Introduction

Driven by the dual imperatives of accelerating global warming and mounting demand for energy transition, low-carbon development of the power system has become a critical pathway to achieving carbon neutrality. As the world’s largest carbon emitter and electricity producer, China’s pursuit of deep decarbonization in the power system is vital for climate governance and the new energy transition (Liu et al., 2026; Myhre & Rosenberg, 2025). It also profoundly impacts macroeconomic stability and social sustainability (Yu et al., 2023).

Since the launch of the target of carbon peaking and carbon neutrality, China has witnessed continuous expansion in its renewable energy installed capacity and a notable rise in consumers’ willingness to adopt green electricity (Yang et al., 2026). Nevertheless, according to the latest data released by the National Energy Administration, China’s installed thermal power capacity has exceeded 1.4 billion kilowatts by the end of December 2024, representing approximately 43% of the total installed power capacity around the world (Wu et al., 2025a, 2025b). Meanwhile, the actual consumption share of non-hydro renewable energy remains far below the projected level (Fumey et al., 2026). The structural transition of the power system remains arduous (Zhang et al., 2024). It urgently requires an efficient evolution toward clean, low-carbon and intelligent energy systems.

Guided by the carbon peaking and neutrality goals, a research framework for low-carbon power transition has been established. This system prioritizes cost-effectiveness as its core and relies on policy mechanisms (Chai et al., 2023). Yuan et al. (2023) constructed a bottom-up energy-economic model, providing a roadmap for the orderly phase-out of coal-fired power. Based on multi-level agent-based modeling, Ghorbani et al. (2024) analyzed global inclusive approaches to the green energy transition. They provide deeper insights into inherent challenges and potential pathways toward sustainability, while aiming to unravel greenhouse gas dynamics. In terms of transition practice, the national energy system has promoted market-oriented institutional innovation via the pilot and promotion model. New carbon footprint requirements have driven the green power direct connection mechanism, with the development of carbon emission trading, low-carbon city pilot programs have achieved initial results (Armstrong, 2026). Green and low-carbon power system has witnessed vigorous development (Huang et al., 2026a). However, existing studies have paid limited attention to analyzing the interactive relationship among power generation enterprises, electricity consumers and competent government departments under the background of green power direct connection. These studies mainly focus on the two-party evolutionary game between the government and power generation enterprises, failing to fully consider the tripartite interaction mechanism involving power consumers in the green power direct connection scenario. There is lack of a systematic depiction of the dynamic evolutionary trajectory and endogenous regulation under policy intervention, leading to the unclear evolutionary laws and paths of the low-carbon transition of the power system.

Evolutionary game theory is a systematic research method that integrates game theoretical analysis with dynamic evolutionary process analysis (Wang et al., 2025a). Based on bounded rationality, it focuses on how groups dynamically adjust strategies and reach stable equilibria in long-term interactions. Li et al. (2026). In contrast to classical game theory, which applies to individual-level games, evolutionary game theory does not require participants to possess full rationality or complete information. It also differs from stochastic learning models, which are suitable for capturing the strategy-switching process of a single individual, as evolutionary game theory focuses on how the proportion of a strategy evolves within a population over time. Therefore, replicator dynamics are adopted in this study as the analytical tool.

The core concept of evolutionary game theory, Evolutionary Stable Strategy, was first proposed in 1973. It is based on Darwin’s theory of biological evolution and Lamarck’s genetic theory, and does not require participants to possess complete rationality or perfect information, thereby abandoning the assumption of complete rationality in traditional game theory. This enables evolutionary game theory to accurately depict the evolutionary patterns in multi-agent interactions, providing a feasible path for studying the dynamic evolutionary trajectories of multi-agent strategies under policy intervention. Later, the replicator dynamic equation was proposed. It quantifies the correlation between the growth rate of pure strategies and relative costs, and has become the core tool for dynamic analysis in evolutionary game theory. It demonstrates the feasibility of evolutionary game theory in the dynamic decision-making analysis of multiple agents from a theoretical perspective, as well as the most widely used dynamic equation of selection mechanism in evolutionary game theory (Khaw et al., 2021). Plenty of studies analyze the low-carbon power system via this theory. Liu and Wu (2024) identified more effective strategies for the micro-level application and diffusion of carbon capture, utilization and storage technology under the carbon trading framework. Yue et al. (2024) explored the impact of the change in the policy of issuing green certificates for hydropower on the strategic choices of the trading parties. Wu et al. (2025a, 2025b) examined how the China certified emission reduction scheme can be effectively diffused within the carbon trading mechanism.

