Coordination of Renewable Energy Integration and Peak Shaving through Evolutionary Game Theory

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This article mainly utilizes the advantages and characteristics of evolutionary game theory and, based on the ideas and methods of evolutionary game theory, describes the relationship between the renewable energy generation enterprise group and the power grid enterprise group as a “learning” progressive evolution system, focusing on the evolution process of the relationship between various stakeholders and the influencing factors of evolutionary stability. It provides a reasonable explanation for the spontaneous formation of interest equilibrium between power generation enterprises and power grid companies and provides theoretical reference and policy recommendations for government regulation of the electricity bidding market. The insights derived from the simulations offer a framework that can inform practical applications, particularly in improving grid stability and promoting renewable energy adoption through effective peak shaving mechanisms and electricity pricing strategies.

Abstract

This paper presents a novel approach to optimizing the coordination between renewable energy generation enterprises and power grid companies using evolutionary game theory. The research focuses on resolving conflicts and distributing benefits between these key stakeholders in the context of large-scale renewable energy integration. A theoretical model based on replicator dynamics is developed to simulate and analyze the evolutionary stable strategies of power generation enterprises and grid companies with particular emphasis on peak shaving services and electricity bidding. These simulations are based on theoretical models and do not incorporate real-world data directly, but they aim to replicate scenarios that reflect realistic behaviors within the electricity market. The model is validated through dynamic simulation under various scenarios, demonstrating that the final strategic choices of both thermal power and renewable energy enterprises tend to evolve towards either high-price or low-price bidding strategies, significantly influenced by initial system parameters. Additionally, this study explores how the introduction of peak shaving compensation affects the coordination process and stability of renewable energy integration, providing insights into improving grid efficiency and enhancing renewable energy adoption. Although the results are simulation-based, they are designed to offer practical recommendations for grid management and policy development, particularly for the integration of renewable energies such as wind power in competitive electricity markets. The findings suggest that effective government regulation, alongside well-designed compensation mechanisms, can help establish a balanced interest distribution between stakeholders. By offering a clear framework for analyzing the dynamics of renewable energy integration, this work provides valuable policy recommendations to promote cooperation and stability in electricity markets. This study contributes to the understanding of the complex interactions in the electricity market and offers practical solutions for enhancing the integration of renewable energy into the grid.

1. Introduction

Human socio-economic development is intrinsically linked to energy, and electricity is an indispensable necessity in modern society, driving progress forward. Currently, power generation primarily relies on traditional fossil fuels, which inevitably leads to pollution, contrary to the principles of sustainable and green development [1]. Furthermore, as fossil fuel reserves become depleted, securing a stable energy supply is increasingly challenging [2]. In recent years, to address the dual crises of environmental degradation and energy scarcity, as well as the growing tension between rising demand and limited supply, numerous countries and regions have placed greater emphasis on the expansion of renewable energy sources [3]. Globally, the proportion of renewable energy in the power generation mix has been steadily increasing year by year [4]. For example, the share of renewables in global electricity generation reached nearly 30% in 2021, reflecting ongoing investment in solar, wind, and hydropower [5].

As part of a new wave of technological advancements, the renewable energy industry has become a strategic focus in China’s global industrial positioning [6]. The Chinese government has introduced a series of policies and incentives to foster the healthy and orderly development of the renewable energy sector, catapulting the nation into a leadership role in global renewable energy investment [7]. However, despite the rapid growth in renewable energy capacity, China’s renewable energy sector faces significant bottlenecks, particularly in terms of grid integration [8]. At present, these challenges are predominantly related to market mechanisms and developmental frameworks, with grid connection for emerging renewable energy technologies, such as wind power, being particularly problematic [9]. The key to overcoming these challenges lies in balancing the interests of grid operators and renewable energy producers [10].

Under the overarching framework of “generation-grid separation and competitive grid access bidding”, coupled with ongoing reforms in the power market, the success of power generation enterprises depends not only on their bidding strategies but also on the prices offered by other participants in the market [11,12].

Renewable energy companies typically seek higher grid electricity prices to recoup their investments quickly, while grid operators, on the other hand, may hesitate to integrate renewable energy into the grid due to uncertainties, such as weather-related fluctuations, which could pose risks to the stability of the power system. Unless the cost of renewable energy becomes comparatively advantageous over traditional energy sources, grid operators are unlikely to prioritize it. Governments, balancing economic development with environmental protection, provide subsidies to encourage renewable energy development, but the absence of comprehensive policies supporting renewable energy undermines the enthusiasm of grid operators to connect renewable energy to the grid [13].

Thus, the core issue for China’s renewable energy development is to establish a mechanism for the coordination and fair distribution of interests between renewable energy producers and grid operators. Game theory, which focuses on the strategic interactions between stakeholders, is particularly well-suited for analyzing the economic dynamics between renewable energy producers and grid operators. Evolutionary game theory (EGT), an extension of classical game theory, incorporates the concepts of bounded rationality and imperfect information, making it a more realistic tool for studying the dynamic interactions and decision-making processes of stakeholders [14,15,16,17,18]. EGT is widely applied in various fields to model and predict the evolution of group behaviors, making it a suitable framework for analyzing the interactions between renewable energy companies and grid operators.

This paper applies EGT principles and theoretical dynamic simulation to explore the evolutionary game process between renewable energy producers and grid operators. The simulated scenarios reflect the challenges faced by power grid companies and renewable energy enterprises in the real world, particularly in peak shaving and grid balancing under renewable energy integration. These scenarios mirror operational challenges such as managing the intermittent nature of wind and solar power and balancing demand with fluctuating renewable energy supply. The findings aim to provide policy recommendations and operational strategies that are applicable in real-world settings. These assumptions model behavior patterns observed in real-world energy markets, providing insights into the strategic interactions between grid operators and renewable energy enterprises. It aims to model and analyze their interactions as an evolving system, emphasizing the dynamic nature of their interest coordination. Through theoretical analysis and simulation, this paper identifies the factors influencing the stability of different scenarios, providing a valuable reference for policy formulation. This paper introduces a novel application of evolutionary game theory to the coordination of renewable energy integration and peak shaving strategies. Unlike previous studies, which often focus on isolated aspects of the electricity market, this research develops a comprehensive game-theoretic model that captures the dynamic interactions between thermal power plants and renewable energy producers. The key advancement lies in the integration of replicator dynamics with practical constraints on grid stability, offering both theoretical insights and policy recommendations for improving grid integration mechanisms.

From a long-term perspective, the sustainable development of the renewable energy sector necessitates the collaboration of renewable energy companies, grid operators, and government entities. Policymakers should take the interests of all major stakeholders into account, establishing a fair and balanced distribution mechanism to promote cooperation between renewable energy producers and grid operators. This paper first introduces key concepts of the electricity market and EGT, followed by the development of a theoretical framework for analyzing the evolutionary dynamics between these two key stakeholder groups. Next, an evolutionary game model is constructed to coordinate the interests of renewable energy integration, supported by theoretical and simulation-based analysis. Finally, the study concludes by exploring the strategies thermal power companies can adopt to participate in grid balancing during large-scale wind power integration. Based on this, the major research contributions of this paper are summarized as follows.

• Development of a benefit coordination game model between renewable energy and power grid enterprises: This study adopts Evolutionary Game Theory (EGT) to construct a game model between renewable energy generation enterprises and power grid companies within the electricity market. The model focuses on analyzing the interaction process between these two stakeholders in terms of grid connection and benefit distribution. This provides a theoretical foundation for understanding how to coordinate interests during the integration of renewable energy into the power grid.
• Introduction of a game analysis for thermal power enterprises participating in peak shaving under large-scale wind power integration: Addressing the challenges posed by large-scale wind power integration, this study investigates the game strategies of thermal power enterprises participating in peak shaving services. The analysis explores how revenue from peak shaving influences the bidding behavior of thermal power enterprises. By modeling the game behavior of thermal power plants in the context of wind power integration, the study further examines the impact of introducing peak shaving services on system evolution outcomes.
• Dynamic simulation and variable parameter analysis: Dynamic simulations are employed to verify the theoretical model, with all findings based on theoretical assumptions rather than empirical data. The simulations explore the relationships between key parameters (e.g., electricity prices, subsidy rates) under various scenarios, replicating potential real-world dynamics in the context of renewable energy integration. This study analyzes how these variables influence the stability of the benefit coordination game between renewable energy generation companies and grid enterprises.
• Stability analysis based on evolutionary game theory: In addition to theoretical modeling, the study integrates Lyapunov stability theory to derive and analyze the stability of the system’s equilibrium points. This approach reveals how different strategy choices can influence the long-term evolution of the system.
• Further, the key innovations of this paper are elaborated as follows:
• Proposal of a novel benefit coordination mechanism in the electricity market based on evolutionary game theory: This research introduces a new analytical framework by applying evolutionary game theory to the problem of benefit distribution in the electricity market. The framework effectively models the dynamic and uncertain interactions between power generation enterprises and grid companies, capturing their evolving strategies over time.
• Incorporation of peak shaving ancillary service revenue into evolutionary game analysis: An innovative aspect of this study is the inclusion of peak shaving ancillary service revenue in the evolutionary game model. This study analyzes how peak shaving revenue impacts the bidding strategies of thermal power enterprises and their interactions with grid enterprises in the context of large-scale wind power integration. This contribution is significant in addressing the challenges of balancing the interests of renewable energy and conventional power generation.
• Construction of a game strategy model for thermal power enterprises under large-scale wind power integration: This study develops a novel game strategy model for thermal power enterprises engaged in peak shaving under conditions of large-scale wind power integration. This model fills a gap in existing research by providing insights into the strategic interactions between thermal power enterprises and renewable energy providers, particularly in relation to peak shaving capacity and system stability.
• Policy recommendations for enhancing cooperation between renewable energy and power grid enterprises: Based on the dynamic evolutionary analysis of the game model, this study offers practical policy recommendations for optimizing renewable energy pricing, enhancing grid company enthusiasm for renewable energy integration, and improving the stability of the electricity market. These recommendations provide theoretical guidance for government departments in formulating relevant policies.

The remaining part of this paper is organized as follows: In Section 2, the current development trend of EGT and its applications to power bidding market and the development of bidding models are elaborated and summarized as preliminaries. In Section 3, a typical two-group asymmetric evolutionary game-theoretic model is established, and several core concepts in EGT are introduced. In Section 4, the evolutionary game analysis of thermal power enterprise groups’ participation in peak shaving is investigated. Based on Section 4, we further conduct research on evolutionary game analysis of renewable energy integration benefits coordination in Section 5. Lastly, Section 6 concludes this paper.

2. Literature Review

Since the start of the 21st century, numerous countries have prioritized the development and deployment of renewable energy industries, implementing a range of pricing policies to incentivize investment in renewable energy. These policies have played an instrumental role in stimulating widespread societal participation in the renewable energy sector. However, over time, certain issues have emerged, particularly in the formulation and assessment of these pricing policies. As a result, scholars have increasingly focused on evaluating these policies, producing a substantial body of research on renewable energy pricing mechanisms.

Existing research has classified renewable energy policies into several categories, such as quota systems, on-grid electricity pricing policies, and green pricing mechanisms. The primary methods used to evaluate these policies include empirical analysis, net present value analysis, and game theory. In this paper, the focus is on the strategic behavior of grid operators and renewable energy producers within the context of on-grid electricity pricing. A fundamental prerequisite for the success of this strategy is overcoming the three major barriers: grid connection, grid integration, and market participation.

2.1. Electricity Market Game Models

Extensive research has been conducted on the electricity market, with many scholars using EGT as a primary tool to study the strategic behavior of power generation companies. The electricity markets in Europe and the U.S. have matured earlier, resulting in a wealth of research on bidding strategies and the application of game theory in these contexts. In China, however, the electricity market is still in its infancy. As competitive bidding mechanisms for grid access continue to develop, Chinese scholars have increasingly focused on the bidding strategies of power companies, yielding a range of findings. The primary research approach involves constructing bidding game models under conditions of incomplete information in the electricity market.

In China, many scholars mainly combine the actual situation of domestic power market development, mostly using classical game theory methods to carry out research work on renewable energy quota systems, bidding Bayesian games, market clearing mechanisms, bidding mechanisms, coordination of multi-party interests, and other aspects. For example, Zhao et al. [19] combines the parameters of the relevant institutional variables of the renewable energy quota system, establishes the game evolution model of power generation enterprises, analyzes the factors affecting the strategic behavior of power generation enterprises and considers that the turnover amount of green certificates, the quota ratio and the amount of the unit penalty in the electricity market, as well as the difference in the marginal cost of power generation enterprises all have an impact on the evolutionary equilibrium.

