A method for generating wind and light output scenarios based on a diffusion model improved by multi-head attention

By constructing a meteorological feature matrix and a historical wind and solar power generation matrix, and combining a denoising diffusion probability model and a multi-head attention mechanism, a high-precision wind and solar power output scene is generated, which solves the problem of inaccurate wind and solar power output scene generation in existing technologies and achieves higher accuracy and diversity of wind and solar power output scenes.

CN122263602APending Publication Date: 2026-06-23MAINTENANCE BRANCH OF STATE GRID FUJIAN ELECTRIC POWER +1

Patent Information

Authority / Receiving Office
CN · China
Patent Type
Applications(China)
Current Assignee / Owner
MAINTENANCE BRANCH OF STATE GRID FUJIAN ELECTRIC POWER
Filing Date
2026-03-12
Publication Date
2026-06-23

AI Technical Summary

Technical Problem

Existing technologies struggle to generate high-precision wind and solar power output scenarios and cannot effectively account for the impact of multi-dimensional meteorological data, resulting in high uncertainty in power system planning and operation.

Method used

A meteorological feature matrix and a historical wind and solar power generation matrix are constructed. By combining a denoising diffusion probability model and a multi-head attention mechanism, a wind and solar power output scenario is generated. The scenario is then optimized using a conditional denoising diffusion probability model and compared with the measured values.

Benefits of technology

It improves the accuracy and diversity of wind and solar power output scenarios, and can better explore the nonlinear coupling relationship between meteorological characteristics and wind and solar power output fluctuations, generating high-fidelity and diverse wind and solar power output scenarios.

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Abstract

The application relates to a wind and light output scene generation method based on a diffusion model improved based on a multi-head attention mechanism and belongs to the technical field of scene generation. First, wind and light historical power generation and corresponding meteorological data are collected, key meteorological factors are screened by using a Pearson correlation coefficient, and a meteorological feature matrix and a wind and light historical power generation matrix are constructed. Second, historical power data are processed by using a denoising diffusion probability model to generate a preliminary output scene. Third, a multi-head attention mechanism is introduced to improve the diffusion model, so that the diffusion model can effectively fuse multi-dimensional meteorological features, thereby constructing a conditional denoising diffusion probability model to generate a high-precision wind and light output scene considering the influence of meteorological fluctuations. Finally, the model is used to generate a wind and light power day-ahead scene, which is compared with measured values for verification. By fusing meteorological conditions and a diffusion model, the application can generate diversified wind and light output scenes more in line with actual statistical characteristics, and the precision of scene generation is improved.
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Description

Technical Field

[0001] This invention belongs to the field of scene generation technology, specifically relating to a method for generating wind and solar power output scenes based on a diffusion model improved by multi-head attention. Background Technology

[0002] With the rapid growth of new energy installed capacity, the penetration rate of renewable energy sources such as wind power and photovoltaics in the power system continues to increase. However, their output is driven by weather conditions and exhibits strong randomness, intermittency, and non-stationarity. Coupled with multiple time scales and significant spatial correlations, power source uncertainty becomes a core constraint for system planning, operation, and safety assessment. To support key issues such as source-grid-load-storage coordinated optimization, reserve and peak-shaving configuration, energy storage capacity design, transmission channel and distribution network expansion, and power quality and resilience assessment of new power systems, it is urgent to construct a set of wind and solar energy scenarios that can reproduce statistical characteristics and cover extreme situations. Wind and solar energy scenario generation has therefore become an important technology for uncertainty modeling. By learning marginal distributions and temporal dependencies from historical output and meteorological data, it generates multiple representative possible trajectories for stochastic optimization and robust optimization of new power systems. Summary of the Invention

[0003] The purpose of this invention is to address the shortcomings of existing technologies by providing a method for generating wind and solar power output scenarios based on an improved diffusion model using multi-head attention. The method includes: constructing a meteorological feature matrix related to power generation and a historical wind and solar power generation matrix; using the historical wind and solar power generation matrix combined with a denoised diffusion probability model to obtain a wind and solar power output scenario; improving the denoised diffusion probability model using a multi-head attention mechanism to obtain a wind and solar power output scenario considering the meteorological feature matrix; and generating a day-ahead wind and solar power scenario using the model and comparing it with measured values. Compared to typical wind and solar power output scenario generation methods, this invention can effectively consider multi-dimensional meteorological data and generate more accurate wind and solar power output scenarios.

