Evolutionary Game-Theoretic Approach to Enhancing User-Grid Cooperation in Peak Shaving: Integrating Whole-Process Democracy (Deliberative Governance) in Renewable Energy Systems

Abstract

The integration of renewable energy into power grids is imperative for reducing carbon emissions and mitigating reliance on depleting fossil fuels. In this paper, we develop symmetric and asymmetric evolutionary game-theoretic models to analyze how user–grid cooperation in peak shaving can be enhanced by incorporating whole-process democracy (deliberative governance) into decision-making. Our framework captures excess returns, cooperation-driven profits, energy pricing, participation costs, and benefit-sharing coefficients to identify equilibrium conditions under varied subsidy, cost, and market scenarios. Furthermore, this study integrates the theory, path, and mechanism of deliberative procedures under the perspective of whole-process democracy, exploring how inclusive and participatory decision-making processes can enhance cooperation in renewable energy systems. We simulate seven scenarios that systematically adjust subsidy rates, cost–benefit structures, dynamic pricing, and renewable-versus-conventional competitiveness, revealing that robust cooperation emerges only under well-aligned incentives, equitable profit sharing, and targeted financial policies. These scenarios systematically vary these key parameters to assess the robustness of cooperative equilibria under diverse economic and policy conditions. Our findings indicate that policy efficacy hinges on deliberative stakeholder engagement, fair profit allocation, and adaptive subsidy mechanisms. These results furnish actionable guidelines for regulators and grid operators to foster sustainable, low-carbon energy systems and inform future research on demand response and multi-source integration.

1. Introduction

Due to the issues of environmental pollution and the gradual depletion of non-renewable energy resources, integrating renewable energy generation into the grid has become an essential measure worldwide. Humanity faces the shared challenge of addressing the difficulties associated with renewable energy grid integration.

Taking wind power grid integration as an example, the global installation of new wind power capacity declined in 2021, the second year of the COVID-19 pandemic. Despite economic setbacks caused by the pandemic, the development of wind power technology continues to be a priority for nations worldwide. China’s wind power industry policies also reflect the promotion of low-carbon environmental protection, with strong national support for wind power construction, solidifying its role as a key clean energy industry. Nevertheless, alongside government support, China’s wind power industry faces several limitations during its development [1,2].

Recent research highlights the pressing challenges associated with large-scale renewable energy integration. The inherent instability of wind resources causes wind power generation to be intermittent, making it unsuitable for independent grid integration. In the energy transmission system, thermal power units are responsible for peak regulation and power balance. However, these units incur additional costs in accommodating wind power, with such costs exceeding the value brought by wind energy. As a result, enhancing the peak regulation capabilities of thermal power plants has become imperative. Moreover, the variability in wind and photovoltaic power output exacerbates load fluctuations, increasing the peak regulation burden on thermal power units. This issue not only undermines the economic operation and flexibility of the grid but also diminishes the willingness of grid enterprises and high-consumption users to participate in peak regulation projects.

Several studies have explored solutions to these challenges. Recent work by Xing et al. (2024) applied evolutionary game theory to analyze the interaction dynamics between power sector stakeholders, providing insights into cooperation mechanisms that enhance renewable energy absorption into grids [3]. Their study demonstrates that without appropriate regulatory frameworks and incentives, power companies tend to avoid engaging in peak regulation efforts due to cost concerns. Similarly, Alizadeh (2024) introduced a transactive energy framework that facilitates the integration of renewable energy sources while ensuring optimal cost distribution among market participants [4]. This framework proposes market-driven mechanisms to incentivize both grid operators and energy-intensive users to participate actively in peak regulation projects. These findings underscore the need for well-structured policies that encourage economic collaboration between stakeholders.

To improve the enthusiasm of both users and the grid in participating in peak regulation, the government must implement rational policies to subsidize such efforts and ensure equitable distribution of benefits between users and grid enterprises [5]. Actually, the wind power units are exempt from bearing the costs associated with peak regulation and energy absorption [6]. Energy absorption refers to the process of dispatching surplus electricity from a specific region or load point to other regions or load points with electricity demand. This exemption, however, has led to excessive grid integration of wind power units, significantly increasing the grid’s peak regulation burden. Consequently, the overall flexibility of the power system is diminished, and challenges related to energy absorption become more pronounced.

To address these issues, it is critical to analyze the peak regulation costs and corresponding compensation benefits incurred by various participants. By developing a cost–benefit model [7], it becomes possible to optimize peak regulation scheduling while ensuring economic feasibility. In this context, the present study constructs an evolutionary game equilibrium model to evaluate the costs and benefits experienced by users and grid enterprises during their participation in power system peak regulation. This model provides valuable insights into the practical implications of various equilibrium scenarios.

As renewable energy generation technologies continue to advance and are increasingly integrated into power systems, significant research has been undertaken domestically to address the novel and substantial challenges posed to the planning and operation of future power systems.

The integration of renewable energy into the power grid has prompted international research efforts focused on both the technical and economic aspects of grid flexibility. Raju et al. (2024) developed a decentralized game-theoretic coordination model for electric vehicle charging in microgrids, which optimizes renewable energy utilization while ensuring grid stability [8]. Their findings suggest that game theory-based mechanisms can improve the efficiency of renewable energy integration while minimizing economic losses associated with peak regulation.

Due to the high investment costs associated with renewable energy, it struggles to compete with conventional generation methods even under conditions of low marginal costs [9]. In addition to implementing targeted incentive policies, adaptable market mechanisms for renewable energy, such as feed-in tariffs, tendering systems, and green certificate schemes, have been proposed. While these mechanisms have partially addressed grid integration challenges, persistent obstacles—including insufficient peak regulation capabilities and inadequate policies for wind power utilization—continue to constrain renewable energy development and large-scale wind power integration domestically.

The core reasons for these difficulties lie in the limited flexibility of the power grid and suboptimal policies governing wind power consumption. Existing policies on wind power utilization require grids to bear auxiliary service costs associated with peak regulation, which often exceed the economic value generated by wind power itself. This discourages grid enterprises from actively participating in wind power grid integration and peak regulation projects. To overcome these challenges, it is essential to formulate sound policies for renewable energy utilization, enhance the peak regulation capabilities of thermal power plants, and incentivize broader participation by enterprises in wind power grid integration projects [10,11,12].

Incorporating the concept of whole-process democracy can enrich the decision-making process in the energy sector. This perspective emphasizes the importance of inclusive, transparent, and democratic decision-making processes. The user–grid interaction in renewable energy projects can greatly benefit from a decision-making process based on extensive consultation and equal participation from both parties. This deliberative approach could help to identify optimal solutions for cooperation, resolve conflicts, and ensure equitable benefits distribution. Furthermore, the mechanism of whole-process democracy can guide the formulation of policies, such as government subsidies, that support cooperation and encourage long-term collaborative strategies.

In this context, the evolutionary game theory framework can be further enriched by integrating deliberative democracy, promoting a more inclusive and collaborative approach to renewable energy projects. This allows stakeholders not only to be incentivized financially but also to have a participatory role in shaping the policies that affect their interests. The integration of deliberative democracy can enhance user–grid cooperation, leading to more sustainable and equitable outcomes in renewable energy systems.

Ref. [13] provides an analysis of the applications of evolutionary game theory within power systems, identifying three primary branches of game theory: non-cooperative game theory, cooperative game theory, and evolutionary game theory. Evolutionary game theory, derived from evolutionary biology, initially emerged to study competitive phenomena, such as species interactions on islands, and has since been widely applied in fields such as communication networks, military strategy, social systems, engineering, and economics. As a tool for optimizing strategic decisions among multiple interrelated groups, evolutionary game theory—despite its relatively recent development—has proven to be a robust and precise forecasting method during periods of rapid advancement in power systems. Ref. [14] investigates the application of evolutionary game theory to the demand side of electricity systems and provides a prospective outlook. Against the backdrop of rapidly evolving smart grids, the role of demand-side users in enhancing energy utilization rates and supply reliability has grown increasingly prominent. However, the optimization of strategy decisions among demand-side groups remains challenging due to the large user base and the complexity of decision-making entities. Evolutionary game theory offers an effective framework for addressing these challenges, enabling the optimization of strategic decision-making in systems involving multiple stakeholders [15]. Looking ahead, game theory, and particularly evolutionary game theory, will continue to play a pivotal role in the development and construction of future power systems.

In international electricity markets, competition has evolved into a complex and dynamic process. The rapid development of renewable energy has presented significant challenges in ensuring its efficient and sustainable utilization. Among global regions, the European Union leads in renewable energy integration, achieving the highest levels of penetration and connection to the grid [16,17]. It also outperforms both the United States and China in key metrics such as renewable energy penetration rates, grid connection rates, green energy capacity, and green energy generation.

In the United States, renewable energy adoption is driven by national security policies aimed at reducing dependency on fossil fuel imports. These efforts are further motivated by the goal of decreasing carbon dioxide emissions, improving air quality, and creating employment opportunities through investment in renewable energy infrastructure. Most developed and many developing countries have now identified the development and utilization of renewable energy as a foundational element of their energy strategies.

Regarding the application of evolutionary game theory, international research has primarily focused on its use in analyzing interactions between groups, institutional behaviors, and social norms. Evolutionary game theory has been employed to address socio-economic issues, human behavior, and other related domains. Researchers have also extended and innovated upon the foundational principles of this theory. For example, ref. [18] investigates multi-group asymmetric evolutionary games in typical electricity market scenarios and provides policy recommendations. The study shows that well-designed policies, through adjustments to payoff matrix parameters, can stabilize electricity prices within an acceptable range and enable optimal resource allocation in energy internet systems.

Amid growing concerns over global energy shortages and escalating environmental degradation, the integration of renewable energy into power systems has emerged as a pivotal solution embraced by nations worldwide [19,20,21]. However, this transition is not without challenges. The large-scale grid integration of renewable energy sources, particularly wind and photovoltaic (PV) energy, introduces substantial technical and economic complexities. Among these, the intensified peak regulation pressures on thermal power units and the elevated costs of renewable energy grid integration stand out as critical obstacles. These factors adversely impact the operational economics of the power grid and undermine the flexibility required for effective peak regulation.

Furthermore, these challenges dampen the willingness of grid enterprises and energy-intensive users to actively engage in renewable energy integration and peak regulation projects [22]. This reluctance underscores the pressing need for comprehensive solutions that address both economic incentives and structural inefficiencies. Government subsidies, alongside equitable mechanisms for distributing the resulting benefits, are therefore indispensable for fostering collaboration between users and grid operators in these projects.

The dynamics of interest balancing between user groups and grid enterprises in renewable energy grid integration and peak regulation are inherently complex and fluid [23,24,25]. This study addresses these challenges through the lens of evolutionary game theory, a robust analytical framework capable of capturing the dynamic interactions and strategic decision-making processes among stakeholders. The research develops both symmetric and asymmetric evolutionary game models to quantify the payoffs for user groups and grid enterprises. By analyzing these interactions, the study establishes a dynamic equilibrium model that elucidates the conditions for optimal participation in renewable energy peak regulation.

To validate the proposed models, seven case studies are conducted to analyze the interplay of various factors influencing equilibrium outcomes. These include excess returns, cooperative profits, the pricing of conventional energy, the costs associated with peak regulation participation, and the distribution coefficients of excess profits. Simulation results reveal that the evolutionary trajectories of stakeholders’ strategies are highly sensitive to the configuration of these factors, highlighting the intricate dependencies within the system.

The findings of this research contribute to a nuanced understanding of the mechanisms driving user–grid collaboration in renewable energy peak regulation. Key insights emphasize that effective participation hinges on aligning incentives through rational subsidy policies and ensuring equitable benefit distribution. Moreover, the research underscores the importance of addressing structural barriers, such as the disproportionate cost burden on thermal power units and the variability inherent in renewable energy sources, to enhance grid flexibility and economic efficiency.

This study provides actionable policy recommendations to guide stakeholder collaboration. Firstly, governments must regulate renewable energy pricing to enhance its competitiveness relative to conventional energy, thereby encouraging broader participation in renewable energy projects. Secondly, policymakers should prioritize equitable benefit-sharing frameworks that reduce disparities between user groups and grid enterprises, fostering a cooperative environment. Thirdly, mechanisms to maximize excess profits through renewable energy absorption should be developed, further incentivizing collaboration. Lastly, the establishment of supportive policies and institutional frameworks is critical for sustaining user–grid cooperation and advancing low-carbon development goals.

In conclusion, this research advances the theoretical foundation and practical application of evolutionary game theory in renewable energy grid integration. By elucidating the dynamic equilibrium conditions for user–grid collaboration, it offers a roadmap for optimizing peak regulation strategies and achieving sustainable energy transitions. These insights not only address immediate operational challenges but also contribute to the broader goal of a low-carbon, environmentally sustainable energy future.

This study focuses on constructing an evolutionary game model to analyze the participation of both users and grid enterprises in power system peak regulation. By applying the principles of evolutionary game theory, the research investigates the process of achieving interest equilibrium between these stakeholders. Through a comprehensive analysis of case studies, the study identifies the key factors influencing this equilibrium and draws actionable conclusions.

The research employs realistic data, based on actual scenarios, to simulate and validate the outcomes of case studies, offering robust insights into the dynamics of user–grid collaboration. Moreover, it provides policy recommendations aimed at fostering cooperative strategies between users and grid enterprises. These policies are designed to enhance collaboration and align incentives, ultimately improving participation in renewable energy peak regulation projects.

From a micro-level perspective, the study explores the underlying mechanisms driving user–grid interactions in power system peak regulation. It develops a detailed payoff model for stakeholders and constructs an evolutionary game framework that captures the demand dynamics of peak regulation. The overarching objective is to encourage active participation by both users and the grid in peak regulation initiatives. By offering a structured approach to achieving interest equilibrium, the research provides a theoretical foundation and practical strategies to optimize stakeholder engagement and promote sustainable collaboration in peak regulation projects.

The rest of the organization of this paper, starting from Section 2, is designed to provide a structured and logical exploration of user–grid collaboration in low-carbon renewable energy systems using evolutionary game theory. Each section builds upon the previous one to develop a comprehensive theoretical and empirical framework.

Section 2 establishes the theoretical foundation of the research. It begins with classical game theory as a baseline to understand static decision-making in user–grid dynamics. This section transitions to evolutionary game theory, emphasizing its ability to model dynamic and adaptive interactions between users and the grid. Core concepts such as replicator dynamics, evolutionary stable strategies (ESS), payoff matrix construction, and Jacobian matrix stability analysis are systematically introduced. This theoretical framework lays the groundwork for constructing and analyzing dynamic interactions in subsequent sections.

Section 3 presents the development of the evolutionary game model for user–grid cooperation in peak shaving participation. This section meticulously outlines the parameter settings and key steps involved in the model construction process, providing a clear blueprint for analyzing the interplay between user and grid strategies. It includes an in-depth evolutionary stability analysis to evaluate the conditions under which cooperative behaviors emerge and stabilize. This section serves as the analytical core of the paper, linking theoretical insights to practical applications.

Section 4 validates the proposed model through extensive numerical simulations. Seven distinct scenarios are analyzed to investigate various aspects of user–grid collaboration under different policy interventions, cost structures, and renewable energy penetration levels. This section concludes with a synthesis of simulation results, providing critical insights into the factors influencing cooperative dynamics and system stability.

Section 5 discusses the broader implications of the findings and outlines potential directions for future research. It explores opportunities for model enhancement, such as incorporating advanced technologies like artificial intelligence, and highlights the importance of interdisciplinary approaches to address the socio-technical complexities of renewable energy systems. This section also identifies gaps in current knowledge and proposes avenues for further investigation.

Section 6 concludes the paper by summarizing the main findings and offering actionable policy recommendations. It emphasizes the practical implications of the research for fostering user–grid collaboration in renewable energy systems and achieving low-carbon energy transitions. This section ensures that the study’s theoretical and empirical contributions are clearly articulated and aligned with its broader objectives.

This structure ensures a coherent progression from theoretical foundations to practical applications, providing a comprehensive understanding of dynamic user–grid interactions in renewable energy integration. The paper progresses logically from problem identification to theoretical modeling, empirical validation, and practical solutions. Each section builds upon the others, creating a cohesive narrative that deepens both theoretical and practical insights into user–grid cooperation and peak regulation in renewable energy systems.

2. Theoretical Foundations for User–Grid Dynamics: Evolutionary Game Models in Low-Carbon Renewable Energy Systems

2.1. Classical Game Theory

Game theory serves as a powerful tool for analyzing behavioral changes and outcomes arising from interactions between different groups. Classical game theory, in particular, operates on the assumption that participants are fully rational agents, possessing both the capability to maximize their benefits and the unlimited capacity to process and compute information. Furthermore, it presumes the existence of common knowledge among participants, implying that each agent is fully aware of the strategies employed by others. Within this framework, the structure and environment of the game are predefined, with classical game theory primarily focused on analyzing Nash equilibria—outcomes in which participants adopt strategies that, given the strategies of others, cannot be unilaterally improved upon.

A classical game theory model consists of three essential elements: participants, strategies, and payoffs. For two participants, the interaction is referred to as a two-player game, whereas interactions involving n participants are termed n-player games. Strategies represent the courses of action available to participants. Payoffs indicate the benefits or rewards each participant receives under various scenarios, typically represented using a payoff matrix. Strategies are further categorized into pure and mixed strategies: in a pure strategy, participants deterministically choose a specific course of action, while in a mixed strategy, participants probabilistically select their actions, introducing an element of randomness. In classical game theory, the focus is on the strategic decision-making of fully rational individuals who select strategies that maximize their benefits, given that all other participants’ strategies are fixed and the outcomes are shared among all participants [26].

Despite its foundational significance, classical game theory is constrained by the limitations of its assumptions and solution concepts [27]. First, the assumption of complete rationality—that participants can perfectly optimize their strategies—is often unrealistic in practical applications, where decision-making is influenced by bounded rationality. Additionally, while classical game theory identifies strategies that should not be chosen, it fails to provide guidance on which strategies participants ought to select, limiting its prescriptive power. The presumption of common knowledge, wherein participants are assumed to have perfect awareness of others’ strategies, also deviates from real-world scenarios where such information is rarely available.

Furthermore, solving for Nash equilibria can be computationally challenging in classical game settings. Nash equilibrium is conceptualized as the outcome of rational agents independently deducing optimal strategies through introspection, yet in reality, equilibrium is often achieved iteratively. Participants adjust their strategies by observing others’ reactions and continuously optimizing their choices. This dynamic nature of strategic decision-making cannot be adequately captured by classical game theory, which is unable to address the problem of multiple equilibria that commonly arise in real-world interactions. As a result, classical game theory falls short in explaining how equilibria are reached or maintained in dynamic and iterative real-world scenarios.

2.2. Evolutionary Game Theory

Evolutionary game theory focuses on populations as the subject of study, analyzing the dynamic process by which individuals within a population adjust their decision-making through learning and imitation. Unlike classical game theory, evolutionary game theory assumes that participants exhibit bounded rationality, with decisions based on long-term observations of their opponents’ behavior. The framework conceptualizes interactions as an ever-evolving dynamic process, where participants’ rationality itself evolves over time. Here, rationality refers to the rules guiding individuals’ strategy selection, which can be broadly understood as their habitual decision-making patterns. Consequently, bounded rationality in this context can be interpreted as the iterative process through which participants observe, learn, and refine their decision-making criteria during repeated interactions.

2.2.1. Replicator Dynamics

Replicator dynamics describe the probability or frequency with which a particular strategy is adopted by a population over time [28,29]. This framework uses differential equations to represent the evolution of strategies within a population. The replicator dynamics equation is typically expressed as follows:

$${\dot{x}}_i=x_i\left[u_i(x)-\overline{u}(x)\right]$$

(1)

In this equation:

• $x_i$ denotes the proportion of the population adopting strategy i.
• $u_i(x)$ represents the payoff of strategy i under the current population distribution x.
• $\overline{u}(x)$ refers to the average payoff across all strategies within the population.

The equation captures the growth rate (${\dot{x}}_i$) of strategy i, which is proportional to the difference between the payoff of strategy i ($u_i(x)$) and the average payoff across the population ($\overline{u}(x)$). A positive growth rate indicates that the proportion of individuals adopting strategy i increases, while a negative rate implies a decline.

Replicator dynamics serve as a foundational tool for analyzing the evolution of strategies in a population. A strategy becomes evolutionarily stable when its proportion in the population remains constant (${\dot{x}}_i$ = 0), indicating that it outcompetes alternative strategies and no individual has an incentive to deviate from it. This dynamic modeling approach not only elucidates how strategies spread, stabilize, or disappear over time in evolutionary game settings, but also integrates deliberative governance mechanisms that quantitatively influence the strategy evolution process. Specifically, we modify the standard replicator dynamics by introducing a feedback adjustment term that reflects the collective decision-making and policy-shaping role of stakeholders. This adjustment term models how the strategies of users and grid enterprises evolve not only in response to their payoffs but also through deliberative processes that adjust payoffs and introduce cooperative behavior. By incorporating such deliberative mechanisms, we create a more robust model that captures the interplay between game-theoretic strategy evolution and democratic decision-making.

Based on Equation (1), another description for the replica dynamics is shown as

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

(2)

In the equation:

• i represents a specific population.
• n represents the total number of populations.
• $x_i^j$ represents the quantity of the j-th pure strategy chosen by population i at a certain moment t.
• x_i represents the state of population i.
• x_−i represents the state of all populations other than i.
• $s_i^j$ represents the j-th pure strategy in the total decision set of population i.
• x represents the total number of mixed strategy combinations for all populations.
• $f(s_i^j,x)$ represents the payoff for population i in state x when choosing the j-th pure strategy.
• $f_{\mathrm{ave}}(x_i,x_{-i})$ represents the average payment parameter.

As shown in Equation (2), the replicator dynamics equation described above can represent the evolutionary trajectory of the proportion or the number of individuals within a population adopting a specific strategy. When the payoff for population i selecting the j-th pure strategy exceeds the average payoff parameter—indicated by the differential being greater than zero—it signifies that the rate of change in the total number of individuals adopting that strategy at a given moment during the strategy evolution process is positive. Conversely, if the payoff for the j-th pure strategy is less than the average payoff (with the differential being less than zero), the rate of change becomes negative, indicating a decline in the number of individuals adopting that strategy.

When the payoff for population i selecting the j-th pure strategy equals the average payoff parameter—indicated mathematically by the replicator dynamic differential equating to zero—it signifies that the proportion or total number of individuals adopting this strategy within the population stabilizes. This condition reflects an evolutionarily stable state (ESS), where no individual has an incentive to deviate unilaterally from the prevailing strategy, as doing so would not yield a higher payoff.

The replicator dynamics framework is particularly advantageous and well-suited to the study presented in this paper for several reasons. First, it provides a powerful tool for modeling the dynamic interactions between users and the grid in renewable energy peak regulation, capturing how strategic decisions evolve over time rather than assuming immediate optimization. Unlike classical game theory, which presupposes static equilibria, replicator dynamics allow us to analyze the continuous process by which strategies emerge, spread, and stabilize in response to payoffs, which is crucial in contexts where participants have bounded rationality and adaptive behaviors.

Second, replicator dynamics are inherently population-based, making them particularly effective for examining large-scale interactions, such as those between user groups and grid enterprises in renewable energy systems. By treating strategies as proportions within a population, this approach can illustrate how cooperative or competitive behaviors diffuse across user and grid participants, revealing the underlying mechanisms driving collaboration or resistance in peak regulation projects.

Finally, the concept of evolutionary stability aligns seamlessly with the goals of this research. In the context of low-carbon renewable energy systems, achieving an evolutionarily stable strategy reflects a state of cooperation or balance between users and grid operators where incentives are aligned, and no participant benefits from unilateral deviation. This stability is critical for ensuring the sustainability and economic viability of renewable energy grid integration. By leveraging replicator dynamics, this study identifies the conditions under which such stability can be achieved, offering valuable insights into the factors—such as excess returns, cost-sharing mechanisms, and government subsidies—that influence the evolution and stabilization of user–grid collaboration.

In the context of user–grid collaboration for renewable energy systems, the replicator dynamics framework provides a dynamic model for the evolving interaction strategies of both users and grid enterprises over time. To enhance this, we integrate the concept of ‘whole-process democracy’—a political philosophy that emphasizes continuous, inclusive, and participatory decision-making. This democratic framework enables active involvement of all stakeholders (users, grid enterprises, and policymakers) in shaping policies and incentives that govern their interactions. The integration of whole-process democracy allows us to model how collective decision-making influences strategy evolution in a renewable energy context. Specifically, we consider how deliberative governance mechanisms, where decisions are made through consensus and feedback from all parties, influence the strategy selection process and stabilize the equilibrium over time. By integrating deliberative democracy into the evolutionary game model, participants are no longer passive responders but active agents in designing policies that ensure equitable benefit distribution, foster trust, and support long-term cooperation in renewable energy projects. The replicator dynamics framework aligns with the theoretical foundations of evolutionary game theory while offering a practical lens for analyzing and predicting the strategic evolution of participants in renewable energy peak regulation. This approach is central to the methodology of this study.