In addition, based on the constructed non-cooperative game model, Wang [20] analyzes the influencing factors affecting the upgrading motivation of power grid enterprises under grid integration of renewable energy power generation (e.g., wind, electricity) from the consideration of the interests of major interest subjects and concludes that the production scale of renewable energy power generators is the decisive factor. Lu et al. [21] outline the typical application scenarios and research ideas of game theory from several aspects, which are four aspects of power market, power system planning, scheduling, and control. Liu et al. [22] summarize the current research progress and results of game theory in this field from three aspects, namely, high energy users, medium and low energy consumption, and distributed energy users, respectively. Based on the constructed Bayesian game model of feed-in tariffs under incomplete information conditions, Zhang et al. [23] analyze the strategy selection research on the feed-in tariffs of power generation enterprises and grid enterprises and finally arrive at an appropriate pricing strategy. Zhao et al. [24] combine the clearing and settlement mechanism of the domestic power market and the offer of power generation enterprises in the spot market, improve and upgrade the existing genetic algorithm, construct a ladder offer model based on genetic algorithm and EGT, and research and analyze the offer decision-making of traditional energy power generation manufacturers under the relationship between supply and demand of different power markets. The author in [25], according to the current renewable energy quota system policy, based on the constructed relevant model, adopts a variety of theoretical methods to conduct research and analysis on the behavioral strategies of the power generation market. The research theory methods involved are EGT, policy network theory, and system dynamics. Gao and Sheng [26] investigate and analyze the influence of governmental regulatory and control measures in the process of power generation enterprises’ bidding for Internet access and believe that the government plays an important role in the spontaneous evolution of bidding strategies. Wang et al. [27] study and analyze the “rationality” in the assumption of “limited rationality” in EGT and explains the physical and economic meanings of “limited rationality” in practical applications. Moreover, Da et al. [28] construct a homogeneous two-group three-strategy symmetric evolutionary game model under general circumstances, analyze the factors affecting its evolutionary equilibrium, and obtain complete qualitative results.

Based on EGT, Wang et al. [29] investigate the coordination relationship between the interests of renewable energy power generation enterprises and grid enterprises and analyze the factors affecting the evolutionary stable equilibrium in different situations; they believe that the spontaneous formation of the equilibrium of their interests is a dynamic evolutionary process. In terms of the power market bidding mechanism, the scholar in [30] uses the idea of a genetic algorithm to carry out the dynamic simulation of evolutionary game on the behavioral decisions of power generation enterprises and believes that in terms of the equilibrium price in the power market, although there is no big difference between the MCP mechanism and the PAB mechanism, the market price will reach equilibrium faster and be more stable under the MCP mechanism. Moreover, Cheng [31] proposes a theoretical research method suitable for analyzing the long-term evolutionary stable equilibrium characteristics of typical two-group and three-group asymmetric evolutionary game scenarios under the electricity market. Zhou et al. [32] analyze and compare the settlement mechanisms used in representative foreign power markets, summarize their respective characteristics, and then combine them with the actual situation of the domestic power market at various stages of development to provide reference suggestions. In the “sell more, buy more” power market model of bilateral transactions between power producers and grid companies, Fei [33] analyzes the behavioral strategies of bidding for Internet access between power producers and grid companies based on the EGT and argues that under this model, it is conducive to fair competition among power producers, and it is also beneficial to the interests of power purchasers with different economic scales and levels of development. The interests of power purchasers with different economic scale and development level can also be guaranteed, thus playing a certain theoretical guidance role for the bidding behavior of enterprises in China’s regional power market.

Under the assumption of finite population, the authors in [34] imitate the actual enterprise behavior strategy, combine the EGT to construct the stochastic evolution model, study and analyze the long-term evolutionary stable equilibrium process of power generation enterprises in the power market, and finally verify that the evolutionary stable equilibrium output of power generation enterprises corresponds to the competitive output of the two populations of enterprises, respectively. After successively comparing and analyzing the three grid connection modes of renewable energy power generation, exploring the problems arising from renewable energy access to the power grid, and studying the interrelationship between renewable energy grid connection, smart grid, and power supply marketing, the scholar in [35] argues that the smart grid will play a certain promotional role in solving the problem of the difficulty of renewable energy power generation enterprises to connect to the grid and that, in the future, renewable energy power generation will be widely used in the grid connection. Distributed generation will be widely used.

2.2. Game Theory and Its Application to Electricity Markets

Internationally, especially in game theory and evolutionary dynamics, Taylor and Jonker proposed replicator dynamics when observing evolutionary phenomena in natural ecosystems, which greatly promoted the development of EGT [36]. Taylor used differential equations to describe the characteristics of replicator dynamics, assuming that the number of individuals in the population is infinite, but he did not incorporate uncertainty factors into the model. Another important concept in EGT, evolutionary stable strategy, was proposed in [37], providing a new perspective for studying EGT. Ref. [38] discusses the branching situations of various evolutionary stable equilibria in replicator dynamic systems with multiple game participants. Oechssler and Riedel [39] analyze the prerequisites for replicator dynamic systems to evolve to equilibrium states in the continuous strategy space.

Based on game theory, especially EGT, relevant scholars have begun to conduct preliminary research on multi-agent or multi-group game issues in the electricity market. For example, in terms of bidding in the electricity market, Peng et al. [40] combine dynamic simulation to analyze the declared electricity prices of power generation enterprises in a modular theoretical manner, believing that the influencing factors include the payment functions of each player in the game and the electricity transmission status of the power system. With the PAB settlement mechanism and enterprise risk avoidance as the research background, Ma et al. [41] conduct theoretical research and analysis on the bidding behavior strategy of power generation enterprises, first evaluating the revenue, risk level, and controllability generated by adopting this strategy, and then using genetic algorithm dynamic simulation verification. Ren and Galiana [42] compare and analyze different bidding strategies of power generation enterprises and conducts simulation verification. Power generation enterprises believe that under the PAB settlement mechanism, they will declare electricity prices deviating from marginal costs in order to pursue more profits, while under the MCP settlement mechanism, even if power generation enterprises declare electricity prices based on actual marginal costs rather than high prices, it is still profitable. Zhang et al. [43] combine the current situation of China’s electricity market to analyze and discuss the opportunities and challenges that may be encountered in the process of implementing renewable energy-related incentive policies and evaluate the applicability of renewable energy quota policies. Moreover, Thurber et al. [44] use game theory to analyze the impact of different energy policies on renewable energy and traditional energy enterprises. Aiming at bidding strategies adopted in generators, Li et al. [45] consider the uncertainty of renewable energy in supply and demand aspects and then investigate the bidding behavior strategy of power generation enterprises. In addition, Mei [46] constructs a game planning model for wind, solar, and storage hybrid power systems and studies the game results under different cooperation scenarios, providing theoretical reference for achieving capacity optimization configuration.

2.3. Development of the Bidding Model Used in Power Generators

For generator bidding, there is currently no electricity spot market based on time-of-day tariffs in most parts of China, and the actual volume traded in the medium- and long-term electricity market is long-term (several months). Such long-term power contracts are different from short-term power contracts: short-term power contracts are used by generators to determine the amount and price of power generation based on market demand, whereas long-term power contracts are used by power users to determine their power consumption by entering into long-term power contracts. In the medium- and long-term market, generating units are awarded contracts to produce a certain amount of electricity (kilowatt-hours, megawatt-hours) over a certain period of time. It is necessary for generators to comply with the regulations. Most regions in China generally stipulate auxiliary charges for generators and make certain distinctions between paid and unpaid services. Basic ancillary services are those that provide the most basic free services without additional price compensation from the generator, while compensated services are those that are compensated on top of the price of their basic services. After the generating unit reaches deep peaking, it will not only damage the generating unit, but also affect its revenue, then at this time the generator will receive compensation for its service [47,48]. In the China region, the peaking rules are generally based on the output level of the generation side.

Taking the Chinese market as an example, firstly, the compensation service is mainly provided to power producers; secondly, within the current compensation rules, it is stipulated that only thermal and hydropower companies can participate in this service, and the division between paid and unpaid services is mainly based on whether it will have an impact on the cost-earnings on the power generation side. This kind of planning fails to satisfy the needs of power producers. In the process of market development mentioned above, auxiliary service, as a public good, contributes from the revenue of the power system, which will lead to the phenomenon that units without peaking capacity can enjoy the peaking service of other units without any compensation, which is an irrational phenomenon. The advantage of ancillary services comes from its power system revenue, but the cost charged for such services should be subsumed into the transmission and distribution tariff for all participants in the power system to bear. In this market environment, generators should be the ones to provide the service, not to share the cost. In mature market mechanisms in foreign countries, ancillary services are mainly planned as basic and remunerated services, and one of the basic conditions for participation in the electricity market is the ability to provide basic ancillary services.

In addition, internationally, the supply of auxiliary services to the power system is mainly divided into three forms: centralized competitive market, long-term contract, and mandatory provision. Among them, take the centralized competitive market as an example; this mode is generally applied in the trading system such as frequency regulation and standby, which is a kind of service that is supplied in sufficient quantity and changes with the time requirement, and it is the system operating organization that builds the market of short-term competition. The cost of the service is difficult to establish, and constant prices do not reflect the diversity of service costs, so the use of market transactions to purchase the service is an efficient way to compensate for the service and increase the incentive to serve. Some ancillary services are intertwined with electricity, and there is a case for integrating the service market with the electricity intermediary, and the way in which the service operates depends on the way in which the market operates between countries. As an example, in the U.S. PJM, in a full feed-in tariff spot market, generators provide data prior to the offer deadline, and the ISO develops a generation plan to constrain units and economic dispatch. In this program, information on units turned on, units turned off, and generation trends are charted, and the generator receives the benefits through the rules of power trading.

Based on the above research results, it can be seen that most of the current research in the electricity market field mainly revolves around the online bidding of power generation companies, mostly focusing on simple two-group evolutionary game stability analysis. There is little research on the online strategies of renewable energy generation companies and grid companies, and the process of equilibrium formation in the game evolution between the two, as well as the analysis of variable parameters affecting the equilibrium results in conjunction with actual situations, is somewhat lacking in existing research. Therefore, this article first studies the peak shaving revenue model of thermal power units, and based on the EGT, analyzes and derives strategic decisions that lead to the existence of evolutionary stable states according to the replicator dynamics equation. Establishing a balance of interests and incentive policies is an effective way to solve this problem, ultimately reducing the high wind curtailment rate to maintain the stable operation of the power grid. Building on this, this article further constructs an evolutionary game model for coordinating the benefits of renewable energy grid connection and conducts theoretical analysis and dynamic simulation verification, specifically exploring the relationships between variable parameters such as initial marginal costs, subsidy rates, electricity prices of conventional energy, and distribution coefficients under different scenarios, and their impact on the equilibrium results of system evolution, providing some theoretical guidance and decision-making references for government departments in implementing renewable energy policies in practice.

2.4. Current Structure of China’s Energy Market

China’s energy market is one of the largest and most complex in the world, reflecting its rapid industrial growth and increasing demand for energy. The market is structured around several key sectors, including fossil fuels (coal, oil, natural gas) and renewable energy sources (wind, solar, hydro). This section provides a detailed description of the current structure of China’s energy market, including the roles of various stakeholders, the regulatory framework, and the influence of recent market reforms.

(1)

Key Stakeholders in China’s Energy Market. The primary stakeholders in China’s energy market can be categorized into the following groups.

State-Owned Enterprises (SOEs): The energy sector in China is dominated by state-owned enterprises, particularly in the areas of coal, oil, and electricity. The most notable among these are the State Grid Corporation of China, Beijing, China (SGCC) and the China Southern Power Grid Company, Beijing, China, which are responsible for electricity transmission and distribution. The China National Petroleum Corporation, Beijing, China (CNPC), China National Offshore Oil Corporation, Beijing, China (CNOOC), and China Petrochemical Corporation, Beijing, China (Sinopec) are the major players in the oil and gas sectors.

Independent Power Producers (IPPs): With market reforms, there has been an increase in participation by independent power producers, especially in the renewable energy sector. Companies such as Goldwind and Longi have become significant players in wind and solar power generation, respectively [49].