[0004] To achieve the above objectives, the technical solution of the present invention is: a method for generating wind and solar power output scenes based on a diffusion model improved by multi-head attention, comprising:

[0005] Construct a meteorological characteristic matrix and a historical wind and solar power generation matrix related to power generation;

[0006] By utilizing the historical power generation matrix of wind and solar power and combining it with a noise-reducing diffusion probability model, a preliminary wind and solar power output scenario is generated.

[0007] By combining the multi-head attention mechanism to improve the denoising diffusion probability model, a conditional denoising diffusion probability model considering the meteorological feature matrix is ​​obtained, and a wind and solar power output scene considering the meteorological feature matrix is ​​generated.

[0008] The daytime scene of wind and solar power was generated by a conditional denoising diffusion probability model, and the generated scene was compared with the measured value.

[0009] Furthermore, a meteorological characteristic matrix and a historical wind and solar power generation matrix related to power generation are constructed, including:

[0010] Historical wind and solar power generation data and corresponding meteorological data were collected. Pearson correlation coefficient was used to screen meteorological factors with high correlation to power generation. The formula for calculating the Pearson correlation coefficient r is as follows:

[0011]

[0012] In the formula, X i Y i Let X be the value of the i-th data group. mean Y mean This represents the mean of the corresponding data.

[0013] Based on the selected meteorological factors, a multidimensional meteorological feature matrix C is constructed. m Historical power generation matrix of wind and solar power C s .

[0014] Furthermore, the multidimensional meteorological feature matrix is ​​represented as follows:

[0015]

[0016] in, Represents the meteorological characteristic matrix. For a point in time, The total radiance, It is in degrees Celsius. For humidity, Atmospheric pressure The wind speed at 10m is Wind direction at 10m;

[0017] The historical power generation matrix of wind and solar power is represented as follows:

[0018]

[0019] in A matrix representing historical wind and solar power generation. This represents the daily power generation value of a photovoltaic power station. This represents the daily power generation of a wind power station.

[0020] Furthermore, the forward diffusion process of the denoising diffusion probability model includes: progressively adding noise to the power data in the historical wind and solar power generation matrix, wherein the noise addition process from step t-1 to step t satisfies:

[0021]

[0022] And the diffusion at any step t satisfies:

[0023]

[0024] in, This represents the conditional probability distribution in the forward process. For the first The noise scheduling step, with a value between 0 and 1, controls the amount of noise added in each step. () represents a Gaussian distribution. Let be the identity matrix, indicating that the noise is independent in all dimensions and has the same variance. It is noise sampled from a Gaussian distribution.

[0025] Furthermore, through reparameterization techniques, the data at any step t in the forward diffusion process... From initial data and noise Represented as:

[0026]

[0027] in, ;

[0028] Its conditional probability distribution is:

[0029] .

[0030] Furthermore, the probability distribution model for the reverse diffusion process of the denoising diffusion probability model is as follows:

[0031]

[0032]

[0033] in, The parameterized distribution for model learning, and Here, T represents the mean and variance predicted by the model, and T represents the total number of steps. This represents the complete data sequence of the diffusion process from step 0 to step T.

[0034] Furthermore, when the initial data is known... At that time, the conditional probability distribution of the reverse diffusion process satisfies:

[0035]

[0036] The mean and variance of the conditional probability distribution for the reverse diffusion process are expressed as follows:

[0037]

[0038] .

[0039] Furthermore, by minimizing the model prediction distribution Compared with the true probability distribution The KL divergence between them is used to construct the optimization objective. :

[0040]

[0041] And predict the mean Reparameterization is as follows:

[0042]

[0043] in, The noise in the model's predictions; the optimization objective ultimately becomes minimizing the prediction noise. With actual noise Distance, simplified optimization objective for:

[0044] .