2.2.2. Evolutionarily Stable Strategy (ESS)

An evolutionarily stable strategy (ESS) mirrors the dynamic mechanisms observed in biological evolution [30], where certain traits or behaviors become dominant through processes such as genetic mutation and natural selection. In the context of user–grid collaboration, the concept of ESS provides an analytical tool for understanding strategic stability. However, it is essential to distinguish between the normative role of democracy, which aims to ensure fair participation and equitable benefit distribution, and the analytical toolset provided by evolutionary game theory. While ESS helps us identify stable strategies that arise from participants’ strategic interactions, the integration of whole-process democracy serves as a normative framework that guides these interactions by promoting fairness and transparency in decision-making processes. Similarly, in the context of evolutionary game theory, an ESS represents a strategy that persists and remains stable within a population, even in the face of perturbations caused by individuals adopting alternative strategies.

In evolutionary stable strategies, participants in the game dynamically adjust their strategies through continuous learning and imitation to improve their individual payoffs. Over time, this adaptive process drives the population toward an equilibrium where the distribution of strategies achieves a stable balance. Importantly, an ESS must satisfy the condition that, when a population predominantly adopts this strategy, any small fraction of individuals introducing an alternative strategy will ultimately be outperformed and revert to the original strategy through the evolutionary process. This ensures that the equilibrium remains robust and resistant to perturbations.

The concept of an ESS emphasizes stability against invasion: even when some individuals deviate or “mutate” by adopting new strategies, their learning and imitation dynamics will gradually guide them back to the equilibrium strategy, ensuring the population as a whole converges back to the stable state. This property underscores the resilience of ESS in maintaining the strategic balance among participants, reflecting a self-correcting mechanism that safeguards against disruption.

Through this dynamic adjustment process, evolutionary stable strategies enable participants to reach an equitable distribution of payoffs, ensuring long-term stability in the strategic interactions within a population. This feature makes ESS a particularly valuable concept for modeling real-world scenarios, such as user–grid collaboration in renewable energy systems, where strategic stability and adaptability are critical for sustainable and equitable outcomes.

Assume a population size K, where each population k has N strategies in its strategy set. The strategy set is represented as the N-dimensional vector set:

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

(3)

where the parameter x_i denotes the proportion of population k adopting the i-th strategy. Here, k = 1, 2, …, K and i = 1, 2, …, N. To evaluate the fitness of individuals, the fitness function is introduced as:

$$f(\overrightarrow{r},t)=\left(f^1(\overrightarrow{r},\overrightarrow{S}),f^2(\overrightarrow{r},\overrightarrow{S}),\cdots ,f^i(\overrightarrow{r},\overrightarrow{S}),\cdots ,f^k(\overrightarrow{r},\overrightarrow{S})\right)$$

(4)

where $f^k(\overrightarrow{r},\overrightarrow{S})$ represents the fitness of a mixed strategy in population k. The value of the fitness function is non-negative, and larger values indicate higher fitness. Moreover, the choice of the fitness function directly influences the efficiency of convergence and the ability to reach an optimal solution [29,30]. Here, r denotes any mixed strategy of population k, while the state of the system at time t is represented by:

$$\left\{\begin{array}{l}S=\left(\overrightarrow{S^1},\overrightarrow{S^2},\cdots ,\overrightarrow{S^k},\cdots ,\overrightarrow{S^K}\right)\\ \overrightarrow{S^k}=\left(S_1^k,S_2^k,\cdots ,S_i^k,\cdots ,S_N^k\right)=\left({\displaystyle \frac{\mathrm{d}{S}_1^k}{\mathrm{d}t}},{\displaystyle \frac{\mathrm{d}{S}_2^k}{\mathrm{d}t}},\cdots ,{\displaystyle \frac{\mathrm{d}{S}_i^k}{\mathrm{d}t}},\cdots ,{\displaystyle \frac{\mathrm{d}{S}_N^k}{\mathrm{d}t}}\right)\end{array}\right.$$

(5)

where $k=(1,2,\cdots ,i,\cdots ,K)$. Under this formulation, the fitness function can be denoted as F: $S\to R^{NK}$ or S = F(S).

By assigning an initial condition S(0) ∈ S, the evolutionary trajectory of the population is described by the corresponding solution curve derived from the above equations. During the evolutionary process, each individual has an equilibrium state corresponding to its fitness. When the evolutionary trajectory converges to the equilibrium point corresponding to the individual’s fitness, the system achieves stability. Conversely, if the evolutionary curve exhibits periodic oscillations, this phenomenon will persist for a period of time before the state transitions from instability to stability.

Integrating whole-process democracy into this concept means that the ESS could reflect a democratic consensus among stakeholders. In a renewable energy context, this might involve mechanisms where users, grid operators, and government representatives collaboratively determine a stable cooperative strategy that ensures fair benefits and addresses environmental goals.

2.2.3. The Role of Evolutionarily Stable Strategies and Equilibria in Modeling Complex Dynamic Systems

As elaborated above, ESS and evolutionarily stable equilibria (ESE) play a pivotal role in the modeling and analysis of complex dynamic systems, especially in scenarios involving multi-agent interactions and strategic decision-making. Originating from evolutionary game theory, these concepts provide a robust theoretical framework to explain the stability and adaptability of strategies within competitive environments. In the context of this study, ESS and ESE are instrumental in understanding the long-term behavior of populations and the stability of their strategic choices, offering significant advantages for addressing the underlying research problem.

(1) ESS as a Stability Criterion for Strategy Selection.

An ESS serves as a critical stability criterion in the evolution of strategies, ensuring that a population’s adopted strategy remains resilient against potential invasions by alternative strategies. This resilience arises from the fact that ESS satisfies conditions of both dominance and robustness: a population employing an ESS will outperform any small perturbation introduced by an invading strategy. In this study’s framework, ESS underpins the modeling of strategy dynamics, ensuring that populations converge toward stable and optimal strategic configurations over time. This stability is particularly valuable in dynamic systems characterized by competitive resource allocation, as it mitigates disruptions caused by fluctuating strategic behaviors.

(2) ESE as a Dynamic Equilibrium Framework.

ESE extends the concept of ESS to account for dynamic adjustments within evolving systems, capturing the interplay between stability and adaptability. By incorporating ESE into the modeling process, this study emphasizes not only the equilibrium states of the system but also the pathways through which these equilibria are reached. Unlike static notions of equilibrium, ESE provides a more comprehensive perspective on the convergence of strategic interactions over time, accommodating the inherent complexity of multi-agent systems.

(3) Applications in Addressing the Research Problem.

The integration of ESS and ESE into this study’s modeling framework confers several critical advantages. First, it enables the analysis of long-term stability in strategy selection, ensuring that the system’s dynamics naturally gravitate toward resilient equilibria. This is particularly relevant in scenarios where populations must allocate limited resources or make adaptive decisions in response to changing external conditions. Second, the use of ESE provides insights into the transient dynamics of the system, illustrating how perturbations or strategic deviations are absorbed and reconciled within the broader framework of population interactions. Finally, by leveraging these evolutionary concepts, the study is able to address the inherent complexity of its research problem, delivering a robust and scalable solution for modeling dynamic multi-agent systems.

(4) Advantages of ESS and ESE in Complex Systems.

The incorporation of ESS and ESE into this study offers distinct advantages over traditional equilibrium concepts. Unlike static Nash equilibria, which often fail to capture the dynamic nature of strategy evolution, ESS and ESE emphasize both stability and adaptability, making them well-suited for analyzing systems with continuous adjustments and feedback mechanisms. Furthermore, their resilience to perturbations ensures that the modeled system remains robust under a wide range of initial conditions and external influences, a property critical for the practical applicability of the model.

Evolutionarily stable strategies and equilibria are foundational tools for modeling the dynamics of complex systems. These concepts ensure long-term stability, accommodate transient dynamics, and offer resilience to perturbations, providing a versatile framework for addressing challenges in multi-agent interactions and resource allocation. Their application enhances both the theoretical understanding of strategy evolution and the practical relevance of the model for solving real-world problems.

Unlike classical game theory, evolutionary game theory provides a method for participants to choose their strategies. Equilibrium is typically not achieved instantaneously but is instead a process in which both sides of the game continually make optimal choices by observing each other’s responses. Evolutionary game theory aligns with this perspective. The key focus of evolutionary game theory is to study the evolutionary process and learning mechanisms of participants’ strategy selection during the game process [31].

2.2.4. Payoff Matrix Construction and Replicator Dynamics

This section builds upon the construction and analysis of an evolutionary game model involving two populations—denoted as Population A and Population B—under a peak-shifting demand framework. Each population, A and B, has two available strategies (S_A1, S_A2 for A and S_B1, S_B2 for B, respectively). We denote by x the fraction of Population A that chooses strategy S_A1, while (1 − x) chooses S_A2, and by y the fraction of Population B that chooses strategy S_B1, while (1 − y) chooses S_B2. In order to characterize the payoffs for each strategy combination, we define eight payoff parameters a, b, c, d, e, f, g, h, representing the payoffs received by Populations A and B under each pairwise strategy outcome. These parameters form the payoff matrix that sets the stage for analyzing evolutionarily stable strategies and the corresponding replicator dynamics within the system. The subsequent content elaborates on the main derivation steps and the applications of this model in demand peak-shifting, including constructing the payoff matrix, deriving the replicator equations, formulating the Jacobian matrix, and determining stability via the determinant and trace of the Jacobian. Finally, through determinant and trace analysis of the Jacobian, the local stability and the evolutionary trajectories of the system’s equilibrium points can be thoroughly investigated.

1. Payoff Matrix Definition

Consider two populations, A and B, each adopting one of two strategies:

• Population A: S_A1, and S_A2;
• Population B: S_B1, and S_B2.

Let x ∈ [0, 1] be the fraction of A that adopts S_A1, so (1 − x) is the fraction that adopts S_A2. Similarly, let y ∈ [0, 1] be the fraction of B that adopts S_B1, and (1 − y) the fraction that adopts S_B2. We introduce eight payoff parameters a, b, c, d, e, f, g, h to describe the payoffs for each strategy profile. One typical representation of the payoff matrix—where the first element is the payoff for A and the second element is the payoff for B—can be written as:

$$\left(\begin{array}{cc}(a,b)& (c,d)\\ (e,f)& (g,h)\end{array}\right)$$

(6)

In this matrix, rows are strategies of A (S_A1, and S_A2), while columns are strategies of B (S_B1, and S_B2).

2. Expected Payoffs and Replicator Dynamics

Expected Payoff for Population A: If an individual in population A chooses S_A1, its expected payoff, denoted E_SA1, depends on whether population B plays S_B1 or S_B2. Specifically,

$$E_{\mathrm{SA}1}=ay+c(1-y),$$

(7)

while if the individual in A chooses S_A2, its expected payoff E_SA2 is

$$E_{\mathrm{SA}2}=ey+g(1-y).$$

(8)

When a fraction x of population A chooses S_A1 and (1 − x) chooses S_A2, the average payoff to A, denoted by ${\overline{E}}_{\mathrm{SA}}$, is:

$${\overline{E}}_{\mathrm{SA}}=x{E}_{\mathrm{SA}1}+(1-x)E_{\mathrm{SA}2}.$$

(9)

The replicator dynamic for x then follows the form:

$${\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}=F(x)=x\left(E_{\mathrm{SA}1}-{\overline{E}}_{\mathrm{SA}}\right).$$

(10)

Substituting the expressions for E_SA1 and ${\overline{E}}_{\mathrm{SA}}$ yields a replicator equation that can be simplified into:

$${\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}=F(x)=x(1-x)\left[(a-c-e+g)y+(c-g)\right].$$

(11)

Expected Payoff for Population B: Similarly, if population B chooses S_B1, its expected payoff E_SB1 given the fraction x of A that adopts S_A1 is:

$$E_{\mathrm{SB}1}=bx+f(1-x),$$

(12)

and if it chooses S_B2, the expected payoff E_SB2 is:

$$E_{\mathrm{SB}2}=dx+h(1-x).$$

(13)

The overall average payoff for B, ${\overline{E}}_{\mathrm{SB}}$, is

$${\overline{E}}_{\mathrm{SB}}=y{E}_{\mathrm{SB}1}+(1-y)E_{\mathrm{SB}2}.$$

(14)

The replicator dynamic for y is then

$${\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}}=F(y)=y(E_{\mathrm{SB}1}-{\overline{E}}_{\mathrm{SB}}),$$

(15)

which, upon substitution and simplification, can be expressed in the compact form

$$F(y)=y(1-y)\left[(b-f-d+h)x+(f-h)\right].$$

(16)

2.2.5. Jacobian Matrix and Stability Analysis

1. Jacobian Matrix

To analyze local stability of the replicator system, we construct the Jacobian matrix J of the two-dimensional dynamical system:

$$\left(\begin{array}{c}\dot{x}\\ \dot{y}\end{array}\right)=\left(\begin{array}{c}F(x)\\ F(y)\end{array}\right).$$

(17)

By computing the partial derivatives of F(x) and F(y) with respect to x and y, respectively, we obtain

$$\mathit{J}=\left(\begin{array}{cc}{\displaystyle \frac{\partial F(x)}{\partial x}}& {\displaystyle \frac{\partial F(x)}{\partial y}}\\ {\displaystyle \frac{\partial F(y)}{\partial x}}& {\displaystyle \frac{\partial F(y)}{\partial y}}\end{array}\right).$$

(18)

In this particular model, after substituting and simplifying, the Jacobian matrix takes the form:

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

(19)

2. Determinant and Trace

The determinant det(J) and the trace tr(J) of J are essential for determining local stability near the equilibrium points. From the original derivation (Equations (18) and (19)), we have:

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

(20)

By evaluating the sign of det(J) and the sign of tr(J) at a particular equilibrium (x^∗, y^∗), one can determine whether that equilibrium is asymptotically stable, unstable, or of a saddle type. Further, to incorporate external shocks or uncertainty—such as measurement errors, environmental fluctuations, or other stochastic forces—we introduce noise terms to the replicator dynamics equations. However, to reflect the impact of deliberative decision-making processes on these uncertainties, we modify the noise terms to include adjustment factors that represent the feedback loops in the democratic process. These factors account for the influence of collective stakeholder decisions on the evolution of strategies. Specifically, we introduce terms such as ΔX_t and ΔY_t in the stochastic equations, which reflect the deliberative adjustments made by users and grid operators based on the participatory inputs from all involved parties. This modification makes the model more responsive to democratic processes while still preserving the core dynamics of stochastic strategy evolution. One straightforward approach is to replace each deterministic term with a stochastic differential equation (SDE). In this framework, x(t) and y(t) become stochastic processes X_t and Y_t. A common choice is to add a diffusion term proportional to Brownian motion dW_t. Concretely, we can write:

$$\left\{\begin{array}{l}\mathrm{d}{X}_t=F(X_t)\mathrm{d}t+{\sigma }_x\mathrm{d}{W}_t^{(1)},\\ \mathrm{d}{Y}_t=F(Y_t)\mathrm{d}t+{\sigma }_y\mathrm{d}{W}_t^{(2)},\end{array}\right.$$

(21)

where

• σ_x and σ_y are constants representing the intensity (amplitude) of the noise in the X- and Y-equations, respectively.
• $W_t^{(1)}$ and $W_t^{(2)}$ are independent (or possibly correlated) standard Wiener processes (Brownian motions).

In some applications, one might generalize further by allowing σ_x and σ_y to depend on X_t and Y_t, or by incorporating cross terms that couple the noise affecting X_t with the dynamics of Y_t, thus yielding a more general diffusion matrix:

$$\left(\begin{array}{c}\mathrm{d}{X}_t\\ \mathrm{d}{Y}_t\end{array}\right)=\left(\begin{array}{c}F(X_t)\\ F(Y_t)\end{array}\right)\mathrm{d}t+\left(\begin{array}{cc}{\sigma }_{xx}(X_t,Y_t)& {\sigma }_{xy}(X_t,Y_t)\\ {\sigma }_{yx}(X_t,Y_t)& {\sigma }_{yy}(X_t,Y_t)\end{array}\right)\left(\begin{array}{c}\mathrm{d}{W}_t^{(1)}\\ \mathrm{d}{W}_t^{(2)}\end{array}\right).$$

(22)

From a modeling standpoint, this stochastic extension captures the fact that real-world replicator processes—whether biological, economic, or in demand-side management—are rarely isolated from random perturbations. These stochastic terms can represent unexpected changes in payoffs, random switching behaviors, or measurement noise in the system. The resulting stochastic replicator dynamics allow analysts to study how fluctuations affect the long-term behavior and stability of equilibria [32], and whether stable states under the deterministic model remain robust when random shocks are considered.

Moreover, when analyzing such stochastic extensions, one may apply tools from stochastic calculus (e.g., Ito or Stratonovich integrals) to derive moment equations, investigate probability distributions of (X_t, Y_t) over time, or assess almost-sure convergence. This enriches the traditional replicator framework by making it more realistic and better suited to uncertain environments.

To integrate the concept of whole-process democracy into the stochastic replicator dynamics framework (as represented by Equation (22)), we can make several adjustments that reflect the importance of inclusive, transparent, and democratic decision-making. In the context of the user–grid interaction, the integration of deliberative democracy could influence the behavior of both populations (users and grid operators) by introducing decision-making processes that incorporate feedback from all parties, resolving conflicts, and ensuring equitable distribution of benefits. This can be modeled by introducing a mechanism in the equations that represents not just stochastic fluctuations but also the deliberative decision-making process that considers inputs from all stakeholders. These inputs can influence the noise terms in the system, making them dynamic and responsive to the collective participation. Here is a proposed modification to Equation (22):

$$\left(\begin{array}{c}\mathrm{d}{X}_t\\ \mathrm{d}{Y}_t\end{array}\right)=\left(\begin{array}{c}F(X_t,Y_t)\\ F(Y_t,X_t)\end{array}\right)\mathrm{d}t+\left(\begin{array}{cc}{\sigma }_{xx}(X_t,Y_t)& {\sigma }_{xy}(X_t,Y_t)\\ {\sigma }_{yx}(X_t,Y_t)& {\sigma }_{yy}(X_t,Y_t)\end{array}\right)\left(\begin{array}{c}\mathrm{d}{W}_t^{(1)}\\ \mathrm{d}{W}_t^{(2)}\end{array}\right)+\left(\begin{array}{c}\Delta X_t\\ \Delta Y_t\end{array}\right),$$

(23)

where

• F(X_t, Y_t) and F(Y_t, X_t) represent the payoff functions, reflecting the strategies adopted by both users (X_t) and grid operators (Y_t).
• σ_xx, σ_xy, σ_yx, σ_yy represent the noise coefficients, which describe the random perturbations due to environmental or system uncertainties.
• dW_t⁽¹⁾ and dW_t⁽²⁾ are the Brownian motions for each of the populations, reflecting their inherent randomness.
• ΔX_t and ΔY_t represent the adjustment terms added to incorporate the effects of deliberative decision-making. These adjustments correspond to the democratic process where the populations adjust their strategies based on feedback and participation from all stakeholders.

Interpretation of ΔX_t and ΔY_t: These terms introduce a new layer of decision-making into the model. They can be defined as follows:

ΔX_t: The adjustment term for users, representing the collective decision-making effect based on inputs from all users. This term could depend on the perceived fairness of the system, the equity of benefit distribution, and the level of cooperation among users. For example: ΔX_t = α₁·(Participation Feedback from Users).

ΔY_t: The adjustment term for grid operators, representing the grid’s response to the collective decisions of users and its participation in the deliberative process. Similar to users, this term could depend on how cooperative the grid feels and how equitably the benefits are being distributed. For example: ΔY_t = α₂·(Grid’s Policy Adjustment Based on User Feedback).

The incorporation of these terms reflects the impact of democratic, deliberative decision-making processes where both parties (users and grid operators) influence the system over time, making it more inclusive and responsive to the needs and feedback from all stakeholders. This enhanced model reflects the dynamic and evolving nature of the system, where the strategies evolve not just in response to payoffs and randomness, but also through cooperative adjustments driven by the interaction between users and grid operators. This aligns with the goals of fostering cooperation, resolving conflicts, and ensuring equitable benefits distribution within renewable energy systems.

2.2.6. Parameter Selection and Numerical Experiments

1. Parameter Assignments

In this study, the specific parameter values chosen—a = 3, b = 5, c = 7, d = 9, e = 6, f = 3, g = 4, and h = 2—were selected to illustrate key dynamics in the context of peak-shifting demand management, which is a common scenario in energy economics. These values are hypothetical and intended to provide a proof-of-concept model for understanding the interaction between users and the grid operator. While these parameters are not directly derived from empirical data, they are calibrated to reflect typical cost–benefit scenarios observed in similar systems. For example, parameters like a, b, and c are representative of the costs associated with user participation, grid incentives, and peak demand penalties, respectively. Future work may consider real-world data to refine these parameters and make the model more closely aligned with empirical observations.

2. Step Sizes in Strategy Fractions

In these simulations, the fraction variables x and y represent the proportion of users (Population A) and grid operators (Population B) adopting specific strategies, respectively. These fractions are discretized over the interval [0, 1], at increments of 1/q (where q = 10, 20, 30, 40, 50, 60). This discretization allows us to simulate a range of initial conditions and observe how these strategies evolve over time. By tracking the evolution of x and y, we can analyze the convergence behavior of user and grid strategies, which reflects how real-world demand-side management programs might evolve under various policy scenarios. The strategy space corresponds to typical decisions in demand-response programs: users deciding whether to participate in DR (S_A1) and grid operators choosing the appropriate tariff structure (S_B1 or S_B2).

3. Interpretation for Peak-Shifting

In the context of demand-side management, we model two populations: (i) users (Population A), who adopt or reject specific strategies, such as peak-shifting demand response (DR) participation (S_A1: participate in DR, S_A2: ignore signals), and (ii) the grid operator (Population B), who implements strategies based on peak tariffs or incentives (S_B1: set high tariffs, S_B2: offer incentives). These strategies reflect typical decisions in demand-response programs, where users decide whether or not to engage with grid incentives, while the grid operator sets policies to manage demand. By incorporating these strategies into the evolutionary game framework, we aim to understand how interactions between these two populations evolve over time, and how their equilibrium is influenced by the payoff structures. By examining the replicator dynamic trajectories, researchers and practitioners can gain insights into how likely it is that a socially efficient or economically favorable equilibrium is reached, and whether that equilibrium is stable in the long run.

Collectively, through payoff matrix formulations, replicator equations, and the Jacobian-based stability analysis, the proposed non-symmetric evolutionary game model provides a comprehensive theoretical tool for examining how user–grid interactions evolve in the presence of various peak-shifting or demand-response mechanisms.

Setting different step sizes indicates that the number of game iterations increases over time. The phase trajectory diagrams for different step sizes are shown in the figures below. As shown in Figure 1, Figure 2 and Figure 3, the red points represent evolutionarily stable points, the blue points represent evolutionarily near-stable points, also known as saddle points, and the green points represent unstable points. The stability analysis of the equilibrium points in Figure 3 is presented in

Table 1. Long-term evolutionary stable equilibrium analysis and summarization.

(x, y)	det(J)	tr(J)	Evolutionary Stability
(0, 0)	3	4	Evolutionarily unstable (not an ESS)
(0, 1)	3	−4	ESS
(1, 0)	12	−7	ESS
(1, 1)	12	7	Evolutionarily unstable (not an ESS)
(2/5, 3/7)	−6/5	0	A saddle point (evolutionarily unstable)

.

Figure 1, Figure 2 and Figure 3 illustrate the phase trajectories of the evolutionary game under varying step sizes (i.e., different numbers of discrete iterations as time progresses). In these figures, each curve represents a dynamic trajectory of the fraction x (for Population A) and y (for Population B) starting from different initial conditions. By comparing trajectories across different step sizes, one can observe how quickly—or slowly—the fractions x and y evolve toward particular equilibrium points.

First, the phase trajectory in x-t (Figure 1) and y-t (Figure 2):

(i)

As time t increases, most trajectories converge toward certain values of x or y, reflecting the system’s tendency to lock onto stable or near-stable equilibria.

(ii)

The red points on the right-hand side of each subplot mark evolutionarily stable strategies (ESS). When trajectories from a wide range of initial conditions cluster around these red points, it indicates that these points exert a strong attracting influence in the evolutionary sense.

(iii)

The blue points (often located in the interior or somewhere within the domain) represent saddle points (also described as “near-stable” in the figures). Trajectories may approach these points along one direction but diverge along another; thus, only a subset of initial conditions actually converges to a saddle, while many others eventually leave its vicinity.

(iv)

The green points indicate unstable equilibria. The trajectories around these points tend to move away rather than settle, making them repelling states from an evolutionary perspective.

Second, the phase trajectory in the (x, y) Plane (Figure 3):

(i)

Plotted in the unit square [0, 1] × [0, 1], each trajectory shows how the pair (x, y) changes jointly over time.

(ii)

The colored corners and interior points correspond to equilibrium candidates: Red corners in these diagrams typically coincide with stable boundary equilibria (ESS). Green corners are unstable equilibria, where trajectories rarely linger. The blue point in the interior (e.g., (2/5, 3/7)) is a saddle-like equilibrium: it may attract trajectories along certain directions yet repel them along others.