Renewable Energy Enterprises: The rapid growth of renewable energy in China has led to the emergence of specialized companies focusing on wind, solar, and hydroelectric power. These companies are often supported by government subsidies and policies promoting green energy [50].

Regulatory Bodies: The National Development and Reform Commission (NDRC) and the National Energy Administration (NEA) are the central regulatory bodies responsible for setting energy policy, pricing mechanisms, and overseeing market reforms. The NDRC also oversees investments in energy infrastructure, while the NEA focuses on energy strategy and regulation [51].

(2)

Energy Generation. China’s energy generation is still heavily dependent on coal, which accounted for approximately 56.8% of total energy consumption in 2020 [52]. However, there has been significant growth in renewable energy generation. By 2021, renewable energy (primarily hydro, wind, and solar power) constituted nearly 30% of China’s electricity generation mix [1].

Coal and Thermal Power: The majority of China’s energy comes from coal-fired power plants. Although coal’s share has been gradually decreasing, it remains a cornerstone of China’s energy system due to its abundance and the infrastructure that supports it [4].

Hydropower: China is the world leader in hydropower generation, with major projects like the Three Gorges Dam. Hydropower plays a critical role in providing renewable, stable electricity, especially in times of peak demand [53].

Wind and Solar: In recent years, China has invested heavily in wind and solar power generation, becoming the global leader in both industries. The government has set ambitious targets for increasing the share of renewables, supported by substantial subsidies and a favorable policy environment [50,54].

(3)

Electricity Transmission and Distribution. Electricity transmission and distribution in China are managed primarily by two state-owned grid companies as follows.

State Grid Corporation of China (SGCC): The largest utility company in the world, SGCC operates the power grid in most of China. It is responsible for the transmission and distribution of electricity across provincial borders, ensuring grid stability, and managing the integration of renewable energy sources [55].

China Southern Power Grid (CSG): CSG operates the grid in the five southern provinces and has a similar role to SGCC, focusing on transmission and distribution.

Both SGCC and CSG are key players in managing China’s ultra-high voltage (UHV) transmission networks, which are critical for long-distance transmission of electricity from renewable energy sources in western China to demand centers in the east. This infrastructure is crucial for integrating wind and solar power into the national grid and addressing regional imbalances in energy supply and demand [55].

(4)

Market Reforms. Over the past decade, China has implemented significant reforms aimed at liberalizing the electricity market and increasing the role of competition. Prior to these reforms, the energy market was characterized by a single-buyer model, where electricity prices were set by the government, and power generators sold electricity directly to grid companies at fixed rates [56].

Electricity Market Liberalization: In 2015, the Chinese government introduced new reforms under its “Document No. 9”, which aimed to separate power generation from transmission and distribution. This reform has introduced competitive bidding mechanisms in certain regions, where electricity prices are now determined by market forces rather than government controls [49].

Renewable Energy Quotas: In order to promote the integration of renewable energy, China introduced a Renewable Portfolio Standard (RPS), which mandates that electricity providers must meet certain quotas for renewable energy usage [57]. This policy is aimed at reducing curtailment rates for wind and solar power and ensuring that renewable energy producers have a stable market for their electricity [58,59].

Green Certificates and Carbon Trading: As part of its commitment to reducing carbon emissions, China has also introduced green certificates for renewable energy producers and has begun to develop a national carbon trading market. These mechanisms are designed to incentivize investment in renewable energy and promote cleaner power generation [60].

(5)

Challenges and Future Directions. While the reforms have made significant progress, there are still challenges in China’s energy market. One major issue is the curtailment of renewable energy, where wind and solar power generation is restricted due to grid limitations. The lack of flexibility in the grid infrastructure and the dominance of coal in the energy mix make it difficult to fully integrate renewable energy [61,62].

Additionally, the regional disparity in energy generation and consumption presents ongoing challenges. Western China, which has abundant renewable energy resources, often generates more power than can be consumed locally, while eastern China, where demand is highest, faces electricity shortages. China’s UHV grid development aims to address this issue by transporting electricity over long distances, but full implementation will take time [63].

In conclusion, China’s energy market is in the midst of a significant transformation, driven by a combination of government reforms, the expansion of renewable energy, and increasing market competition. While coal still plays a dominant role, the transition toward a more diversified and sustainable energy mix is well underway, supported by state-owned enterprises, private companies, and robust regulatory frameworks [64].

3. Evolutionary Game-Theoretic Model

In classical game theory, we usually require participants to be divided into at least two populations, and the benefits they obtain are divided based on quantity. The participant who first selects a strategy needs to acquire information about the other party, and then make decisions based on the quality of the selected strategy. The participant who selects a strategy later than the first participant will receive more information than the first participant, leading to a more comprehensive strategy. Therefore, the latter will receive more strategic benefits than the former. Classical game theory can generally be divided into three assumptions: the first is that all individuals in the game are 100% rational; the second is that the participating individuals have the same background conditions; and the third is that all information and elements can be known before participating in the game. Obviously, the “complete rationality” assumption used in most classical games cannot explain the limited rationality people show when making decisions on complex real-world problems. When games encounter multiple equilibria, traditional game theory also struggles to determine which equilibrium will be reached. In contrast, the introduction of EGT has effectively overcome the above limitations. EGT starts from limited rationality and is based on limited information (often under asymmetric information conditions), focusing on groups as the main research object. It emphasizes that games are dynamic and gradual processes and believes that reaching equilibrium in a system takes time and there may be multiple equilibrium points, with the final convergence direction depending on the initial conditions and path of evolution. In addition, the concepts of evolutionary stable strategies and evolutionary stable equilibria in EGT can effectively describe the local dynamic properties of complex differential systems. Next, this paper will briefly introduce the classic evolutionary game models and the core concepts involved.

3.1. Core Concepts in Evolution Game Theory

3.1.1. Bounded Rationality Assumption

EGT refers to the game theory that studies the final strategy selection trends of participants under the premise of bounded rationality, taking into account the mutual influence of all participants [65,66,67,68,69,70,71]. The equations established, the historical and regulatory factors, and some minor details of the equilibrium process will all have an impact on the choices and outcomes of the game. Based on reality, this theory is of practical significance in biology, economics, finance, and securities studies.

In the theoretical basis of classical game theory, participants are 100% rational, and participants make choices under the condition of having all information, but in actual economic life, participants cannot reach the conditions of complete rationality and complete information. Individuals in the population who cooperate and compete are all different, and factors of individuals themselves as well as the influence brought by the environment and the inability to be completely rational will cause the information obtained to be incomplete. The main research content of EGT is the evolution of group strategies over time, mainly to explain the reasons for achieving this result in the specific process of evolution. However, in real life, individuals are often in a certain random environment, and their behavior is influenced by multiple factors. These factors have different degrees of impact on the evolution of the population, with the most important being the conditions inherent in the system itself. When analyzing complex systems, stochastic differential equations are often used to describe their dynamic behavior; cooperation and conflict issues among individuals are rarely considered.

Most EGT focuses on the analysis and prediction of outcomes determined by a group’s strategies, a process that often has a certain inertia and may also mutate into new features or characteristics. Actually, we often need to describe co- and non-cooperative behavior strategies among individuals containing high computational complexity. EGT is widely used to solve economic problems and has achieved great success. It is a very successful model. The application of EGT in economics and biology is quite different. For example, concepts such as gender and mating, chromosomes, and generations, are almost impossible to apply in economics. EGT is not only suitable for analyzing macroeconomic systems but can also be used to address micro-level business decision-making issues. The basic assumption of EGT is individual rationality.

General evolutionary game models are determined by two elements: selection and mutation. Mutation refers to the phenomenon of changing one’s strategy in a specific situation to achieve greater gains; while selection refers to a strategy of taking different actions among members of a group when there is conflict to achieve the desired goal. Selection means that more participants will adopt strategies that increase rewards; mutation means that some individuals randomly choose strategies different from the group (strategies with higher or lower rewards). Mutation is a process of repeated experimentation, continuous learning, imitation, adaptation, and improvement. However, in practical applications, many problems often arise due to a lack of in-depth understanding and knowledge of these characteristics.

3.1.2. Evolutionary Game Theory Framework

In practical application, the establishment of evolutionary game contains four steps.

• Step I: Parameter setting. This includes four main parameters: participants, strategy set, payment matrix, and order of decision selection.
• Step II: Strategy change. The result of the game cannot achieve a balance in one or two times, but can only reach the equilibrium of evolutionary game through repeated games. Games played under different parameter conditions often yield different equilibrium points.
• Step III: Parameter dynamic solution. Solve to obtain fixed time and probability.
• Step IV: Equilibrium strategy solution. Different games have different solution processes. For example, deterministic evolutionary games are solved using RD dynamic equation, and stochastic evolutionary games usually require the consideration of various stochastic factors and are usually solved based on stochastic differential equation theory.

3.1.3. Evolutionarily Stable Strategy (ESS)

The environment of an evolutionary game can be divided into a large group and some other small groups; the small groups choose different strategies from the large group, the name of the group formed by these two groups is called a mixed group. When the strategies adopted by these small groups yield higher returns than those adopted by these large groups, this large group is unstable; conversely, when this large group yields higher returns than any of the strategies adopted by any of the small groups, i.e., the stable equilibrium point (ESS). If one ESS is chosen among most of the participants, the other small group of participants who choose other strategies cannot change the outcome of this evolutionary game. The mathematical model of ESS can be explained as follows. We assume that there are K populations in an evolutionary game, where each specific population, denoted by k, has a set of N strategies. Then, we further assume that this population is defined by an N-dimensional vector set, namely

$$S^{\overrightarrow{k}}=\left\{\overrightarrow{x}=\left(x_1,x_2,\cdots ,x_i,\cdots ,x_N\right)|x_i\ge 0,{\displaystyle \sum x_i=1}\right\}$$

(1)

where x_i represents the percentage of the i-th strategy in a specific population out of the entire strategy set, and k ∈ {1, 2, …, K}, I ∈ {1, 2, …, N}. Based on this, the fitness function is described as a mapping: $f^k:S^k\times S\to R$, where k ∈ {1, 2, …, K}, and this function can be also denoted by $f\left(\overrightarrow{r},t\right)=\{f^1\left(\overrightarrow{r},s\right),f^2\left(\overrightarrow{r},s\right),..{.,f}^i\left(\overrightarrow{r},s\right),...,f^k\left(\overrightarrow{r},s\right)\}$, where r is a mixed strategy for any individual in population k. Moreover, the change of state S with respect to time t is generally represented using derivatives. For example, for the case with continuous time t, the derivative of S with respect to time is denoted by $S=\left(S^{\overrightarrow{1}},S^{\overrightarrow{2}}.{..S}^{\overrightarrow{i}}.,S^{\overrightarrow{K}}\right)$, where $S^{\overrightarrow{K}}$ is shown as $S^{\overrightarrow{K}}=\left(S_1^k,S_2^k,...,S_i^k,...,S_N^k\right)=\left(\mathrm{d}{S}_1^k/\mathrm{d}t,\mathrm{d}{\mathrm{S}}_2^{\mathrm{k}}/\mathrm{d}t,...,\mathrm{d}{S}_i^k/\mathrm{d}t,...,{\mathrm{d}S}_N^k/\mathrm{d}t\right)$, and k ∈ {1, 2, …, K}.

Finally, the function of an evolutionary game can be represented as F:S→R^NK or S = F(S). Obviously, this is a system of differential equations, and given certain initial conditions S(0) ∈ S, the evolutionary process of all groups can be expressed by the corresponding curves of the two equations solved above. In this process, each individual has a corresponding equilibrium point, and when it reaches the equilibrium position, a one-period oscillatory behavior occurs; and when that chaotic motion lasts for a certain period of time, its state will be transformed from unstable to steady state. If the system solution is stable according to the stability analysis, the solution corresponds to the evolutionary stabilization strategy.

Based on the above explanation, the stability determination rules are explained as follows. At an internal equilibrium whose corresponding determinant and trace of the Jacobi matrix J, which are denoted by det(J) and tr(J), respectively, are simultaneously positive, then the equilibrium remains locally asymptotically unstable, i.e., it is an evolutionarily unstable equilibrium point. Moreover, if det(J) is positive and meanwhile tr(J) is negative, then the corresponding equilibrium remains locally asymptotically stable, i.e., it is an evolutionarily stable equilibrium point. Otherwise, if there is a special case where the determinant det(J) is positive but the trace tr(J) is zero, the corresponding internal equilibrium point is called a saddle point. Obviously, the saddle point reveals critical stability and remains an evolutionarily unstable equilibrium point.