[0045] Furthermore, the denoising diffusion probability model is improved by combining a multi-head attention mechanism. Specifically, a multi-head self-attention mechanism is introduced to process the meteorological feature matrix. For the input feature matrix X∈RN×D, where N is the sequence length and D is the feature dimension, the query, key, and value of h heads are first obtained through linear projection.

[0046]

[0047] in, , , , The learnable parameter matrix for each head, where d is the dimension of each head;

[0048] Then calculate the scaled dot product attention for each head:

[0049]

[0050] Finally, the outputs of h heads are concatenated and a linear transformation is performed to obtain the final output:

[0051]

[0052] Concat is a function that concatenates h attention results along the feature dimension; The output projection matrix is ​​trainable and optimized through backpropagation.

[0053] Furthermore, a day-ahead scene of wind and solar power is generated using a conditional denoising diffusion probability model, and the generated scene is compared with the measured values, including:

[0054] During model training, historical power curves are sampled from historical wind and solar power output as input. For each sample, a time step t is randomly selected, and noise is added according to a noise schedule to obtain noisy data xt. The meteorological feature matrix is ​​used as a condition c and input into the noise prediction network. By minimizing prediction noise With actual noise The mean square error between them is used for training;

[0055] During scene generation, initial noise is sampled from a standard Gaussian distribution. The trained noise prediction network Substituting into the formula for the reverse diffusion process, the wind and solar power output scene x0 is generated through iterative denoising;

[0056] Using the mean absolute error E MAE Root mean square error and relative entropy D KL The generated scene is evaluated as a performance metric, and the calculation formulas are as follows:

[0057]

[0058]

[0059]

[0060] Where x(i) is the measured value, Generate values ​​for the model. Let P be the average value, and Q be the distribution of the measured data and the distribution of the generated data, respectively. Let N be the total number of samples, i represent the i-th data point, p(x) be the probability density function of the measured data distribution, and q(x) be the probability density function of the generated data distribution.

[0061] Compared with the prior art, the present invention has the following beneficial effects:

[0062] 1. The wind and solar power output scene generation method based on the conditional denoising diffusion probability model proposed in this invention can generate high-fidelity and diverse wind and solar power output scenes.

[0063] 2. This invention integrates meteorological features and diffusion models through a multi-head self-attention mechanism, which can effectively explore the nonlinear coupling relationship between the fluctuation of meteorological features and the fluctuation of wind and solar power output. Attached Figure Description

[0064] Figure 1This is a flowchart of the method of the present invention.

[0065] Figure 2 This is the photovoltaic power output scenario set generated by the present invention.

[0066] Figure 3 This is a set of wind power output scenarios generated by the present invention. Detailed Implementation

[0067] The technical solution of the present invention will now be described in detail with reference to the accompanying drawings.

[0068] It should be noted that the following detailed descriptions are exemplary and intended to provide further explanation of this application. Unless otherwise specified, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application pertains.

[0069] It should be noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to limit the exemplary embodiments according to this application. As used herein, the singular form is intended to include the plural form as well, unless the context clearly indicates otherwise. Furthermore, it should be understood that when the terms "comprising" and / or "including" are used in this specification, they indicate the presence of features, steps, operations, devices, components, and / or combinations thereof.

[0070] This invention provides a method for generating wind and solar power output scenes based on an improved diffusion model with multi-head attention, comprising:

[0071] Construct a meteorological characteristic matrix and a historical wind and solar power generation matrix related to power generation;

[0072] By utilizing the historical power generation matrix of wind and solar power and combining it with a noise-reducing diffusion probability model, a preliminary wind and solar power output scenario is generated.

[0073] By combining the multi-head attention mechanism to improve the denoising diffusion probability model, a conditional denoising diffusion probability model considering the meteorological feature matrix is ​​obtained, and a wind and solar power output scene considering the meteorological feature matrix is ​​generated.

[0074] The daytime scene of wind and solar power was generated by a conditional denoising diffusion probability model, and the generated scene was compared with the measured value.