To this end, based on the strategic context of the target of carbon peaking and carbon neutrality, this study comprehensively incorporates key factors, which include new energy generation deviation assessment, the proportion of green electricity consumption, electric energy trading, carbon trading and subsidy supervision. It innovatively establishes a dynamic evolutionary game model among enterprises, consumers and the government, systematically reveals the strategic evolution paths and equilibrium mechanisms of multiple agents, and further provides theoretical support for the scientific formulation of low-carbon transition policies. This study contributes to the United Nations’ Sustainable Development Goals on affordable and clean energy, while also advancing China's carbon peaking and neutrality targets.

The rest is organized as follows. Section "Model construction and parameter setting" elaborates on model construction and parameter setting. Section "Stability analysis of evolutionary game among three parties" constructs the payoff matrix for three game parties, and calculates the results of their replicator dynamic equations. Section "Numerical simulation analysis" carries out numerical simulation analysis with empirical evidence, including system evolution simulation, model variable assignment, evolutionary game analysis, and verification of model parameter influence and robustness. Section "Conclusion" summarizes the research findings and puts forward corresponding policy recommendations.

Model construction and parameter setting

Game model construction

The dynamic evolutionary game model established in this study comprises three parties, namely the power generation enterprises (PGE), the power consumers (PC) and the power regulatory government (PRG). They are, respectively, on the supply side, the demand side and the neutral regulatory party. The evolutionary game mechanism of low-carbon transition in power systems is illustrated in Fig. 1.

Fig. 1
Fig. 1
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Evolutionary game mechanism of low-carbon transition in power systems

Evolutionary game theory takes policy guidance as its orientation and focuses on the source–load interaction between the supply side and demand side of the power system, aiming to explore the low-carbon transition pathways for both the generation side and the load side. The supply side PGE, which can choose to invest in the research and development of clean energy power generation technologies, dismantle or retrofit existing power generation facilities with high carbon emission intensity, and raise the share of green power (Gumber et al., 2026). Alternatively, they may maintain the status quo. These two strategies are hereinafter referred to as choose to develop new energy and choose not to develop new energy, respectively.

The demand-side PC can choose to replace equipment with high-efficiency and energy-saving alternatives to reduce costs and improve efficiency (Ke et al., 2024). It participates in demand response programs, meets the requirements of green energy utilization, and supports the low-carbon transition of the power system (Ke et al., 2023a, 2023b; Liu et al., 2025). Alternatively, it may maintain the status quo. These two strategies are hereinafter referred to as choose to update equipment and choose not to update equipment, respectively.

As the main body responsible for macro policy implementation, the PRG can choose to implement relevant policy support and industrial subsidies to advance the low-carbon transition of the power system and achieve environmentally sustainable development (Algers et al., 2025; Campiglio et al., 2024). Based on whether it adopts strategic intervention in the transition process, the corresponding strategies are hereinafter referred to as choose to intervene and choose not to intervene, respectively (Marshall et al., 2026).

Basic assumptions of game theory

Assumption 1: All three game parties are in the initial stage, exhibiting bounded rationality in their strategic choices and behaviors.

Assumption 2: The costs and benefits incurred by all parties during the dynamic evolutionary game can be quantified and are positive values.

Assumption 3: None of the three parties adopt mixed strategies. Specifically, PGE will only select one of the two strategies: develop new energy or not develop new energy; PC will only select one of the two strategies: update equipment or not update equipment; and the PRG will only select one of the two strategies: intervene or not intervene.

Assumption 4: Following the agent-based modeling approach, the three game players are conceptual clusters representing the three main parties in the power system. Although PGE, PC, and PRG are constructed as single entities in the model, they serve as representative agents of their respective categories, and the replicator dynamics framework is then applied to describe how their strategic choices evolve over time.

Assumption 5: Considering that household consumers have relatively low electricity consumption and their decisions have a limited impact on the low-carbon transition of the power system, this study assumes that the PC are large industrial consumers. Furthermore, the benefits discussed in this study include economic benefits, environmental benefits, and social benefits.