3.1.4. Replicator Dynamics (RD)

The replication dynamics, i.e., the participants still choose the most profitable of the strategies. Based on this, a differential equation describing the number of times a strategy is selected versus time is generally used to describe as follows.

$${\displaystyle \frac{\mathrm{d}{x}_i^j}{\mathrm{d}t}}=\left[f\left(s_i^j,x\right)-f\left(x_i-x_{-i}\right)\right]x_i^j$$

(2)

where i represents the group number and n is the total number of groups; $x_i^j$ is the percentage of the number of individuals in the given group i that selected the j-th pure strategy out of the total number of individuals in that given group i; $x_i$ represents one group state, while $x_{-i}$ is another completely relative group state in the group i; $s_i^j$ denotes the j-th pure strategy out of the total number of decision combinations in a given group i; x represents the mixed strategy combination of all groups, and then $f\left(s_i^j,x\right)\mathrm{a}\mathrm{n}\mathrm{d}f\left(x_i-x_{-i}\right)$ represent the payoff of group i choosing a specific pure strategy when the state is x, and the average payment parameter, respectively.

Based on the evolutionary stability (asymptotic stability) theorem of the RD equations, the final stabilization decision scheme can be derived by calculating the equilibrium point of the RD equations and performing stability analysis.

Overall, based on [72], we propose a complete methodology system for theoretical analysis of the long-term evolutionary stable equilibrium characteristics of multi-subject multi-strategy games in general situations and the system flowchart of dynamic simulation research methods, as demonstrated in Figure 1.

3.2. A Basic Model of Two-Group Asymmetric Evolutionary Game

In this study, we model the interaction between power grid companies and renewable energy producers as a two-group two-strategy evolutionary game. The key assumptions of the model include bounded rationality of the agents and the availability of incomplete information regarding the opposing party’s strategy. The model is built on a payoff matrix, where the two strategies for each group are cooperation (i.e., peak shaving or integrating renewable energy) and non-cooperation. The replicator dynamics equations are derived to track the evolution of strategies over time, with stability conditions defined by the Jacobi matrix. The parameters used in the model include peak shaving revenue, marginal costs, and external subsidies, which are grounded in both theoretical simulations and empirical observations from real-world data.

Based on this, in this section, we introduce the evolutionary game-theoretic model used to analyze the interaction between power grid companies and renewable energy generation enterprises. The model is based on a two-group, two-strategy asymmetric game structure. We begin by outlining the basic assumptions, model parameters, and the structural components of the evolutionary game. Core assumptions of the model are elaborated as follows.

(a)

Bounded Rationality. The participants (i.e., power grid companies and renewable energy generation enterprises) are assumed to have bounded rationality. This means that while they aim to maximize their respective payoffs, they do so under limited information and cognitive constraints. Therefore, instead of making fully rational decisions, they adapt their strategies based on historical payoffs and observed behaviors.

(b)

Groups and Strategies. The first group consists of the renewable energy generation enterprises (Group A), and the second group consists of the power grid companies (Group B). Both groups are provided with two pure strategies: Group A (Renewable Energy): Strategy 1 (Cooperate) and Strategy 2 (Not Cooperate); Group B (Grid Companies): Strategy 1 (Cooperate) and Strategy 2 (Not Cooperate).

(c)

Payoff Parameters. a, b, c, d, e, f, g, h are the payoff parameters when different strategies are selected by the groups. For example, the payoff of Group A selecting Strategy 1 (Cooperate) when Group B selects Strategy 2 (Not Cooperate) is denoted by a, while the reverse scenario would be denoted by another payoff parameter.

(d)

Structure of the Evolutionary Game. The interaction between the two groups is represented as a two-player asymmetric evolutionary game. The strategies selected by both groups result in certain payoffs, and the model tracks how the strategies evolve over time. To evaluate the long-term behavior of the system, we calculate the Jacobian matrix from the replicator dynamic equations, which helps to assess the local stability of equilibrium points.

(e)

Stability Analysis and Simulation. By analyzing the determinant and trace of the Jacobian matrix, we can classify the stability of the equilibrium points: Evolutionarily stable strategies (ESS), which mean that if the determinant of the Jacobian is positive and the trace is negative, the equilibrium point is evolutionarily stable; Unstable Equilibria, which mean that if the trace is positive, the equilibrium is unstable; and Saddle Points, which mean that if the trace is zero and the determinant is positive, the equilibrium is a saddle point.

3.2.1. Mathematical Modeling

Assuming there are two groups A and B, each with two pure strategies S_A1 and S_A2, S_B1 and S_B2, respectively, the corresponding selection probabilities are x and (1 − x) and y and (1 − y). Obviously, there are a total of four game combinations in this evolutionary game system, as shown in Formula (3), where a, b, c, d, e, f, g, and h are the payoff parameters when different groups choose different strategies. The asymmetric evolutionary payment matrix of two populations in general situations is shown as follows.

$$\begin{array}{ll}& y\hspace{1em}\hspace{1em} 1-y\\ & {\mathrm{S}}_{\mathrm{B}1}\hspace{1em}\hspace{1em}{\mathrm{S}}_{\mathrm{B}2}\\ \mathrm{A}\left\{\begin{array}{c}x\\ 1-x\end{array}\right.\begin{array}{c}{\mathrm{S}}_{\mathrm{A}1}\\ {\mathrm{S}}_{\mathrm{A}2}\end{array}& \left[\begin{array}{cc}(a,b)& (c,d)\\ (e,f)& (g,h)\end{array}\right]\end{array}$$

(3)

Based on Formula (3), assume that the expected payoff of group A and the average payoff of the entire group A when the individuals adopt strategy S_A1 are $E_{{\mathrm{S}}_{\mathrm{A}1}}$ and ${\overline{E}}_{{\mathrm{S}}_{\mathrm{A}}}$, respectively. They are obtained as follows:

$$\left\{\begin{array}{l}{E}_{{\mathrm{S}}_{\mathrm{A}1}}=ay+c(1-y)\hfill \\ {\overline{E}}_{{\mathrm{S}}_{\mathrm{A}}}=axy+cx(1-y)+e(1-x)y+g(1-x)(1-y)\hfill \end{array}\right.$$

(4)

Similarly, we assume that the expected payoff of group B and the average payoff of the entire group B when the individuals adopt strategy S_B1 are $E_{{\mathrm{S}}_{\mathrm{B}1}}$ and ${\overline{E}}_{{\mathrm{S}}_{\mathrm{B}}}$, respectively. They are obtained as follows:

$$\left\{\begin{array}{l}{E}_{{\mathrm{S}}_{\mathrm{B}1}}=bx+f(1-x)\hfill \\ {\overline{E}}_{{\mathrm{S}}_{\mathrm{B}}}=bxy+f(1-x)y+d(1-x)y+h(1-x)(1-y)\hfill \end{array}\right.$$

(5)

Further, we can obtain the dynamic equations of the system RD when individuals in group A and group B adopt strategies S_A1 and S_B1, respectively, as follows:

$$\begin{array}{ll}F(x)& ={\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}=x(E_{{\mathrm{S}}_{\mathrm{A}1}}-{\overline{E}}_{{\mathrm{S}}_{\mathrm{A}}})\\ & =x(1-x)(ay+c-cy-ey-g+gy)\\ & =x(1-x)[(a-c-e+g)y+(c-g)]\end{array}$$

(6)

$$\begin{array}{ll}F(y)& ={\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}}=y(E_{{\mathrm{S}}_{\mathrm{B}1}}-{\overline{E}}_{{\mathrm{S}}_{\mathrm{B}}})\\ & =y(1-y)(bx+f-fx-dx-h+hx)\\ & =y(1-y)[(b-f-d+h)x+(f-h)]\end{array}$$

(7)

Based on Equations (6) and (7), the corresponding Jacobian matrix J is obtained as

$$\mathit{J}=\left[\begin{array}{cc}(1-2x)[(a-c-e+g)y+(c-g)]& x(1-x)(a-c-e+g)\\ y(1-y)(b-f-d+h)& (1-2y)[(b-f-d+h)x+(f-h)]\end{array}\right]$$

(8)

For the Jacobian matrix J shown in (8), we can further obtain its determinant and trace, which are denoted by det(J) and tr(J) as follows, respectively:

$$\left\{\begin{array}{l}\det (\mathit{J})=(1-2x)(1-2y)(b-f-d+h)x+(f-h)[(a-c-e+g)y+c-g]\\ \hspace{1em}\hspace{1em} -xy(1-x)(1-y)(a-c-e+g)(b-f-d+h)\\ tr(\mathit{J})=(1-2x)[(a-c-e+g)y+c-g]+(1-2y)[(b-f-d+h)x+(f-h)]\end{array}\right.$$

(9)

As mentioned above, through the Lyapunov stability theorem, we can determine the asymptotic stability of the corresponding internal equilibrium point by judging the determinant and trace of the Jacobian matrix J in (9) at a certain equilibrium point, that is, whether it can maintain a long-term evolutionarily stable state.

3.2.2. Simulation Verification

According to the replicator dynamic equations obtained in the previous section, we assign values of 2, 5, 7, 8, 6, 3, 4, 1 to the above payoff parameters in (3) in order. In the evolutionary game with x and y ranging from 0 to 1, we take 1/q as the interval step, where q = 10, 20, 30, 40, 50, and 60. The phase trajectory diagrams obtained with different step sizes are demonstrated in Figure 2, where the red dots shown in the figures are evolutionarily stable points, the blue dots are evolutionarily near-stable points (saddle points), and green dots are unstable points. Furthermore, we conducted stability analysis on each equilibrium point in Figure 2 separately, and the analysis results are shown in

Table 1. Results of long-term evolutionary stability analysis for each equilibrium point.

(x, y)	det(J)	tr(J)	Evolutionary Stability
(0, 0)	6	5	Evolutionarily unstable strategy
(0, 1)	8	−6	Evolutionarily stable strategy (ESS)
(1, 0)	9	−6	ESS
(1, 1)	12	7	Evolutionarily unstable strategy
(2/5, 3/7)	−35/72	0	Saddle point (it also remains evolutionarily unstable)

. The analysis results show that the simulation results can effectively verify the aforementioned theoretical RD model and its theoretical derivation process.

4. Evolutionary Game Analysis of Thermal Power Enterprise Groups’ Participation in Peak Shaving

The ultimate goal of power plants is to maximize profits. In each round of peak shaving game, it is impossible for each thermal power plant to fully consider the external environment and the choices of competitors. With the deepening of power market reform, power generation enterprises will gradually participate in market competition. In the new situation, there have been significant changes in the interest pattern of thermal power plants, with increasingly fierce competition among interest entities and more non-cooperative relationships. The game behavior of thermal power plants exhibits characteristics of “bounded rationality” and “learning adjustment strategies” [72]. In order to analyze the dynamic competitive behavior and influencing factors of different types of thermal power enterprises under different market structures, this article first classifies thermal power enterprises and then establishes a dynamic game model of thermal power enterprises based on multi-stage game theory.

The evolutionary game model of peak regulation in thermal power set in this paper belongs to a non-cooperative nature of evolutionary game. The composition of power plant revenue consists of bidding profit and peak regulation compensation. The factors affecting the bidding price and ancillary service prices and their interrelationships were analyzed, and then a power generation bidding model was established and a solution algorithm was provided; based on this, a peak optimization method for thermal power units based on multi-objective planning was proposed. Power plants should formulate power supply strategies for units based on the range of power generation and other historical pricing strategies for local power supply. In order for a thermal power plant to obtain maximum benefits, it must ensure the economic efficiency of its own power generation. With the acceleration of the marketization process in China’s power industry, the competition among power plants is becoming increasingly fierce, and how to improve economic efficiency has become a common concern for all power enterprises. Due to the differences in equipment unit models, even if the prices are the same, the operating costs of equipment units will vary due to different manufacturers. In the current electricity market environment, power generation companies face pressure from competitive grid access. In market competition, each power plant must quote reasonably according to its own actual situation in order to enhance its competitiveness. Driven by profits, each plant must adjust its game strategy to increase or maintain its revenue.