[0075] The following is a detailed implementation process of the present invention.

[0076] like Figure 1 As shown in Figure 2, this embodiment provides a method for generating wind and solar power output scenes based on a diffusion model improved by multi-head attention, including the following steps:

[0077] S1. Construct a meteorological characteristic matrix and a historical wind and solar power generation matrix related to power generation;

[0078] S2. The wind and solar power output scenario is obtained by combining the historical power generation matrix of wind and solar power with the noise-reducing diffusion probability model.

[0079] S3. Improve the denoising diffusion probability model by combining multi-head attention mechanism to obtain a wind and solar power output scenario that considers meteorological feature matrix;

[0080] S4. Generate daytime wind and solar power scenarios using the model and compare them with measured values.

[0081] In this example, step S1 includes:

[0082] Step S11: Collect historical power generation data of wind and solar power and corresponding meteorological data, and use Pearson correlation coefficient to screen meteorological factors:

[0083] Among the numerous meteorological factors affecting power generation, the Pearson correlation coefficient is used to measure the correlation between these factors and power generation, thus identifying meteorological factors with high correlation. The Pearson correlation coefficient ranges from -1 to 1; the closer it is to 1, the stronger the positive correlation, and similarly, the closer it is to -1, the stronger the negative correlation. The formula for calculating the Pearson correlation coefficient is as follows:

[0084]

[0085] In the formula: r—correlation coefficient; X i Y i —The i-th data set value; X mean Y mean —Mean.

[0086] Step S12: Construct the meteorological feature matrix and the historical wind and solar power generation matrix:

[0087] Constructing a multidimensional meteorological feature matrix:

[0088]

[0089] in Represents the meteorological characteristic matrix. For a point in time, The total radiance, It is in degrees Celsius. For humidity, Atmospheric pressure The wind speed at 10m is The wind direction is at 10m.

[0090] Constructing a historical power generation matrix for wind and solar power:

[0091]

[0092] in A matrix representing historical wind and solar power generation. This represents the daily power generation value of a photovoltaic power station. This represents the daily power generation of a wind power station.

[0093] In this example, step S2 includes:

[0094] Step S21: Forward diffusion process of the denoising diffusion probability model:

[0095] First, noise is gradually added to the power data in the historical wind and solar power generation matrix, as the noise in the matrix increases... As the noise level continues to increase, the power data is eventually converted into Gaussian noise. The specific noise addition formula is as follows:

[0096]

[0097] In the formula: For the first The noise scheduling step, with a value between 0 and 1, controls the amount of noise added in each step. () represents a Gaussian distribution. Let be the identity matrix, indicating that the noise is independent and has the same variance across all dimensions. The diffusion at any step satisfies the following formula:

[0098]

[0099] in, It is noise sampled from a Gaussian distribution. Since the diffusion process is often fixed, a pre-selected noise value is necessary so that it can be directly applied to any given data based on the original data. Step Sampling is performed. This is achieved through reparameter resampling techniques. It can be used and The linear combination of is expressed by the following formula:

[0100]

[0101] In the formula After deparameterization, the above formula can be written in conditional probability form to obtain:

[0102]

[0103] Step S22: Reverse process of the denoising diffusion probability model:

[0104] The reverse diffusion process starts with Gaussian noise and gradually diffuses the noise-added data. To denoise and restore the original image, the probability distribution of backdiffusion is modeled as follows:

[0105]

[0106]

[0107] When the formula is Given that the conditional probability distribution of the reverse diffusion process satisfies the formula:

[0108]

[0109] By transforming Bayes' formula, we get:

[0110]

[0111] Substituting formulas (3) and (7) from step S31, the mean and variance of the conditional probability distribution of the reverse diffusion process are derived:

[0112]

[0113]

[0114] Predicting probability distribution by minimizing the model Compared with the true probability distribution The distance between them is used to construct the optimization objective. The KL divergence is used to measure the distance between two distributions:

[0115]