Assumption 6: When a PGE chooses not to develop new energy and a PC chooses to update equipment, the consumer cannot rely entirely on traditional electricity due to carbon emission targets. Therefore, in addition to using traditional electricity supplied by the PGE, the PC will purchase a portion of electricity from the green power market as a supplement.

Assumption 7: This study uses photovoltaic power generation as a representative of new energy, excluding other green power sources such as wind power.

Assumption 8: Considering that new energy projects undertaken by PGEs are typically large-scale centralized infrastructure and limited in number, their electricity output can be directly cross-verified through metering data from the grid company, fraudulent behavior is relatively easy to detect and carries severe consequences. In contrast, PCs' new electricity consumption involves long verification chains and fragmented data sources; the large number of consumers and the decentralized nature of their operations make regulatory coverage difficult, relying mainly on sample-based inspections. Consequently, the probability of detecting fraudulent declarations by individual users is relatively low, and the risk of subsidy fraud on the consumer side is significantly higher than on the generation side. Based on these differences, our model incorporates fraud-related parameters only on the consumer side, while not considering PGE-side fraud. It should be noted, however, that as renewable energy subsidy auditing becomes increasingly stringent, and as the electricity market continues to develop with growing installed capacity, the possibility of PGE-side fraud, though low, cannot be completely ruled out in extreme cases.

Assumption 9: Let \(x\) be the probability that a PGE chooses developing new energy, so the probability of choosing not developing new energy is \(1 - x\). Let \(y\) be the probability that a PC chooses updating equipment, so the probability of choosing not updating equipment is \(1 - y\). Let \(z\) be the probability that the PRG chooses interference, so the probability of choosing non-interference is \(1 - z\). Here, \(x,y,z \in \left[ {0,1} \right]\), and all are functions of time \(t\).

Model variables

The variables involved in this study are presented in Table 1.

Table 1 Variables of the evolutionary game model
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Stability analysis of evolutionary game among three parties

Payoff matrix

Based on the aforementioned game relationship and main parameters, the game payoff matrix of the three parties, namely PRG, PGE, and PC, under the background of the low-carbon transition of the power system is presented in Table 2.

Table 2 Payoff matrix of evolutionary game model
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Analysis of tripartite game

Let the utilities of PGE choosing to develop new energy or not be \(U_{11}\) and.\(U_{12}\), respectively.

$$\begin{aligned} U_{11} =& (1 - y)(1 - z)( - A_{1} - A_{2} - B + G_{2} H) + z(1 - y)( - A_{1} - A_{2} - B + G_{2} H + N) \hfill \\ & + y(1 - z)( - A_{1} - A_{2} - B + G_{2} J) + yz( - A_{1} - A_{2} - B + G_{2} J + N), \hfill \\ \end{aligned}$$
$$\begin{aligned} U_{12} =& (1 - y)(1 - z)[ - C + (1 - p)G_{1} H] + z(1 - y)[ - C - D + (1 - p)G_{1} H] \hfill \\ & + y(1 - z)[ - C + (1 - p)G_{1} J] + yz[ - C - D + (1 - p)G_{1} J]. \hfill \\ \end{aligned}$$

Furthermore, the replicator dynamic equation \(G_{A} (x)\) can be derived as follows:

$$\begin{aligned} G_{A} (x) =& x(x - 1)[A_{1} + A_{2} + B - C - Dz + G_{1} H(py - p - y + 1) \hfill \\ & - G_{1} J(py - y) + G_{2} (Hy - H - Jy) - Nz]. \hfill \\ \end{aligned}$$

Let the utilities of PC choosing to update equipment or not be \(U_{21}\) and \(U_{22}\), respectively.

$$\begin{aligned} U_{21} =& (1 - x)(1 - z)[ - E + F - pG_{2} J - (1 - p)G_{1} J + K] + (1 - x)z[ - E + F - pG_{2} J - (1 - p)G_{1} J \hfill \\ &\, + K + O] + x(1 - z)( - E + F - G_{2} J + K) + xz( - E + F - G_{2} J + K + O), \hfill \\ \end{aligned}$$
$$\begin{aligned} U_{22} =& (1 - x)(1 - z)[(p - 1)G_{1} H - pG_{2} H - L] + xz( - G_{2} H - qrM + q\left( {1 - r} \right)O - L) \, \hfill \\ &\, + (1 - x)z[(p - 1)G_{1} H - pG_{2} H - qrM + q(1 - r)O - L] + x(1 - z)( - G_{2} H - L). \hfill \\ \end{aligned}$$