4.1. General Evolutionary Game Model of Thermal Power Peak Shaving in Normal Situations

4.1.1. Evolutionary Game Model without Considering Peak Shaving Auxiliary Service Revenue for Thermal Power Plant Group

Driven by the continuous improvement of wind power technology, the global wind power generation capacity has been increasing. In recent years, various countries around the world have begun to gradually pay attention to the construction of wind power generation, constituting a significant increase in the global installed capacity of wind power. Before constructing the evolutionary game model, a simple data establishment is performed first. It is assumed that there are two types of thermal power units in the grid: the first type is 300 MW (k = 1), and the second type is 600 MW (k = 2). Afterwards, it is defined that thermal power companies are categorized into two types of offers: high quoting price strategy and low quoting price strategy. The high quotation is defined as a high-bid strategy S_price1 (i = 1) and the low quotation as denoted as a low-bid strategy S_price2 (i = 2). The following is the revenue matrix and corresponding payoff parameters for the two types of thermal power plants without considering the peak shaving scenario [23], as demonstrated in

Table 2. The revenue matrix and corresponding payoff parameters for the two types of thermal power plants without con-sidering the peak shaving scenario.

Quotation Strategy	Sprice1 (High-Bid Strategy by Group 2)	Sprice2 (Low-Bid Strategy by Group 2)
Sprice1 (high-bid strategy by group 1)	(a1 + b + d, a2 + e + d)	(a1 − e, a2 + e)
Sprice2 (low-bid strategy by group 1)	(a1 + b, a2 − b)	(a1, a2)

.

In Table 2, a₁ represents the profit of the first type of thermal power plant group when choosing the base price quotation (S_price2); a₂ represents the profit of the second type of thermal power plant when choosing the base price quotation (S_price2); when the first type of thermal power plant group adopts S_price1 and the second type of thermal power plant group adopts S_price2, the latter’s excess profit or the former’s loss is defined as e; when the first type of power generators adopt S_price2 and the second type of power generators adopt S_price1, the additional profit obtained by the former compared to both when they adopt S_price2 is defined as b; when both types of thermal power generation groups adopt quotation S_price1, the additional profit compared to when they both choose quotation S_price2 is defined as d.

4.1.2. Evolutionary Game Model of Thermal Power Plant Group Considering Peak Shaving Auxiliary Service Revenue

Large-scale wind power plants are more volatile than thermal power plants, and it is difficult to store a large amount of electricity. Existing research work shows that wind power can only provide electricity to the grid and cannot provide efficient generating capacity. In addition, as wind power is integrated on a large scale into the power system, the fluctuations it brings to the entire power system are increasing. Currently, most regions in China are facing serious power shortages. At the same time, as the scale of wind power integration increases, the issue of its accommodation is becoming increasingly prominent. Therefore, in order to ensure the stable operation of the grid, a certain number of thermal power plants are needed to provide peak shaving auxiliary services after the large-scale integration of wind power. We assume that C_j is peak shaving benefit, which is calculated based on the following equations.

$$\left\{\begin{array}{l}{C}_j={\eta }_j{\alpha }_j{S}_{\mathrm{jzz}}(P_j-F_j-P_j{Y}_{iC}^{\prime })T\\ {\eta }_j=\left\{\begin{array}{cc}0\hfill & F_j\le W_j(where W_j=T{F}_{ij}(1-\delta ))\\ 1\hfill & F_j>W_j\end{array}\right.\\ {\alpha }_j=B_j/B , where B_j=(P_j-F_j)/P_j(1-{\gamma }_j), B=({\displaystyle {\sum }_{j=1}^N{B}_j})/n\end{array}\right.$$

(10)

where C_j is the peak shaving auxiliary cost subsidy, η_j represents the start-stop status of units, α_j is the peak load capability level of each unit within the same power grid system, S_jzz is the benchmark compensation standard for peak load regulation behavior stipulated by the system, P_j represents the unit capacity, F_j represents the average output size of the unit during a certain period, $Y_{iC}^{\prime }$ is the compensation for peak shaving limit, F_ij is the minimum technical output of the unit, B_j is the actual peaking behavior of the unit, T represents the number of peak shaving in the power system, B represents the average peak shaving capability of the unit, γ_j is the minimum load rate of unit j, $\delta$ represents the power plant’s factory electricity consumption rate, and W_j is the critical output value of unit j.

The peak shaving compensation standards for thermal power plants are shown in

Table 3. The peak shaving compensation standards for thermal power plants.

Unit Category	Capacity Range/MW	Compensation for Peak Shaving Limit/%
1	$\le 300$	29
2	$\ge 600$	40

(where the peak shaving limit ratio equals 1 minus the value of unit output ratio). For convenience, the following represent the peak shaving benefits of the first type of thermal power plant and the second type of thermal power plant, respectively. The compensated peak shaving limits of thermal power enterprises are divided into two categories, with higher unit peak shaving limits indicating stronger peak shaving capabilities. After introducing peak shaving compensation, the payment matrix for considering peak shaving can be obtained, as shown in

Table 4. Payoff distribution matrix of bidding game among thermal power plants considering peak shaving auxiliary benefits.

Quotation Strategy	Sprice1 (High-Bid Strategy) from the Second Type of Unit Group	Sprice2 (Low-Bid Strategy) from the Second Type of Unit Group
Sprice1 (high-bid strategy) from the first type of unit group	$(a_1+b+d+C_{1j},a_2+e+d+C_{2j})$	$(a_1-e+C_{1j}^{\prime },a_2+e+C_{2j}^{\prime })$
Sprice2 (low-bid strategy) from the first type of unit group	$(a_1+b+C_{1j}^{″},a_2-b+C_{2j}^{″})$	$(a_1+C_{1j}^{‴},a_2+C_{2j}^{‴})$

.

Based on Table 3 and Table 4 above, we further assume that x and y represent the proportions of generators in the first and second type of thermal power plant quoting the high-bid strategy S_price1, respectively. Then, we can obtain the system state S, which is denoted by $S=\{(s_1^1,s_2^1),(s_1^2,s_2^2)\}=\{(x,1-x),(y,1-y)\}$, where (x, y) can represent the dynamic evolution of the quotation strategies adopted by the thermal power plants. Furthermore, for the first type of power plant group, when the generators in this group adopt the pricing strategy S_price1 and S_price2, their payments are calculated as $f^1(r^1,t)$ and $f^1(r^2,t)$, respectively.

$$\left\{\begin{array}{l}{f}^1(r^1,t)=(a_1+b+d+C_{1j})y+(a_1-e+C_{1j}^{\prime })(1-y)\\ f^1(r^2,t)=(a_1+b+C_{1j}^{″})y+(a_1+C_{1j}^{‴})(1-y)\end{array}\right.$$

(11)

Based on (11), we can obtain the average fitness as

$$f^1(x,t)=x{f}^1(r^1,t)+(1-x)f^1(r^2,t)$$

(12)

Similarly, for the second type of power plant group, when the generators in this group adopt the pricing strategy S_price1 and S_price2, their payments are calculated as $f^2(r^1,t)$ and $f^2(r^2,t)$, respectively.

$$\left\{\begin{array}{l}{f}^2(r^1,t)=(a_2+e+d+C_{2j})x+(a_2-b+C_{2j}^{″})(1-x)\\ f^2(r^2,t)=(a_2+e+C_{2j}^{\prime })x+(a_2+C_{2j}^{\prime })(1-x)\end{array}\right.$$

(13)

Based on (13), we can obtain the average fitness as

$$f^2(y,t)=y{f}^2(r^1,t)+(1-y)f^2(r^2,t)$$

(14)

Based on Formula (14), as mentioned earlier, it can be seen that when the fitness of any strategy in the population is greater than the average fitness of the population, this strategy is an evolutionarily stable strategy. Based on this, the replicator dynamics equation for the first type of power generators group when selecting S_price1 for bidding is:

$$\begin{array}{ll}\frac{\mathrm{d}x}{\mathrm{d}t}& =x\left(f^1(r^1,t)-f^1(x,t)\right)\\ & =x(1-x)\{y(d+C_{1j}-C_{1j}^{″}+e-C_{1j}^{\prime }+C_{1j}^{‴})-(e-C_{1j}^{\prime }+C_{1j}^{‴})\}\end{array}$$

(15)

Similarly, the replicator dynamics equation for the second type of power generators group when selecting S_price1 for bidding is:

$$\begin{array}{ll}\frac{\mathrm{d}y}{\mathrm{d}t}& =y\left(f^2(r^2,t)-f^2(y,t)\right)\\ & =y(1-y)\{x(d+C_{2j}-C_{2j}^{\prime }+b-C_{2j}^{″}+C_{2j}^{‴})-(b-C_{2j}^{″}+C_{2j}^{‴})\}\end{array}$$

(16)

Based on the RD equations shown in (15) and (16), the dynamic differential equations of this evolutionary game system can be obtained as:

$$\left\{\begin{array}{l}{\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}=x(1-x)\{y(d+C_{1j}-C_{1j}^{″}+e-C_{1j}^{\prime }+C_{1j}^{‴})-(e-C_{1j}^{\prime }+C_{1j}^{‴})\}\\ {\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}} =y(1-y)\{x(d+C_{2j}-C_{2j}^{\prime }+b-C_{2j}^{″}+C_{2j}^{‴})-(b-C_{2j}^{″}+C_{2j}^{‴})\}\end{array}\right.$$

(17)

We can see from Formula (17) that the percentage of units in the first type of thermal power plants selecting the high-bid strategy S_price1 will be evolutionarily stable only when x = 0 or 1 and at the same time $y=(e-C_{1j}^{\prime }+C_{1j}^{‴})/(d+C_{1j}-C_{1j}^{″}+e-C_{1j}^{\prime }+C_{1j}^{‴})$. Similarly, only when y = 0 or 1 and meanwhile $x=(b-C_{2j}^{″}+C_{2j}^{‴})/(d+C_{2j}-C_{2j}^{\prime }+b-C_{2j}^{″}+C_{2j}^{‴})$, the percentage of generators in the second type of thermal power plants selecting the high-bid strategy S_price1 will be evolutionarily stable. To this end, we can separately calculate the first-order partial derivatives of x and y with respect to the strategic bidding time t to obtain the corresponding Jacobian matrix J as follows.

$$\mathit{J}=\left[\begin{array}{cc}{J}_{11}& J_{12}\\ J_{21}& J_{22}\end{array}\right]$$

(18)

where the four elements in J are $J_{11}=(1-2x)[y(d+C_{1j}-C_{1j}^{″}+e-C_{1j}^{\prime }+C_{1j}^{‴})-(e-C_{1j}^{\prime }+C_{1j}^{‴})]$, $J_{12}=x(1-x)(d+C_{1j}-C_{1j}^{″}+e-C_{1j}^{\prime }+C_{1j}^{‴})$, $J_{21}=y(1-y)(d+C_{2j}-C_{2j}^{\prime }+b-C_{2j}^{″}+C_{2j}^{‴})$, and $J_{22}=(1-2y)[x(d+C_{2j}-C_{2j}^{\prime }+b-C_{2j}^{″}+C_{2j}^{‴})-(b-C_{2j}^{″}+C_{2j}^{‴})]$.

4.2. Long-Term Evolutionary Stability Analysis for Peak Regulation Involving Thermal Power Units

4.2.1. Peak Shaving Ancillary Service Revenue Not Considered

In this scenario, both the first type of thermal power plant group and the second type of thermal power plant group have not reached the compensation standard limit for peak regulation rates, nor have they met the peak regulation standard, so C_ij equals 0. At this time, the RD dynamic equation can be simplified to the following form:

$$\left\{\begin{array}{c}{\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}=x(1-x)\{y(d+e)-e\}\\ {\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}} =y(1-y)\{x(d+b)-b\}\end{array}\right.$$

(19)

Correspondingly, the Jacobian matrix J can be simplified as:

$$\mathit{J}=\left[\begin{array}{cc}(1-2x)\{y(d+e)-e\}& x(1-x)(d+e)\\ y(1-y)(d+b)& (1-2y)\{x(d+b)-b\}\end{array}\right]$$

(20)

Based on (20), we can further analyze the local asymptotic stability of each internal equilibrium point, that is, the stability analysis results of the long-term competitive bidding evolution game of various thermal power plants without peak shaving compensation, as summarized in

Table 5. The asymptotic stability analysis results of two types of thermal power plant groups in a long-term bidding evolution game without peak shaving compensation.

