The CEG framework is illustrated in Table 1. Here, p is described as the fraction of the risk unit group that bids with low risk and q as the fraction of the regulator who applies a light punishment, so the fraction of the risk unit group that bids with high risk as 1-p and the fraction of the regulator who applies a severe punishment as 1-q. a ₁–a ₄ and b ₁–b ₄ are payoffs. For the sake of the formulas’ simplicity, we define four relative net payoffs for four strategy combinations as follows (Cheng and Yu, 2018): (1) α = a ₁–a ₃, (2) β = a ₂–a ₄, (3) γ = b ₁–b ₂, and (4) δ = b ₃–b ₄.

Replicator dynamics equations (RDEs) for risk units and the regulator are Eq. (1) and Eq. (2), respectively (Amann and Possajennikov, 2009). d p d t = p 1 − p q α − β + β , (1) d q d t = q 1 − q p γ − δ + δ , (2) where 1-p and 1-q are non-negative and have no substantial effect on the evolutionary results of strategy selection. For the sake of discussion, Eq. (1) and Eq. (2) are simplified to the following equations. d p d t = p q α − β + β , (3) d q d t = q p γ − δ + δ . (4)

Introducing Gaussian white noise into Eq. (3) and Eq. (4) (Xu et al., 2015), the one-dimensional Itô stochastic replicator dynamic equations (SRDEs) for players are as follows: d p t = p t q t α − β + β d t + μ 1 p t d ω t , (5) d q t = q t p t γ − δ + δ d t + μ 2 q t d ω t , (6) where ω(t) is a one-dimensional standard Brownian motion, representing a random fluctuation phenomenon, that effectively reflects how players are affected by stochastic disturbance factors. dω(t) obeys the normal distribution N (0, h ). μ ₁ and μ ₂ represent the intensity of stochastic disturbance. The strategy probabilities at time t for the low-risk bidding and mild punishment are denoted as p(t) and q(t), respectively.

By combining the stochastic Taylor expansion and the Itô stochastic formula, the nonlinear Itô SRDE can be expanded and solved with Gaussian random disturbances (Hu et al., 2008).

Considering the following general Itô stochastic differential equation (SDE) (Kamrani, 2015), d x t = f t , x t d t + g t , x t d ω t , (7) where t∈[t ₀,T], x(t ₀) = x _0, x ₀∈R, h=(T-t ₀)/r, and t _s = t ₀+sh. r is the number of sampling times, t ₀ is the initial time point, and t _s is the sth sampling time point. s∈{0,1,…,r}, performing a stochastic Taylor expansion on Eq. (7), we obtain x t n + 1 = x t n + J 0 f x t n + J 1 g x t n + J 11 G 1 g x t n + J 00 G 0 f x t n + R , (8) where J ₀ = h, J ₁ = ∆ω _n , J ₁₁ = [(∆ω _n )²-h]/2, G⁰ = ∂ g x / ∂ x + 1 / 2 ∂ 2 g x / ∂ x 2 , G ¹ = ∂ g x / ∂ x , J ₀₀ = h ²/2, and R is the remainder term of the expansion.

In practical applications, the Milstein method can be used to numerically iterate and solve the SDE by truncating some terms in the stochastic Taylor expansion (Kai and Guiding, 2022). The format of the Milstein method is as follows: x t n + 1 = x t n + h f x t n + Δ ω n 2 − h g x t n g ′ x t n / 2 . (9)

Based on Eq. (9), the equilibrium solutions for SRDEs Eq. (5) and Eq. (6) can be calculated. The stability analysis of their stochastic evolution is as follows (Hu et al., 2008; Kamrani, 2015):

Given an SDE, suppose there exists a function V(t, x) such that c ₁|x|l ≤ V(t, x)≤c ₂|x|l, where c ₁, c ₂, and l are all positive constants. If there exists a positive constant σ such that V*(t, x)≤-σV(t, x), then the l-order moment of the zero-solution of Eq. (7) is said to be exponentially stable, and it holds: E x t , x 0 l < c 2 / c 1 x 0 l e − η t , t ≥ 0 , (10) V * t , x = V t t , x + V x t , x f t , x + g 2 t , x V x x t , x / 2 . (11)