Furthermore, the replicator dynamic equation \(G_{{\text{B}}} (x)\) can be derived as follows:

$$\begin{aligned} G_{B} (y) =& y(y - 1)[ - G_{1} (H - J)(1 - p)(1 - x) - G_{2} (H - J)(p - px + x) \hfill \\& + E - F - K - L - z(Mqr + Oqr - Oq + O)]. \hfill \\ \end{aligned}$$

Let the utilities of PRG choosing to intervene or not be \(U_{31}\) and \(U_{32}\), respectively.

$$\begin{aligned} U_{31} =& (1 - x)(1 - y)[ - S - U + qrM - q(1 - r)O + D] + x(1 - y)[ - N - S - U + qrM - q(1 - r)O] \hfill \\& + (1 - x)y( - O - S - U + D) + xy( - N - O - S - U + V), \hfill \\ \end{aligned}$$
$$U_{32} = - (1 - x)(1 - y)T - x(1 - y)W - (1 - x)y(T + W).$$

Furthermore, the replicator dynamic equation \(G_{{\text{C}}} (x)\) can be derived as follows:

$$\begin{aligned} G_{C} (z) =& z(z - 1)[S + U - D(1 - x) - Mqr(1 - y) + (N - W)x \hfill \\ & + Oq(1 - r)(1 - y) - T(1 - x) + y(O + 2Wx - Vx - W)]. \hfill \\ \end{aligned}$$

According to Lyapunov’s First Method, the eigenvalues of the eight pure strategy equilibrium points of the tripartite evolutionary game system can be calculated using the Jacobian matrix, as shown in Table 3.

Table 3 Eigenvalues of pure strategy equilibrium points in the tripartite game
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In order to simplify the presentation of the eigenvalue expressions in Table 3, we define the following composite parameters. \(a = A_{1} + A_{2} + B - C\), represents the total cost for PGE to develop new energy. \(b = G_{2} - G_{1} \left( {1 - p} \right)\), denotes the electricity price difference for PC between using green electricity and traditional electricity. \(c = E - F - K - L\), represents the comprehensive cost for PC to update their equipment. \(d = \left( {1 - p} \right)G_{1} \left( {H - J} \right)\), denotes the additional electricity cost that PC must bear if they do not update their equipment. \(e = G_{2} \left( {H - J} \right)\), represents the electricity cost savings achieved by PC after updating their equipment. \(f = q\left( {M - Or - V} \right)\), denotes the expected net income of PC under the condition that they may engage in subsidy fraud. \(g = S + U\), represents the PRG’s policy costs related to the low-carbon transition of the power system.

In addition, the notations \(pe\), \(bH\), and \(bJ\) used in Table 3 represent products of the corresponding parameters rather than independent parameters. Here, the parameter \(p\), \(H\), and \(J\) are independent parameters defined in Table 1, while \(e\) and \(b\) are the composite parameters defined above.

Numerical simulation analysis

This study considers a PGE with an installed capacity of 350 MW. By adjusting the game parameters and initial strategy combinations of the agents, it quantitatively analyzes the variation law of convergence speed of the system’s evolutionary path under different parameter conditions.

Model variable assignment analysis

In terms of the parameters of PGE, the cost for PGE to dismantle and retrofit traditional power stations \(A_{1}\) can be calculated as 11.27 million CNY (Wang et al., 2024b). Based on the "Roadmap for the Development of China's Photovoltaic Industry (2024–2025)" released by the China Photovoltaic Industry Association (China Photovoltaic Industry Association, 2025). Taking the photovoltaic power station construction cost of 2.8 CNY/W and operation and maintenance cost of 0.038 CNY/W as the calculation basis, through equivalent conversion of power generation hours, and based on a discount rate of 7% and a 20-year calculation period to convert to present value using the annuity present value factor, the total cost for PGE to construct, operate, and maintain new energy power stations \(A_{2}\) is estimated to be 212.76 million CNY (Zhang et al., 2022). For consistency, all multi-year cost parameters presented below are discounted using this same approach. According to the deviation assessment rules of the electricity market, considering that electricity exceeding the 3% deviation threshold is penalized at 10% of the coal-fired electricity price, the loss due to power generation deviation of new energy power stations \(B\) is approximately 2.57 million CNY (Ke et al., 2022). Based on the unit electricity cost of thermal power units and their operating costs, and assuming an electricity price of 0.2659 CNY/kWh, the material, operation, and maintenance costs for PGE do not develop new energy \(C\), based on 1,000 million kWh of electricity generation, are estimated at approximately 265.9 million CNY (Feng et al., 2025; Wang et al., 2025b). Then, as for the electricity penalty standards, the loss incurred by PGE that do not develop new energy and are urged by the PRG to transform \(D\) is 2.5 million CNY (Du et al., 2025).