Internal Equilibrium Points	det (J)	tr(J)	Asymptotic Stability Analysis Results
(0, 0)	be (positive)	−b − e (negative)	ESS
(0, 1)	db (positive)	d + b (positive)	Evolutionarily unstable
(1, 0)	de (positive)	d + e (positive)	Evolutionarily unstable
(1, 1)	d2 (positive)	−2d (negative)	ESS
$({\displaystyle \frac{b}{d+b}},{\displaystyle \frac{e}{e+d}})$	$-{\displaystyle \frac{be{d}^2}{(d+b)(d+e)}}$ (negative)	0	Saddle point (also evolutionarily unstable)

. From Table 5, it can be seen that the internal points (0, 0) and (1, 1) are long-term evolutionarily stable states, which means that the bidding strategies of both types of thermal power plants will simultaneously quote high prices and low prices, respectively. At this time, regardless of how the initial data (namely their initial pricing strategy probability distribution) change, the bidding strategies of these two groups of power generators in the electricity bidding market will tend towards these two possibilities, that is, the probability that the groups of power generators will adopt these two bidding strategies will be the same. On this basis, we will next further study the evolution of the bidding strategy of the generator groups when considering the benefits of the peak shaving ancillary services.

4.2.2. Peak Shaving Ancillary Service Revenue Is Considered

In this case, at least one type of thermal power plant group has reached the limit of the compensation standard for peak shaving regulation rate. Then, the corresponding RD equations at this time are shown as follows.

$$\left\{\begin{array}{c}{\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}=x(1-x)\{y(d+C_{1j}-C_{1j}^{″}+e-C_{1j}^{\prime }+C_{1j}^{‴})-e+C_{1j}^{\prime }-C_{1j}^{‴}\}\\ {\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}} =y(1-y)\{x(d+C_{2j}-C_{2j}^{\prime }+b-C_{2j}^{″}+C_{2j}^{‴})-b+C_{2j}^{″}-C_{2j}^{‴}\}\end{array}\right.$$

(21)

Based on Formula (21), we can further obtain its Jacobian matrix shown as follows.

$$\mathit{J}=\left[\begin{array}{cc}{J}_{11}^{\prime }& J_{12}^{\prime }\\ J_{21}^{\prime }& J_{22}^{\prime }\end{array}\right]$$

(22)

where the four elements in J are represented as

$$\left\{\begin{array}{l}{J}_{11}^{\prime }=(1-2x)\{y(d+C_{1j}-C_{1j}^{″}+e-C_{1j}^{\prime }+C_{1j}^{‴})-e+C_{1j}^{\prime }-C_{1j}^{‴}\}\\ J_{12}^{\prime }=x(1-x)(d+C_{1j}-C_{1j}^{″}+e-C_{1j}^{\prime }+C_{1j}^{‴})\\ J_{21}^{\prime }=y(1-y)(d+C_{2j}-C_{2j}^{\prime }+b-C_{2j}^{″}+C_{2j}^{‴})\\ J_{22}^{\prime }=(1-2y)\{x(d+C_{2j}-C_{2j}^{\prime }+b-C_{2j}^{″}+C_{2j}^{‴})-b+C_{2j}^{″}-C_{2j}^{‴}\}\end{array}\right.$$

(23)

Further, the evolution stability analysis results for these 5 equilibrium points are shown in

Table 6. The asymptotic stability analysis results of these 5 equilibrium points when considering peak shaving compensation.

Internal Equilibrium Points	det (J)	tr(J)	Asymptotic Stability Analysis Results
(0, 0)	${\pi }_1$	${\pi }_2$	ESS
(0, 1)	${\pi }_3$	${\pi }_4$	Uncertain
(1, 0)	${\pi }_5$	${\pi }_6$	Uncertain
(1, 1)	${\pi }_7$	${\pi }_8$	ESS
$({\displaystyle \frac{b}{d+b}},{\displaystyle \frac{e}{e+d}})$	${\Delta }_1{\Delta }_2$	0	Saddle point (also evolutionarily unstable)

, where ${\pi }_1=(e-C_{1j}^{\prime }+C_{1j}^{‴})(b-C_{2j}^{″}+C_{2j}^{‴})$, ${\pi }_2=-(e-C_{1j}^{\prime }+C_{1j}^{‴})-(b-C_{2j}^{″}+C_{2j}^{‴})$,${\pi }_3=(d+C_{1j}-C_{1j}^{″})\times (b-C_{2j}^{″}+C_{2j}^{‴})$, ${\pi }_4=(d+C_{1j}-C_{1j}^{″})+(b-C_{2j}^{″}+C_{2j}^{‴})$, ${\pi }_5=(d+C_{2j}-C_{2j}^{\prime })\times (e-C_{1j}^{\prime }+C_{1j}^{‴})$, ${\pi }_6=(d+C_{2j}-C_{2j}^{\prime })+(e-C_{1j}^{\prime }+C_{1j}^{‴})$, ${\pi }_7=(-d-C_{1j}+C_{1j}^{″})\times (-d-C_{2j}+C_{2j}^{\prime })$, ${\pi }_8=(-d-C_{1j}+C_{1j}^{\prime })+(-d-C_{2j}+C_{2j}^{\prime })$, ${\Delta }_1={\mathcal{l}}_1/\left\{({\mathcal{l}}_2-{\mathcal{l}}_1){\mathcal{l}}_3\right\}$, ${\Delta }_2={\mathcal{l}}_4/\left\{({\mathcal{l}}_5-{\mathcal{l}}_4){\mathcal{l}}_6\right\}$, ${\mathcal{l}}_1=d+C_{2j}-C_{2j}^{\prime }+b-C_{2j}^{″}+C_{2j}^{‴}$, ${\mathcal{l}}_2=b-C_{2j}^{″}+C_{2j}^{‴}$, ${\mathcal{l}}_3=d+C_{1j}-C_{1j}^{″}+e-C_{1j}^{\prime }+C_{1j}^{‴}$, ${\mathcal{l}}_4=d+C_{1j}-C_{1j}^{″}+e-C_{1j}^{\prime }+C_{1j}^{‴}$, ${\mathcal{l}}_5=e-{C^{\prime }}_{1j}+C_{1j}^{‴}$, and ${\mathcal{l}}_6=d+C_{2j}-C_{2j}^{\prime }+b-C_{2j}^{″}+C_{2j}^{‴}$.

We can see from Table 6 that, since there are no detailed numerical values, it is not possible to determine the positive and negative values of the parameters related to the equilibrium point, and the stability of the equilibrium point cannot be judged. It is necessary to combine actual specific numerical values to determine the stable strategy of evolutionary game under peak shaving conditions.

4.3. Example Verification

4.3.1. Peak Shaving Ancillary Service Revenue Not Considered

Based on the theoretical derivation and mathematical model mentioned above, the stability analysis of the five internal equilibrium points in the system is carried out through specific examples. Firstly, taking a certain region’s electricity bidding market as an example, a mathematical model of the market is constructed, and then the rated output of each unit and their proportion in the grid are calculated based on the model parameters. It should be noted that in the process of example analysis, it is necessary to clearly specify that the minimum stable load limit of thermal power plants in the region is within 40% of the rated power, otherwise they will not be able to participate in grid bidding. On this basis, we assume that the required capacity of the regional grid is 700 MW. According to the parameters in references [68,69,70,71,72], we can establish the cost functions and bidding schemes of the two types of thermal power plants mentioned above, with specific parameter settings as shown in

Table 7. The on-grid bidding schemes adopted by the thermal power generator groups.

300 MW	Generating Capacity Bidding Sections
	(120 150]	(150 180]	(180 210]	(210 240]	(240 270]	(270 300]
Sprice1 (high-bid strategy)	280	290	300	310	315	320
Sprice2 (low-bid strategy)	144.23	154.23	164.23	174.23	179.23	184.23
600 MW	Generating capacity bidding sections
	(240 300]	(300 360]	(360 420]	(420 480]	(480 540]	(540 600]
Sprice1 (high-bid strategy)	282	292	302	312	316	320
Sprice2 (low-bid strategy)	106.49	116.49	126.49	136.49	140.49	144.49

. Here, aiming at the calculation of power generation costs of power plants, for the first type of thermal power unit (denoted by group 1), with a power generation capacity of 300 MW, characterized by low power generation capacity and high power generation costs, the corresponding power generation cost is $C_1(q_1)=6700+110q_1-0.18q_1^2$; for the second type of thermal power unit (denoted by group 2), characterized by large power generation capacity and low power generation costs, with a power generation capacity of 600 MW, its generation cost is $C_2(q_2)=8000+91.4q_2-0.076q_2^2$.

Based on Table 7, we set the upper limit of the electricity price for thermal power generation at 320 yuan/MWh. The electricity bidding market in this region adopts a settlement mechanism based on a unified market clearing price. Thus, the revenue of the first type of thermal power plants (group 1) reporting base price (i.e., the low-bid strategy S_price2) is calculated $350\ast 184.23-(6700+110*175-0.18\ast {175}^2)\ast 2=44,043$ (yuan), and the revenue of the second type of generators (group 2) quoting the base price strategy S_price2 is obtained as $116.49\ast 350-(8000+91.4\ast 350-0.076\ast {350}^2)=10,091.5$ (yuan). This is the profit obtained by the two types of thermal power plants when they quote the base price, and at this time, the two groups share 700 MW of electricity. If one power generation company group chooses to report a high price and the other company group chooses the base price strategy, then the online electricity volume of the second company will be more than both companies reporting the base price strategy. The second-generation power generation company will pay more corresponding costs, which is parameter e and calculated as $e=116.49\ast 540-(8000+91.4\ast 540-0.076\ast {540}^2)-10,091.5=17,618.7$ (yuan).

The power generation companies connected to the grid have less electricity, which inevitably leads to lower payments. When the first company adopts a base price strategy and the second company chooses a higher price, the former will first connect to the grid for power generation and pay more fees, that is, parameter b, which is calculated as $b=184.23\ast 540-(6700+110\ast 270-0.18\ast {270}^2)\ast 2-44,043=8885.2$ (yuan). In addition, the reduction in feed-in tariffs in the second generation directly leads to a decrease in revenue. The spread between overpriced pricing and underpricing is an additional gain for the firms, and when both generating firms offer the overpricing strategy at the same time, they both gain additional profit, i.e., the parameter d, which is calculated as $d=320\ast 350-(6700+110\ast 175-0.18\ast {175}^2)\ast 2-44,043-8885.2=41,196.8$ (yuan).

When the peak shaving ancillary service revenues are not considered, we substitute the calculated parameters into the payment matrix (as shown in Table 2) without considering the peak shaving ancillary service revenue, and we can obtain the specific payment matrix as shown in

Table 8. Distribution of on-grid electricity price quotation strategy for thermal power plants without considering peak shaving ancillary service revenue.

Quotation Strategy	Sprice1 (High-Bid Strategy by Group 2)	Sprice2 (Low-Bid Strategy by Group 2)
Sprice1 (high-bid strategy by group 1)	(94,125, 68,907)	(26,424.3, 23,710.2)
Sprice2 (low-bid strategy by group 1)	(52,928.2, 1206.3)	(44,043, 10,091.5)

and then proceed to stability analysis and other steps. Here, group 1 means the first type of thermal power generators and group 2 represents the second type of thermal power generators.

Based on Table 8, the above data are respectively taken as interval step lengths in the decision space [0, 1] × [0, 1], where q = 10, 20, 30, 40, 50, and 60. Then, we assign values to x and y from 0 to 1 for dynamic simulation, and the simulation results are demonstrated in Figure 3 as follows:

Here, Figure 3a shows the trajectory of the proportion x over time t, Figure 3b shows the trajectory of the proportion y over time t, and Figure 3c is the trajectory of (x, y) over time t in the entire evolutionary game system. In Figure 3, green points represent unstable points, blue points represent saddle points, and red points represent stable points (ESS). From the simulation results in Figure 3, it can be seen that the long-term dynamic evolution simulation ultimately tends towards the strategy of high price bidding by thermal power companies and low price bidding together, which belongs to an asymptotically stable strategy, consistent with the conclusion analysis in Table 5.

4.3.2. Consider Peak Shaving Ancillary Service Revenue

In this case, the actual online electricity generation and single-unit power generation capacity of the two types of power plant groups during peak load are shown in

Table 9. Distribution of on-grid electricity price quotation strategy for thermal power plants considering peak shaving ancillary service revenue.