According to the aforementioned theorem, for Eq. 5 and Eq. 5, taking V(t, p(t)) = p(t), where p(t)∈(0,1],V(t, q(t)) = q(t), where q(t)∈(0,1], c ₁ = c ₂ = 1, l = 1, η = 1, then V*(t, p(t)) = f(t, p(t)) = p(t)[q(t)(α-β)+β],V*(t, q(t)) = f(t, q(t)) = q(t)[p(t)(γ-δ)+δ]. If the l-order moment of the zero-solution is stable, it must satisfy the condition that p t q t α − β + β ≤ − p t , (12) q t p t γ − δ + δ ≤ − q t . (13) Subsequently, the proposition of Eq. (12) can be expressed as follows:

C₁, when p(t)∈(0,1], α-β>0, it holds that q(t)≤(-β-1)/(α-β), and α+1 ≥ 0.

C_2, when p(t)∈(0,1], α-β<0, it holds that q(t)≥(-β-1)/(α-β), and α+1 ≤ 0.The conditions to satisfy Eq. (13) are as follows:

C_3, when q(t)∈(0,1], γ-δ>0, it holds that q(t)≤(-δ-1)/(γ-δ), and γ+1 ≥ 0.

C₄, when q(t)∈(0,1], γ-δ<0, it holds that q(t)≥(-δ-1)/(γ-δ), and γ+1 ≤ 0.

The paper randomly sets the value of α, β, γ, and δ, which satisfied the aforementioned propositions C₁ and C₃. Then, the zero-solution exponential stability of Eq. 1–Eq.2 and Eq. 5–Eq. 6 is simulated. The trend in Figure 1 demonstrates the validity of the SEG model and the differences between SEG and CEG models. It is evident that SEG and CEG models show different features. The SEG model can simulate randomness and uncertainty in the real world, whereas the CEG model is more suitable for problems with well-defined strategies and rules. Figure 1A illustrates a more realistic evolutionary path of players’ strategies. In the period of 0 h–39 h, strategy probabilities experience significant fluctuations in the SEG model. The fluctuations better simulate players’ decision-making interfered with by various external or internal factors. By contrast, the CEG path remains smooth, with players evolving undisturbed in an ideal state, as presented in Figure 1B.

The pure strategy game matrix of players is shown in Table 2. A _d,i and B _d,i are all positive normalized indicators, which respectively denote the payoffs of each strategy combination on trading day d _i .

The optimal market-clearing model implements the paper’s risk prevention mechanism by increasing the bid prices of risk units during market clearing, which uses market-based measures to constrain the winning capacity of risk units rapidly and accurately. Our earlier work on this model is detailed in Xie et al. (2023a). In this paper, the optimal market-clearing model is used as the same as our previous work.

The “stochastic evolutionary game–optimal market-clearing” model follows a top–down iterative process, as depicted in Figure 3. The specific steps are as follows.

(1) On trading day d _i , the equilibrium stability analysis is conducted through the SEG model to determine a stable equilibrium point (x, y) and the stochastic evolutionary stabilization strategy (SESS) of players. Then, the risk unit group’s SESS and the regulator’s SESS will be outputted. Additionally, data regarding four strategy combinations of players are computed.

In the market-clearing process, risk units are subject to bid price adjustments based on their risk levels. It aims to reduce their winning capacity as a form of “penalty.” Penalties are positively correlated with the magnitude of bid price revisions. The development of the regulator’s risk prevention strategy considers both subjective and objective factors.

When considering objective factors in the power spot market, several indicators are formulated based on the aspect of risk units’ power generation incomes, risk units’ historical performances, and risk units’ bidding behaviors.

The electricity market has been shown to involve both renewable and conventional energy units in this paper. Compared to conventional energy units, renewable energy units have a distributed nature. It imposes requirements on units’ output stability and regulatory capabilities. In this paper, it can be approximated that their game strategy selection and formulation are assumed to be identical.