Third, the equilibrium stability and Table 1: Table 1 compiles the determinant det(J) and trace tr(J) of the Jacobian for each identified equilibrium point, alongside the corresponding stability classification. A brief interpretation of the listed cases is as follows:

(i)

(0, 0) and (1, 1): Both have a positive determinant but a positive trace, leading to instability under the replicator dynamics. Hence, these corner points are labeled green in Figure 3.

(ii)

(0, 1) and (1, 0): These corners exhibit negative trace with positive determinant, indicating ESS (red points). They attract a wide range of trajectories that begin in their “basin” of attraction.

(iii)

(2/5, 3/7): The determinant is negative (or only marginally positive/negative depending on parameter variations), and the trace is zero. Such a condition typically signifies a saddle or “soft” equilibrium. Consequently, it is shown in blue in the phase plots, reflecting its partial attraction and partial repulsion properties.

Overall, the figures confirm that different step sizes (i.e., different iteration frequencies) do not alter the fundamental stability classification of the equilibrium points but can affect the speed and manner in which the trajectories converge or diverge. The red, green, and blue markers concisely indicate the stable, unstable, and saddle-like natures of the equilibria, while Table 1 provides a rigorous analytical verification based on det(J) and tr(J). Such consistency between numerical simulations and analytical results reinforces confidence in the evolutionary game framework and its predictive validity for strategy selection in demand-side management or other application domains.

This study presents a proof-of-concept model for user–grid co-participation in peak-shifting demand management, leveraging a stochastic replicator dynamics framework. While this model is not based on specific empirical data, it serves to demonstrate the potential dynamics between users and grid operators under various noise scenarios. In comparison to real-world demand-response behavior, our model predicts that user and grid operator populations tend to polarize toward extreme strategies (either full participation or no participation) under strong incentives. This finding aligns with trends observed in certain demand-response programs, where users tend to adopt either extreme strategy depending on tariff structures and incentives offered by grid operators. However, as real-world programs often exhibit more gradual transitions, future work could further refine the model by incorporating real-world data to validate these theoretical predictions and account for additional complexities such as time-of-use pricing or dynamic incentive schemes. The goal is to explore how evolutionary game theory can provide insights into demand-side management under uncertainty. Future iterations of this model could incorporate real-world data to validate the proposed theoretical framework and enhance its predictive power. Specifically, two populations (users and the grid) each choose one of two strategies, and their evolutionary interactions are governed by a set of payoff parameters {a, b, c, d, e, f, g, h}. The motivation is to understand how random disturbances—captured by a general diffusion matrix—affect the convergence and stability of strategies within a demand-response setting. The primary objectives are to (i) illustrate how stochastic effects may alter or reinforce stable strategies, (ii) identify evolutionarily stable equilibria (ESE) and near-stable (saddle-like) points, and (iii) provide a theoretical basis for practical policies in peak-shifting demand management. Key parameters include:

(a)

a = 3, b = 5, c = 7, d = 9, e = 6, f = 3, g = 4, and h = 2, which define payoff outcomes in different user–grid strategic interactions;

(b)

A simulation horizon t_max = 50 (in, for instance, dimensionless time units or hours, depending on context);

(c)

An integration step size Δt = 0.01 (unit), and two hundred independent runs (n_trajectories = 200) per noise “case” to explore the variability of outcomes under distinct noise intensities and coupling structures.

Through this setup, we seek to elucidate how evolutionary game methods can guide decision-making processes, offering robust insights into both deterministic and noise-perturbed conditions. To enhance the understanding of model robustness, we have conducted a sensitivity analysis to examine how small changes in key parameters, such as the g/h ratio (the relative importance of grid incentives versus user behavior), affect the equilibria. Our results indicate that while the overall dynamics of the system remain largely stable, variations in these parameters can shift the location of equilibrium points, especially under stochastic conditions. This highlights the sensitivity of the model to parameter changes and provides important implications for real-world applications where such parameters may vary over time.

Figure 4: Stochastic temporal evolution of x in the user–grid peak-shifting game. This figure depicts how the fraction x (representing, for example, the proportion of users adopting a particular load-shifting strategy) evolves over time under six different noise configurations. Regardless of the initial conditions, most trajectories converge toward either the upper boundary (x ≈ 1) or remain near lower values (x ≈ 0), with each subfigure reflecting the impact of distinct diffusion intensities and off-diagonal couplings. This strongly suggests that while random perturbations can briefly delay convergence, they do not fundamentally dislodge the system from gravitating toward one of the stable edges. Notably, higher coupling (e.g., Cases 4 and 6) introduces more transient volatility, yet still converges reliably to boundary-dominated equilibria.

Figure 5: Stochastic temporal evolution of y in the user–grid peak-shifting game. This figure tracks the fraction y (e.g., the proportion of grid operators or aggregators favoring a certain tariff or incentive policy) under the same six noise cases. Similar to x, the trajectories exhibit rapid convergence to either near 0 or near 1, indicating that the grid’s chosen policy tends to polarize in response to user strategies and payoff incentives. Random fluctuations—manifest as diffusion in the replicator dynamic equations—may cause minor excursions away from the deterministic path, but the overall drift remains strongly attracted to boundary states. These results imply that once the grid population collectively starts shifting toward a dominant strategy, the presence of stochastic effects rarely reverses the overall direction of change.

Figure 6: Two-dimensional stochastic phase trajectories for (x, y). This figure provides a joint phase-plane view, showing how pairs (x, y) traverse the state space over time. Each panel corresponds to a different noise scenario, capturing the interplay between user strategies and grid policies under stochastic forces. One observes that trajectories fan out from diverse initial conditions yet funnel toward corners (0, 1) or (1, 0), marked as ESS points with red stars. These corners act as robust attractors, indicating that both the user population and the grid operator population jointly prefer complementary strategies in equilibrium. Even when off-diagonal coupling is introduced (e.g., Cases 3, 4, 6), interior saddle points do not hold the population permanently; instead, they merely slow or deflect transient dynamics on the route to boundary-dominated equilibria.

Taken together, these results underscore the resilience of evolutionarily stable equilibria even in the face of stochastic perturbations. The replication of boundary equilibria (x, y) ∈ {(1, 0),(0, 1)} points to a marked tendency for user–grid systems to become polarized under strong payoff incentives. However, we also observe that the model exhibits robust behavior under varying noise and parameter settings, particularly when examining the g/h ratio and other key payoff parameters. Even with fluctuations in these parameters, the system tends to converge to one of the boundary equilibria. This suggests that the model’s findings are relatively insensitive to small perturbations in parameter values, providing confidence in its robustness for policy design in dynamic environments. This finding supports the premise that evolutionary game theory, enriched with stochastic replicator models, offers a powerful framework for analyzing demand-response strategies in which uncertainties and environmental fluctuations are inevitable. As a natural extension, future work could refine the payoff structures to reflect real-world pricing mechanisms, introduce correlation in noise terms, or scale the approach to larger populations or multiple strategy sets. These enhancements would further reinforce the theoretical underpinnings of user–grid peak-shifting analyses and provide a robust platform for designing more adaptive and resilient energy management policies.

3. Evolutionary Game Model of User–Grid Cooperation in Peak Shaving Participation

In the context of renewable energy grid integration, both the grid operator and energy users participate in peak shaving with the ultimate objective of maximizing their respective benefits. This process inherently involves a conflict of interest. When one party seeks to maximize its benefits without regard for the other, it compromises the opposing party’s ability to achieve its optimal outcome. For instance, if the grid enforces strict demand management policies or dynamic pricing to stabilize supply–demand balance, it may burden users with higher costs or reduce their flexibility in electricity consumption. Conversely, if users prioritize their cost savings by minimizing consumption during peak hours, the grid may face challenges in maintaining operational stability or recovering infrastructure costs. This mutual contention may lead to a breakdown of cooperation, undermining the shared goals of system efficiency and renewable energy integration.

In this context, the interaction between users and the grid must adapt under specific conditions that maximize mutual benefits. This balance, known as the ‘equilibrium of mutual interests,’ is crucial for the long-term sustainability of cooperation in peak shaving and renewable energy adoption. Equilibrium is reached when both parties pursue their individual goals while finding a strategic middle ground that aligns their interests, thus supporting the system’s stability.

In the evolutionary game process, the strategies of users and the grid are not static; they are dynamic and continuously adapt based on the actions of the other party. The interaction is shaped by a range of factors, including economic incentives, technological constraints, and policy interventions. For instance, government agencies play a pivotal role in influencing this dynamic through regulatory frameworks, such as subsidies for renewable energy technologies, penalties for excessive emissions, or dynamic electricity pricing. These interventions serve as external constraints that shape the strategic choices of both parties, guiding them toward cooperative behaviors that benefit the entire energy ecosystem.

In each round of peak shaving within the power system, the strategic choices of users and the grid are interdependent. For example, if the grid adopts a cooperative strategy by offering attractive incentives for demand response programs, users are more likely to participate in these programs, thereby helping to flatten peak demand and improve system efficiency. On the other hand, if users adopt non-cooperative strategies—such as refusing to adjust consumption patterns during peak hours—the grid may respond by imposing punitive measures or higher tariffs to enforce compliance. This cyclical interplay of strategy and counter-strategy underscores the complexity of achieving a stable equilibrium where both parties can maximize their benefits without undermining the other’s objectives.

Analyzing the equilibrium of user–grid interactions in renewable energy peak shaving requires a rigorous methodology. Evolutionary game theory provides an effective tool for this purpose, offering insights into how users and the grid can adapt their strategies over time to achieve mutual benefit. By examining the cooperation and non-cooperation strategies of both parties, researchers can identify conditions that lead to stable, mutually beneficial outcomes. This involves constructing payoff matrixes that account for the costs and benefits associated with each strategic choice, as well as analyzing the evolutionary stability conditions (ESCs) that determine whether these strategies are sustainable in the long run.

Moreover, external factors like government policies, market mechanisms, and technological advancements must be included in the analysis to fully capture user–grid interactions. For example, subsidies for home-based renewable energy systems, such as solar panels, may incentivize users to adopt cooperative strategies by reducing the upfront costs of participation. Similarly, advancements in smart grid technologies and data analytics can enable more precise demand forecasting and dynamic pricing, fostering a more adaptive and efficient interaction between users and the grid.

To analyze the equilibrium process of mutual benefits when users and the grid participate in renewable energy peak shaving projects, this section investigates the evolutionary game model between users and the grid. This is achieved by analyzing the feasibility of their cooperation in power system peak shaving and constructing an evolutionary game model based on their strategic choices of cooperation and non-cooperation. The gains of both users and the grid are associated with the normal revenue under no renewable energy peak shaving integration, the surplus revenue generated by renewable energy peak shaving integration, and the distribution of this surplus revenue. If either party attempts to maximize its benefits unilaterally, it inevitably impedes the other party’s ability to achieve its own maximum benefits. The relationship between the revenue of users and the grid and the costs incurred by cooperation or non-cooperation is analyzed, followed by the development of an asymmetric evolutionary game model. Through this, the ESCs are derived, which provide a foundation for case analysis and simulations.

Ultimately, achieving equilibrium in user–grid interactions during peak shaving is not a one-time event but a continuous process of negotiation, adaptation, and optimization. It requires both parties to recognize the interdependence of their actions and to embrace strategies that balance individual gains with collective benefits. This equilibrium not only ensures the efficient operation of the power system but also lays the foundation for a more sustainable and resilient energy future.

The proposed model incorporates 20 distinct parameters, each serving a critical role in capturing the dynamics of user–grid interactions in renewable energy systems. These parameters include:

• Economic parameters, such as revenue, costs, subsidies, and penalties.
• Operational parameters, like efficiency, dynamic pricing, and user–grid interaction costs.
• Behavioral parameters, such as risk aversion and responsiveness to incentives.

This categorization ensures clarity in understanding the role of each parameter in influencing the outcomes of the game-theoretic model.

3.1. Evolutionary Game Model of User–Grid Participation in Power System Peak Shaving

3.1.1. Parameter Settings and Key Steps in Model Construction

The renewable energy grid integration peak shaving project requires the cooperation and collaboration of both the user group and the grid group, encompassing the distribution of benefits between the two. When either party attempts to maximize its benefits, it naturally hinders the other party’s ability to achieve their maximum benefits. Thus, the construction of an evolutionary game model is necessary to analyze the user–grid collaboration in peak shaving. Before establishing this model, it is critical to account for the relationship between the revenue and costs of users and the grid. In constructing the payoff matrix for the evolutionary game model, it is assumed that both the user group and the grid group have two strategic options: cooperation and non-cooperation. Below are the payoff matrixes, related parameter descriptions, Jacobian matrix, and ESCs. These are presented in

Table 2. User–grid joint participation in peak-shaving benefit matrix.

Strategy Selection		The Grid Group
		Cooperation	Non-Cooperation
The user group	Cooperation	(${\pi }_{\mathrm{g}}+(1-\alpha )\Delta S-l_{\mathrm{g}}\Delta Q$, ${\pi }_{\mathrm{f}}+\alpha \Delta S-l_{\mathrm{f}}\Delta Q$)	(${\pi }_{\mathrm{g}}+{\gamma }_{\mathrm{f}}\Delta Q-l_{\mathrm{g}}\Delta Q$, ${\pi }_{\mathrm{f}}$)
	Non-cooperation	(${\pi }_{\mathrm{g}}$, ${\pi }_{\mathrm{f}}+{\gamma }_{\mathrm{f}}\Delta Q-l_{\mathrm{f}}\Delta Q$)	(${\pi }_{\mathrm{g}}$, ${\pi }_{\mathrm{f}}$)

and

Table 3. Explanation of parameters in user–grid joint participation in peak-shaving game.

Parameter	Definition	Description
${\pi }_{\mathrm{g}}$	Basic revenue obtained by the grid when no renewable energy peak-shaving project is implemented (yuan)	The fundamental profit for the grid without implementing renewable energy generation and peak-shaving integration projects.
${\pi }_{\mathrm{f}}$	Basic revenue obtained by the user when no renewable energy peak-shaving project is implemented (yuan)	The fundamental profit for users without participating in renewable energy generation and peak-shaving integration projects.
$\Delta S_{\mathrm{g}}$	Additional profit generated by the grid from adopting a cooperative strategy (yuan)	Extra profit achieved by the grid when choosing a cooperative strategy.
$\Delta S_{\mathrm{f}}$	Additional profit generated by users from adopting a cooperative strategy (yuan)	Extra profit achieved by users when choosing a cooperative strategy.
$\Delta S$	Total additional profit generated by implementing a renewable energy peak-shaving project (yuan)	The cumulative additional profit achieved through implementing renewable energy peak-shaving projects with user participation.
$\alpha$	Grid’s allocation coefficient for the total additional profit	The proportion of total additional profit allocated to the grid.
$p$	Price of conventional energy electricity (yuan)	The price per unit of electricity generated by conventional energy sources.
$l_{\mathrm{g}}$	Cost increment generated per unit of renewable energy consumption for users (yuan)	The additional cost incurred for users per unit of renewable energy consumption.
$l_{\mathrm{f}}$	Cost increment generated per unit of renewable energy consumption for the grid (yuan)	The additional cost incurred for the grid per unit of renewable energy consumption.
${\gamma }_{\mathrm{f}}$	Additional profit per unit of renewable electricity supply for the grid (yuan)	The extra profit achieved by the grid per unit of renewable electricity supply.
${\gamma }_{\mathrm{g}}$	Additional profit per unit of renewable electricity supply for users (yuan)	The extra profit achieved by users per unit of renewable electricity supply.
$\Delta Q$	Incremental power consumption for users under a cooperative strategy (MW)	The additional electricity consumption by users when adopting a cooperative strategy.
$Q$	Incremental power consumption for users under a non-cooperative strategy (MW)	The additional electricity consumption by users when adopting a non-cooperative strategy.

. Based on this, the key steps in model construction are summarized as follows.

(i)

Strategies of Cooperation and Non-Cooperation: Each party, users and the grid, has the option to adopt either a cooperative or non-cooperative strategy during peak shaving. Cooperation involves mutual participation in renewable energy grid integration, while non-cooperation reflects unilateral actions or inaction.

(ii)

Payoff Matrix Construction: The payoff matrix defines the potential revenue for each party under each strategic combination (cooperation/cooperation, cooperation/non-cooperation, etc.). These revenues are calculated based on their respective benefits and associated costs of participation or abstention.

(iii)

Jacobian Matrix and Evolutionary Stability Conditions: Using the constructed payoff matrix, the Jacobian matrix is derived to analyze the dynamic interactions between users and the grid. The conditions for ESCs are then formulated, identifying the stable strategies that ensure mutual benefit maximization under given circumstances.

(iv)

Case Analysis and Simulation: The derived evolutionary stability conditions are applied to specific case scenarios, analyzing various equilibrium outcomes. Simulations are conducted to illustrate the impacts of strategic choices on the revenue and cost dynamics of users and the grid.

In Table 2, for each strategy combination, the first payoff in the parentheses represents the benefit gained by the user group when selecting the corresponding strategy, while the second payoff indicates the benefit obtained by the grid group under the same strategy. The benefit matrix presented in Table 2 highlights the strategic interaction between the user group and the grid group in the context of peak-shaving operations during the integration of renewable energy. Both participants—users and the grid—can select between two strategies: cooperation or non-cooperation. The payoff for each strategy is influenced by variables such as base revenue, surplus profit generated by cooperation, and the additional costs associated with renewable energy integration.

(i)

Cooperation by Both Sides: When both the user and the grid choose cooperation, they collectively maximize the net benefits derived from renewable energy integration projects. The grid’s revenue increases due to shared surplus profit allocation (1 − α)ΔS, while the user benefits through its surplus profit allocation αΔS, minus the respective additional costs related to renewable energy consumption or generation.

(ii)

One-Sided Cooperation: If one party opts for cooperation while the other does not, the cooperative party faces additional costs while receiving no benefit from surplus profit allocation, leading to an imbalanced distribution of benefits.

(iii)

Non-Cooperation by Both Sides: When both parties choose non-cooperation, the payoffs revert to the baseline level of benefit without any surplus profit or additional cost incurred, resulting in a stagnant scenario for the overall system’s advancement.

This payoff distribution matrix effectively demonstrates the interdependent nature of decision-making in the energy system. It also highlights that the greatest gains arise from mutual cooperation, while imbalanced strategies or mutual non-cooperation lead to suboptimal outcomes for both parties.

Table 3 provides a detailed explanation of the parameters used in the payoff matrix of Table 2. These parameters quantify the economic and operational dynamics between the user group and the grid group during peak-shaving operations with renewable energy integration.

(i)

Base Benefits (π_g and π_f): These represent the foundational economic returns for the grid and users, respectively, under normal conditions without implementing peak-shaving programs.

(ii)

Surplus Profit (ΔS_g, ΔS_f, and ΔS): These variables quantify the additional profit generated through cooperative strategies. ΔS represents the total surplus profit shared between the grid (1 − α) and users (α).

(iii)

Cost Parameters (l_g, l_f): These represent the incremental costs incurred by the grid and users for handling renewable energy integration during peak-shaving programs.

(iv)

Incremental Revenue (γ_g, γ_f): These factors reflect the unit revenue increases for the grid and users due to renewable energy adoption.

(v)

Energy Variables (Q and ΔQ): These specify the additional electricity consumption or production changes depending on the user’s cooperative or non-cooperative behavior.

Overall, Table 2 and Table 3 collectively provide a structured analytical framework for understanding the benefit distribution and cost implications of renewable energy integration during peak-shaving operations. Table 2 demonstrates the payoff dynamics between cooperative and non-cooperative strategies, while Table 3 specifies the parameters essential for quantifying these payoffs. Together, these tables underscore the importance of collaboration between users and the grid to maximize mutual benefits and promote the efficient integration of renewable energy. Furthermore, they highlight the critical role of cost allocation and profit-sharing mechanisms in achieving strategic equilibrium in the system.

3.1.2. Model Construction

Based on the aforementioned benefit matrix, the incremental profits generated by adopting a cooperative strategy between the power grid and the users are defined as follows. The incremental profit from the grid’s perspective is expressed as: ΔS_g = αΔS, and the incremental profit from the user’s perspective is expressed as: ΔS_f = (1 − α)ΔS. It follows that the total incremental profit ΔS can be written as ΔS = ΔS_g + ΔS_f.

Here, it is assumed that users’ electricity consumption has a significant impact on their revenues and that the user’s revenue is positively correlated with electricity consumption. Additionally, it is presumed that the power grid balances electricity supply and demand, ensuring that supply matches user demand.

Assuming both the user group and the power grid adopt cooperative strategies, the total incremental profit gained by the power grid and the user group will exceed the production cost of electricity. Based on this assumption, the following revenue and constraint relationships can be derived.

(1) Incremental profits of the power grid under a cooperative strategy:

$$\Delta S_{\mathrm{g}}=pQ+{\gamma }_{\mathrm{g}}\Delta Q-l_{\mathrm{g}}\Delta Q,$$

(24)

where p represents the electricity price, Q is the total electricity demand, γ_g is the benefit coefficient for the power grid, and l_g represents the production cost coefficient for the grid.

(2) Incremental profits of users under a cooperative strategy:

$$\Delta S_f=pQ+{\gamma }_{\mathrm{f}}\Delta Q-l_{\mathrm{f}}\Delta Q,$$

(25)

where γ_f is the benefit coefficient for users and l_f is the cost coefficient for users.

(3) Total incremental profits from the implementation of renewable energy projects and peak shaving strategies:

$$\Delta S=2pQ+({\gamma }_{\mathrm{g}}+{\gamma }_t-l_t-l_t)\Delta Q,$$

(26)

Let x and y represent the proportion of users and the power grid, respectively, that adopt a cooperative strategy. Conversely, 1 − x and 1 − y represent the proportion of users and the power grid that do not adopt a cooperative strategy. The state is shown as:

$$S=\{(s_1^1,s_2^1),(s_1^2,s_2^2)\}=\{(x,1-x),(y,1-y)\}$$

(27)

Equation (27) describes the strategic behaviors of the user group and the power grid in participating in the system’s peak shaving. Based on this, the expected payoffs for users are calculated as follows:

When adopting a cooperative strategy:

$$f^1(r^1,t)=\left({\pi }_{\mathrm{g}}+(1-\alpha )\Delta S-l_{\mathrm{g}}\Delta Q\right)y+\left({\pi }_{\mathrm{g}}+{\gamma }_{\mathrm{f}}\Delta Q-l_{\mathrm{g}}\Delta Q\right)(1-y),$$

(28)

When not adopting a cooperative strategy:

$$f^1(r^2,t)={\pi }_{\mathrm{g}}y+{\pi }_{\mathrm{g}}(1-y),$$

(29)

Average fitness (expected payoff) for users:

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

(30)

(5) Expected payoffs for the power grid:

When adopting a cooperative strategy:

$$f^2(r^1,t)=\left({\pi }_{\mathrm{f}}+\alpha \Delta S-l_{\mathrm{f}}\Delta Q\right)x+\left({\pi }_{\mathrm{f}}+{\gamma }_{\mathrm{f}}\Delta Q-l_{\mathrm{f}}\Delta Q\right)(1-x),$$

(31)

Average fitness (expected payoff) for the power grid:

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

(32)

When the fitness of a specific strategy surpasses the average fitness of the population, that strategy tends to dominate within the system.

(6) Replicator dynamic equations:

Based on the profit functions, the replicator dynamics for the user group and the power grid are constructed as follows:

$${\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}=x(1-x)\left[f^1(r^1,t)-f^1(r^2,t)\right],$$

(33)

Or equivalently,

$${\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}=x(1-x)\left[y\left((1-\alpha )\Delta S-{\gamma }_{\mathrm{g}}\Delta Q\right)+({\gamma }_{\mathrm{g}}-l_{\mathrm{g}})\Delta Q\right].$$

(34)

For the power grid:

$${\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}}=y(1-y)\left[f^2(r^1,t)-f^2(r^2,t)\right],$$

(35)

Or equivalently,

$${\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}}=y(1-y)\left[x\left(\alpha \Delta S-{\gamma }_{\mathrm{f}}\Delta Q\right)+({\gamma }_{\mathrm{f}}-l_{\mathrm{f}})\Delta Q\right].$$

(36)

(7) Dynamic equilibrium conditions

The system reaches equilibrium when:

$${\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}=0,{\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}}=0.$$

(37)

This occurs under the following conditions:

$$x=0,x=1,y=0,y=1,$$

(38)

where we can find four pure-strategy equilibrium points as (0, 0), (0, 1), (1, 0), and (1, 1). In addition, there is also a mixed strategy equilibrium point can be found:

$$x={\displaystyle \frac{({\gamma }_{\mathrm{f}}-l_{\mathrm{f}})\Delta Q}{{\gamma }_{\mathrm{f}}\cdot \Delta Q-\alpha \Delta S}},y={\displaystyle \frac{({\gamma }_{\mathrm{g}}-l_{\mathrm{g}})\Delta Q}{{\gamma }_{\mathrm{g}}\Delta Q+\alpha \Delta S-\Delta S}}.$$

(39)

The mixed strategy equilibrium point shown in Equation (39) represents a state where the proportions x and y of users and the power grid, respectively, adopting cooperative strategies are determined by the balance of incremental profits and costs. Specifically, x and y are functions of the relative benefit coefficients (γ_g, γ_f) and cost coefficients (l_g, l_f), as well as the total incremental profit (ΔS) and changes in electricity demand (ΔQ).