In terms of the parameters of PC, based on relevant policy requirements, the parameter is assigned in accordance with the 21% green electricity consumption ratio, so the value of \(p\) is 0.21 (Dong et al., 2019). Combined with the existing equipment renewal data in “Compilation of Typical Cases of Energy Green and Low Carbon Transformation (2024)” (National Energy Administration, 2024), the value of \(E\) is set at 78.55 million CNY (Hu et al., 2025). According to case data from the examples such as the green smart energy utilization practices, the electric energy demand when PC do not update green energy-saving equipment \(H\) is approximately 100,000,000 kWh, and the electric energy demand after PC update their equipment \(J\) is 80,000,000 kWh (Wang et al., 2024a). The economic benefits brought by PC’s research and development and equipment renewal to improve productivity \(F\) are about 73.85 million CNY. According to publicly released electricity price data, \(G_{1}\) and \(G_{2}\) are, respectively, valued at 0.367 CNY/kWh and 0.414 CNY/kWh. Based on the data released in 2025, taking the carbon emissions from 1 kWh of electricity produced by Southern Power Grid Corporation are approximately 0.572 kg and the carbon trading market price of 70.45 CNY/ton as the calculation benchmark, the corresponding parameter \(K\) is calculated to be approximately 8 million CNY (Huang et al., 2026b). Combined with environmental operation cost accounting, the relevant parameter \(L\) is approximately 7.2 million CNY (Ke et al., 2023a, 2023b).

In terms of the parameters of PRG, assuming that 100 out of 400,000 PGE engage in subsidy fraud, the value of \(q\) is 0.00025. Based on “The Special Action Plan for Energy Conservation and Carbon Reduction in the Iron and Steel Industry (NDRC Huanzi [2024] No. 730)”, which was released on May 27, 2024, and according to the rule that the penalty for subsidy fraud is three times the amount of the illegal funds, \(O\) and \(M\) can be calculated as 11.78 million CNY and 35.34 million CNY, respectively. Combined with the subsidy policy of Guangdong Province at 0.15 CNY/kWh, \(N\) can be calculated as 12.54 million CNY (Song et al., 2021). The cost for the PRG to formulate and implement favorable policies such as tax incentives can be measured by the funds of PRG research and consulting projects, so the value of \(S\) is approximately 2 million CNY (Zhang et al., 2025). The additional environmental governance cost borne by the PRG when the power system fails to achieve successful transition is the annual special fund for the critical battle of pollution prevention and control, and \(T\) is approximately 3 million CNY (Wang et al., 2021). According to carbon emission verification reports of regulated PGE in the national and Guangdong carbon markets, the PRG inspection coverage rate \(r\) is 0.90, and the inspection cost \(U\) is approximately 100,000 CNY. The parameter about social credibility \(V\) gained by the PRG from promoting the successful transition of the power system is 15 million CNY, while the social credibility \(W\) lost due to the PRG’s failure to actively promote the transition is 177 million CNY.

Furthermore, the stability judgment parameters can be calculated as follows. \(a\) = − 39.3 million CNY; \(b\) = 0.1241 CNY/kWh; \(c\) = − 10.5 million CNY; \(d\) = 57.99 million CNY; \(e\) = 82.8 million CNY; \(f\) = − 0.7657 million CNY; \(g\) = 2.1 million CNY.

Evolutionary game analysis

Let \(x\),\(y\) and \(z\) be 0.1, 0.3, 0.5, 0.7 and 0.9, respectively. The simulation diagram of the three-party evolutionary game is obtained, as shown in Fig. 2. According to the eigenvalue distribution of the eight evolutionary equilibrium points, it can be concluded that \(E_{2} \left( {0,0,1} \right)\), \(E_{3} \left( {0,1,0} \right)\), \(E_{4} \left( {0,1,1} \right)\), \(E_{5} \left( {1,0,0} \right)\), \(E_{6} \left( {1,0,1} \right)\) and \(E_{8} \left( {1,1,1} \right)\) are saddle points; \(E_{1} \left( {0,0,0} \right)\) is an unstable point, \(E_{7} \left( {1,1,0} \right)\) is the stable point. The tripartite game system finally evolves to the equilibrium state, which refers to the strategies that PGE develop new energy, PC update equipment, and the PRG does not interfere.