Risk units adjust bid strategies based on their own benchmark prices. Risk units, being the group with bounded rationality and asymmetric information compared to the regulator, rely more on subjective risk preferences when formulating a bid strategy (Jiao et al., 2017). Specifically, the high-risk bid strategy and low-risk bid strategy are defined as shown in Eq. (37) and Eq. 38, respectively: p t , n h = p t , n b S 1 x , (37) p t , n l = p t , n b S 2 x , (38) where S ₁(x) and S ₂(x) are risk-modifying factors that represent the subjective risk preference of risk units. p t , n b represents the benchmark bid price for risk unit n at time t. According to reports of China’s National Development and Reform Commission (National Development and Reform Commission, 2021a; National Development and Reform Commission, 2021b), the range of risk bid price is defined as a 20% fluctuation around the benchmark price. The values of S ₁(x) and S ₂(x) can be found in Section 5.1.

To calculate the strategy data for players in the SEG, it is necessary to establish data fuzzy classification rules. Additionally, strategy modifying factors C ₁, C ₂, S ₁(y), and S ₂(y) and risk preference factors S ₁(x) and S ₂(x) need to be determined.

The fuzzy classification rules for probability and indicator data are as follows.

(1) Strategy selection probabilities

Probabilities {x, y, 1-x, 1-y} are divided into four classes, as shown in Table 3.

According to market trading results, the upper, middle, and lower quartiles, j _i , g _i , and e _i , are chosen to classify I RUGR n , d and I RUOB n , d in a fuzzy manner. See Table 4 and Table 5.

The membership functions for the modifying factors C ₁ and C ₂ can be represented by Eq. (50) and Eq. 51, respectively. C 1 = φ 1 ⁡ exp p 1 + p 2 + ν 1 , (50) C 2 = φ 2 / q 1 + q 2 + ν 2 . (51)

When I RUGR n , d is smaller, small, large, and larger, p ₁ can be taken as 1, 2, 3, and 4, respectively, and the same relationship exists with I RUOB n , d and p ₂. When the historical anomaly level is better, good, bad, and worse, q ₁ can be taken as 1, 2, 3, and 4, respectively, and the same relationship exists between credit level and q ₂. φ ₁, ν ₁, φ ₂, and ν ₂ that are coefficients to be determined.

S ₁(y) and S ₂(y) are determined by values of τ, C ₁, and C ₂. They are divided into four parts equally, each forming a membership function with probabilities y and 1-y, respectively, as shown in Table 6.

S ₁(x) and S ₂(x) are determined by the upper and lower limits of risk units’ bid price. They are divided into four parts equally, each forming a membership function with probabilities x and 1-x, respectively, as shown in Table 7.

Once the set of risk units is determined, a simulation analysis can be performed using the MATLAB platform. The initial strategy selection probabilities are set as x(0) = 0.7 and y(0) = 0.6. The initial time is t ₀ = 0, and the number of samples, r, is set to 1,500. The simulation time, T, is 240 h. The random disturbance intensity index, a, is taken as 0, 1, or 2. In this section, a detailed analysis of the evolutionary dynamics will be conducted specifically for trading day d ₁. The analysis principles for the remaining trading days (d ₂ to d ₁₀) are the same and will not be reiterated. The detailed analysis for trading day d ₁ is as follows.

1. Because of the random disturbance, the evolutionary game system exhibits fluctuations as it converges to the equilibrium point, as is shown in Figure 5. Finally, the combination of players’ SESS is the risk units’ low-risk bid and the regulator’s light punishment.

Due to random factors such as risk awareness or risk preference, the strategy probabilities of the risk group fluctuate, and the fluctuation increases with a stronger disturbance. It is evident that various factors, including risk awareness and risk preference, introduce randomness into the strategy probabilities of risk groups. Moreover, this randomness becomes more pronounced with increasing external disturbances. As the game unfolds, risk units tend to gravitate toward stable, low-risk bidding strategies in order to safeguard their baseline incomes. From the regulatory standpoint, decision-making is influenced by stochastic elements such as market-clearing outcomes, appeals, and public sentiment. When the level of random disruption intensifies, regulatory decisions exhibit more substantial fluctuations in the initial stages of evolution.