While this equilibrium point may appear stable mathematically, it cannot serve as a long-term evolutionary stable equilibrium (ESS) due to the dynamic nature of the replicator equations governing the system [33,34,35]. The key reasons for its instability are [36,37,38,39]:

(i)

Instability of Fitness Gradients: The replicator dynamic equations are driven by the differences in fitness (or payoffs) between strategies. At the mixed strategy equilibrium, any small deviation in either x or y caused by external perturbations will lead to a fitness imbalance. This imbalance drives the system away from the equilibrium point, as the proportions of individuals adopting each strategy will adjust dynamically to maximize their respective payoffs.

(ii)

Absence of Evolutionary Stability: For a point to be evolutionarily stable, any deviation from the equilibrium should result in forces that restore the system back to the equilibrium. However, at this mixed strategy equilibrium, small deviations in x or y create reinforcing feedback loops in the fitness gradients, pushing the system further away instead of returning to the equilibrium. This lack of restorative forces means the mixed strategy equilibrium cannot maintain long-term stability.

(iii)

Competitive Dynamics Between Strategies: The coexistence of cooperative and non-cooperative strategies inherently creates competition between these strategies. The payoff structures for both the user group and the power grid indicate that as one strategy becomes slightly more dominant, it starts to outcompete the other, further destabilizing the equilibrium.

(iv)

Path Dependency and System Sensitivity: The equilibrium values of x and y are highly sensitive to the parameters ΔS, ΔQ, γ_g, γ_f, l_g, and l_f. Any changes in these parameters, such as fluctuations in demand or shifts in cost structures, can disrupt the equilibrium state, leading to an evolutionary trajectory away from the mixed strategy equilibrium.

The mixed strategy equilibrium point is inherently unstable in the long-term evolutionary dynamics of the system. This instability results from the replicator dynamics’ inability to counter deviations and the reinforcing competition between strategies, making it a transient state rather than a sustainable evolutionary outcome.

Further, the Jacobian matrix for stability analysis can be obtained. The Jacobian matrix for the system is given as:

$$\mathit{J}=\left[\begin{array}{cc}(1-2x)[y((1-\alpha )\Delta S-{\gamma }_{\mathrm{g}}\Delta Q)+({\gamma }_{\mathrm{g}}-l_{\mathrm{g}})\Delta Q]& x(1-x)((1-\alpha )\Delta S-{\gamma }_{\mathrm{g}}\Delta Q)\\ y(1-y)(\alpha \Delta S-{\gamma }_{\mathrm{f}}\Delta Q)& (1-2y)[x(\alpha \Delta S-{\gamma }_{\mathrm{f}}\Delta Q)+({\gamma }_{\mathrm{f}}-l_{\mathrm{f}})\Delta Q]\end{array}\right].$$

(40)

Let:

$$\left\{\begin{array}{l}a=(1-\alpha )\Delta S,b=\alpha \Delta S,c={\gamma }_{\mathrm{g}}\Delta Q,d=h=l_{\mathrm{f}}\Delta Q,e=g=l_{\mathrm{g}}\Delta Q,f={\gamma }_{\mathrm{f}}\Delta Q\\ {\gamma }_1=a-c-e+g\\ {\gamma }_2=c-g\\ {\gamma }_3=b-f-d+h\\ {\gamma }_4=f-h\end{array}\right.$$

(41)

for simplifying subsequent calculations.

3.1.3. Evolutionary Stability Analysis

Based on the previously constructed model, this section examines the evolutionary stability of the system during the peak shaving process involving both users and the power grid. The results are summarized in

Table 4. Stability analysis of user and power grid participation in peak shaving process.

Equilibrium Point	det(J)	tr(J)	Evolutionary Stability Conditions
(0, 0)	$({\gamma }_{\mathrm{g}}-l_{\mathrm{g}})\Delta Q({\gamma }_{\mathrm{f}}-l_{\mathrm{f}})\Delta Q$	$({\gamma }_{\mathrm{g}}-l_{\mathrm{g}})\Delta Q+({\gamma }_{\mathrm{f}}-l_{\mathrm{f}})\Delta Q$	${\gamma }_{\mathrm{g}}<l_{\mathrm{g}},{\gamma }_{\mathrm{f}}<l_{\mathrm{f}}$
(0, 1)	$-((1-\alpha )\Delta S-l_{\mathrm{g}}\Delta Q)({\gamma }_{\mathrm{f}}-l_{\mathrm{f}})\Delta Q$	$(1-\alpha )\Delta S-l_g\Delta Q-({\gamma }_{\mathrm{f}}-l_{\mathrm{f}})\Delta Q$	$(1-\alpha )\Delta S<l_{\mathrm{g}}\Delta Q,r_{\mathrm{f}}>l_{\mathrm{f}}$
(1, 0)	$-({\gamma }_{\mathrm{g}}-l_{\mathrm{g}})\Delta Q(\alpha \Delta S-l_{\mathrm{f}}\Delta Q)$	$-({\gamma }_{\mathrm{g}}-l_{\mathrm{g}})\Delta Q+\alpha \Delta S-l_{\mathrm{f}}\Delta Q$	$\alpha \Delta S<l_{\mathrm{f}}\Delta Q,{\gamma }_{\mathrm{g}}>l_{\mathrm{g}}$
(1, 1)	$((1-\alpha )\Delta S-l_g\Delta Q)(\alpha \Delta S-l_{\mathrm{f}}\Delta Q)$	$-((1-\alpha )\Delta S-l_g\Delta Q)-(\alpha \Delta S-l_{\mathrm{f}}\Delta Q)$	$(1-\alpha )\Delta S>l_g\Delta Q,\alpha \Delta S<l_{\mathrm{f}}\Delta Q$
(x′, y′)	${\displaystyle \frac{(e-a)(b-d){\gamma }_4{\gamma }_2}{{\gamma }_1{\gamma }_3}}$	0	A saddle point (still evolutionarily unstable)

. By analyzing the incremental profits and costs of the power grid and users, it can be inferred that the benefits derived from the cooperative strategy and the associated costs depend on the initial revenues of both parties, as well as their respective participation rates in the renewable energy and peak shaving projects.

The five equilibrium points listed in Table 4 are derived by setting the replicator dynamic equations to zero. In this table, $x^{\prime }={\displaystyle \frac{({\gamma }_{\mathrm{f}}-l_{\mathrm{f}})\Delta Q}{{\gamma }_{\mathrm{f}}\cdot \Delta Q-\alpha \Delta S}},y^{\prime }={\displaystyle \frac{({\gamma }_{\mathrm{g}}-l_{\mathrm{g}})\Delta Q}{{\gamma }_{\mathrm{g}}\Delta Q+\alpha \Delta S-\Delta S}}$. From the table, we can determine the Jacobian matrix for each equilibrium point and calculate its determinant (det(J)) and trace (tr(J)). The stability conditions of each equilibrium point are judged based on the signs of these parameters. Combining theoretical analysis with specific numerical values allows us to assess the stability of each equilibrium point and infer the evolutionary stable strategies for the peak shaving process under different system scenarios. Aiming at Table 4, the analysis and insights are shown as follows.

(1) Pure Strategy Equilibria

The points (0, 0), (0, 1), (1, 0), and (1, 1) represent scenarios where either none, one, or both parties (users and the power grid) adopt the cooperative strategy. Stability at these points depends on the relationships between incremental profits (ΔS), benefit coefficients (γ_g, γ_f), and cost coefficients (l_g, l_f). For example, at (0, 0), both the power grid and users avoid cooperation. This equilibrium is stable only if the benefit coefficients are smaller than the respective cost coefficients (γ_g < l_g, γ_f < l_f), meaning cooperation is not economically advantageous. At (1, 1), both parties cooperate. This equilibrium is stable if the incremental profits outweigh the costs (αΔS > γ_fΔQ, γ_g > l_g), signifying mutual benefits from cooperation.

(2) Mixed Strategy Equilibrium

The mixed strategy equilibrium point involves partial cooperation, where the proportions x (users) and y (power grid) are determined by the specific economic parameters of the system. While this point mathematically satisfies the replicator dynamic equations, it is inherently unstable. The determinant (det(J)) and trace (tr(J)) indicate that deviations from this equilibrium lead to system divergence. This instability arises because the mixed strategy does not provide strong enough feedback to restore equilibrium, making it a transient state rather than a long-term stable outcome.

(3) Practical Implications

The stability of the pure strategy equilibria highlights the conditions under which cooperation between the power grid and users is economically viable. Policymakers can use this insight to design incentives or subsidies to ensure that the system evolves toward the (1, 1) equilibrium, where both parties benefit from mutual cooperation. The mixed strategy point serves as a transitional phase, indicating a potential tipping point in the system. However, its instability underscores the need for mechanisms to guide the system toward stable and desirable cooperative outcomes.

(4) Evolutionary Mechanisms

The evolution of strategies in the system depends heavily on the relative values of ΔS, ΔQ, γ_g, γ_f, and l_g, l_f. Policymakers should focus on reducing costs (l_g, l_f) and increasing benefits (γ_g, γ_f) associated with cooperation to drive the system toward stable cooperation. Numerical simulations could further validate these theoretical findings and offer insights into real-world implementations.

Overall, the analysis of evolutionary stability reveals that cooperation between the power grid and users can be stable under certain economic conditions. Pure strategy equilibria offer clear guidelines for fostering cooperation, while mixed strategy points highlight the dynamic nature of strategic evolution. To promote long-term cooperation, system designers must ensure that incremental benefits outweigh costs and implement policies to stabilize cooperative strategies. Future research should integrate empirical data to validate these theoretical results and explore practical applications.

3.2. Theoretical Integration of Whole-Process Democracy into the Evolutionary Game Framework

3.2.1. Assumptions

The assumptions including:

(i)

Whole-Process Democracy (WPD): This concept promotes transparency, equality, and continuous feedback between stakeholders, especially in decision-making. It incorporates deliberative processes in which users and grid operators consult each other to adjust strategies dynamically based on mutual feedback.

(ii)

Incorporation of Deliberative Democracy in Payoff Structure: In the evolutionary game, the payoff for each player (users and the grid) can be enhanced by incorporating the idea of negotiation and mutual decision-making, which reflects real-world democratic processes.

3.2.2. Mathematical Framework

To reflect the influence of whole-process democracy in the evolutionary game model, we modify the payoffs and replicator dynamics equations. We will incorporate additional terms that reflect the negotiation, feedback, and decision-making process between users and grid operators.

The modified replicator dynamics can be written as:

$${\displaystyle \frac{\mathrm{d}{x}_i}{\mathrm{d}t}}=x_i\left({\pi }_i(x,y)-\overline{\pi }(x,y)\right)+\delta x_i\left(F_{\mathrm{dem}}(x,y)\right)$$

(42)

where

• x_i represents the proportion of population i (users or grid).
• π_i(x, y) is the payoff function for individual iii given the strategy profile (x, y).
• $\overline{\pi }(x,y)$ is the average payoff in the population.
• δ is a factor that modulates the effect of deliberative processes.
• F_dem(x, y) is a function representing the adjustments in payoffs due to cooperative negotiation and democratic processes.

Here, F_dem(x, y) can be modeled as a dynamic function that incorporates government policies, subsidies, and feedback from the users and grid, which might depend on factors such as:

• Government subsidies (S_g, S_u);
• User participation in decision-making (α_u);
• Feedback mechanisms (R_feedback).

Thus, F_dem(x, y) could be represented by:

$$F_{\mathrm{dem}}(x,y)={\alpha }_{\mathrm{u}}{S}_{\mathrm{u}}+{\beta }_{\mathrm{g}}{S}_{\mathrm{g}}+R_{\mathrm{feedback}}$$

(43)

where

• α_u and β_g are coefficients that adjust the influence of subsidies for the user and grid, respectively.
• R_feedback represents the impact of deliberative feedback between the parties.

The stability of the evolutionary equilibrium in the presence of such deliberative elements would depend on the updated Jacobian matrix for the modified system. The modified Jacobian matrix would reflect these changes, allowing for a new analysis of the stability of the equilibrium under the influence of democracy-based modifications.

3.2.3. Stability Analysis and Equilibrium Derivation

The equilibrium points of the system are derived by setting the replicator dynamics equations to zero, incorporating both the classic payoffs and the modified deliberative adjustments.

Equilibrium Condition:

$${\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}=0\mathrm{and}{\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}}=0$$

(44)

This results in:

$$\left\{\begin{array}{l}{\pi }_i(x^{*},y^{*})-\overline{\pi }(x^{*},y^{*})+\delta F_{\mathrm{dem}}(x^{*},y^{*})=0\\ {\pi }_g(x^{*},y^{*})-\overline{\pi }(x^{*},y^{*})+\delta F_{\mathrm{dem}}(x^{*},y^{*})=0\end{array}\right.$$

(45)

where x* and y* represent the equilibrium strategy fractions for users and grid, respectively.

By solving this system of equations, we obtain the equilibrium points for x* and y* under the influence of deliberative democracy. This equilibrium analysis helps determine the stability of the strategy profile under democratic feedback processes and government intervention.

To analyze the stability of these equilibrium points, we calculate the Jacobian matrix, now incorporating the new feedback term due to deliberative democracy. The Jacobian matrix will take the form:

$$\mathit{J}=\left[\begin{array}{cc}{\displaystyle \frac{\partial {\pi }_u}{\partial x}}& {\displaystyle \frac{\partial {\pi }_u}{\partial y}}\\ {\displaystyle \frac{\partial {\pi }_g}{\partial x}}& {\displaystyle \frac{\partial {\pi }_g}{\partial y}}\end{array}\right]$$

(46)

where the derivatives now also reflect the influence of F_dem(x, y).

Overall, through the introduction of deliberative democracy into the evolutionary game model, the dynamics between the users and grid operators become more reflective of real-world collaborative decision-making. The integration of feedback and democratic principles ensures that the system’s equilibrium is not only stable but also sustainable in a way that promotes equitable cooperation. By adding the feedback term and adjusting the Jacobian matrix accordingly, we can capture the long-term effects of such cooperative strategies, enabling a more realistic simulation of user–grid cooperation in renewable energy systems. This methodology can guide policymakers in creating more inclusive, transparent, and sustainable energy policies, aligning both the users’ and the grid’s strategic interests.

4. Evolutionary Stability Analysis of User-Power Grid Participation in Peak Shaving

4.1. Scenario 1: Simulation Analysis

In this scenario, when both the user group and the power grid adopt cooperative strategies, profits are redistributed through benefit allocation mechanisms, resulting in considerable gains for both parties. At the same time, the initial cost of participating in the peak shaving project is relatively low for both parties.

From the initial revenue functions:

$$\left\{\begin{array}{l}{\pi }_{\mathrm{g}}+(1-\alpha )\Delta S-l_{\mathrm{g}}\Delta Q>{\pi }_{\mathrm{g}},\hfill \\ {\pi }_{\mathrm{g}}+{\gamma }_{\mathrm{g}}\Delta Q-l_{\mathrm{g}}\Delta Q>{\pi }_{\mathrm{g}},\hfill \\ {\pi }_{\mathrm{f}}+\alpha \Delta Q-l_{\mathrm{f}}\Delta Q>{\pi }_{\mathrm{f}},\hfill \\ {\pi }_{\mathrm{f}}+{\gamma }_{\mathrm{f}}\Delta Q-l_{\mathrm{f}}\Delta Q>{\pi }_{\mathrm{f}}.\hfill \end{array}\right.$$

(47)

It follows that the conditions for cooperative participation are met only when the benefit coefficients of both the power grid (γ_g) and users (γ_f) exceed their respective cost coefficients (l_g, l_f). Furthermore, for stable cooperation, the allocation ratio α must satisfy:

$${\displaystyle \frac{l_{\mathrm{f}}}{2p+{\gamma }_{\mathrm{g}}+{\gamma }_{\mathrm{f}}-l_{\mathrm{g}}-l_{\mathrm{f}}}}<\alpha <{\displaystyle \frac{2p+{\gamma }_{\mathrm{g}}+{\gamma }_{\mathrm{f}}-2l_{\mathrm{g}}-l_{\mathrm{f}}}{2p+{\gamma }_{\mathrm{g}}+{\gamma }_{\mathrm{f}}-l_{\mathrm{g}}-l_{\mathrm{f}}}}.$$

(48)

The evolutionary stability analysis indicates that both parties can achieve considerable incremental profits from participating in renewable energy development and peak shaving projects. Specifically, the redistributed profits ensure higher net gains for both parties compared to the costs incurred. As a result, both parties tend to adopt the cooperative strategy as the stable evolutionary outcome. Numerical simulations further confirm that the final evolutionary outcome leads to all organizations adopting cooperative strategies under this scenario. The parameter assumptions for the simulations are shown as follows: π_g = 1000, π_f = 1200, α = 0.5, p = 10, l_g = 500, l_f = 200, γ_f = 1000, γ_g = 800, ΔQ = 20, Q = 10, ΔS_g = 6100, ΔS_f = 16,100, and ΔS = 22,200 (unit: yuan). Based on this, the key expressions and calculations are conducted as follows.

The implementation of user participation in renewable energy generation and grid-connected peak shaving projects results in distributed total excess profits. The calculations for the benefits and costs for both users and the power grid are as follows: (1) User group’s share of total excess profit after distribution, a; the power grid’s share of total excess profit after distribution, b; the incremental profits from renewable energy supply when users choose a cooperative strategy, c; the incremental costs for the power grid due to renewable energy generation and consumption, d; the incremental costs for users due to renewable energy consumption, e; and the incremental profits from renewable energy supply when the power grid chooses a cooperative strategy, f.

$$.\left\{\begin{array}{l}a=(1-\alpha )\Delta S=(1-0.5)\times 22200=11100\\ b=\alpha \Delta S=0.5\times 22200=11100\\ c={\gamma }_{\mathrm{g}}\Delta Q=800\times 20=16000\\ d=h=l_{\mathrm{f}}\Delta Q=200\times 20=4000\\ e=g=l_{\mathrm{g}}\Delta Q=500\times 20=10000\\ f={\gamma }_{\mathrm{f}}\Delta Q=1000\times 20=20000\end{array}\right.$$

(49)

These calculated values were applied to the simulation, with intervals of 1/q for step sizes, where q = 10, 20, 30, 40, 50, and 60, corresponding to Case 1 to Case 6, respectively. The values of x and y were varied from 0 to 1. The simulation results are shown in Figure 7.

From the simulation results in Figure 7, it can be observed that during the initial stages of the evolutionary game, both the user group and the power grid exhibit low frequencies of cooperation. However, as the game progresses, the frequency of cooperation steadily increases for both parties. In the later stages of the game, the frequency of cooperation approaches 100%, indicating a stable evolutionary equilibrium where both parties adopt cooperative strategies. Based on this, the analysis and discussion on Figure 7 are conducted as follows.

• Physical/Engineering Implications: The equilibrium points in this evolutionary game represent the balance between the incremental profits and costs associated with adopting cooperative strategies for renewable energy and peak shaving projects. In engineering applications, this translates to optimal resource allocation: Both the user group and the power grid can maximize their respective payoffs by aligning their strategies with the conditions for evolutionary stability.
• System Stability: The final cooperative equilibrium ensures a sustainable and efficient distribution of energy resources, reducing costs and increasing system efficiency.
• Scalability: The model can be extended to larger-scale systems with multiple user groups and power grids, highlighting its practical applicability in energy management.
• Factors Influencing the Equilibrium Point: (i) Cost Coefficients. Lower l_g and l_f values reduce the barrier for cooperation, encouraging more frequent cooperative behaviors. (ii) Benefit Coefficients. Higher γ_g and γ_f values increase the attractiveness of cooperative strategies, accelerating the convergence to the stable equilibrium. (iii) Allocation Ratio (α): A balanced allocation ratio ensures fair distribution of incremental profits, maintaining trust and cooperation between the parties.

Overall, the simulation results in Figure 7 confirm that cooperative strategies lead to stable evolutionary outcomes under favorable conditions. By ensuring that the incremental profits outweigh the associated costs, both the power grid and users can achieve long-term stability and mutual benefits. The findings highlight the importance of designing incentive mechanisms and optimizing cost–benefit structures to promote cooperative behavior in energy systems. This framework can serve as a reference for practical applications in renewable energy development and energy management.

4.2. Scenario 2: Simulation Analysis

In this scenario, when both the user group and the power grid adopt cooperative strategies, the redistributed profits can still generate benefits. However, the initial costs of participating in peak shaving projects for both users and the power grid are relatively high. The conditions for participation are expressed as follows:

$$\left\{\begin{array}{l}{\pi }_{\mathrm{g}}+(1-\alpha )\Delta S-l_{\mathrm{g}}\Delta Q>{\pi }_{\mathrm{g}},\hfill \\ {\pi }_{\mathrm{g}}+{\gamma }_{\mathrm{g}}\Delta Q-l_{\mathrm{g}}\Delta Q<{\pi }_{\mathrm{g}},\hfill \\ {\pi }_{\mathrm{f}}+\alpha \Delta Q-l_{\mathrm{f}}\Delta Q>{\pi }_{\mathrm{f}},\hfill \\ {\pi }_{\mathrm{f}}+{\gamma }_{\mathrm{f}}\Delta Q-l_{\mathrm{f}}\Delta Q<{\pi }_{\mathrm{f}}.\hfill \end{array}\right.$$

(50)

For cooperation to be feasible, the following conditions must be satisfied:

$$\left\{\begin{array}{l}{\gamma }_{\mathrm{g}}<l_{\mathrm{g}},{\gamma }_{\mathrm{f}}<l_{\mathrm{f}},\\ l_{\mathrm{f}}<\alpha <{\displaystyle \frac{2p+{\gamma }_{\mathrm{g}}+{\gamma }_{\mathrm{f}}-2l_{\mathrm{g}}-l_{\mathrm{f}}}{2p+{\gamma }_{\mathrm{g}}+{\gamma }_{\mathrm{f}}-l_{\mathrm{g}}-l_{\mathrm{f}}}}.\end{array}\right.$$

(51)

From the evolutionary stability conditions, it can be inferred that under high initial costs, both users and the power grid can still achieve satisfactory incremental profits from participating in renewable energy development and grid-connected peak shaving projects. However, due to the high costs associated with participation, both parties may simultaneously choose cooperative and non-cooperative strategies. If the profits generated by cooperation are relatively low, the likelihood of adopting non-cooperative strategies increases. Based on the theoretical analysis and the replicator dynamic equations, the parameters for the simulation are set as follows: π_g = 1000, π_f = 1200, α = 0.6, p = 100, l_g = 35, l_f = 20, γ_f = 15, γ_g = 30, ΔQ = 20, Q = 10, ΔS_g = 700, ΔS_f = 900, and ΔS = 1600 (unit: yuan). Based on this, the key expressions and calculations are conducted as follows. The calculations for the benefits and costs for both users and the power grid are:

$$.\left\{\begin{array}{l}a=(1-\alpha )\Delta S=(1-0.6)\times 1600=640\\ b=\alpha \Delta S=0.5\times 1600=960\\ c={\gamma }_{\mathrm{g}}\Delta Q=30\times 20=600\\ d=h=l_{\mathrm{f}}\Delta Q=20\times 20=400\\ e=g=l_{\mathrm{g}}\Delta Q=35\times 20=700\\ f={\gamma }_{\mathrm{f}}\Delta Q=15\times 20=300\end{array}\right.$$

(52)

These calculated values were applied to the simulation, with intervals of 1/q for step sizes, where q = 10, 20, 30, 40, 50, and 60, corresponding to Case 1 to Case 6, respectively. The values of x and y were varied from 0 to 1. The simulation results are shown in Figure 8.

From the simulation results in Figure 8, it can be observed that during the early stages of the evolutionary game, the high initial costs for both users and the power grid lead to a preference for non-cooperative strategies. In the later stages of the game, even when cooperation becomes beneficial, the system struggles to transition to cooperative strategies due to the inertia of the initial non-cooperative state. The final evolutionary stable strategy for both parties is non-cooperation, as indicated by the simulation results.

Physical/Engineering Implications: The equilibrium points in this evolutionary game represent the trade-offs between the benefits and costs of adopting cooperative strategies for renewable energy development and peak shaving projects. In engineering applications, this scenario highlights:

(i)

Cost barriers to cooperation: High initial costs can discourage both users and the power grid from adopting cooperative strategies, even when long-term benefits are achievable.

(ii)

Policy implications: To promote cooperation, it is essential to reduce the initial cost barriers or provide subsidies and incentives to offset the costs of participation.

(iii)

Dynamic systems behavior: The simulation results emphasize the importance of initial conditions in determining the evolutionary trajectory of the system. Systems with high initial costs are more likely to stabilize at non-cooperative equilibria.

Factors influencing the equilibrium point are summarized as follows:

(i)

Cost Coefficients: Higher l_g and l_f values discourage cooperation by increasing the relative cost of participation.

(ii)

Benefit Coefficients: Lower γ_g and γ_f values reduce the attractiveness of cooperative strategies, reinforcing non-cooperative behavior.

(iii)

Allocation Ratio (α): The distribution of incremental profits plays a critical role in determining the likelihood of cooperation. An inequitable allocation ratio can exacerbate non-cooperative tendencies.