[0116] Simplify equation (8) and let the variance The optimization objective is transformed into minimizing the predicted value. Compared with the actual mean The distance. From the training process ,Will Reparameterization:

[0117]

[0118] In the formula: To predict noise, the optimization objective ultimately becomes minimizing the prediction noise. With actual noise The distance. Based on this, from the first Step to The reverse diffusion process of the step satisfies the formula:

[0119]

[0120] Optimization objective after reparameterization and simplification of weight terms As shown in the following formula:

[0121]

[0122] In this example, step S3 includes:

[0123] To process meteorological data, a multi-head self-attention mechanism is introduced. First, the multi-head self-attention mechanism divides the input feature matrix X∈RN×D (N is the sequence length, D is the feature dimension) into h subspaces (heads). Then, through linear projection, h sets of queries, keys, and values ​​are obtained:

[0124]

[0125] In the formula: , , , Given a learnable parameter matrix for each head, where d is the dimension of each head. Then, independently compute the scaled dot-product attention for each head:

[0126]

[0127] In the formula: Softmax is an activation function that normalizes a numerical vector into a probability distribution vector. This is a scaling factor to prevent the gradient from vanishing due to an excessively large dot product. Finally, the outputs of h heads are concatenated and then subjected to a linear transformation:

[0128]

[0129] In the formula: Concat is a function that concatenates h attention results along the feature dimension; The output projection matrix is ​​trainable and optimized through backpropagation.

[0130] In this example, step S4 includes:

[0131] The effectiveness of this invention is demonstrated by using the following three different models to predict photovoltaic power.

[0132] During model training, this invention first samples historical power curves from historical wind and solar power generation as input to the model. For each sample, a time step t is randomly selected, and noise is added to the input according to a noise schedule, resulting in noisy data xt. Next, the meteorological feature matrix is ​​used as input condition c. Then, the noisy data xt, time step t, and condition c are input into the noise prediction network. In the middle. By minimizing prediction noise. With actual noise The mean squared error between the two is used to train the denoising diffusion probability model. During scene generation, noise is first sampled from a standard Gaussian distribution. Then the trained prediction noise Substituting into the formula for the reverse diffusion process, the scene from the previous step is generated through the reverse diffusion process. After repeating T steps, the wind power or solar power output scenario x0 can be obtained.

[0133] At the same time, this invention uses two other different models for comparison with the model of this invention:

[0134] Model 1: Latin cube sampling (LHS) modeled using Copula;

[0135] Model 2: Monte Carlo simulation (MC) modeled using a Gaussian mixture model (GMM);

[0136] This invention selects mean absolute error (MAE), root mean square error (RMSE), and relative entropy (Kullback–Leibler divergence, KL) as performance indicators, and their calculation formulas are as follows:

[0137]

[0138]

[0139]

[0140] The system operation results under the above scenarios are shown in Table 1.

[0141] Table 1. Model Indicators

[0142]

[0143] Based on the results in Table 1, compared with Model 1 and Model 2, the cDDPM model used in this invention has better prediction performance, smaller error indicators, and the predicted value is closer to the measured power value, thus improving the accuracy of the wind and solar power output scenario set.

[0144] Figure 2 This is the photovoltaic power output scenario set generated by the present invention. Figure 3 This is a set of wind power output scenarios generated by the present invention.

[0145] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention in any other way. Any person skilled in the art may make changes or modifications to the above-disclosed technical content to create equivalent embodiments. However, any simple modifications, equivalent changes, and modifications made to the above embodiments based on the technical essence of the present invention without departing from the scope of the present invention shall still fall within the protection scope of the present invention.

Claims

1. A method for generating wind and solar power output scenes based on an improved diffusion model with multi-head attention, characterized in that, include: Construct a meteorological characteristic matrix and a historical wind and solar power generation matrix related to power generation; By utilizing the historical power generation matrix of wind and solar power and combining it with a noise-reducing diffusion probability model, a preliminary wind and solar power output scenario is generated. By combining the multi-head attention mechanism to improve the denoising diffusion probability model, a conditional denoising diffusion probability model considering the meteorological feature matrix is ​​obtained, and a wind and solar power output scene considering the meteorological feature matrix is ​​generated. The daytime scene of wind and solar power was generated by a conditional denoising diffusion probability model, and the generated scene was compared with the measured value.