Quotation Strategy	Sprice1 (High-Bid Strategy by Group 2)	Sprice2 (Low-Bid Strategy by Group 2)
Sprice1 (high-bid strategy by group 1)	(350, 350)	(160, 540)
Sprice2 (low-bid strategy by group 1)	(540, 160)	(350, 350)

and

Table 10. Distribution of single-unit power generation capacity quotation strategy of thermal power plants considering peak shaving ancillary service revenue.

Quotation Strategy	Sprice1 (High-Bid Strategy by Group 2)	Sprice2 (Low-Bid Strategy by Group 2)
Sprice1 (high-bid strategy by group 1)	(175, 350)	(160, 350)
Sprice2 (low-bid strategy by group 1)	(270, 350)	(175, 350)

.

By analyzing Table 9 and Table 10, it can be seen that the single-unit power generation capacity of the second type of thermal power plant is 350 MW, with a peak shaving rate of 41.6%, meeting the peak shaving standard; meanwhile, the peak shaving rate of the first type of power plant is also 41.6%, not meeting the peak shaving standard. To this end, we set S = 0.19, the average peak shaving rate is set as 50%, and T = 1000. According to the evolutionary game model established in this paper, we can obtain ${\alpha }_j=1.4$ and ${\eta }_j=1$, and then we substitute them into the following equation $C_{2j}={\eta }_j{\alpha }_{j}S(P_j-F_j-P_j{Y}_{iC}^{\prime })T$, we can obtain $C_{2j}=1\ast 1.4\ast 0.19\ast (600-350-600\ast 0.4)\ast 1000=2639$. For the above parameters, we take iterative steps for simulation verification, and here we take q = 10, 20, 30, 40, 50, and 60. By assigning initial values from 0 to 1 for x and y, we can obtain the following 18 phase trajectory curves, as demonstrated in Figure 4.

The rules are as shown in the previous figures, and based on these rules, we can see from Figure 4 that the power bidding market containing two types of thermal power generator groups will finally converge to two internal equilibrium points: (0, 0) and (1, 1), that is, the market will eventually reach a long-term evolutionarily stable state at (0, 0) and (1, 1). At this time, the two types of generator groups will both tend to quote high-bid strategy or low-bid strategy to maximize their revenues. The evolution stability analysis is shown in

Table 11. The asymptotic stability analysis results of all internal equilibrium points for the two types of generator groups in bidding when considering peak shaving compensation.

Internal Equilibrium Points	det (J)	tr(J)	Asymptotic Stability Analysis Results
(0, 0)	(+)	(−)	ESS
(0, 1)	(+)	(+)	Evolutionarily unstable
(1, 0)	(+)	(+)	Evolutionarily unstable
(1, 1)	(+)	(−)	ESS
(0.177, 0.23)	(−)	0	Saddle point (also evolutionarily unstable)

. As seen in Table 11, by analyzing the phase trajectory diagram, the system will directly determine different evolutionary stable strategies in different initial states. When both generation companies initially adopt a high-price bidding strategy, the final evolutionarily stable strategy of the two types of thermal power companies will also tend towards the initial strategy. In addition, when the two power generation plant groups initially quote low prices, the final stable strategy point will also tend towards the initial strategy. Even if new strategies emerge, this result will not change. This indicates that the market has reached a long-term evolutionarily stable equilibrium state at this point, and any variant pricing strategy will not be able to penetrate this stable equilibrium state. Overall, we simulate and draw conclusions through the evolutionary game model by inputting actual data, and we find that when we analyze the thermal power plant model without considering peak shaving, the analysis results are similar to the general model situation, and moreover, the stability analysis considering peak shaving is similar to the stability peak shaving analysis without considering peak shaving, and the two cases are correlated.

5. Evolutionary Game Analysis of Renewable Energy Integration Benefits Coordination

5.1. Parameters Setting

The payment matrix of the evolutionary game model between renewable energy power generation enterprises and power grid enterprises is similar to the payment matrix of the typical deer hunting game. The main idea of the deer hunting game is that two hunters can only hunt rabbits if they hunt alone, but can hunt deer if they cooperate together, and the benefit of cooperating to hunt deer is naturally much larger than that of hunting rabbits. It is well known that cooperation is win-win, but in practice, hunters have high or low ability, and often seek to maximize their own interests. In the process of benefit distribution, the party with strong ability and big contribution will put forward higher requirements on the distribution of benefits and lead to disagreements resulting in difficulty in achieving cooperation, so cooperation should be based on mutual full respect for each other’s interests.

Prior to modeling, the parameters involved in the following are defined as follows (Game Player 1 denotes renewable energy generators and Game Player 2 denotes grid companies). Concretely, R₁ represents the normal returns earned by renewable energy generators when they adopt a non-cooperative strategy, R₂ represents the normalized benefits received by grid companies when they adopt a non-cooperative strategy, S_all is the total excess profit earned by both parties when they adopt a cooperative strategy, Q_n is the output of renewable energy generators when they choose a non-cooperative strategy, Q_y is the yield of renewable energy generators when choosing a cooperative strategy, l₁ is the incremental cost incurred by adding one unit of renewable electricity supply (the marginal cost of the renewable energy generator), l₂ is the incremental cost incurred by adding one unit of renewable electricity service (the grid company’s marginal cost), V₁ is the amount of unit subsidy for renewable energy electricity supply (renewable energy power producers), V₂ is the amount of unit subsidy for renewable electricity services (grid companies), λ is the grid firm’s allocation coefficient for the total excess profit S_all, and P is the price at which grid companies sell electricity from conventional energy sources. Furthermore, we assume that x is the proportion of renewable energy generation companies adopting a cooperative strategy within the group, and y is the proportion of grid companies adopting a cooperative strategy; correspondingly, 1 − x represents the proportion of renewable energy generation companies adopting a non-cooperative strategy, and while 1 − y represents the proportion of grid companies adopting a non-cooperative strategy. The payment matrix for renewable energy generation companies and grid companies is as follows:

$$\begin{array}{c}\mathrm{Game} \mathrm{Player} 1\begin{array}{cc}& \stackrel{\mathrm{Game} \mathrm{Player} 2}{\overbrace{\begin{array}{cc}y& 1-y\\ \mathrm{cooperation}& \mathrm{not} \mathrm{cooperation}\end{array}}}\end{array}\\ \left\{\begin{array}{cc}x& \mathrm{cooperation}\\ 1-x& \mathrm{not} \mathrm{cooperation}\end{array}\right.\left[\begin{array}{cc}({\Theta }_1, {\Theta }_2)& ({\Theta }_3, {\Theta }_4)\\ ({\Theta }_5, {\Theta }_6)& ({\Theta }_7, {\Theta }_8)\end{array}\right]\end{array}$$

(24)

In this evolutionary game payoff distribution matrix in (24), we observe two players (Game Player 1 and Game Player 2) and their corresponding strategies, which are divided into cooperative and non-cooperative behaviors. The eight variables ${\Theta }_1, {\Theta }_2, {\Theta }_3, {\Theta }_4, {\Theta }_5, {\Theta }_6, {\Theta }_7$, and ${\Theta }_8$ represent the respective payoffs for each combination of strategies between the two players, which are explained as follows.

• ${\Theta }_1$: The payoff for Game Player 1 when both players choose cooperation (x and y).
• ${\Theta }_2$: The payoff for Game Player 2 when both players choose cooperation (x and y).
• ${\Theta }_3$: The payoff for Game Player 1 when Game Player 1 cooperates (x), but Game Player 2 does not cooperate (1 − y).
• ${\Theta }_4$: The payoff for Game Player 2 when Game Player 1 cooperates (x), but Game Player 2 does not cooperate (1 − y).
• ${\Theta }_5$: The payoff for Game Player 1 when Game Player 1 does not cooperate (1 − x), but Game Player 2 cooperates (y).
• ${\Theta }_6$: The payoff for Game Player 2 when Game Player 1 does not cooperate (1 − x), but Game Player 2 cooperates (y).
• ${\Theta }_7$: The payoff for Game Player 1 when both players choose not to cooperate (1 − x and 1 − y).
• ${\Theta }_8$: The payoff for Game Player 2 when both players choose not to cooperate (1 − x and 1 − y).

This matrix outlines the dynamic interactions in an evolutionary game framework, where each player’s payoff is contingent on the combination of cooperative or non-cooperative strategies employed by both participants. Each $\Theta$ value reflects a unique scenario of cooperative or non-cooperative interaction. Concretely, the eight variables ${\Theta }_1, {\Theta }_2, {\Theta }_3, {\Theta }_4, {\Theta }_5, {\Theta }_6, {\Theta }_7$, and ${\Theta }_8$ in (24) are described as

$$\left\{\begin{array}{l}{\Theta }_1=R_1+(1-\lambda )S_{\mathrm{all}}-l_1{Q}_y\\ {\Theta }_2=R_2+\lambda S_{\mathrm{all}}-l_2{Q}_y\\ {\Theta }_3=R_1+V_1{Q}_y-l_1{Q}_y\\ {\Theta }_4=R_2\\ {\Theta }_5=R_1\\ {\Theta }_6=R_2+V_2{Q}_y-l_2{Q}_y\\ {\Theta }_7=R_1\\ {\Theta }_8=R_2\end{array}\right.$$

(25)

5.2. Simulation Analysis

Renewable energy generation companies and grid companies engage in repeated games in a space of limited information and bounded rationality. In this process, the strategies chosen by both parties may not necessarily be optimal, leading to different equilibrium outcomes. Under the guidance of relevant policies, the initial state of the game will change, affecting the final evolutionary equilibrium outcome of the game system. We assume that, for renewable energy generation companies, S_A1 is the cooperative strategy, S_A2 is the non-cooperative strategy; for grid companies, S_B1 is the cooperative strategy, and S_B2 is the non-cooperative strategy. The following section will simulate three sets of different parameters that satisfy the premise conditions (payment matrix inequality) in different scenarios. Since this paper mainly explores the benefit coordination and distribution relationship between renewable energy generation companies and grid companies, the values of different sets of parameters in each scenario mainly change the payoffs of both sides in the completely cooperative scenario.

The simulation is conducted all within the decision space of [0, 1] × [0, 1] with a step size of 1/20, taking initial values for x and y from 0 to 1, that is, repeating the dynamic game simulation for 441 rounds in the above decision space. The phase trajectory graph obtained through simulation can demonstrate the dynamic evolution characteristics of the system well, where the first subgraph of each row shows the phase trajectory graph of the proportion of individuals in population A adopting strategy S_A1, denoted by x, against decision time t. The second subgraph shows the phase trajectory graph of the proportion of individuals in population B adopting strategy S_B1, denoted by y, against decision time t. The third subgraph shows the phase trajectory graph of (x, y) against decision time t in the entire evolutionary game system. In addition, the meanings represented by the solid-colored dots in the phase trajectory graph are explained as follows. For a certain internal equilibrium point corresponding to the replicator dynamic equation of the evolutionary game model, the meanings represented by each color solid circle are as follows (and are universally applicable throughout the text, not repeated): green indicates that the internal equilibrium point is evolutionarily stable; red indicates that the internal equilibrium point is evolutionarily unstable; and blue indicates that the internal equilibrium point is evolutionarily critical stable (i.e., a saddle point); correspondingly, at the above equilibrium points, the evolutionary game model will achieve long-term evolutionarily stable equilibrium states (i.e., evolutionarily stable strategies), evolutionarily unstable equilibrium states, and evolutionarily critical equilibrium states. According to the different behavioral parameters affecting the evolution of the game system, the evolution results are analyzed and discussed as follows. Here, we discuss a typical scenario. In this case, regardless of the strategy adopted by the other party, when renewable energy generation companies and grid companies choose S_A1 and S_B1, respectively, their respective profits are greater than when they choose S_A2 and S_B2, respectively. These conditions need to be met simultaneously as follows, namely $R_1+(1-\lambda )S-l_1{Q}_y>R_1$, $R_1+V_1{Q}_y-l_1{Q}_y>R_1$, $R_2+\lambda S_{\mathrm{all}}–l_2{Q}_y>R_2$, and $R_2+V_2{Q}_y-l_2{Q}_y>R_2$. At this time, we can obtain

$$\left\{\begin{array}{l}{V}_1>l_1\\ {\displaystyle \frac{l_2}{2P+V_1+V_2-l_1-l_2}}<\lambda <{\displaystyle \frac{2P+V_1+V_2-2l_1-l_2}{2P+V_1+V_2-l_1-l_2}}\\ V_2>l_2\end{array}\right.$$

(26)

From Figure 5, it can be seen that the phase trajectory diagrams of the system all converge at the vertex (1, 1) in the upper right corner, spontaneously forming an ESS at that point and eventually reaching the ESE state. It can be inferred that in this scenario, it means that both parties have lower initial costs l_iQ_y, or higher excess profits S_all and subsidies V_iQ_y. Under the cooperation strategy, both renewable energy generation companies and grid companies will achieve higher profits, and the distribution of excess benefits between the two parties will be fairer and more reasonable. Therefore, both renewable energy generation companies and grid companies tend to adopt a cooperative strategy, and the game evolution system will eventually evolve into a stable state of complete cooperation, as shown by the green solid circle at (1, 1).