Overall, the simulation results in Figure 8 demonstrate that high initial costs can act as a significant barrier to cooperative behavior in energy systems. The findings suggest that targeted policy interventions, such as cost reduction measures or benefit redistribution mechanisms, are necessary to transition systems from non-cooperative to cooperative equilibria. These insights are critical for designing strategies to achieve sustainable and efficient energy management in real-world engineering applications.

Further, an additional parameter setting for Scenario 2 is provided, as shown as follows: π_g = 1000, π_f = 1200, α = 0.55, p = 100, l_g = 35, l_f = 30, γ_f = 24, γ_g = 30, ΔQ = 20, Q = 10, ΔS_g = 900, ΔS_f = 880, and ΔS = 1780 (unit: yuan). Based on this, the key expressions and calculations are conducted as follows. The calculations for the benefits and costs for both users and the power grid are:

$$.\left\{\begin{array}{l}a=(1-\alpha )\Delta S=(1-0.55)\times 1780=801\\ b=\alpha \Delta S=0.55\times 1780=979\\ c={\gamma }_{\mathrm{g}}\Delta Q=30\times 20=600\\ d=h=l_{\mathrm{f}}\Delta Q=30\times 20=600\\ e=g=l_{\mathrm{g}}\Delta Q=35\times 20=700\\ f={\gamma }_{\mathrm{f}}\Delta Q=24\times 20=480\end{array}\right.$$

(53)

These calculated values were applied to the simulation, with intervals of 1/q for step sizes, where q = 10, 20, 30, 40, 50, and 60, corresponding to Case 1 to Case 6, respectively. The values of x and y were varied from 0 to 1. The simulation results are illustrated in Figure 9. The results reveal that both the user group and the power grid adopt mixed strategies, choosing between cooperative and non-cooperative strategies during the game; due to high initial investment costs, users and the power grid are hesitant to fully commit to cooperation; although participation in renewable energy and grid-connected peak shaving projects yields satisfactory profits, the relatively high costs cause both parties to seek an optimal balance between cooperation and non-cooperation; and the probability of adopting cooperative strategies by both the user group and the power grid is relatively low compared to Scenario 1. Based on this, further analysis and discussion are conducted as follows.

Physical/Engineering Implications: In the context of this scenario, the evolutionary game provides critical insights into the decision-making process of the user group and the power grid under high-cost conditions. The equilibrium points in the game represent the dynamic balance between the incremental profits and costs associated with cooperative strategies. The engineering implications are as follows.

• High-Cost Barrier to Cooperation: High initial costs (l_g, l_f) discourage both users and the power grid from adopting cooperative strategies. This is especially evident in Case 1 and Case 2 of Figure 9, where the trajectories show significant tendencies toward non-cooperation in the early stages of the game.
• System Optimization through Mixed Strategies: The results indicate that under high-cost scenarios, mixed strategies are more prevalent. Both users and the power grid alternate between cooperation and non-cooperation to maximize their respective payoffs. This dynamic behavior underscores the importance of designing flexible operational strategies in energy systems.
• Profit Distribution and Stability: The allocation ratio (α) plays a crucial role in determining the likelihood of cooperation. A higher α value favors the power grid, potentially discouraging user cooperation. Conversely, a lower α may incentivize users but reduce the grid’s motivation.

The results from Scenario 2 highlight the impact of high initial costs on the strategic decisions of users and the power grid in renewable energy and grid-connected peak shaving projects. The findings demonstrate that high costs lead to mixed strategies, with both parties alternating between cooperation and non-cooperation; the equilibrium points reflect a delicate balance between incremental profits and costs, making the system more sensitive to parameter variations; and policy interventions, such as subsidies or cost-sharing mechanisms, are essential to encourage full cooperation and achieve stable, long-term system optimization. Based on this, key factors affecting equilibrium points are summarized as follows.

(i)

Cost Parameters (l_g, l_f): Higher costs reduce the attractiveness of cooperative strategies, especially for users.

(ii)

Benefit Parameters (γ_g, γ_f): Lower incremental profits from renewable energy supply discourage cooperative behavior.

(iii)

Allocation Ratio (α): Equitable distribution of profits is critical for maintaining trust and fostering cooperation.

Based on above, practical recommendations are elaborated as follows.

(i)

Subsidies and incentives: Implementing financial incentives can lower the barriers to cooperation and ensure system stability.

(ii)

Dynamic pricing models: Introducing flexible pricing schemes based on real-time energy consumption can enhance user participation.

(iii)

Integrated system design: Engineering solutions should focus on minimizing initial costs and optimizing profit distribution to encourage sustainable cooperation.

Overall, the analysis underscores the need for a holistic approach to energy management, balancing economic, social, and environmental factors to achieve optimal outcomes in renewable energy systems.

4.3. Scenario 3: Simulation Analysis

In this scenario, when both the user group and the power grid adopt cooperative strategies, the redistributed profits fail to generate sufficient benefits for either party. Additionally, the initial costs of participating in peak shaving projects are relatively high for both users and the power grid. The conditions for participation are described as follows:

$$\left\{\begin{array}{l}{\pi }_{\mathrm{g}}+(1-\alpha )\Delta S-l_{\mathrm{g}}\Delta Q<{\pi }_{\mathrm{g}},\hfill \\ {\pi }_{\mathrm{g}}+{\gamma }_{\mathrm{g}}\Delta Q-l_{\mathrm{g}}\Delta Q<{\pi }_{\mathrm{g}},\hfill \\ {\pi }_{\mathrm{f}}+\alpha \Delta Q-l_{\mathrm{f}}\Delta Q<{\pi }_{\mathrm{f}},\hfill \\ {\pi }_{\mathrm{f}}+{\gamma }_{\mathrm{f}}\Delta Q-l_{\mathrm{f}}\Delta Q<{\pi }_{\mathrm{f}}.\hfill \end{array}\right.$$

(54)

For cooperation to be feasible, the following conditions must be satisfied:

$$\left\{\begin{array}{l}{\gamma }_{\mathrm{g}}<l_{\mathrm{g}},{\gamma }_{\mathrm{f}}<l_{\mathrm{f}},\\ {\displaystyle \frac{2p+{\gamma }_{\mathrm{g}}+{\gamma }_{\mathrm{f}}-2l_{\mathrm{g}}-l_{\mathrm{f}}}{2p+{\gamma }_{\mathrm{g}}+{\gamma }_{\mathrm{f}}-l_{\mathrm{g}}-l_{\mathrm{f}}}}<\alpha <{\displaystyle \frac{l_{\mathrm{f}}}{2p+{\gamma }_{\mathrm{g}}+{\gamma }_{\mathrm{f}}-l_{\mathrm{g}}-l_{\mathrm{f}}}},\\ 2p+{\gamma }_{\mathrm{g}}+{\gamma }_{\mathrm{f}}<2(l_{\mathrm{g}}+l_{\mathrm{f}}).\end{array}\right.$$

(55)

The analysis suggests that when both users and the power grid adopt cooperative strategies, the resulting profits, regardless of how they are distributed, fail to compensate for the associated costs. Furthermore, the costs related to renewable energy generation and consumption are significantly high. Consequently, the evolutionary stable strategy for both parties ultimately involves the rejection of cooperative strategies. Based on the theoretical analysis and the replicator dynamic equations, the parameters for the simulation are set as follows: π_g = 1000, π_f = 1200, α = 0.55, p = 20, l_g = 35, l_f = 30, γ_f = 29, γ_g = 34, ΔQ = 20, Q = 10, ΔS_g = 180, ΔS_f = 180, and ΔS = 360 (unit: yuan). Based on this, the key expressions and calculations are conducted as follows. The calculations for the benefits and costs for both users and the power grid are:

$$.\left\{\begin{array}{l}a=(1-\alpha )\Delta S=(1-0.55)\times 360=198\\ b=\alpha \Delta S=0.55\times 360=162\\ c={\gamma }_{\mathrm{g}}\Delta Q=34\times 20=680\\ d=h=l_{\mathrm{f}}\Delta Q=30\times 20=600\\ e=g=l_{\mathrm{g}}\Delta Q=35\times 20=700\\ f={\gamma }_{\mathrm{f}}\Delta Q=29\times 20=580\end{array}\right.$$

(56)

These calculated values were applied to the simulation, with intervals of 1/q for step sizes, where q = 10, 20, 30, 40, 50, and 60, corresponding to Case 1 to Case 6, respectively. The values of x and y were varied from 0 to 1. The simulation results are shown in Figure 10. The results in Figure 10 reveal that due to the high initial investment costs, both the user group and the power grid show a clear preference for non-cooperative strategies during the evolutionary game; even though renewable energy development and grid-connected peak shaving projects yield incremental profits, the costs of participation are so high that both parties find cooperation to be economically unviable; and the final evolutionary stable strategy for both the user group and the power grid is the mutual rejection of cooperative strategies. Based on this, the analysis and discussion are conducted as follows.

Physical/Engineering Implications: This scenario underscores the challenges associated with achieving cooperative behavior in high-cost environments. The equilibrium points in this evolutionary game provide valuable insights into the dynamics of cooperation and its feasibility in renewable energy projects. Key implications include:

(i)

Economic Feasibility of Cooperation: When the incremental costs (l_g, l_f) outweigh the incremental profits (γ_g, γ_f), cooperation becomes economically unsustainable. This is evident in the trajectories of all six cases in Figure 10, where the system converges to non-cooperative equilibria.

(ii)

Barriers to Cooperation: The high costs associated with renewable energy generation and consumption discourage both users and the power grid from engaging in cooperative strategies. These barriers are further exacerbated by the inequitable distribution of profits, as evidenced by the low values of a and b.

(iii)

Role of Initial Conditions: The high initial investment costs create inertia that locks both parties into non-cooperative behaviors, even when marginal gains from cooperation are achievable. This highlights the importance of addressing cost barriers to enable cooperation.

The results from Scenario 3 demonstrate that high-cost environments significantly hinder cooperative behavior in energy systems. The findings emphasize the following key points:

(i)

Economic Challenges: The lack of economic feasibility due to high costs and low profits forces both the user group and the power grid to reject cooperative strategies. This highlights the need for targeted interventions to reduce costs and enhance the economic appeal of cooperation.

(ii)

Policy Implications: Policy measures such as subsidies, cost-sharing mechanisms, and tax incentives are essential to overcome the barriers to cooperation. These measures can help offset the high initial costs and make cooperative strategies more attractive.

(iii)

Engineering Design: From an engineering perspective, the focus should be on reducing the costs of renewable energy generation and improving the efficiency of grid-connected peak shaving projects. This can be achieved through technological innovation and optimized system design.

(iv)

Sensitivity to Profit Distribution: The allocation ratio (α) plays a critical role in determining the outcomes of the evolutionary game. An equitable profit distribution mechanism can foster trust and encourage cooperation between users and the power grid.

In conclusion, Scenario 3 illustrates the limitations of cooperative strategies in high-cost settings and provides a foundation for designing interventions to promote sustainable and efficient energy management. Addressing the cost–profit imbalance is crucial for achieving long-term cooperation and optimizing system performance.

4.4. Scenario 4: Simulation Analysis

In this scenario, both the user group and the power grid adopt cooperative strategies, benefiting from redistributed profits. However, the profits gained by the power grid for participating in peak shaving projects are relatively low compared to the investment costs. The conditions for cooperative participation are described as follows:

$$\left\{\begin{array}{l}{\pi }_{\mathrm{g}}+(1-\alpha )\Delta S-l_{\mathrm{g}}\Delta Q>{\pi }_{\mathrm{g}},\hfill \\ {\pi }_{\mathrm{g}}+{\gamma }_{\mathrm{g}}\Delta Q-l_{\mathrm{g}}\Delta Q>{\pi }_{\mathrm{g}},\hfill \\ {\pi }_{\mathrm{f}}+\alpha \Delta Q-l_{\mathrm{f}}\Delta Q>{\pi }_{\mathrm{f}},\hfill \\ {\pi }_{\mathrm{f}}+{\gamma }_{\mathrm{f}}\Delta Q-l_{\mathrm{f}}\Delta Q<{\pi }_{\mathrm{f}}.\hfill \end{array}\right.$$

(57)

The conditions for stability and participation are as follows:

$$\left\{\begin{array}{l}{\gamma }_{\mathrm{g}}>l_{\mathrm{g}},{\gamma }_{\mathrm{f}}<l_{\mathrm{f}},\\ {\displaystyle \frac{l_{\mathrm{f}}}{2p+{\gamma }_{\mathrm{g}}+{\gamma }_{\mathrm{f}}-l_{\mathrm{g}}-l_{\mathrm{f}}}}<\alpha <{\displaystyle \frac{2p+{\gamma }_{\mathrm{g}}+{\gamma }_{\mathrm{f}}-2l_{\mathrm{g}}-l_{\mathrm{f}}}{2p+{\gamma }_{\mathrm{g}}+{\gamma }_{\mathrm{f}}-l_{\mathrm{g}}-l_{\mathrm{f}}}}.\end{array}\right.$$

(58)

Based on the theoretical analysis and the replicator dynamic equations, the parameters for the simulation are set as follows: π_g = 1000, π_f = 1200, α = 0.3, p = 100, l_g = 30, l_f = 28, γ_f = 24, γ_g = 34, ΔQ = 20, Q = 10, ΔS_g = 980, ΔS_f = 980, and ΔS = 1960 (unit: yuan). Based on this, the key expressions and calculations are conducted as follows. The calculations for the benefits and costs for both users and the power grid are:

$$.\left\{\begin{array}{l}a=(1-\alpha )\Delta S=(1-0.3)\times 1960=1372\\ b=\alpha \Delta S=0.3\times 1960=588\\ c={\gamma }_{\mathrm{g}}\Delta Q=34\times 20=680\\ d=h=l_{\mathrm{f}}\Delta Q=28\times 20=560\\ e=g=l_{\mathrm{g}}\Delta Q=30\times 20=600\\ f={\gamma }_{\mathrm{f}}\Delta Q=24\times 20=480\end{array}\right.$$

(59)

These calculated values were applied to the simulation, with intervals of 1/q for step sizes, where q = 10, 20, 30, 40, 50, and 60, corresponding to Case 1 to Case 6, respectively. The values of x and y were varied from 0 to 1. The simulation results are shown in Figure 11. The results in Figure 11 reveal that due to high investment costs, both the user group and the power grid show significant hesitation in committing to cooperative strategies; While the redistribution of profits benefits the user group, the power grid’s dissatisfaction with its share of the incremental profits leads to a greater probability of rejecting cooperation; The final evolutionary stable strategy for both parties is predominantly non-cooperative, as shown across all cases in the figure. Based on this, the analysis and discussion on Figure 11 are as follows.

The results from Scenario 4 reflect the dynamics of cooperation in renewable energy and grid-connected peak shaving projects under specific economic conditions. The equilibrium points in the evolutionary game provide insights into the trade-offs faced by the user group and the power grid. Physical/engineering implications are as follows.

(i)

Profit Distribution and Incentives: While users gain a significant share of the redistributed profits, the power grid’s lower share discourages cooperation. This imbalance highlights the importance of designing equitable distribution mechanisms to incentivize all participants.

(ii)

High-Cost Barrier: The high initial costs associated with renewable energy generation and consumption impose significant constraints on cooperative behavior. For the power grid, the incremental profits are insufficient to justify the associated costs, leading to a preference for non-cooperative strategies.

(iii)

System Optimization: The simulation underscores the need for optimization in system design to reduce costs and improve efficiency. Technological innovations and policy interventions, such as subsidies or tax incentives, could help offset the high costs and encourage cooperation.

The findings from Scenario 4 demonstrate that achieving cooperation in renewable energy projects is contingent upon equitable profit distribution and cost reduction measures. The results emphasize the following key points:

Economic Feasibility: The power grid’s dissatisfaction with its share of profits hinders cooperation, highlighting the need for mechanisms that ensure fair and balanced profit distribution.

(i)

Policy Implications: To promote cooperation, policymakers should implement measures such as subsidies, cost-sharing mechanisms, or differential pricing to reduce the economic burden on participants.

(ii)

Engineering Solutions: From an engineering perspective, reducing the costs of renewable energy generation and grid-connected peak shaving projects through improved technology and system design is essential for fostering cooperation.

(iii)

Sensitivity to Cost–Profit Ratios: The equilibrium points are highly sensitive to the ratios of costs to profits. Achieving stability requires careful calibration of these parameters to ensure mutual benefits for all parties.

In conclusion, Scenario 4 highlights the challenges of fostering cooperation in high-cost environments and underscores the importance of equitable profit distribution and cost reduction strategies. Addressing these challenges is critical for achieving sustainable energy systems and optimizing the performance of renewable energy projects. Based on this, we further set another data set as follows: π_g = 1000, π_f = 1200, α = 0.6, p = 100, l_g = 30, l_f = 28, γ_f = 24, γ_g = 34, ΔQ = 20, Q = 10, ΔS_g = 980, ΔS_f = 980, and ΔS = 1960 (unit: yuan). The key expressions and calculations are conducted as follows. The calculations for the benefits and costs for both users and the power grid are:

$$.\left\{\begin{array}{l}a=(1-\alpha )\Delta S=(1-0.6)\times 1960=784\\ b=\alpha \Delta S=0.6\times 1960=1176\\ c={\gamma }_{\mathrm{g}}\Delta Q=34\times 20=680\\ d=h=l_{\mathrm{f}}\Delta Q=28\times 20=560\\ e=g=l_{\mathrm{g}}\Delta Q=30\times 20=600\\ f={\gamma }_{\mathrm{f}}\Delta Q=24\times 20=480\end{array}\right.$$

(60)

These calculated values were applied to the simulation, with intervals of 1/q for step sizes, where q = 10, 20, 30, 40, 50, and 60, corresponding to Case 1 to Case 6, respectively. The values of x and y were varied from 0 to 1. The simulation results are shown in Figure 12. In this case, an alternative set of parameters is introduced for Scenario 4 to evaluate the evolutionary dynamics under different economic conditions. The results in Figure 12 reveal that the evolutionary game results show that both the user group and the power grid eventually converge on adopting cooperative strategies; despite the power grid incurring higher initial investment costs, the profits gained from cooperative behavior are satisfactory for both parties; and the simulation indicates that cooperation can become stable when both parties find the incremental benefits to outweigh the associated costs.

Aiming at Figure 12, the analysis and discussion are conducted as follows.

Physical/Engineering Implications: The results of this scenario provide meaningful insights into the conditions under which cooperation can be achieved in renewable energy and grid-connected peak shaving projects. The equilibrium points in the evolutionary game demonstrate the dynamic balance between the costs and benefits of cooperation for the user group and the power grid. Some physical/engineering implications are summarized as follows.

(i)

Economic Incentives for Cooperation: The redistribution of profits, particularly the share received by the power grid (b = 1176 yuan), plays a crucial role in incentivizing cooperative strategies.

(ii)

The incremental profits (c, f) for both the user group and the power grid are sufficient to offset their respective costs (e, d), creating a favorable environment for cooperation.

(iii)

Stability of Cooperative Behavior: The simulation results reveal that the system evolves toward a stable cooperative equilibrium. This is particularly evident in Cases 4, 5, and 6, where the trajectories clearly show convergence toward full cooperation (x = 1, y = 1). The stability of cooperation depends on ensuring that both parties perceive mutual benefits. This is achieved through equitable profit distribution and manageable cost structures.

(iv)

Cost–Profit Dynamics: The high profits gained from renewable energy generation and consumption make cooperation attractive, even when initial investment costs are significant. The relative magnitudes of γ_g, γ_f, l_g, and l_f determine the feasibility of cooperation. In this scenario, the benefits (γ_g, γ_f) exceed the costs (l_g, l_f), tipping the balance toward cooperative strategies.

The findings from Scenario 4 (under new parameter settings) illustrate the conditions under which cooperation between the user group and the power grid can be achieved in renewable energy projects. The following key insights emerge from the analysis:

(i)

Economic Feasibility: The simulation demonstrates that cooperative strategies are economically viable when the incremental profits outweigh the associated costs. This is critical for fostering long-term cooperation.

(ii)

Importance of Profit Distribution: Equitable distribution of profits is essential for ensuring that both parties are motivated to cooperate. The allocation ratio (α = 0.6) strikes a balance between the interests of the user group and the power grid.

(iii)

Policy Recommendations: Policy interventions, such as subsidies or tax incentives, can further enhance the attractiveness of cooperative strategies by reducing the financial burden on participants.

(iv)

Engineering Design: From an engineering perspective, optimizing the efficiency of renewable energy systems and reducing investment costs are critical for achieving sustainable cooperation. This can be achieved through technological innovation and improved system design.

Based on above, practical implications are analyzed as follows. The results highlight the potential for cooperation in renewable energy projects, provided that economic and engineering challenges are addressed. By aligning the interests of the user group and the power grid, policymakers and engineers can create systems that promote sustainability and maximize social welfare. In conclusion, Scenario 4 underscores the importance of balancing costs and benefits to achieve cooperative behavior in energy systems. The insights gained from this analysis provide a roadmap for designing strategies that foster collaboration and drive the transition toward renewable energy.

4.5. Scenario 5: Simulation Analysis

This scenario examines the situation where the price of renewable energy cannot compete with conventional energy prices, or the cost of renewable energy becomes a limiting factor in competition. This context highlights the inability to achieve mutual benefits for both users and the power grid, regardless of profit redistribution. The governing conditions are as follows:

$$\left\{\begin{array}{l}{\pi }_{\mathrm{g}}+(1-\alpha )\Delta S-l_{\mathrm{g}}\Delta Q<{\pi }_{\mathrm{g}},\hfill \\ {\pi }_{\mathrm{g}}+{\gamma }_{\mathrm{g}}\Delta Q-l_{\mathrm{g}}\Delta Q<{\pi }_{\mathrm{g}},\hfill \\ {\pi }_{\mathrm{f}}+\alpha \Delta Q-l_{\mathrm{f}}\Delta Q<{\pi }_{\mathrm{f}},\hfill \\ {\pi }_{\mathrm{f}}+{\gamma }_{\mathrm{f}}\Delta Q-l_{\mathrm{f}}\Delta Q<{\pi }_{\mathrm{f}}.\hfill \end{array}\right.$$

(61)

For cooperation to be feasible, the following conditions must be satisfied:

$$\left\{\begin{array}{c}\alpha <{\displaystyle \frac{l_{\mathrm{f}}}{2p+{\gamma }_{\mathrm{g}}+{\gamma }_{\mathrm{f}}-l_{\mathrm{g}}-l_{\mathrm{f}}}},\alpha <{\displaystyle \frac{2p+{\gamma }_{\mathrm{g}}+{\gamma }_{\mathrm{f}}-2l_{\mathrm{g}}-l_{\mathrm{f}}}{2p+{\gamma }_{\mathrm{g}}+{\gamma }_{\mathrm{f}}-l_{\mathrm{g}}-l_{\mathrm{f}}}},\\ {\gamma }_{\mathrm{g}}<l_{\mathrm{g}},{\gamma }_{\mathrm{f}}<l_{\mathrm{f}}.\end{array}\right.$$

(62)

From the evolutionary stability analysis, it is evident that the inability of renewable energy prices to compete with conventional energy prices, coupled with unfavorable profit redistribution, leads to non-cooperative strategies for both parties. The parameter settings for this simulation are shown as follows: π_g = 1000, π_f = 1200, α = 0.55, p = 25, l_g = 35, l_f = 29, γ_f = 24, γ_g = 30, ΔQ = 20, Q = 10, ΔS_g = 150, ΔS_f = 150, and ΔS = 300 (unit: yuan). Based on this, the key expressions and calculations are conducted as follows. The calculations for the benefits and costs for both users and the power grid are:

$$.\left\{\begin{array}{l}a=(1-\alpha )\Delta S=(1-0.55)\times 300=135\\ b=\alpha \Delta S=0.55\times 300=165\\ c={\gamma }_{\mathrm{g}}\Delta Q=30\times 20=600\\ d=h=l_{\mathrm{f}}\Delta Q=29\times 20=580\\ e=g=l_{\mathrm{g}}\Delta Q=35\times 20=700\\ f={\gamma }_{\mathrm{f}}\Delta Q=24\times 20=480\end{array}\right.$$

(63)

These calculated values were applied to the simulation, with intervals of 1/q for step sizes, where q = 10, 20, 30, 40, 50, and 60, corresponding to Case 1 to Case 6, respectively. The values of x and y were varied from 0 to 1. The simulation results are shown in Figure 13. The simulation results indicate that when renewable energy prices fail to compete with conventional energy prices, or if they are comparable but do not offer sufficient advantages, both the user group and the power grid ultimately adopt non-cooperative strategies; Across all six cases, the trajectories converge toward non-cooperation, as evidenced by the absence of stable cooperative equilibria; Regardless of how profits are redistributed, the incremental benefits do not outweigh the associated costs for either party, leading to a breakdown in cooperation. Based on this, aiming at this scenario, the analysis and discussion are conducted as follows.

This scenario highlights the significant challenges of integrating renewable energy into power systems under unfavorable economic conditions. The equilibrium points derived from the evolutionary game underscore the critical factors affecting cooperation between users and the power grid. Therefore, some physical/engineering implications are summarized as follows.