Fig. 2
Fig. 2
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Evolution simulation results

Then, further analyze the impact of different initial values of \(x\), \(y\) and \(z\) on the evolution speed of the system. We consider that the evolutionary system reaches stability when the probability value hits 0.999 or 0.001.

Firstly, the initial values of \(x\),\(y\) and \(z\) are all set to 0.5. Figure 3 depicts the evolutionary diagram of the strategy selection probabilities of the three parties under this scenario. It can be observed that the convergence speed of the strategy selection probability for PGE is the fastest, reaching a convergent state at \(t = 4.8126\). The strategy selection probability of PC shows a characteristic of being fast at first and then slow, and tends to converge at \(t = 7.3887\). The convergence speed of the PRG’s strategy selection probability follows an S-shaped curve (Dai et al., 2024). The convergence speed is relatively slow when \(t \in [0,0.5]\), while it increases significantly when \(t \in [0.5,2.5]\), and slows down when \(t \in [2.5,5]\), finally, converge at \(t = 5.2450\).

Fig. 3
Fig. 3
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Evolution diagram with an initial selection probability of 0.5 for the three parties

Secondly, only change the value of \(x\). The initial probability \(x\) for PGE to develop new energy is set to 0.2, 0.5 and 0.8, respectively, and the impact of PGE’s initial strategy choice on the evolutionary results of the three-party game is observed. Figure 4 depicts the evolutionary diagram of the strategy selection probabilities of the three parties under this scenario. Under the three cases with different initial user strategy proportions, the PGE's strategy converges to the stable state at 5.8262, 4.8126 and 3.8000, respectively, it can be observed that with the increase of the initial \(x\) value, the time for PGE to evolve to a stable state is shortened, indicating that PGE with a higher awareness of green power generation will take the lead in the market. The PC’s convergence times are 7.4696, 7.3887, and 7.4763 for the three cases, while the PRG’s are 5.5066, 5.2450, and 5.0331, respectively. Therefore, the change in the initial \(x\) value has weak impact on the evolutionary speed of PC, revealing the problem of insufficient coordination mechanism between the power generation side and the power consumption side. The increase of the initial \(x\) value has a slight accelerating effect on the PRG’s decision-making behavior.

Fig. 4
Fig. 4
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Evolution diagrams with different initial \(x\)

Thirdly, only change the value of \(y\). The initial probability \(y\) for PC to update equipment is set to 0.2, 0.5 and 0.8, respectively, to observe the impact of PC’s initial strategy choice on the evolutionary results of the three-party game. Figure 5 depicts the evolutionary diagram of the strategy selection probabilities of three parties under this scenario. Under the three cases with different initial user strategy proportions, the PGE's strategy converges to the stable state at 4.5263, 4.8126 and 4.8792, respectively, it can be observed that the change in the initial \(y\) value has almost no impact on the evolutionary speed of PGE, which is consistent with the analysis result that the change in the initial \(x\) value affects PC. With the increase of the initial \(y\) value, the time for PC to evolve to a stable state is shortened; they are 8.8120, 7.3887, and 5.2450, respectively, indicating that the PC with green electricity consumption awareness have greater advantages in sustainable development. The increase in the initial \(y\) value exerts an obvious reverse incentive effect on the PRG’s decision-making behavior at the initial stage, and shortens the time for the PRG to evolve to a stable state, they are 6.6867, 5.2450, and 4.4554, respectively, which shows that green electricity consumers play a more important role in the implementation and landing of relevant policies.

