electricity to meet load demands. The analysis for the remaining periods from 00:00 to 15:00 is similar and not extensively detailed here.

From Figure 10c, it can be observed that on a typical winter day, LTHS converts part of the electrical energy into hydrogen energy through energy conversion, storing it in SHSS for later use on a typical spring day. This enables energy storage across two seasons. The cross-seasonal energy interaction is achieved through the mutual conversion of electricity and hydrogen facilitated by LTHS in SHSS, indirectly improving the seasonal characteristics of power load in the system while enhancing ESO’s economic benefits.

5.3. Comparative Analysis of Different Game Models

In this section, we analyze the impact of different game-theoretic models on the economic benefits and resource allocation of an IES that includes components such as EL, STHS, LTHS, and FC. The goal is to compare the Stackelberg, Nash, and Cournot models in terms of optimizing resource allocation, economic benefits, and renewable energy utilization.

The Stackelberg model stands out due to its hierarchical structure, where the ESO plays the role of the leader, setting prices and dispatch decisions to guide the responses of other participants, such as EP, ESP, and LA. This anticipatory approach allows the ESO to optimize system performance across multiple energy carriers. In our analysis, the Stackelberg model achieves a significant annual profit of CNY 10.869 million for the ESO, with a notable increase in the REAR to 93.86%. This result demonstrates that the Stackelberg model has a distinct advantage in terms of resource optimization and economic benefits, effectively coordinating the interests of all participants.

In contrast, the Nash model involves simultaneous decision making by all participants, lacking foresight into future market changes. This results in resource allocation that is relatively less efficient. The experimental results show that under the Nash framework, the ESO’s annual profit is CNY 10.143 million, with an REAR of 89.67%, reflecting the relative inefficiencies of this model in dynamic markets.

Similarly, the Cournot model, based on quantity competition with endogenous prices, leads to a supply-side reduction and does not align well with our regulated price-band design. This model tends to induce price volatility and instability in power supply. Under this model, the experimental results show that the ESO’s annual profit is CNY 9.892 million, with an REAR of 86.73%, highlighting the limitations of the Cournot model in ensuring stable energy supply and efficient market operations.

Table 9. Comparison of indicators across different models.

Indicator	Stackelberg Model	Nash Model	Cournot Model
ESO’s Annual Profit (CNY million)	10.869	10.143	9.892
LA’s Annual Expenditure (CNY million)	43.272	44.351	45.034
EP’s Annual Profit (CNY million)	8.660	8.452	8.292
REAR	93.86%	89.67%	86.73%
TNI (kWh)	51,679	54,073	55,267

below summarizes the economic benefits and system performance of the key participants under each game-theoretic model:

The experimental results indicate that the integration of SHSS into the Stackelberg model not only maximizes profits for the ESO but also significantly enhances the overall efficiency of the energy system. Within this model, the ESO proactively manages resource allocation, leading to a more reliable energy supply and better utilization of renewable resources. For instance, research shows that the REAR is 93.86%, indicating that a substantial portion of energy needs is met through renewable sources, effectively reducing reliance on fossil fuels and promoting environmental sustainability.

Moreover, the increase in profits and the reduction in expenditures for LA and EP within the Stackelberg model suggest that good coordination among stakeholders can create a more favorable economic ecosystem. This interaction fosters positive feedback in terms of benefit sharing among parties, thereby enhancing the overall efficiency of the market. This also underscores the importance of hierarchical decision-making frameworks in driving economic benefits and promoting collaboration among market players, which is crucial for addressing the challenges of energy transition and strengthening energy security.

In particular, when comparing the three models, the experimental results indicate that the Stackelberg model significantly outperforms the others. For example, the annual profit for the ESO reaches CNY 10.869 million, while the profits for the Nash and Cournot models are CNY 10.143 million and CNY 9.892 million, respectively. These results emphasize the importance of adopting the Stackelberg model, which strategically configures multiple energy devices to not only enhance economic benefits but also improve the efficiency of renewable energy utilization.

In conclusion, the clear differences in performance metrics among the three models highlight the necessity of adopting the Stackelberg approach for regional integrated energy systems. As energy markets continue to evolve, the strategic insights gained from this study should inform future policymaking and operational strategies, aiming to foster a more collaborative and efficient energy landscape.

6. Conclusions

Addressing issues such as the single profit model of grid-side operators and seasonal energy supply–demand imbalances in current research, a novel ESO model incorporating an SHSS is proposed. This model considers constraints on the start-stop operations of EL units and the operational states of seasonal hydrogen storage, enhancing its practical applicability. Furthermore, a Stackelberg game model is developed with ESO as the leader and LA, EP, and ESP as the followers, along with an algorithm proposed for its solution. The main conclusions drawn are as follows:

(1)

The configuration of the SHSS in the ESO is helpful in improving the economic benefits for each subject in the IES. Compared with the traditional ESO model, the operating income of the ESO, LA, EP, and ESP in the proposed model increased by 38.60%, 4.04%, 6.10%, and 108.75%, respectively.

(2)

Configuring both STHS and LTHS in ESO is more economical than considering only one storage mode. By implementing LTHS and corresponding electric–hydrogen coupling equipment, electric energy is converted into hydrogen energy for long-term storage, facilitating cross-seasonal energy interaction. This approach effectively promotes the local integration of renewable energy, achieving a renewable energy accommodation rate of 93.86%, which represents a 20.60 percentage point improvement over previous models.

(3)

The configuration of an SHSS in the ESO enhances its initiative in addressing energy supply and demand imbalance, significantly reducing associated risks. The total net energy imbalance decreases by 55.70%, and heat-loss load is reduced by 31.74%. Price signals are employed strategically to guide energy consumers in responding to demand pressures. This approach mitigates system energy pressures and achieves a win-win effect by lowering the overall cost of energy purchases for users.

We demonstrate notable economic gains and higher renewable accommodation with SHSS under a Stackelberg setting. Due to space constraints, comprehensive robustness tests are deferred. Future work will assess sensitivities to price spreads, hydrogen prices, load perturbations, and key techno-economic parameters. Preliminary numerical indications and the model structure suggest that profits remain positive over realistic ranges, and that FC becomes viable only beyond threshold spreads or efficiency gains. Higher renewable utilization implies CO2e abatement when standard emission factors are applied to avoided curtailment and load shifting.

These findings provide policy implications for regional energy planning. Enhanced renewable utilization and stakeholder profitability suggest that ESO-side long-duration storage should be treated as a regulated resource in integrated energy systems. Supportive measures such as capacity remuneration, green financing, and availability payments can aid cost recovery for SHSS investment. Coordinated cross-carrier price bands aligned with market signals would prevent distortions and maintain system synergy. Incorporating demand response and aggregator participation in planning and adequacy evaluation, together with pilot projects in high-renewable regions, would facilitate validation and gradual policy implementation.

However, several limitations in the current research must be acknowledged. The model’s structure may not fully account for the intricate dynamics of market fluctuations and policy changes, which could significantly influence the results. Additionally, the chosen data sets might limit the generalizability of the findings to other geographical contexts or operational scenarios. The assumptions regarding load demand and the availability of renewable resources were simplified and may overlook certain real-world complexities and thus impact the practical implications of our results. Therefore, future research will need to address these limitations by expanding the scope of the data and incorporating more detailed market dynamics.

We will also investigate performance under prevailing time-of-use tariffs, green certificate arrangements, and CAPEX parameters, and we will quantify the impacts of carbon prices and seasonal peak-shaving on dispatch and revenues.