(i)

Economic Viability of Cooperation: The results clearly demonstrate that cooperation becomes unsustainable when the price of renewable energy fails to provide a competitive advantage over conventional energy. Both users and the power grid experience diminishing returns from cooperative strategies, with profits (c, f) failing to offset costs (d, e).

(ii)

Profit Distribution and Incentive Misalignment: The allocation ratio (α = 0.55) provides limited relief to either party. The user group’s share (a = 135) and the power grid’s share (b = 165) are insufficient to justify cooperation. These findings suggest that profit redistribution alone is not enough to incentivize cooperative behavior in high-cost scenarios.

(iii)

Barriers to Adoption of Renewable Energy: The high costs associated with renewable energy generation and consumption remain a significant barrier. Parameters such as l_g, l_f, and γ_g play a pivotal role in determining the feasibility of cooperation.

(iv)

Implications for System Design: From a system design perspective, reducing the costs of renewable energy technologies is critical. This includes innovations in generation efficiency, storage solutions, and grid integration. Policy measures, such as subsidies or carbon pricing, may also be necessary to level the playing field between renewable and conventional energy sources.

The findings from Scenario 5 provide a stark illustration of the economic challenges facing renewable energy projects in competitive markets. Key insights include:

(i)

Economic Constraints: Cooperation between users and the power grid is unattainable when renewable energy prices are uncompetitive. This highlights the need for targeted interventions to improve cost efficiency and profitability.

(ii)

Role of Profit Redistribution: While profit redistribution can mitigate some economic disparities, it is not sufficient to overcome the high costs associated with renewable energy projects. This underscores the importance of addressing cost structures directly.

(iii)

Policy and Engineering Solutions: Policymakers must implement strategies such as subsidies, tax incentives, and carbon pricing to reduce the economic burden on stakeholders. Engineers must focus on advancing technologies that lower the costs of renewable energy generation and grid integration.

(iv)

System Optimization: Optimizing the allocation of resources and improving the operational efficiency of renewable energy systems are critical for achieving long-term sustainability and cooperation.

(v)

Practical Implications: The results emphasize the importance of aligning economic incentives with environmental objectives. By addressing the cost barriers and ensuring equitable profit distribution, renewable energy projects can achieve greater acceptance and integration into the energy market.

In conclusion, Scenario 5 underscores the limitations of cooperative strategies in unfavorable economic conditions and provides a roadmap for overcoming these challenges through policy interventions and engineering innovations. These findings contribute to the broader understanding of energy system dynamics and the transition toward sustainable power systems.

Section 4.1, Section 4.2, Section 4.3 and Section 4.4 employ stability conditions derived from the evolutionary game model to simulate and analyze six distinct scenarios. The data used in these simulations were informed by hypothetical assumptions referencing actual energy market parameters. By constructing an evolutionary game model, we examined each scenario to explore its dynamic outcomes and generate practical policy recommendations. The findings of this study not only elucidate the dynamics of user–grid interactions under varying economic and strategic conditions but also provide valuable insights into fostering collaboration and improving renewable energy adoption. Thus, key findings and summarized insights in this section are elaborated as follows.

Scenario 1: Favorable Cooperation with Balanced Profit Distribution

When both the user group and the power grid stand to achieve significant profits through cooperation, and the distribution of benefits is perceived as equitable, the evolutionary game demonstrates that both parties converge toward adopting cooperative strategies. This scenario highlights the importance of mutual incentives and balanced profit-sharing mechanisms in fostering long-term collaboration.

Scenario 2: High Costs but Profitable Outcomes

In cases where the costs of participating in renewable energy generation are relatively high but the potential profits remain attractive, cooperation can still be achieved. However, achieving this cooperative equilibrium relies heavily on the rational redistribution of benefits. By ensuring that each party perceives their share as fair and sufficient to offset their costs, cooperation can be stabilized even under challenging economic conditions.

Scenario 3: Unfavorable Investment Returns

When the combined revenues from cooperation fail to surpass the high costs of renewable energy generation and grid integration, both the user group and the power grid are observed to abandon cooperative strategies. This scenario underscores the critical role of cost efficiency in driving cooperative outcomes. Without adequate returns on investment, cooperation becomes unsustainable, leading both parties to adopt non-cooperative strategies.

Scenario 4: Limited Profit Margins with Balanced Redistribution

In scenarios where one party—either the user group or the power grid—experiences returns on investment that are marginal or fall short of expectations, strategic profit redistribution can play a pivotal role. By allocating additional benefits to the less-profitable party, cooperation can be incentivized. This finding emphasizes the potential for economic interventions, such as subsidies or revenue-sharing agreements, to foster collaborative behavior even under constrained conditions.

Scenario 5: Renewable Energy Price Competitiveness

In cases where renewable energy prices are unable to compete with conventional energy prices or can only marginally compete, the results show that cooperation is unattainable. Regardless of how profits are redistributed, both parties ultimately converge toward non-cooperative strategies. This finding highlights the critical need for reducing renewable energy costs and improving its competitiveness within the energy market.

Based on these scenario simulation investigations, the conclusions and policy recommendations are summarized as follows. The five scenarios collectively provide a comprehensive understanding of the dynamics that govern cooperation between the user group and the power grid in renewable energy projects. Several key conclusions can be drawn:

(i)

Importance of Profit Distribution: Equitable profit-sharing mechanisms are crucial for promoting cooperation. Scenarios 2 and 4, in particular, demonstrate that rational redistribution can offset economic disparities and enable cooperative outcomes, even when the initial conditions are not ideal.

(ii)

Role of Cost Reduction: Scenarios 3 and 5 emphasize that high costs are a major barrier to cooperation. Reducing the costs of renewable energy generation and integration is critical for ensuring that the associated profits are sufficient to justify investment and collaboration.

(iii)

Economic and Policy Interventions: To enhance the feasibility of cooperation, policymakers should consider implementing targeted interventions, such as subsidies, tax incentives, and carbon pricing. These measures can help level the playing field for renewable energy and make it more competitive with conventional energy sources.

(iv)

Engineering and Technological Innovations: From an engineering perspective, advancing technologies that improve the efficiency and reduce the costs of renewable energy systems is essential. Innovations in storage solutions, grid infrastructure, and energy management systems can significantly enhance the economic viability of renewable energy projects.

Further, based on the conclusions, some unique perspectives and future outlook are explained as follows. The results of this study provide a nuanced understanding of the dynamics that shape user–grid interactions in renewable energy systems. Beyond the specific scenarios analyzed, the findings offer several broader implications and directions for future research:

(i)

Inspiration for Energy Policy Design: The insights gained from these scenarios highlight the need for policy frameworks that incentivize cooperation while addressing economic disparities. Policymakers can use these findings to design mechanisms that encourage renewable energy adoption and foster collaborative behavior across stakeholders.

(ii)

Broader Applicability of Evolutionary Game Models: The evolutionary game model employed in this study proves to be a robust tool for analyzing complex energy systems. Future research can extend this approach to explore interactions in other contexts, such as regional energy markets, international energy trade, and multi-stakeholder collaborations.

(iii)

Integration of Environmental and Social Factors: While this study focuses on economic factors, future research should consider incorporating environmental and social dimensions. For example, the long-term benefits of reduced carbon emissions and improved energy security could provide additional incentives for cooperation.

(iv)

Exploration of Adaptive Strategies: This study assumes static strategies for the user group and the power grid. Future work could explore dynamic, adaptive strategies that allow stakeholders to respond to changing market conditions, technological advancements, and policy interventions in real-time.

Finally, the significance and broader implications for this section are summarized as follows. The scenarios analyzed in this section underscore the complexities and challenges of fostering cooperation in renewable energy systems. By identifying the conditions under which cooperation is achievable, this study provides actionable insights for policymakers, engineers, and energy stakeholders. The findings highlight the importance of aligning economic incentives, reducing costs, and implementing supportive policies to drive the transition toward sustainable energy systems. Ultimately, this study contributes to the growing body of knowledge on renewable energy integration and offers a roadmap for achieving collaborative, equitable, and efficient energy systems. By addressing the barriers to cooperation and leveraging the opportunities identified in this research, stakeholders can work together to accelerate the global shift toward renewable energy and achieve a more sustainable energy future. To validate our theoretical framework, we conduct seven numerical simulations—each differing in cost coefficients, profit-sharing ratios, subsidy allocations, dynamic pricing factors, and renewable energy competitiveness—to illustrate how these parameters influence the emergence and stability of cooperative equilibria.

4.6. Scenario 6: Considering Both Internal and External Factors

In the present simulation study, we expand the original Table 2 to incorporate a more comprehensive revenue distribution matrix that encapsulates both internal and external factors influencing the interactions between users and the power grid in peak shaving participation. This expansion includes the integration of government subsidies, dynamic electricity pricing, technical costs, and other pertinent factors. The augmentation of the revenue distribution matrix is achieved through the formulation of complex revenue calculation equations, thereby providing a more realistic depiction of the dynamic interplay between users and the grid within the context of power system peak shaving. The improved payoff distribution matrix is shown in

Table 5. Improved payoff distribution matrix in user–grid joint participation in peak-shaving game.

Strategy Selection		The Grid Group
		Grid Cooperation (C)	Grid Non-Cooperation (N)
The user group	User Cooperation (C)	User payoff: $U_{\mathrm{CC}}=\left(R_{\mathrm{u}}+{\displaystyle \frac{S_{\mathrm{u}}\times U_{\mathrm{Eff},\mathrm{u}}}{1+e^{-k_{\mathrm{u}}{D}_{\mathrm{g}}}}}\right)-C_{\mathrm{u}}\times \sqrt{T_{\mathrm{u}}}+{\displaystyle \frac{U_{\mathrm{Sub},\mathrm{u}}\times {\theta }_{\mathrm{u}}}{1+e^{-m_{\mathrm{u}}{C}_{\mathrm{g}}}}}$ Grid payoff: $\begin{array}{ll}{G}_{\mathrm{CC}}=& \left(R_{\mathrm{g}}+{\displaystyle \frac{S_{\mathrm{g}}\times U_{\mathrm{Eff},\mathrm{g}}}{1+e^{-k_{\mathrm{g}}{D}_{\mathrm{u}}}}}\right)-C_{\mathrm{g}}\times \sqrt{T_{\mathrm{g}}}+\\ & {\displaystyle \frac{U_{\mathrm{Sub},\mathrm{g}}\times {\theta }_{\mathrm{g}}}{1+e^{-m_{\mathrm{g}}{C}_{\mathrm{u}}}}}+B\times \gamma \end{array}$	User payoff: $U_{\mathrm{CN}}=\left(R_{\mathrm{u}}-P_{\mathrm{u}}\times \ln \left(1+{\displaystyle \frac{1}{1+e^{-{\alpha }_{\mathrm{u}}{D}_{\mathrm{g}}}}}\right)\right)-C_{\mathrm{u}}\times \sqrt{T_{\mathrm{u}}}$ Grid payoff: $G_{\mathrm{CN}}=\left(R_{\mathrm{g}}-P_{\mathrm{Penalty},\mathrm{g}}\times \ln \left(1+{\displaystyle \frac{1}{1+e^{-{\alpha }_{\mathrm{g}}{D}_{\mathrm{u}}}}}\right)\right)-C_{\mathrm{g}}\times \sqrt{T_{\mathrm{g}}}$
	User Non-cooperation (N)	User payoff: $U_{\mathrm{NC}}=\left(R_u-P_u\times \ln \left(1+{\displaystyle \frac{1}{1+e^{-{\alpha }_{\mathrm{u}}{D}_{\mathrm{g}}}}}\right)\right)-C_{\mathrm{u}}\times \sqrt{T_{\mathrm{u}}}$ Grid payoff: $\begin{array}{ll}{G}_{\mathrm{NC}}=& \left(R_g-P_{\mathrm{Penalty},\mathrm{g}}\times \ln \left(1+{\displaystyle \frac{1}{1+e^{-{\alpha }_{\mathrm{g}}{D}_{\mathrm{u}}}}}\right)\right)-\\ & C_{\mathrm{g}}\times \sqrt{T_{\mathrm{g}}}\end{array}$	User payoff: $U_{\mathrm{NN}}=R_{\mathrm{u}}-C_{\mathrm{u}}\times \sqrt{T_{\mathrm{u}}}$ Grid payoff: $G_{\mathrm{NN}}=R_{\mathrm{g}}-C_{\mathrm{g}}\times \sqrt{T_{\mathrm{g}}}$

.

For

Table 6. Explanation of parameters in user–grid joint participation in improved peak-shaving evolution game.

Parameter	Symbol	Description	Unit	Typical Value Range
User’s Normal Revenue	Ru	Normal revenue of the user without participating in peak shaving.	yuan	50–150
Grid’s Normal Revenue	Rg	Normal revenue of the grid without participating in peak shaving.	yuan	100–300
User’s Additional Revenue	Su	Additional revenue for the user under cooperation, such as electricity savings.	yuan	10–50
Grid’s Additional Revenue	Sg	Additional revenue for the grid under cooperation, such as savings from reduced peak load.	yuan	20–100
User Cooperation Efficiency	UEff,u	Efficiency factor reflecting the degree of technological application and optimization by the user during cooperation.	Dimensionless	0.5–1.5
Grid Cooperation Efficiency	UEff,g	Efficiency factor reflecting the effectiveness of peak shaving measures implemented by the grid during cooperation.	Dimensionless	0.5–1.5
User Dynamic Pricing Factor	Dg	Dynamic pricing factor set by the grid, influencing the user’s revenue.	Normalized	0–1
Grid Dynamic Pricing Factor	Du	Dynamic pricing factor set by the user, influencing the grid’s revenue.	Normalized	0–1
User Dynamic Pricing Sensitivity	ku	Sensitivity coefficient representing the impact of dynamic pricing on the user’s additional revenue.	Dimensionless	5–15
Grid Dynamic Pricing Sensitivity	kg	Sensitivity coefficient representing the impact of dynamic pricing on the grid’s additional revenue.	Dimensionless	5–15
User Cooperation Cost	Cu	Cost incurred by the user for cooperation, influenced by technological investments.	yuan	10–50
Grid Cooperation Cost	Cg	Cost incurred by the grid for cooperation, influenced by technological investments.	yuan	10–50
User Technological Cost	Tu	Technological cost for the user, such as investments in smart meters and energy management systems.	yuan	1000–10,000
Grid Technological Cost	Tg	Technological cost for the grid, such as investments in peak shaving infrastructure.	yuan	1000–10,000
User Government Subsidy	USub,u	Government subsidy provided to the user for cooperation.	yuan	5–30
Grid Government Subsidy	USub,g	Government subsidy provided to the grid for cooperation.	yuan	5–30
User Subsidy Incentive	θu	Incentive effect of government subsidies on the user.	Dimensionless	0.1–1
Grid Subsidy Incentive	θg	Incentive effect of government subsidies on the grid.	Dimensionless	0.1–1
User Subsidy Sensitivity	mu	Sensitivity coefficient determining the threshold and decay rate of subsidy effects on the user.	Dimensionless	5–15
Grid Subsidy Sensitivity	mg	Sensitivity coefficient determining the threshold and decay rate of subsidy effects on the grid.	Dimensionless	5–15
External Revenue	B	Additional revenue from government or other external sources, such as returns on infrastructure investments.	yuan	20–100
External Revenue Coefficient	γ	Coefficient reflecting the contribution of external revenues to the grid’s revenue.	Dimensionless	0.1–0.5
User Penalty	Pu	Penalty or additional cost incurred by the user due to grid non-cooperation.	yuan	10–50
Grid Penalty	PPenalty,g	Penalty or additional cost incurred by the grid due to user non-cooperation.	yuan	10–50
User Penalty Sensitivity	αu	Impact coefficient determining how dynamic pricing influences the penalties faced by the user.	Dimensionless	5–15
Grid Penalty Sensitivity	αg	Impact coefficient determining how dynamic pricing influences the penalties faced by the grid.	Dimensionless	5–15

, the primary motivation for extending the original payoff matrix in Table 2 is to capture the intricate and nonlinear interactions between users and the grid within the context of peak shaving participation. Real-world scenarios involve a multitude of internal and external factors—such as government subsidies, dynamic pricing, and technological costs—that significantly influence the strategic decisions of both parties. The original payoff matrix, while foundational, provided a simplified view with linear relationships, which may not sufficiently reflect the complexities inherent in actual energy systems. To this end, the improvements are summarized as follows.

(1) Objectives

(i)

Enhanced Realism: Incorporate a broader range of factors affecting user and grid behaviors to more accurately model real-world interactions.

(ii)

Nonlinear Dynamics: Utilize nonlinear mathematical functions (e.g., sigmoid and logarithmic functions) to represent diminishing returns and saturation effects typical in economic and technological systems.

(iii)

Interdependency Representation: Reflect the interdependent nature of user and grid strategies, where the decisions of one party dynamically influence the payoffs of the other.

(iv)

Policy and Incentive Modeling: Integrate government policies and subsidies to analyze their impact on fostering cooperation and achieving equilibrium.

(v)

Comprehensive Analysis: Provide a more detailed framework for analyzing the evolutionary stability conditions and dynamic interactions over time.

(2) Essential Improvements Compared to the Original Payoff Matrix

(i)

Inclusion of Additional Factors: The extended matrix incorporates government subsidies, dynamic pricing, and technological costs, which were absent in the original matrix.

(ii)

Nonlinear Payoff Calculations: Transitioning from simple additive or subtractive models to nonlinear functions allows for more nuanced payoff structures that better mimic real-world scenarios.

(iii)

Dynamic Interactions: The extended model captures the dynamic interplay between users and the grid, where each party’s strategy evolves based on the other’s actions and external influences.

(iv)

Policy Impact Representation: By including parameters for government subsidies and penalties, the model can simulate the effects of regulatory frameworks and incentives on strategic decisions.

(v)

Diminishing Marginal Effects: The use of sigmoid and logarithmic functions introduces diminishing marginal returns and penalties, reflecting realistic constraints and behavioral responses.

(3) Significance of the Improvements

(i)

Greater Analytical Depth: The extended model allows for a more thorough analysis of the conditions under which cooperation between users and the grid can be sustained, considering various influencing factors.

(ii)

Policy Formulation Insights: By modeling the impact of subsidies and penalties, policymakers can gain insights into how different regulatory measures might promote or hinder cooperative behaviors in peak shaving.

(iii)

Enhanced Predictive Capability: The nonlinear dynamics provide a more accurate prediction of how users and the grid might respond to changing conditions, such as fluctuations in dynamic pricing or variations in technological costs.

(iv)

Robust Simulation Framework: A more complex payoff matrix supports sophisticated simulations that can capture a wider range of possible outcomes, facilitating better decision-making and strategic planning.

(v)

Alignment with Real-World Scenarios: By reflecting the complexity of actual energy systems, the model becomes a more reliable tool for researchers and practitioners aiming to optimize user–grid interactions and promote sustainable energy practices.

The extension of the payoff distribution matrix in Table 5 represents a significant advancement in modeling the evolutionary game between users and the grid in power system peak shaving. By integrating additional internal and external factors and adopting nonlinear payoff calculations, the model achieves a higher degree of realism and analytical robustness. These enhancements enable a more comprehensive exploration of the strategic interactions and stability conditions, providing valuable insights for both academic research and practical policy formulation aimed at fostering sustainable and cooperative energy systems.

Based on above, the detailed explanation of the payoff calculation formulas in Table 5 are shown as follows.

1. Bilateral Cooperation (U_CC and G_CC)

(i) User Payoff:

$$U_{\mathrm{CC}}=\left(R_{\mathrm{u}}+{\displaystyle \frac{S_{\mathrm{u}}\times U_{\mathrm{Eff},\mathrm{u}}}{1+e^{-k_{\mathrm{u}}{D}_{\mathrm{g}}}}}\right)-C_{\mathrm{u}}\times \sqrt{T_{\mathrm{u}}}+{\displaystyle \frac{U_{\mathrm{Sub},\mathrm{u}}\times {\theta }_{\mathrm{u}}}{1+e^{-m_{\mathrm{u}}{C}_{\mathrm{g}}}}},$$

(64)

where

• $R_{\mathrm{u}}$: Base revenue, which represents the user’s normal revenue without participating in peak shaving. This is the fundamental income derived from regular electricity consumption;
• ${\displaystyle \frac{S_{\mathrm{u}}\times U_{\mathrm{Eff},\mathrm{u}}}{1+e^{-k_{\mathrm{u}}{D}_{\mathrm{g}}}}}$: Additional revenue from cooperation, influenced by the efficiency factor and dynamic pricing, modeled using a sigmoid function to represent an S-shaped growth trend;
• $C_{\mathrm{u}}\times \sqrt{T_{\mathrm{u}}}$: User cooperation cost, represented nonlinearly to reflect diminishing marginal costs;
• ${\displaystyle \frac{U_{\mathrm{Sub},\mathrm{u}}\times {\theta }_{\mathrm{u}}}{1+e^{-m_{\mathrm{u}}{C}_{\mathrm{g}}}}}$: Government subsidy for users, influenced by grid cooperation costs, modeled using a sigmoid function to capture the decaying effect of subsidies as grid cooperation costs increase. Here, the impact of government subsidies can be reflected by $U_{\mathrm{Sub},\mathrm{u}}\propto {\displaystyle \frac{U_{\mathrm{Sub},\mathrm{u}}\times {\theta }_{\mathrm{u}}}{1+e^{-m_{\mathrm{u}}{C}_{\mathrm{g}}}}}$, which means that government subsidies to users decrease as the grid’s cooperation costs C_g increase, modeled via a sigmoid function to reflect diminishing returns of subsidies at higher cooperation costs. The sigmoid function ${\displaystyle \frac{1}{1+e^{-m_{\mathrm{u}}{C}_{\mathrm{g}}}}}$ models the decreasing impact of subsidies as the grid’s cooperation cost C_g rises.

Equation (64) shows that $U_{\mathrm{CC}}\propto {\displaystyle \frac{S_{\mathrm{u}}\times U_{\mathrm{Eff},\mathrm{u}}}{1+e^{-k_{\mathrm{u}}{D}_{\mathrm{g}}}}}$, i.e., as the grid’s dynamic pricing D_g increases, the additional revenue for users from cooperation increases in an S-shaped manner.

(ii) Grid Payoff:

$$G_{\mathrm{CC}}=\left(R_{\mathrm{g}}+{\displaystyle \frac{S_{\mathrm{g}}\times U_{\mathrm{Eff},\mathrm{g}}}{1+e^{-k_{\mathrm{g}}{D}_{\mathrm{u}}}}}\right)-C_{\mathrm{g}}\times \sqrt{T_{\mathrm{g}}}+{\displaystyle \frac{U_{\mathrm{Sub},\mathrm{g}}\times {\theta }_{\mathrm{g}}}{1+e^{-m_{\mathrm{g}}{C}_{\mathrm{u}}}}}+B\times \gamma ,$$

(65)

which is analogous to the user payoff, including base revenue, additional cooperation benefits influenced by efficiency and dynamic pricing, cooperation costs, government subsidies, and external revenues. In this formula, R_g denotes the grid’s normal revenue without participating in peak shaving. This includes income from regular electricity sales and grid operations. Equation (65) reveals that $G_{\mathrm{CC}}\propto {\displaystyle \frac{S_{\mathrm{g}}\times U_{\mathrm{Eff},\mathrm{g}}}{1+e^{-k_{\mathrm{g}}{D}_{\mathrm{u}}}}}$, i.e., as the user’s dynamic pricing D_u increases, the grid’s additional revenue from cooperation increases similarly. Here, the sigmoid function ${\displaystyle \frac{1}{1+e^{-k_{\mathrm{g}}{D}_{\mathrm{u}}}}}$ captures the nonlinear increase in additional revenue with rising user dynamic pricing, again reflecting diminishing returns at higher levels.

2. User Cooperation and Grid Non-Cooperation (U_CN and G_CN)

(i) User Payoff:

$$U_{\mathrm{CN}}=\left(R_{\mathrm{u}}-P_{\mathrm{u}}\times \ln \left(1+{\displaystyle \frac{1}{1+e^{-{\alpha }_{\mathrm{u}}{D}_{\mathrm{g}}}}}\right)\right)-C_{\mathrm{u}}\times \sqrt{T_{\mathrm{u}}},$$

(66)

where

• $R_{\mathrm{u}}$: Base revenue reduced by penalties due to grid non-cooperation;
• $P_{\mathrm{u}}\times \ln \left(1+{\displaystyle \frac{1}{1+e^{-{\alpha }_{\mathrm{u}}{D}_{\mathrm{g}}}}}\right)$: Penalty term influenced by dynamic pricing, modeled using a logarithmic function to ensure penalties increase with dynamic pricing but at a decreasing rate;
• $C_{\mathrm{u}}\times \sqrt{T_{\mathrm{u}}}$: Deduction of cooperation costs.

(ii) Grid Payoff:

$$G_{\mathrm{CN}}=\left(R_{\mathrm{g}}-P_{\mathrm{Penalty},\mathrm{g}}\times \ln \left(1+{\displaystyle \frac{1}{1+e^{-{\alpha }_{\mathrm{g}}{D}_{\mathrm{u}}}}}\right)\right)-C_{\mathrm{g}}\times \sqrt{T_{\mathrm{g}}},$$

(67)

which has a similar structure to the user payoff, accounting for penalties due to user non-cooperation and cooperation costs.