Fig. 5
Fig. 5
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Evolution diagrams with different initial \({\text{y}}\)

Fourthly, only change the value of \(z\). The initial probability \(z\) of PRG intervention is set to 0.2, 0.5 and 0.8, respectively, to observe the impact of the PRG’s initial strategy choice on the evolutionary results of the three-party game. Figure 6 depicts the evolutionary diagram of the strategy selection probabilities of the three parties under this scenario. Under the three cases with different initial user strategy proportions, the PGE's strategy converges to the stable state at 4.8694, 4.8126 and 4.7073, while the PC's converges at 7.4738, 7.3887 and 7.2460, respectively, It can be observed that the change in the initial \(z\) value has weak impact on the evolutionary speed of both PGE and PC. As the initial \(z\) value increases, the time for the PRG to evolve to a stable state is prolonged; they are 4.5426, 5.2450, and 5.8963, respectively, reflecting the implementation inertia of intervention policies. When the PRG’s initial willingness to intervene is excessively high, it is often difficult to withdraw flexibly to ensure the stable operation of the market.

Fig. 6
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Evolution diagrams with different initial \({\text{z}}\)

Upon comprehensive comparison of the three sets of graphs mentioned above, the following conclusions can be drawn as follows.

First, the change in the willingness of PGE to develop new energy is slightly affected by the other two parties in the game, but depends more on their own initial willingness. This indicates that the core driving force for PGE to promote the low-carbon transition of the power system is their own interests and the long-term development trend of the industry. With the advancement of energy-saving and emission reduction targets, the transition and development of new energy is inevitable, resulting in a weak correlation between the decision-making behavior of PGE and external initial willingness. At the same time, it can be found that the willingness of PGE to develop new energy is slightly affected by the PRG’s willingness to intervene, which is consistent with the inference in reference. This phenomenon may be attributed to the large difference in transition subsidies between PGE and PC, leading to significantly lower transition benefits for PGE than for PC. This reduces the enthusiasm of PGE to develop new energy when the PRG intervenes in the low-carbon transition of the power system (Yang & Feng, 2026).

Second, the change in PC’s willingness to update equipment is also slightly affected by the other two parties in the game. It is speculated that this phenomenon is due to PC’s downstream position in the electricity market, limited policy attention, and the long cost recovery period for new energy systems and energy-saving facilities. As a result, PC’s willingness to update equipment is more affected by their own awareness of green electricity consumption, and has no obvious correlation with the initial willingness of PGE and the PRG.

Third, the change in PRG intervention willingness is related to the initial willingness of all three parties in the game, but is most significantly affected by PC’s willingness to update equipment. Higher initial willingness of PGE and PC to transition indicates lower resistance, prompting the PRG to accelerate its withdrawal and promote market independence. For the PRG, the increase in PC’s willingness to update equipment can better represent the smooth progress of the low-carbon transition of the power system than the increase in the probability of PGE developing new energy, so it is more significantly affected by PC.

The above findings do not imply that government policies are mathematically ineffective. Rather, under the specific parameter configuration of this study, the effects of subsidies and penalties on PGE and PC decisions are relatively small compared to other cost components, such as construction costs and equipment updating costs. In other words, under the current parameter configuration, these effects are masked by other cost components, rather than being fundamentally absent. When the intensity of government intervention exceeds certain thresholds, its influence may become significant. This also suggests that policy instruments need to reach sufficient intensity to effectively steer the low-carbon transition. Conversely, in the context of our policy goal where the government eventually withdraws from intervention, a phased subsidy withdrawal strategy would be a reasonable approach, as it would not cause significant fluctuations in the behavior of the other two parties.

Model parameter influence and robustness

In the game model, the values of relevant parameters may affect the evolutionary results. To verify the robustness and reliability of the simulation results, parameter estimation deviations including cost fluctuations, market price changes and project differences are taken into account. A 10% fluctuation threshold is set to calculate the variation of eigenvalues under extreme scenarios. Taking the three eigenvalues of \(E_{7} \left( {1,1,0} \right)\) as an example, its first eigenvalue is \(- G_{2} J\). Positive parameters are set to 110% of their original values and negative parameters to 90%. If multiple parameters are involved, their product is adopted. For example, \(- G_{2} J\) becomes \(- G_{2} J*0.9*0.9\) under extreme scenarios. From this, the first eigenvalue under extreme scenarios can be calculated as − 0.2259, which still satisfies the requirement of negative eigenvalues. Similarly, the second and third eigenvalues under extreme scenarios are calculated as − 0.0119 and − 0.0728, respectively. Therefore, within a 10% fluctuation range, the evolutionary equilibrium point remains unchanged, indicating that the model results have a certain degree of robustness.