When applying the combination of a low-risk bid and light punishment strategy, the market risk level decreases and the overall market environment tends to stabilize. Therefore, there is no significant fluctuation in the evolution path of players on trading days d ₂ and d ₃, as shown in Figure 6.

It is worth noting that the tight electricity supply–demand on trading days d ₅ to d ₁₀ creates the willingness for risk units to increase their risk level. The risk behavior also shifts from intentional withholding to extremely high offers. As is shown in Figure 7, on trading day d ₄, there is a change in the evolutionary process due to the decrease in the supply–demand ratio. This indicates that even with the implementation of risk prevention strategies, there is still a motivation for risk units to increase their bid risk level in the game with the regulator. There is a lag in the decision-making process of the regulatory authority.

The evolutionary process of trading day d ₅ is depicted in Figure 8. The light punishment strategy appears to have insufficient deterrence on the risk group, resulting in risk units continuing to adopt a high-risk bid strategy. Consequently, the regulator experiences a decline in its income, leading to a shift back to the strong penalty strategy.

The evolutionary paths shown in Figure 9 illustrates that risk units rapidly change to a low-risk bid strategy on trading day d ₆ due to the regulator’s imposition of strong penalties. In contrast, the regulator’s strategy evolves at a slower pace, indicating concerns about the possibility of risk units reverting to a high-risk bid behavior. Random disturbances accelerate the evolution time for players with a maximum reduction of over 12 h when the intensity coefficient a is set to 2.

After the SEG of multiple trading days, the regulatory mechanism has successfully suppressed the willingness of risk units to increase their bid risk. Meanwhile, in an effort to minimize administrative intervention in the power market, the regulator has maintained a light punishment strategy. As a result, the market operation has entered a virtuous cycle. The evolutionary paths of players on trading days d ₇ to d ₁₀ are depicted in Figure 6.

Due to penalties imposed by the regulator, the winning capacities of risk units on each day have decreased. Among them, the decrease in the awarded capacities of thermal power units is significantly greater than those of renewable energy units, as detailed in Table 10. The observed phenomenon can be attributed to the following reasons: Thermal power units typically have higher declared capacities and are more sensitive to changes in prices compared to renewable energy units. Therefore, the impact of different bid risk levels on winning capacities of thermal power units is much greater than that of renewable energy units.

Obviously, with the increase in the level of bid risk or the intensity of punishments, the decline rate of risk units’ winning capacities becomes larger. Table 9 presents changing rates of winning capacities for different SESS between the regulator and risk units, highlighting this effect. In more extreme market conditions, where players adopt a strategy combination of severe punishment and a high-risk bid, the average change rate for risk units approaches 50%. When players adopt a combination of the mild penalty and low-bidding risk strategy in milder market environments, this value does not exceed 14%.

After the regulator has imposed penalties on risk units, regardless of the strength of punishment, the average market-clearing price has been found to decrease. The maximum decrease has reached 156.9 yuan/MWh, with a maximum change rate of 17.3%.

Given the constrained supply–demand conditions on trading days d ₅ –d ₁₀, the reduction in electricity prices during this period is less pronounced compared to days d ₁ to d ₄, when supply and demand are more balanced. When comparing the average electricity price decrease of 139.1 yuan/MWh and an average price volatility of 17.2% during trading days d ₁ to d ₄, a slightly lower average decrease can be found during trading days d ₅ to d ₁₀, i.e., 98.5 yuan/MWh, accompanied by an average volatility of 9.8%. It is evident that the prevailing supply and demand conditions significantly influence the effectiveness of the risk prevention mechanism in the spot market, with more favorable risk prevention outcomes observed during periods of lesser supply and demand. The specific situation of market power risk prevention is shown in Table 10.

In summary, through the proposed risk prevention mechanism on the generation side of the electricity spot market, market risks brought by market power are effectively mitigated. Price guidance is achieved, which leads to lower trading prices and further improvements in social welfare in electricity transactions.