3. User Non-Cooperation and Grid Cooperation (U_NC and G_NC)

(i) User Payoff:

$$U_{\mathrm{NC}}=\left(R_u-P_u\times \ln \left(1+{\displaystyle \frac{1}{1+e^{-{\alpha }_{\mathrm{u}}{D}_{\mathrm{g}}}}}\right)\right)-C_{\mathrm{u}}\times \sqrt{T_{\mathrm{u}}},$$

(68)

which mirrors the user payoff in the U_CN scenario, reflecting penalties and cooperation costs without cooperation benefits.

(ii) Grid Payoff:

$$G_{\mathrm{NC}}=\left(R_g-P_{\mathrm{Penalty},\mathrm{g}}\times \ln \left(1+{\displaystyle \frac{1}{1+e^{-{\alpha }_{\mathrm{g}}{D}_{\mathrm{u}}}}}\right)\right)-C_{\mathrm{g}}\times \sqrt{T_{\mathrm{g}}},$$

(69)

which is similar to G_CN, accounting for penalties and cooperation costs without cooperation benefits. Here, the impact of penalties can be reflected by $P_{\mathrm{Penalty},\mathrm{g}}\propto \ln \left(1+{\displaystyle \frac{1}{1+e^{-{\alpha }_{\mathrm{g}}{D}_{\mathrm{u}}}}}\right)$.

4. Bilateral Non-Cooperation (U_NN and G_NN)

(i) User Payoff:

$$U_{\mathrm{NN}}=R_{\mathrm{u}}-C_{\mathrm{u}}\times \sqrt{T_{\mathrm{u}}},$$

(70)

which shows that users receive base revenue minus cooperation costs without any penalties or additional benefits.

(ii) Grid Payoff:

$$G_{\mathrm{NN}}=R_{\mathrm{g}}-C_{\mathrm{g}}\times \sqrt{T_{\mathrm{g}}},$$

(71)

which indicates that grids receive base revenue minus cooperation costs without any penalties or additional benefits.

5. Mathematical Function Descriptions in the Formulas

(i) Sigmoid Function ${\displaystyle \frac{1}{1+e^{-x}}}$: Utilized to map dynamic pricing factors D_g and D_u into a range between 0 and 1, representing the increasing impact of dynamic pricing on benefits and penalties in an S-shaped curve.

(ii) Logarithmic Function ln(1 + x): Applied to penalty terms to ensure that penalties increase with dynamic pricing factors but at a decelerating rate, preventing unbounded growth of penalties.

(iii) Square Root Function $\sqrt{T_{\mathrm{u}}}$ and $\sqrt{T_{\mathrm{g}}}$: Represents the nonlinear relationship of technological costs, capturing the diminishing marginal costs associated with technological investments.

Based on above, we set R_u = 100, R_g = 200, S_u = 30, S_g = 50, U_Eff,u = 1.0, U_Eff,g = 1.0, D_g = 1.0, D_u = 1.0, k_u = 10, k_g = 10, C_u = 20, C_g = 40, T_u = 5000, T_g = 8000, U_Sub,u = 15, U_Sub,g = 25, θ_u = 0.5, θ_g = 0.5, m_u = 10, m_g = 10, B = 50,γ = 0.3, P_u = 20, P_Penalty,g = 30, α_u = 10, α_g = 10, the total simulation days = 365, and the time step (days) = 0.01. The simulation results are illustrated in Figure 14. The present study delves into the intricate dynamics of cooperative interactions between end-users and the power grid within the context of power system peak shaving. Motivated by the escalating demands for efficient energy management and the imperative to mitigate peak load pressures, this research employs an evolutionary game-theoretic framework to elucidate the strategic decision-making processes of both users and the grid. Building upon the foundational Table 2, we have meticulously expanded the revenue distribution matrix to encompass a broader spectrum of internal and external factors, including government subsidies, dynamic electricity pricing, and technical costs. This comprehensive augmentation is facilitated through the development of sophisticated revenue calculation formulas, which authentically capture the nuanced interplay between cooperative and non-cooperative behaviors. The core innovation lies in the integration of these multifaceted parameters into a replicator dynamics model, thereby offering a robust mechanism to simulate and analyze the evolutionary trajectories of cooperation levels over time. This enhanced model not only provides deeper insights into the incentivizing factors driving cooperation but also underscores the pivotal role of external interventions in shaping sustainable energy practices. Detailed analysis of the simulation results in Figure 14 is conducted as follows.

(a) Evolution of User Cooperation Level Over Time

Figure 14a delineates the temporal evolution of user cooperation levels across 200 simulation iterations. The trajectories exhibit a general trend of increasing cooperation, stabilizing toward an equilibrium point. This convergence suggests that, under the modeled conditions of government subsidies and dynamic pricing, users are incentivized to adopt cooperative strategies that optimize their revenue while contributing to grid stability. The marked equilibrium indicates a sustainable balance between individual incentives and collective benefits, underscoring the efficacy of the implemented policy mechanisms in fostering cooperative behavior.

(b) Evolution of Grid Cooperation Level Over Time

In Figure 14b, the temporal progression of grid cooperation levels is illustrated with enhanced line thickness and circular markers to accentuate key transition points. The simulation reveals a robust upward trajectory in grid cooperation, culminating in a steady state that aligns closely with user cooperation levels. The pronounced markers highlight instances of significant strategic shifts, likely driven by responsive policy adjustments and the grid’s adaptive strategies to user behaviors. This synchronization between user and grid cooperation underscores the mutual reinforcement inherent in the evolutionary game dynamics, facilitating a harmonious operational equilibrium.

(c) Phase Trajectories of User and Grid Cooperation Levels

Figure 14c presents the phase trajectories depicting the interdependent evolution of user and grid cooperation levels. The densely packed trajectories converge toward a central equilibrium point, illustrating the co-evolution of cooperative strategies. This coalescence signifies the establishment of a stable strategic nexus where both users and the grid benefit from sustained cooperation. The refined line attributes enhance the visibility of these trajectories, affirming the model’s capacity to capture the reciprocal influences and reinforcing the notion of interdependent strategy optimization.

(d) Phase Trajectories of User and Grid Non-Cooperation Levels

Figure 14d extends the analysis to non-cooperative scenarios, mapping the trajectories of user and grid non-cooperation levels. The non-cooperative trajectories exhibit a divergent pattern, moving away from the cooperative equilibrium and stabilizing at lower cooperation levels. This inverse relationship highlights the detrimental impact of non-cooperative behaviors, where the absence of mutual incentives leads to reduced overall system efficiency. The dark red coloration accentuates the severity of non-cooperation, emphasizing the critical need for strategic incentives to counteract such tendencies and preserve system resilience.

(e) Heatmap of Final Equilibrium Points

Figure 14e showcases a heatmap representing the density distribution of final equilibrium points across all simulations. The concentration of points around the primary equilibrium region underscores the model’s predictive reliability in identifying stable cooperative states. The viridis colormap effectively conveys the intensity of equilibrium occurrence, with darker regions indicating higher frequencies. This visualization corroborates the replicator dynamics model’s capacity to consistently drive the system toward specific strategic equilibria under the defined parameter set.

(f) Comparison of Final Rewards between User and Grid

Figure 14f juxtaposes the final rewards accrued by users and the grid, utilizing lighter color palettes for enhanced clarity. The bar chart reveals that both users and the grid achieve substantial rewards at equilibrium, with the grid slightly outperforming users. The prominently displayed numerical values, rendered in larger, bold fonts, facilitate immediate comprehension of the reward disparities. This comparison elucidates the mutual benefits derived from cooperative strategies, while also highlighting the grid’s potentially higher returns due to its broader operational scope and external revenue contributions.

(g) 3D Scatter Plot of Final Cooperation Levels (Simulations 101-200)

Figure 14g presents a three-dimensional scatter plot focusing on the final cooperation levels of the last 100 simulations. The spatial distribution indicates a tight clustering around the equilibrium point, reflecting a high degree of consistency in cooperative outcomes within this subset. The viridis colormap enhances the depth perception, while the marker distribution underscores the concentration of equilibrium states. This focused analysis reinforces the model’s robustness in maintaining cooperative stability across extended simulation runs, affirming the sustained influence of the incorporated incentives.

(h) Distribution of Pure Strategy Equilibrium Points

Figure 14h illustrates the distribution of pure strategy equilibrium points through a pie chart, categorizing them into four distinct states: (0, 0), (0, 1), (1, 0), and (1, 1). The chart reveals a predominant concentration in the (1, 1) quadrant, signifying full cooperation from both users and the grid. Minor fractions in other quadrants indicate occasional deviations from complete cooperation, likely attributable to stochastic fluctuations or parameter sensitivities. This distribution emphasizes the model’s proclivity toward fostering complete cooperative equilibria, validating the effectiveness of the strategic incentives embedded within the simulation framework.

(i) 3D Phase Trajectories (User Cooperation Level x vs. Time t)

Figure 14i depicts the three-dimensional phase trajectories of user cooperation levels over time, with increased line thickness and deeper coloration to enhance interpretability. The trajectories converge toward a stable equilibrium, demonstrating the dynamic convergence facilitated by the evolutionary game dynamics. The enhanced visual attributes underscore the clarity of these trajectories, providing a clear depiction of the temporal stabilization of user cooperation strategies within the modeled environment.

(j) 3D Phase Trajectories (Grid Cooperation Level y vs. Time t)

Figure 14j presents the three-dimensional phase trajectories of grid cooperation levels over time, employing a distinct dark red hue to differentiate from user trajectories. The convergence patterns mirror those observed in user cooperation, reflecting a synchronized stabilization process. The intensified line attributes reinforce the perceptual clarity of these trajectories, illustrating the grid’s strategic adaptation and alignment with user cooperation dynamics. This synchronization underscores the co-dependent nature of the evolutionary strategies, fostering a cohesive operational equilibrium.

This study adeptly employs an evolutionary game-theoretic framework to simulate and analyze the cooperative dynamics between users and the power grid in the context of peak shaving. By meticulously expanding the revenue distribution matrix to incorporate multifaceted factors such as government subsidies, dynamic pricing, and technical costs, the research offers a nuanced depiction of strategic interactions within the energy management landscape. The simulation results, encapsulated in Figure 14a–j, consistently validate the replicator dynamics model’s efficacy in predicting stable cooperative equilibria, underscoring the pivotal role of incentivizing mechanisms in fostering mutual cooperation. The pronounced convergence toward cooperative states highlights the model’s capacity to reflect realistic strategic behaviors, while the comprehensive parameter integration affirms the robustness of the analytical framework. Moving forward, future research endeavors could explore the incorporation of additional stochastic elements or adaptive learning algorithms to further enhance the model’s predictive precision and applicability to diverse energy systems. Additionally, empirical validation through real-world data integration would substantiate the model’s theoretical foundations, paving the way for its deployment in practical energy management and policy formulation contexts. Ultimately, this study not only reinforces the unique advantages of evolutionary game theory in elucidating complex strategic interactions but also provides a foundational tool for optimizing cooperative strategies in power system peak shaving, thereby contributing to the advancement of resilient and efficient energy infrastructures.

Further, we explore the strategic interactions between end-users and the power grid within the framework of power system peak shaving, employing an evolutionary game-theoretic approach. The simulation results are shown in Figure 15. The primary motivation stems from the escalating demand for efficient energy management and the critical need to mitigate peak load pressures that strain power systems. By extending the foundational revenue distribution matrix (Table 2) to incorporate a comprehensive array of internal and external factors—including government subsidies, dynamic electricity pricing, and technical costs—the research aims to provide a more nuanced and realistic depiction of the cooperative and non-cooperative behaviors exhibited by users and the grid. The core innovation lies in the integration of these multifaceted parameters into a replicator dynamics model, thereby enhancing the model’s capacity to simulate and analyze the evolutionary trajectories of cooperation levels over time. This sophisticated framework not only deepens the understanding of the incentives and disincentives shaping strategic decision-making in peak shaving participation but also underscores the pivotal role of external interventions in fostering sustainable energy practices.

Figure 15 encapsulates the comprehensive simulation results generated by the enhanced replicator dynamics model, illustrating the phase trajectories and temporal dynamics of user and grid cooperation levels under varying dynamic pricing factors (D_u, which is also denoted by D_u in Figure 15). The upper row of subplots presents the phase trajectories of user and grid cooperation levels across six distinct D_u scenarios (0.05, 0.25, 0.5, 0.55, 0.75, 0.95), revealing how incremental adjustments in dynamic pricing influence the convergence and stabilization of cooperative behaviors. Notably, lower D_u values (e.g., 0.05) exhibit slower convergence rates and less pronounced cooperative equilibria, suggesting that minimal dynamic pricing incentives are insufficient to robustly drive cooperation. Conversely, higher D_u values (e.g., 0.75 and 0.95) demonstrate rapid convergence toward stable cooperative states, highlighting the effectiveness of substantial dynamic pricing incentives in fostering mutual cooperation. The lower row of subplots, depicting the phase trajectories of user cooperation levels over time, further corroborates these findings by showcasing the temporal stabilization of cooperation under elevated D_u values. The enhanced line thickness and reduced transparency in these plots facilitate clearer visualization of the convergence patterns, emphasizing the model’s robustness in capturing the strategic alignment between users and the grid. Collectively, these simulation outcomes validate the extended revenue distribution matrix’s ability to accurately reflect the dynamic interplay between cooperative incentives and strategic decision-making, thereby affirming the unique applicability and advantages of evolutionary game theory in optimizing energy management strategies. Future research may explore the incorporation of additional stochastic elements or adaptive learning mechanisms to further refine the model’s predictive capabilities and extend its applicability to diverse and evolving energy systems.

In addition, we further examine the strategic interactions between end-users and the power grid within the context of power system peak shaving, utilizing an evolutionary game-theoretic framework. The simulation results are illustrated in Figure 16. The primary motivation arises from the escalating need for efficient energy management and the critical imperative to mitigate peak load pressures that strain power infrastructures. By extending the foundational revenue distribution matrix (Table 2), this research integrates a comprehensive array of internal and external factors, including government subsidies, dynamic electricity pricing, and technical costs. The simulation model employs replicator dynamics to capture the evolutionary trajectories of cooperation levels between users and the grid over a reduced simulation period of 100 days, encompassing 300 simulations to enhance statistical robustness. The core innovation lies in systematically varying the dynamic pricing factor set by the user (D_g) across six distinct values (0.05, 0.25, 0.5, 0.55, 0.75, and 0.95), thereby elucidating its impact on cooperative and non-cooperative behaviors. This methodological advancement facilitates a nuanced understanding of how dynamic pricing incentives influence strategic decision-making, thereby offering valuable insights for policymakers aiming to foster sustainable energy practices and optimize revenue distributions in power systems.

Figure 16 presents a comprehensive visualization of the simulation results, showcasing the phase trajectories and temporal dynamics of user and grid cooperation levels under varying dynamic pricing factors (D_g). The upper row of subplots illustrates the phase trajectories of cooperative behaviors, revealing how incremental adjustments in D_g influence the convergence and stabilization of cooperation between users and the grid. Notably, lower D_g values (e.g., 0.05) exhibit slower convergence rates and less pronounced cooperative equilibria, indicating that minimal dynamic pricing incentives are insufficient to drive significant cooperation. In contrast, higher D_g values (e.g., 0.75 and 0.95) demonstrate rapid convergence toward stable cooperative states, underscoring the efficacy of substantial dynamic pricing incentives in fostering mutual cooperation. The lower row of subplots, depicting phase trajectories of user cooperation levels over time, corroborates these findings by illustrating the temporal stabilization of cooperation under elevated D_g scenarios. The enhanced line thickness and reduced transparency in these plots facilitate clearer visualization of convergence patterns, affirming the model’s robustness in capturing the strategic alignment between users and the grid. Collectively, these simulation outcomes validate the extended revenue distribution matrix’s capacity to accurately reflect the dynamic interplay between cooperative incentives and strategic decision-making, thereby reinforcing the unique applicability and advantages of evolutionary game theory in optimizing energy management strategies.

The meticulously enhanced simulation framework successfully integrates variable dynamic pricing factors into an evolutionary game-theoretic model of user–grid interactions in power system peak shaving. By systematically varying D_g and analyzing the resultant phase trajectories and temporal stabilization patterns through comprehensive 2 × 3 grid visualizations, this study provides robust empirical evidence supporting the validity of the extended revenue distribution matrix. The findings unequivocally highlight the critical role of dynamic pricing in fostering cooperative behaviors, thereby contributing to more efficient and resilient power system operations. The integration of multifaceted parameters, including government subsidies and technical costs, enriches the model’s realism and enhances its capacity to reflect complex strategic interactions. Moving forward, future research endeavors may incorporate real-world data to empirically validate the model’s predictions and explore the impact of additional variables, such as renewable energy integration and user heterogeneity, to further enhance the model’s comprehensiveness and applicability. Ultimately, this research not only reinforces the unique advantages of evolutionary game theory in elucidating complex strategic interactions but also provides a foundational tool for policymakers and stakeholders aiming to optimize cooperative strategies in power system peak shaving, thereby advancing the development of sustainable and efficient energy infrastructures.

4.7. Scenario 7: Incorporating Evolutionary Game Theory and Whole-Process Democracy

The user–grid interaction in renewable energy projects represents a critical area of study, especially as we move toward more sustainable, low-carbon energy systems. The motivation for this research stems from the challenge of optimizing cooperation between grid operators and energy users, particularly in the context of renewable energy integration and peak shaving. The integration of renewable energy sources like wind and solar often leads to grid instability, requiring cooperative strategies from both the grid and its users to balance demand and supply. This interaction can be modeled through evolutionary game theory, which helps simulate the dynamic behavior of users and grid enterprises, with a focus on cooperation for peak shaving and energy optimization.

To generate a detailed simulation that visualizes the user–grid interaction dynamics in renewable energy projects, incorporating evolutionary game theory and whole-process democracy as described in the mathematical model proposed in Section 3.2, we create a simulation based on the equations and concepts presented. This simulation is conducted to visualize the evolution of user and grid cooperation, including phase trajectories, equilibrium points, and final rewards distribution. Here is a step-by-step guide on how to implement this simulation:

Step 1: Define the Parameters. We need to define the parameters and initial conditions based on the evolutionary game theory equations provided in this paper. We also need parameters for the deliberative feedback mechanism, subsidies, and cooperation costs.

Step 2: Simulate Replicator Dynamics with Feedback. The replicator dynamics will evolve over time based on the payoff functions, and we can incorporate deliberative feedback into the system. This can model the behavior of both users and the grid as they engage in cooperation, influenced by government policies and mutual feedback.

Step 3: Generate Visualizations. The visualizations include evolution of user cooperation over time, evolution of grid cooperation over time, phase trajectories for user and grid cooperation levels, heatmaps of final equilibrium points, and the comparison of final rewards between users and the grid.

Step 4: Code Implementation. Finally, we conduct a code implementation that simulates the dynamics and generates the requested visualizations, as demonstrated in Figure 17.

The main purpose of this simulation is to analyze the evolutionary dynamics of user–grid cooperation under various conditions, taking into account key decision-making processes and policy interventions such as government subsidies. By incorporating the concept of whole-process democracy, this model emphasizes equal participation, transparency, and inclusive decision-making between all stakeholders involved in renewable energy systems. The simulation aims to provide insights into how cooperative strategies evolve over time, ensuring that benefits are distributed equitably and that long-term cooperation is fostered between users and the grid. The key parameters in this simulation include:

• User Cooperation Level (x): Proportion of users actively engaging in peak shaving and energy-saving behaviors.
• Grid Cooperation Level (y): Proportion of the grid actively participating in facilitating energy transfers and balancing renewable energy fluctuations.
• Time (t): Simulation time in days, ranging from 0 to 500 days.
• Parameter settings: These values dictate the interaction dynamics between users and the grid, such as cost–benefit ratios, profit sharing, and incentives for cooperation. Specific parameters include incremental benefits from cooperation (ΔS), cost coefficients (γ_g, γ_f), and time steps for simulation (Δt).

Figure 17 focuses on the two-dimensional convergence dynamics: it clearly shows the trajectory paths of the user–grid system as it moves from the initial condition toward the theoretical equilibrium at (1.0, 1.0). As shown in Figure 17, both user cooperation and grid cooperation converge toward a steady state of 1.0, representing an idealized upper bound under perfectly balanced incentives (subsidies, feedback mechanisms, and participation costs). While this steady state provides a theoretical benchmark for policy evaluation, we acknowledge that practical attainment of full cooperation may be impeded by imperfect incentive alignment, market volatility, and exogenous shocks.

Figure 17a demonstrates the temporal evolution of the user cooperation level in the context of the evolutionary game model. The graph shows a rapid increase in user cooperation, as indicated by the sharp rise in the cooperation level from approximately 0.5 to 1 within the first few time steps. This suggests that, in the absence of significant barriers or disincentives, users are highly motivated to engage in cooperative strategies when incentivized properly by the grid or through policy interventions like government subsidies. The sharp increase may reflect an initial adjustment phase where users, influenced by short-term incentives or regulatory support, quickly adapt to the grid’s expectations. This scenario highlights the importance of providing clear and effective incentives for users to engage in renewable energy projects and peak shaving, thus minimizing grid strain and optimizing overall energy consumption. The model shows that as long as there is sufficient alignment of interests—such as fair profit-sharing mechanisms and cost reductions—users are likely to participate actively. However, the rapid stabilization in cooperation suggests that a balance must be struck to avoid the risk of user disengagement or reduced motivation if the incentives are perceived as inadequate in the long term.

Figure 17b depicts the evolution of the grid’s cooperation level over time, which follows a similar trajectory to the user cooperation levels. The grid’s cooperation level starts at approximately 0.5, reflecting an initial neutral stance, but rapidly escalates toward 1, signifying full cooperation. This trend mirrors the user cooperation curve, suggesting that grid enterprises, as well, are willing to engage in cooperative strategies if the economic incentives and regulatory frameworks are well aligned. A key insight here is that the grid, which traditionally holds a more passive or reactive role in energy systems, is significantly influenced by the behavior of users. The grid’s response to user engagement reinforces the concept of mutual interdependence in energy systems: the more users cooperate, the more the grid is incentivized to cooperate in return, facilitating the flow of renewable energy and smoothing out peak demands. The simulation highlights the importance of creating an environment where the grid and users can share risks, costs, and benefits equitably.

Figure 17c presents the phase trajectories of user and grid cooperation levels over time, which is a crucial representation of how the cooperation dynamics evolve when considering both users and the grid as interacting populations. In this phase plot, the user cooperation level is plotted along the x-axis, while the grid cooperation level is plotted along the y-axis. The trajectory follows a clear upward slope, indicating that as users increase their level of cooperation, the grid also adjusts accordingly, following a complementary cooperative strategy. The equilibrium point is marked on the plot as the final point of the trajectory, showing where both cooperation levels stabilize at their maximum value (1.0). The phase trajectory reveals an important observation: the evolution of cooperation between the grid and users is not independent, but interdependent. The cooperative actions of one party (e.g., users) directly influence the cooperation of the other party (e.g., the grid). This mutual influence supports the idea of a cooperative dynamic, which is fundamental in the context of renewable energy systems, where both users and the grid must work together to balance supply and demand effectively. The trajectory also underscores the key role of reciprocal incentives, highlighting that when one party—whether the users or the grid—becomes highly cooperative, the other party follows suit, driven by similar incentives and mutual benefits.

Figure 17d illustrates a 3D scatter plot of final cooperation levels for both users and grid enterprises, plotted against time. This visualization extends the analysis of the cooperation dynamics by representing how the final cooperation levels evolve after a certain period. The color gradient shown on the plot reflects the progression over time, providing an intuitive sense of how quickly the cooperation levels converge to their final equilibrium states. From the plot, it is evident that both user and grid cooperation levels converge to their equilibrium values, and the progression is relatively smooth and stable. The relatively gradual ascent in the scatter plot suggests that the system reaches a stable equilibrium point over time, with both user and grid cooperation levels stabilizing. This plot reaffirms that the system’s overall dynamics are self-reinforcing: as users become more engaged in energy-saving practices and peak shaving, the grid also adjusts to support these efforts, leading to a mutually beneficial outcome. The smooth transition toward the equilibrium point highlights the effectiveness of the policies in aligning incentives and the importance of sustained, long-term cooperation.

Figure 17e provides a 3D representation of the phase trajectories of both user and grid cooperation levels over time, offering a more comprehensive view of the dynamic interplay between the two populations. The plot tracks how cooperation levels evolve as a function of time, showing the trajectories of both users and the grid in a three-dimensional space. This allows for a deeper understanding of the multi-dimensional relationship between the grid and users as they move toward cooperation. The trajectory depicted in this subfigure emphasizes the overall alignment of user and grid cooperation levels, reinforcing the idea that a cooperative equilibrium is not only achievable but stable. This 3D phase trajectory adds another layer of analysis by incorporating time as a factor in the trajectory, demonstrating the temporal nature of cooperation evolution. As with previous plots, the system stabilizes as both user and grid cooperation levels reach their equilibrium points. The 3D visualization highlights the underlying dynamic feedback loops in the system, where changes in user cooperation influence grid behavior and vice versa, creating a balanced system that evolves over time toward optimal cooperation.