As shown in Fig. 7, from the perspective of evolutionary speed, with the continuous expansion of the fluctuation range, the strategic evolution rate of PGE and PC slows down under extreme scenarios. Among them, PC has the highest sensitivity to parameter values. When the parameters change by 10%, the evolutionary curve of their strategy selection probability is approximately a horizontal line. This indicates that PC tend to choose not to update equipment when the comprehensive cost of equipment renewal rises by more than 10% or the electricity cost savings after equipment renewal drop by more than 10%. Only when the technological effects of equipment are continuously improved, production costs are steadily reduced, and the benefits of green electricity are realized can PC be motivated to update equipment spontaneously. When parameters fluctuate within the range of 0–10%, the strategic evolution rate of PRG departments remains unchanged. This shows that under the current comprehensive environment of the new energy market, policy subsidies, environmental values, and government governance, the government can choose to withdraw from intervention gradually, allowing the market to drive the sustainable development of new energy. In terms of evolutionary time, the system tends to converge after the fifth period, and government intervention can be withdrawn at this stage.

Fig. 7
Fig. 7
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Evolution diagrams under fluctuations of different eigenvalues

Conclusion

This study constructs a dynamic evolutionary game model involving PGE, PC and the PRG, revealing the multi-agent cost game mechanism and strategy evolution rules in the low-carbon transition of the power system. Simulation results indicate that under the tripartite game framework, the system eventually converges to the ideal equilibrium of PGE developing new energy, electricity consumers updating equipment, and the PRG withdrawing from intervention. The PRG’s initial intervention intention exerts a reverse incentive effect, and it tends to adopt the non-intervention strategy in the later stage. Based on the above conclusions and the eigenvalues in Table 3, the following policy recommendations are proposed.

First, a phased subsidy withdrawal strategy can be implemented in an orderly manner, with priority given to phasing out subsidies on the power generation side. In the early stage of the transition, to achieve the stability condition \(V < N + O + g\), generation-side subsidies and consumer-side subsidies must be maintained at sufficiently high levels to overcome transition resistance. However, since the PRG's equilibrium strategy is non-intervention, these subsidies should be gradually phased out as the system approaches the stable state.

Then, optimize the consumer-side subsidy scheme. On the one hand, one-time subsidies can be shifted to dynamic grants based on actual energy savings, so as to reduce the risk of subsidy fraud. On the other hand, consumer-side subsidies can be gradually phased out according to the feedback of power consumers. This ensures that the PRG can evolve toward non-intervention.

Besides, strengthen supervision and punishment. Blockchain technology can be integrated to realize green electricity traceability, link subsidy issuance with environmental compliance, and carry out dynamic verification and penalty for violations. This suggestion is to increase the value of the composite parameter \(f\), to destabilize the unideal equilibrium and confirm that the PRG's equilibrium strategy is non-intervention instead of intervention.

Last but not least, promote the demand-side response mechanism. Given the stable conditions \(a < bJ\) and \(c < e\), increasing the electricity price gap between PCs who update equipment or not will accelerate the system's convergence toward the low-carbon equilibrium. Thus, establish a linkage mechanism between the green electricity consumption ratio and electricity price, and incentivize consumers to save energy and reduce consumption through price signals such as peak–valley electricity prices.

The theoretical contributions of this study are threefold. First, in terms of the modeling framework, existing studies have mainly focused on two-party games between the government and power generators, while the active role of electricity consumers in the low-carbon transition has not been fully captured. In this study, consumers are incorporated into a tripartite evolutionary game framework, and a "PGE–PC–PRG" interaction model is constructed to fully characterize the strategic interplays among the supply side, demand side, and regulatory side. Second, in terms of theoretical findings, through numerical simulation and eigenvalue analysis, the transmission mechanism through which consumer willingness accelerates government withdrawal is identified, revealing that consumer willingness is a key driver in pushing the government from intervention toward non-intervention along the equilibrium path, which is a mechanism that has not been systematically discussed in previous literature. Third, in terms of policy design, policy recommendations are directly linked to model parameters and stability conditions, and quantifiable policy instruments such as a phased subsidy strategy, optimized supervision, and demand-side price signals are proposed, providing clear theoretical foundations and operational guidance for policy implementation. Finally, in practical terms, actionable references for the pathway design of low-carbon transition in the power system are offered through these policy recommendations, with particular value in guiding the pace of subsidy withdrawal and the design of regulatory mechanisms.