From Figure 6, we can see that the phase trajectory plots of the system both converge at (0, 0) and (1, 1), i.e., the ESS is spontaneously formed there and will eventually reach the ESE state. This suggests that the lower subsidy V_iQ of both parties relative to the initial cost l_iQ_y results in lower benefits when cooperating unilaterally, and thus both parties either cooperate together or neither of them cooperates, as indicated by the green solid dots at (1, 1) and (0, 0), respectively. If the excess profit S_all generated by cooperation is larger, i.e., the respective benefits ($\mathrm{i}.\mathrm{e}.,{\Theta }_1,{\Theta }_2$) of both parties are larger when they cooperate, then the probability of the game evolution system converging to the equilibrium point (1, 1) will increase, i.e., the saddle point will be shifted to (0, 0), implying that the probability of both parties adopting a cooperative strategy at the same time is increased as shown by the blue solid dots.

Figure 7 reveals that the phase trajectories of the system all converge at (0, 0), spontaneously forming an ESS at that point and eventually reaching the ESE state. It can be seen that at this point, compared to the marginal cost l_i of renewable energy generation companies and grid companies, the price P of conventional energy generation is lower. Therefore, in the competition with conventional power sources, renewable energy electricity prices are at a disadvantage. Furthermore, the subsidy amount V_i is lower than the marginal cost l_i, and the benefits generated by cooperation are too low for both parties. Regardless of how the benefits are distributed, the cooperative strategy is not attractive to either party. Therefore, the game evolution system will eventually evolve into a stable state of complete non-cooperation, as shown by the green solid dot at (0, 0).

In addition, we can analyze other scenarios and conduct simulation verification. Considering all these situations, we can find that there are only two long-term evolutionary equilibrium results between the two major interest entities in strategic games, which are either all cooperation or all non-cooperation. This is related to the saddle point (which is also an evolutionarily critical equilibrium point). It reveals that if the saddle point is close to the internal equilibrium point (0, 0), the evolutionary game system will tend to evolve along the path of complete cooperation. The position of the saddle point is jointly determined by parameters such as subsidy rate V, conventional energy price P, allocation coefficient λ, and marginal cost l of initial investment. The magnitude of the excess benefits brought by cooperation between renewable energy generation companies and grid companies is determined by V, P, and l, while the proportion of sharing the excess benefits by both parties is determined by λ. If the other prerequisites for the system to evolve along the path of complete cooperation are already met (i.e., parameters such as V, P, and l are already satisfied), then the attitude of the game parties towards the distribution scheme of excess profits (i.e., λ) will directly determine the evolution direction of the game evolution system.

5.3. Discussion

In the context of our study, which utilizes evolutionary game theory to analyze the dynamics between renewable energy generation enterprises and power grid companies, future environmental projections serve as a backdrop that could significantly influence the strategic decisions and stability dynamics modeled in our simulations. While our primary focus has been on the interactions and economic outcomes based on current and theoretical data, the incorporation of future-oriented environmental data indeed adds a valuable dimension to understanding the potential real-world applicability and robustness of our model. To address this point, we propose to augment our discussion in the following ways in the future.

(1)

Literature Integration: We will integrate forecasts from credible sources such as the Intergovernmental Panel on Climate Change (IPCC) and the World Meteorological Organization (WMO) that provide data on future environmental conditions likely to impact renewable energy production. These forecasts will be used to contextualize the assumptions of our evolutionary game model, especially in scenarios related to the availability and variability of solar and wind energy.

(2)

Model Sensitivity Analysis: We propose to extend our simulations to include sensitivity analyses that explore how varying projections of wind speed, solar radiation, and river flows could affect the evolutionarily stable strategies of the stakeholders involved. This would help in understanding the resilience and adaptability of grid and generation enterprises to likely future environmental scenarios.

(3)

Policy Implications: By embedding these environmental forecasts into our discussion, we can offer more nuanced policy recommendations that account for potential future shifts in energy production capabilities and their impact on market dynamics and regulatory needs.

We believe these enhancements will not only strengthen the manuscript but also provide a more comprehensive outlook on the sustainable integration of renewable energy into national grids under varying future conditions. Actually, the integration of evolutionary game theory into the coordination of renewable energy and power grid companies offers a sophisticated approach to understanding the dynamics of energy markets. This paper has successfully modeled the interactions between these stakeholders using replicator dynamics, providing insights into their strategic behavior within the framework of peak shaving and grid stability. Here, we further elaborate on these findings in relation to existing studies and highlight the contributions of this work to the broader field of energy economics.

(1)

Comparison with Other Studies.

(a)

Stakeholder Interaction: Prior research has predominantly focused on isolated aspects of market interactions without a comprehensive understanding of how multiple players dynamically influence each other. For instance, while some studies have emphasized the role of government policies in influencing grid operations, they have not fully captured the adaptive strategies of energy companies in response to evolving market conditions. This paper’s application of evolutionary game theory uniquely contributes by illustrating how the strategies of power grid companies and renewable energy providers evolve towards an equilibrium that optimally balances the interests of both parties, reflecting a deeper understanding of market dynamics.

(b)

Role of Subsidies and Pricing: The influence of peak shaving compensation on bidding strategies, as explored in this study, has been less emphasized in traditional analyses, which tend to focus on static pricing strategies. This research extends the analysis by integrating dynamic incentives such as peak shaving benefits, showing how they can modify the strategic behavior of energy companies. This aligns with findings from Ref. [4] but goes further by quantifying the impacts within a replicator dynamics framework.

(c)

System Stability and Efficiency: The stability analysis provided through Lyapunov functions and the derived conditions for evolutionary stability offer a novel contribution to the literature. Most previous studies do not provide a methodological framework for assessing the stability of strategic outcomes in energy markets, which is crucial for policy formulation and ensuring grid reliability.

(2)

Implications for Policy and Practice.

The insights derived from this research have significant implications for both policymakers and industry practitioners:

(a)

Policy Recommendations: Effective regulatory frameworks can be designed to encourage cooperation between renewable energy firms and grid operators. The evolutionary game theory model suggests that policies should focus on adjusting the initial conditions and incentives to guide the system towards desirable equilibria.

(b)

Operational Strategies: For power grid companies, the strategic implications involve adjusting their bidding and operational strategies based on the expected behavior of renewable energy firms. This includes considering the long-term benefits of cooperative strategies over aggressive competitive tactics, which may lead to suboptimal outcomes.

(c)

Market Design: Enhancing the design of electricity markets to accommodate the nuances of renewable integration, as revealed by the game-theoretic analysis, can improve both the economic efficiency and the reliability of energy supply.

Overall, this discussion underscores the relevance of the presented model in enhancing our understanding of the strategic interactions in energy markets. By bridging theoretical models with practical implications, this paper contributes to a more resilient and adaptive energy market, particularly in the context of increasing renewable energy integration. Further research could explore the application of similar models in other regional markets and under different regulatory conditions to broaden the applicability of these findings.

6. Conclusions

This paper has presented a comprehensive analysis of the coordination of renewable energy integration and peak shaving through the lens of EGT. Through detailed simulations and theoretical analysis, we demonstrated how EGT provides a robust framework for understanding the dynamic interactions between power generation enterprises and grid companies under various market conditions and policy settings. In summary, the main research conclusions and innovations of this paper are summarized as follows.

(i)

Evolution of strategic choices for peak shaving and coordination of renewable energy integration. The strategic choices of thermal power plants in peak shaving tend to collectively adopt either high-price or low-price bidding strategies, which are closely related to initial parameters. The formation of an interest equilibrium between thermal power plants and grid companies is a dynamic process. Increasing regulatory oversight is crucial in improving the peak shaving capacity of thermal power units.

(ii)

Dynamic evolution of interest equilibrium between renewable energy and grid companies. The balance of interests between renewable energy generation companies and grid enterprises evolves gradually. The evolution of strategies is influenced by the payment matrices and initial conditions, with factors such as electricity prices, revenues, and system load levels playing a pivotal role in determining the peak shaving capability and stability of the electricity market.

(iii)

Impact of government policy on market stability and renewable energy integration. Government policies and incentives play a significant role in stabilizing the electricity market and promoting the integration of renewable energy. This study recommends that policymakers consider equitable interest distribution among stakeholders to ensure cooperation between renewable energy generation companies and grid companies, thereby enhancing the long-term stability and development of the electricity market.

(iv)

In the context without considering peak shaving, the specific setting of the evolutionary strategy and parameters of thermal power generation is irrelevant and has similarities. In the context of peak shaving, the final strategy choices of thermal power plants tend to collectively adopt high prices or collectively adopt low prices and are closely related to the initial choices. The stable strategy choices with and without considering peak shaving have similar results and are interrelated.

(v)

In the game evolution process, the evolution direction of the strategies of both sides of the game depends on their payoff matrix. The initial values and changes of certain parameters in the payoff functions of the participating parties will lead the game evolution system to evolve along different paths, converging to different equilibrium points. The parameter variables that affect the cooperative evolution results between renewable energy generation companies and grid companies include: the excess profit S generated by cooperation, the conventional energy price P, the subsidy rate V, the initial input marginal cost l, and the distribution coefficient λ, where λ is a function of P, V, and l. If the conventional energy price P, subsidy rate V, and marginal cost l meet the prerequisite conditions for the system to evolve towards complete cooperation, then whether the distribution of excess profits between renewable energy companies and grid companies is fair and reasonable will directly determine the direction of the system’s evolution.

In the research process, this paper roughly divides the on-grid power generation into two single bidding modes of high price and low price, selects only two groups for the classification of power plants for analysis, and the data analysis of experimental results does not fully consider the impact of environmental factors.

The limitations of this paper are elaborated as follows. Modeling Assumptions: The primary limitation arises from the theoretical and simulation-based nature of our model. Real-world complexities and unpredictable market behaviors may lead to deviations from predicted outcomes. Data and Parameter Sensitivity: The outcomes are highly sensitive to initial conditions and parameter settings, which, though based on plausible assumptions, may not accurately capture all real-world scenarios. Scope of Strategies: The analysis is confined to two strategic groups—peak shaving and non-peak shaving—which may not encompass all potential strategic behaviors in the electricity markets.

In the future development prospects, there is a significant gap between non-peak online electricity and peak online electricity, and this situation will continue to exist for some time. Therefore, it is necessary to further improve relevant policies and mechanisms to promote the process of power market reform. In addition, the healthy development of the renewable energy generation industry cannot be achieved without the joint efforts of renewable energy generation enterprises, grid enterprises, and the government. To solve the problem of renewable energy integration, policymakers should consider the interests of all stakeholders as a prerequisite and, based on the current development of the power market, establish a fair and reasonable interest distribution and coordination mechanism to guide renewable energy enterprises and grid enterprises towards cooperation.

Future research prospects for this paper are summarized as follows. Empirical Validation: Future studies should aim to empirically validate the theoretical predictions through real-world data and case studies to enhance the model’s applicability and reliability. Extended Strategic Models: Expanding the model to include more diverse strategic groups and more complex interactions can provide deeper insights into the dynamics of power markets. Policy Impact Analysis: Further research is needed to quantitatively assess the impact of specific policy interventions on market stability and renewable energy integration, facilitating more informed policymaking.

In conclusion, while EGT offers significant insights into the dynamics of renewable energy integration and peak shaving, the inherent limitations of theoretical modeling necessitate ongoing refinement and validation of the strategies and conclusions drawn. Continued research is essential for advancing our understanding of these complex interactions and for developing more effective strategies and policies for sustainable energy development.