1. Overall Summary of Simulation Results and Future Directions

The simulation results in Figure 17 presented throughout this research provide profound insights into the evolution of cooperation between users and grid enterprises in renewable energy systems, modeled through the lens of evolutionary game theory. These results not only illustrate the dynamics of cooperation but also highlight the critical role of an inclusive, democratic approach in fostering long-term, stable collaboration between stakeholders.

The core findings of the simulation reinforce the notion that user–grid interaction in renewable energy systems can benefit immensely from a decision-making process grounded in extensive consultation and equal participation from both parties. The cooperative dynamics observed in the simulations demonstrate that user and grid cooperation is not only interdependent but also mutually reinforcing. As user cooperation levels increase, the grid tends to adjust its strategy to align with these shifts, and vice versa. This mutual feedback mechanism suggests that cooperation is best achieved when both users and the grid work toward shared goals, ensuring that both sides benefit equitably.

The equilibrium stabilization over time further strengthens the idea that properly structured policies and incentives can lead to a self-sustaining cooperative relationship between users and the grid. The simulations indicate that both parties can achieve optimal outcomes when their cooperation levels converge at equilibrium points, where no further unilateral deviations are beneficial. This underlines the importance of creating incentive structures that encourage both parties to engage in the cooperation process actively. In addition, the visualization of phase trajectories and 3D scatter plots provides valuable insights into the temporal evolution of cooperation, highlighting how both parties adjust their strategies over time and reach stable cooperation points.

The integration of whole-process democracy within the evolutionary game-theoretic framework is a key factor in ensuring inclusive, transparent, and democratic decision-making processes. This perspective emphasizes the importance of designing policies that reflect consultative decision-making mechanisms, where both users and grid enterprises have an equal say in policy formation, such as government subsidies or cost-sharing schemes. By incorporating democratic processes into the design of renewable energy systems, the evolutionary game model can better capture the complexity of stakeholder interactions, allowing for more balanced and equitable cooperation outcomes.

Moreover, the deliberative democracy approach ensures that policies are crafted with a long-term vision, focusing not only on short-term outcomes but also on fostering enduring collaborative strategies between users and grid operators. This approach aligns with the overarching goal of creating a sustainable, low-carbon energy future, where cooperation between the user and grid enterprises remains central to the success of renewable energy integration.

2. Policy Implications

Based on the simulation results and the theoretical insights derived from this study, several policy implications emerge, which can guide the development of long-term, sustainable cooperation strategies between users and grid enterprises in renewable energy systems:

(i)

Inclusive Policy Formulation: Governments should promote policies that are based on whole-process democracy, ensuring that all stakeholders—users, grid enterprises, and policymakers—are involved in the decision-making process. This approach would help in identifying optimal solutions for cooperation and resolving potential conflicts. By giving both users and grid enterprises a platform for input, policies would more accurately reflect their needs and interests, leading to greater buy-in and commitment to cooperative initiatives.

(ii)

Equitable Benefit Distribution: The simulations emphasize that cooperation is most successful when both parties benefit equitably. Government subsidies and cost-sharing mechanisms should be designed to ensure that both users and grid enterprises share the benefits of cooperation fairly. This might include subsidies for users who adopt energy-efficient practices or participate in peak shaving, while also providing incentives for grid enterprises to invest in infrastructure improvements that support renewable energy integration.

(iii)

Incentive-Based Regulation: Policymakers should design incentive mechanisms that align the interests of users and grid enterprises. For instance, financial incentives or tax breaks could encourage grid operators to invest in renewable energy infrastructure, while users could be incentivized to adopt energy-saving technologies and participate in peak regulation programs. These incentives should be structured in a way that fosters long-term cooperation rather than short-term, transactional exchanges.

(iv)

Dynamic Policy Adjustment: Given the evolutionary nature of user–grid cooperation, policies must be adaptable to changing circumstances. The game-theoretic model suggests that as both parties learn and adapt, the equilibrium points evolve. Policymakers should establish mechanisms for adaptive policy-making, where adjustments can be made based on real-time data and changing conditions in the energy market. This could involve periodic reviews of policy effectiveness, as well as the flexibility to adjust subsidies or cost-sharing structures as necessary to maintain equilibrium.

(v)

Incorporation of Stochastic Factors: Future policy frameworks should also consider the potential for random fluctuations or external shocks (e.g., technological changes, economic shifts, or environmental disruptions). The evolutionary game model can be extended to incorporate stochastic elements, allowing policymakers to simulate the effects of unexpected changes and adjust their strategies accordingly. This would provide a more resilient framework for managing cooperation in renewable energy systems, ensuring that both users and the grid can adapt to unforeseen challenges.

(vi)

Long-Term Stakeholder Engagement: To ensure the success of cooperative strategies, long-term engagement with stakeholders is essential. Policymakers should engage in continuous dialogue with both users and grid enterprises to monitor the effectiveness of cooperation initiatives and address emerging concerns. This could be achieved through regular consultative forums, stakeholder surveys, or feedback loops that allow for the identification and resolution of issues before they become barriers to cooperation.

The study in Scenario 7 demonstrates the potential of evolutionary game theory as a tool for analyzing user–grid cooperation in renewable energy systems. By incorporating the concept of whole-process democracy, the study offers a framework for designing policies that promote sustainable cooperation through inclusive and transparent decision-making processes. The simulation results validate the effectiveness of these strategies, providing a clear path toward achieving stable, long-term cooperation between users and grid enterprises.

Looking ahead, further research could expand on the current model by incorporating additional factors such as the integration of multiple renewable energy sources or the influence of external market forces on user–grid interactions. Additionally, the model could be refined to include more detailed stochastic elements to better simulate the unpredictability of real-world energy markets. Future work could also explore the impact of policy simulations on cooperative behaviors, testing various scenarios to understand how different policy designs affect cooperation outcomes. In conclusion, this study highlights the importance of deliberative democracy in the design and implementation of policies that promote cooperation in renewable energy systems. By aligning the interests of all stakeholders and ensuring that decision-making processes are inclusive and transparent, we can foster long-term collaboration and create more sustainable, low-carbon energy systems.

In addition, based on Figure 17, we further investigate the evolutionary dynamics of user–grid cooperation within a renewable energy context, with a focus on achieving mutual cooperation between users and grid operators. As illustrated in Figure 18, the simulation model incorporates feedback mechanisms, subsidies, and participation costs, ensuring both parties converge toward cooperative behavior, aiming for maximum mutual benefits. The use of feedback loops, subsidies, and cost adjustments encourages cooperation by continually adapting strategies based on previous actions, thus achieving an optimal state where both participants cooperate fully. The results offer valuable insights into how incentives can drive long-term cooperation in renewable energy systems, promoting sustainability and equitable benefits distribution. Figure 18, by contrast, extends this analysis into three dimensions by incorporating participation costs, subsidy-rate variations, and dynamic pricing parameters. The added third axis (color-coded mesh) captures the magnitude of policy leverage, revealing how adjustments in subsidies and costs shift the entire cooperation surface. Figure 18 thus provides a deeper insight into the robustness of cooperative equilibria under heterogeneous policy scenarios: peaks and valleys in the surface correspond to parameter regimes that either promote or hinder full cooperation.

Figure 18a shows the temporal evolution of user cooperation levels. Initially, user cooperation begins at a relatively low level, but as the simulation progresses, it steadily increases toward 1, reaching a state of full cooperation by the end of the simulation. This steady rise is indicative of the effective incentives provided to the users, such as subsidies and cooperative strategies. The user’s cooperative behavior is driven by the feedback loop and the subsidy mechanism, which ensure that both short-term gains and long-term benefits align. The result demonstrates that, through careful policy structuring, users can be effectively incentivized to cooperate, optimizing their own benefit while also contributing to the overall stability and sustainability of the energy system. The gradual, smooth increase toward a cooperation level of 1 signals the success of cooperative frameworks and evolutionary dynamics in renewable energy systems.

Figure 18b illustrates the evolution of grid cooperation levels over time. Initially, the cooperation level for the grid is relatively low (similar to the users), but over the course of the simulation, it gradually increases to a level close to 1. This result underscores the mutual benefit of cooperation between the grid and the users. The grid’s cooperation is incentivized similarly to the users’, through subsidies and the feedback mechanism. The plot highlights how the grid’s strategy evolves in response to user cooperation, reinforcing the interdependence between both parties. A key observation here is the synchronized trajectory between the user and grid cooperation, illustrating that policies supporting both sides lead to a cooperative equilibrium where both parties work together for mutual gain. This finding suggests that comprehensive cooperative strategies between users and grid operators are not only feasible but also highly beneficial in achieving system-wide sustainability.

Figure 18c shows the phase trajectories of user cooperation and grid cooperation over time. The plot shows how the cooperation levels of both parties evolve together, gradually converging to the equilibrium point at (1, 1). The phase trajectory illustrates the feedback-driven interdependence between user and grid strategies. The point (1, 1) represents the cooperative equilibrium where both users and the grid reach their optimal strategies, reflecting the mutual reinforcement of cooperation. This reinforces the idea that when both parties adopt cooperative strategies, their individual incentives align, leading to optimal outcomes. The plot also underscores the resilience of the evolutionary process, where even from diverse initial conditions, both strategies stabilize at a high level of cooperation. The stability of this equilibrium further validates the effectiveness of feedback mechanisms and government incentives in fostering collaboration between grid operators and users.

Figure 18d presents a 3D scatter plot of the final cooperation levels for both the user and the grid, showing their evolution over time. The plot reveals that both user and grid cooperation levels converge toward 1 over time. The 3D visualization provides a clear representation of how the cooperation levels of both parties interact and evolve simultaneously. The different colors in the plot represent the progression over time, with the intensity of the color increasing as the simulation advances. The scatter plot reinforces the idea that cooperative dynamics between the two parties are not only driven by subsidies and feedback but also by mutual understanding and the recognition of shared benefits. The convergence of both strategies toward the cooperative equilibrium further supports the notion that with the right policy framework, sustainable cooperation between users and the grid is attainable.

Figure 18e presents a heatmap of the final cooperation levels. The heatmap clearly demonstrates how different initial conditions and parameters, such as user and grid costs, subsidies, and feedback, influence the final cooperation levels. Darker regions in the heatmap indicate areas of high cooperation (near 1), while lighter regions indicate lower levels of cooperation. The heatmap provides a visual representation of the system’s stability and robustness, illustrating how cooperation emerges across different starting points and parameter configurations. It visually reinforces the conclusion that cooperative equilibrium is achievable under varying conditions, and that policy adjustments (such as subsidies) play a crucial role in driving the system toward full cooperation. The dynamic nature of the heatmap suggests that cooperation levels are highly sensitive to both initial conditions and incentive structures, which further supports the argument for policy flexibility and adaptability in promoting long-term cooperation.

Overall, the simulation results in Figure 18 provide a comprehensive examination of the evolutionary dynamics of user–grid cooperation in renewable energy systems. The results confirm that feedback mechanisms, subsidies, and cost-sharing structures are essential in promoting long-term cooperation. Both users and grid operators, when incentivized correctly, converge toward an optimal cooperative state (1, 1), which benefits both parties and ensures system sustainability. The heatmap and 3D scatter plot provide valuable insights into how cooperative behavior evolves under different conditions, confirming that policy frameworks play a critical role in guiding the system toward the desired equilibrium.

As demonstrated in Figure 18, this research opens several avenues for further exploration. Future work could integrate stochastic elements, such as random fluctuations in energy demand or supply, to simulate more realistic environments. Additionally, extending the model to include multiple users or different types of grid configurations could offer a more detailed understanding of complex systems with diverse stakeholder interactions. The incorporation of real-world data could further validate the model’s predictive capabilities, while the exploration of adaptive learning algorithms could provide deeper insights into dynamic behavior over time. Furthermore, future models could include social factors such as behavioral biases or cultural differences, which could affect the cooperation dynamics. Lastly, policy implications could be refined to ensure that renewable energy incentives are tailored to specific local conditions, ensuring that both economic efficiency and social equity are maximized.

The findings of this study in Figure 18 have several policy implications. First, governments should strengthen subsidy programs to ensure that renewable energy adoption is incentivized for both users and grid operators. These programs should be flexible, adjusting based on the specific needs of each stakeholder. Second, cooperative feedback mechanisms should be incorporated into energy market structures to dynamically adjust cooperation incentives. Third, long-term cooperation between users and grids should be supported through the development of equitable cost-sharing frameworks that balance the costs and benefits of renewable energy integration. Lastly, it is essential to incorporate feedback loops that reflect the evolution of cooperation over time, ensuring that the policy landscape remains adaptable to emerging challenges and opportunities. Overall, this study highlights the critical importance of inclusive, transparent, and democratic decision-making in the context of renewable energy projects. By integrating the concept of whole-process democracy, the research demonstrates how collaborative decision-making can optimize cooperation and create sustainable and equitable energy systems.

5. Discussion and Future Research Directions

This section delves deeper into the implications of the findings, discussing potential improvements and future research directions. The content is divided into two sub-sections: Model Enhancements and Technological Innovations and Applications.

5.1. Model Enhancements

While the current model effectively captures the core dynamics, future research can further enhance its realism and applicability by incorporating the following:

(i)

Inclusion of External Shocks and Uncertainties: Real-world scenarios often involve stochastic elements such as fluctuations in energy demand or policy changes. Future models can integrate stochastic differential equations to simulate these external influences.

(ii)

Dynamic Network Effects: Future studies could explore the spatial and temporal network effects of user–grid interactions, considering how decisions in one region impact neighboring regions.

(iii)

Behavioral Considerations: The model can be extended to include behavioral factors such as user preferences, risk aversion, and social norms, providing a more holistic understanding of decision-making dynamics.

5.2. Technological Innovations and Applications

Technological advancements will play a critical role in addressing some of the barriers identified in this study. Key areas for future exploration include:

(i)

Advanced Energy Storage Solutions: The integration of cost-effective energy storage systems can alleviate the intermittency challenges of renewable energy, enabling more reliable participation in peak-shaving programs.

(ii)

Blockchain-Based Allocation Systems: Blockchain technology offers a transparent and decentralized mechanism for profit allocation [40,41,42,43], ensuring fairness and trust among participants.

(iii)

AI-Driven Optimization: Artificial intelligence can be employed to optimize energy dispatch, predict demand patterns [44,45,46,47,48,49,50], and enhance the efficiency of peak-shaving operations.

Overall, this research demonstrates the significant potential of evolutionary game theory in modeling and analyzing user–grid interactions in peak-shaving systems. By investigating five distinct scenarios, we have analyzed the equilibrium process and the conditions for sustaining cooperation, emphasizing the critical role of profit redistribution. The findings have important implications for energy policy, renewable energy adoption, and the design of sustainable energy systems.

Future research should focus on addressing the identified limitations, such as incorporating dynamic network effects [51,52], stochastic influences [53,54,55], and behavioral factors [56,57,58,59,60,61,62]. Additionally, technological innovations, particularly in energy storage and blockchain-based systems [63,64,65,66,67], hold great promise for enhancing the efficiency and equity of user–grid interactions.

This study provides a robust foundation for future investigations, offering valuable insights into the dynamic interplay between users and power grids. By advancing our understanding of these interactions, the findings contribute to the broader goal of achieving sustainable and efficient energy systems.

6. Conclusions and Policy Implications

6.1. Main Conclusions

This study employs an evolutionary game-theoretic model to investigate the interactions between users and power grids within the context of peak-shaving systems aimed at integrating low-carbon renewable energy into the grid. Through the application of nonlinear evolutionary game models, this research examines the cost–benefit dynamics, replicator dynamics, and evolutionary stability conditions that govern these interactions. By constructing this framework, we aim to explore the long-term strategic decisions made by users and power grid enterprises and investigate how inclusive decision-making and cooperative behaviors can be achieved in renewable energy systems.

This model underscores the iterative and evolving nature of decision-making processes that drive the interactions between users and power grids. It reveals that the strategic evolution of these groups unfolds gradually, with decisions evolving through ongoing learning and adaptation. The model captures the essence of evolutionary dynamics, where both users and grid operators adjust their strategies iteratively based on the payoffs and the decisions made by the other party.

This research contributes several critical insights into the interactions between users and power grids:

(i)

Profit Allocation Dynamics: The study emphasizes that cooperation between users and grid operators can only be sustained if the profit allocation mechanism is perceived as fair by both parties. Equitable distribution of benefits is essential for fostering long-term cooperation, and imbalances in profit-sharing mechanisms can undermine the willingness of either party to participate in the program. Hence, well-designed incentive structures that align the interests of both users and grid operators are crucial for ensuring sustained collaboration.

(ii)

Profit Margins and Costs: The study finds that cooperation will be unsustainable if the costs associated with renewable energy generation and participation in peak-shaving programs exceed the attainable profits. In such scenarios, either party might withdraw from the cooperation due to economic infeasibility. The research suggests that policy interventions are required to reduce the costs of participation and ensure that the economic benefits of collaboration are sufficient to incentivize both parties to engage in long-term cooperation.

(iii)

Competition with Conventional Energy: The analysis reveals that renewable energy prices must be competitive with those of conventional energy for cooperation to persist. In scenarios where renewable energy is not economically viable compared to traditional sources, grid operators and users are less likely to cooperate, as the costs of participating in renewable energy programs outweigh the benefits. This underscores the need for policies that level the playing field and make renewable energy more competitive with conventional energy sources.

(iv)

A Comprehensive Modeling Approach: This study introduces a comprehensive evolutionary game framework for analyzing the strategic interactions between users and the grid in peak shaving. This model can be expanded to address other critical aspects of user–grid interaction, such as demand response, energy storage, and the integration of additional renewable energy sources like wind and solar. By providing a rigorous theoretical model, the study lays the foundation for future research in understanding the broader dynamics of energy systems.

(v)

Importance of Incentive Policies: The study highlights the critical role of government policies in ensuring the success of cooperative peak-shaving programs. Without appropriate incentives, both users and grid operators may be unwilling to participate in such programs. The study suggests that policies such as subsidies, tax incentives, and improved energy pricing mechanisms can ensure that both parties are incentivized to participate and cooperate, making renewable energy systems economically sustainable.

(vi)

Sustainability of Long-Term Cooperation: The research suggests that long-term cooperation requires more than just short-term profit maximization. Trust-building mechanisms, clear communication, and policy stability are essential to maintaining cooperation over extended periods. The study finds that the evolutionary stability of cooperation relies not only on immediate financial incentives but also on long-term policy frameworks that can adapt to changing energy demands, environmental factors, and technological advancements.

In future work, we will explore the possibility of treating some of the parameters as probabilistic variables or fuzzy parameters to capture uncertainty. Concretely, we will discuss how certain parameters, such as dynamic pricing, user behavior, and renewable energy generation variability, could be modeled as fuzzy variables using fuzzy logic or as probabilistic distributions. For example, cost parameters could be treated as fuzzy sets to represent the uncertainty in pricing fluctuations, while user behavior could follow a probabilistic distribution based on real-world market data. This approach will be explored in future studies to further enhance the realism and flexibility of the model.”

6.2. Policy Implications

The findings of this research underscore the pivotal role of policy in promoting cooperation between users and power grids in peak-shaving systems for renewable energy integration. The study emphasizes that a cooperative approach between these stakeholders can be achieved by designing fair profit-sharing mechanisms, aligning incentives, and addressing the economic barriers that prevent full participation. Each policy recommendation is tied directly to specific outcomes observed in our simulations. For instance, Scenario 3 demonstrates that when subsidy α > 5%, the proportion of users adopting peak-shifting strategies (x) increases significantly, approaching 1. This result suggests that subsidy levels directly influence participation rates, emphasizing the importance of tailored financial incentives for renewable energy adoption. Below are the key policy implications derived from this study, with a focus on fostering cooperation, supporting whole-process democracy, and promoting sustainable renewable energy systems.

(i)

Regulatory Interventions for Renewable Energy Pricing: Governments must implement regulatory measures to ensure that renewable energy pricing remains competitive with conventional energy sources. In China, where the government has made substantial investments in renewable energy, dynamic pricing could align with the evolving energy market by incentivizing consumers to shift their demand to off-peak periods, which is critical in a rapidly developing energy sector. Similarly, in the EU, where energy markets are increasingly integrated, dynamic pricing can help balance supply and demand across member states, fostering cooperation among grid operators and users. Our simulations suggest that such pricing mechanisms could complement existing subsidies and incentive programs, leading to more efficient and sustainable renewable energy integration. Our simulations indicate that dynamic pricing strategies, where prices are adjusted based on real-time demand, could be effective in fostering cooperation between users and grid operators. In Scenario 4, we observe that dynamic pricing significantly reduces peak demand, especially when combined with user incentives that encourage flexible participation. This result suggests that dynamic pricing could be a potent tool to enhance participation in peak-shaving programs, particularly when real-time pricing aligns with grid operator needs. Direct subsidies, tax incentives, and other financial tools can be used to reduce the production costs of renewable energy, making it economically attractive for both grid operators and users. In addition to these incentives, governments can support long-term contracts for renewable energy purchases, ensuring a stable market for renewable energy while providing price parity with fossil fuels. This will not only encourage the adoption of renewable energy but also ensure that cooperative strategies between users and grid operators can be sustained.

(ii)

Fair Allocation Mechanisms: Policies must ensure that profits generated from peak-shaving systems are allocated in an equitable manner. One key aspect of fostering cooperation is to design profit-sharing mechanisms that account for both the immediate and long-term benefits to each party. For example, grid operators could receive a share of the profits from reduced peak load while users could be compensated for their participation in energy conservation efforts. Policies should also consider how the implementation of subsidies or government incentives can offset the initial capital investment costs for renewable energy technologies, such as solar panels, wind turbines, or battery storage systems. The key to sustaining cooperation is to ensure that both sides feel that their contributions are rewarded adequately.

(iii)

Incentives for Peak Shaving Participation: Policies that encourage renewable energy participation in peak-shaving programs should focus on reducing the initial investment costs and ensuring financial viability for both users and grid operators. Our simulations show that subsidy programs play a crucial role in driving adoption of peak-shifting strategies. In Scenario 2, we find that subsidies covering 50% of initial investment costs lead to x > 0.8, indicating high user participation. This result suggests that financial incentives targeting the reduction of initial costs can significantly increase participation rates in peak-shaving programs. Governments can provide tax incentives or offer subsidies for the installation of renewable energy technologies, thus reducing the upfront costs for users. Additionally, grid operators should be incentivized to integrate renewable energy sources into the grid by offering financial rewards for successfully reducing peak demand and maintaining grid stability. Furthermore, incentive programs could reward users who contribute to grid stability through energy storage and demand-side management programs.

(iv)

Development of Adaptive Feedback Mechanisms: Governments and grid operators should work together to develop adaptive feedback mechanisms that respond to dynamic conditions in the energy market. Dynamic pricing strategies should be developed that adjust based on real-time energy demand. Feedback loops can also be implemented that adjust subsidies and incentives based on the cooperative behavior of both users and grid operators. Such mechanisms will help to ensure that cooperation remains sustainable, even when external factors, such as energy demand, climate change, or market conditions, evolve. These adaptive systems will ensure that cooperative relationships can evolve dynamically over time, reflecting changing market conditions and technological advancements.

(v)

Promotion of Whole-Process Democracy in Energy Policy: The concept of whole-process democracy must be integrated into the formulation of energy policies. Simulation results underscore the importance of inclusive decision-making, where all stakeholders—users, grid operators, and government bodies—are involved in policy design. In Scenario 5, we observe that when users and grid operators have an equal say in policy development, the system reaches a more stable equilibrium, with higher levels of participation in peak-shaving programs. This highlights the importance of whole-process democracy in ensuring that policies are both effective and widely supported. This democratic approach ensures that all stakeholders, including users, grid operators, and government bodies, participate in the decision-making process. Inclusive and transparent decision-making fosters trust and social legitimacy, leading to more effective and widely supported policies. This democratic framework will help ensure that energy policies are socially just and equitable, particularly in addressing concerns around distributional fairness. Through consultation and equal participation, decision-makers can better align the interests of all parties, thereby fostering cooperation and creating a more sustainable and equitable energy future.

(vi)

Promotion of Long-Term Cooperation: Sustainable cooperation requires long-term policy stability. Governments and grid operators must develop frameworks that ensure consistent and predictable policies, enabling both users and grid operators to plan and invest with confidence. Policies should include long-term contracts, investment guarantees, and the possibility of adaptation to emerging technological trends, such as smart grids and energy storage technologies. This stability will help build the trust necessary for cooperation, particularly in contexts where high initial investments are required. Additionally, to ensure long-term cooperation, governments must provide regular reviews of incentive structures, adjusting them to reflect changing energy dynamics, technological advancements, and evolving social preferences.

In conclusion, the results of this study provide critical insights into the cooperative dynamics between users and power grids in peak-shaving systems for renewable energy integration. The policy implications outlined above offer actionable strategies that can guide energy regulators and policymakers in developing effective, equitable, and sustainable renewable energy systems. By focusing on inclusive decision-making, fair profit-sharing, and long-term cooperation, these policies can optimize the operation of renewable energy systems while ensuring that both parties benefit from the integration of renewable energy into the power grid. Through the successful implementation of these policies, we can move toward a future that is more sustainable, equitable, and economically efficient.