2. The method for generating wind and solar power output scenes based on a diffusion model improved by multi-head attention, as described in claim 1, is characterized in that... Construct a meteorological characteristic matrix and a historical wind and solar power generation matrix related to power generation, including: Historical wind and solar power generation data and corresponding meteorological data were collected. Pearson correlation coefficient was used to screen meteorological factors with high correlation to power generation. The formula for calculating the Pearson correlation coefficient r is as follows: In the formula, X i Y i Let X be the value of the i-th data group. mean Y mean This represents the mean of the corresponding data. Based on the selected meteorological factors, a multidimensional meteorological feature matrix C is constructed. m Historical power generation matrix of wind and solar power C s .

3. The method for generating wind and solar power output scenes based on a diffusion model improved by multi-head attention, as described in claim 2, is characterized in that... The multidimensional meteorological feature matrix is ​​represented as follows: in, Represents the meteorological characteristic matrix. For a point in time, The total radiance. It is in degrees Celsius. For humidity, Atmospheric pressure The wind speed at 10m is... Wind direction at 10m; The historical power generation matrix of wind and solar power is represented as follows: in A matrix representing historical wind and solar power generation. This represents the daily power generation value of a photovoltaic power station. This represents the daily power generation of a wind power station.

4. The method for generating wind and solar power output scenes based on a diffusion model improved by multi-head attention, as described in claim 1, is characterized in that... The forward diffusion process of the denoising diffusion probability model includes: progressively adding noise to the power data in the historical wind and solar power generation matrix, wherein the noise addition process from step t-1 to step t satisfies: And the diffusion at any step t satisfies: in, This represents the conditional probability distribution in the forward process. For the first The noise scheduling step, with a value between 0 and 1, controls the amount of noise added in each step. () represents a Gaussian distribution. Let be the identity matrix, indicating that the noise is independent in all dimensions and has the same variance. It is noise sampled from a Gaussian distribution.

5. The method for generating wind and solar power output scenes based on a diffusion model improved by multi-head attention, as described in claim 4, is characterized in that... By employing reparameterization techniques, the data at any step t during the forward diffusion process are obtained. From initial data and noise Represented as: in, ; Its conditional probability distribution is: 。 6. The method for generating wind and solar power output scenes based on a diffusion model improved by multi-head attention, as described in claim 4, is characterized in that... The reverse diffusion process of the denoising diffusion probability model is modeled by the following probability distribution: in, The parameterized distribution for model learning, and Here, T represents the mean and variance predicted by the model, and T represents the total number of steps. This represents the complete data sequence of the diffusion process from step 0 to step T.

7. The method for generating wind and solar power output scenes based on a diffusion model improved by multi-head attention, as described in claim 6, is characterized in that... When the initial data is known At that time, the conditional probability distribution of the reverse diffusion process satisfies: The mean and variance of the conditional probability distribution for the reverse diffusion process are expressed as follows: 。 8. The method for generating wind and solar power output scenes based on a diffusion model improved by multi-head attention, as described in claim 7, is characterized in that... Predicting distribution by minimizing the model Compared with the true probability distribution The KL divergence between them is used to construct the optimization objective. : And predict the mean Reparameterization is as follows: in, The noise in the model's predictions; the optimization objective ultimately becomes minimizing the prediction noise. With actual noise Distance, simplified optimization objective for: 。 9. The method for generating wind and solar power output scenes based on a diffusion model improved by multi-head attention according to claim 1, characterized in that, The denoising diffusion probability model is improved by incorporating a multi-head attention mechanism. Specifically, a multi-head self-attention mechanism is introduced to process the meteorological feature matrix. For the input feature matrix X∈RN×D, where N is the sequence length and D is the feature dimension, the query, key, and value of h heads are first obtained through linear projection. in, , , , The learnable parameter matrix for each head, where d is the dimension of each head; Then calculate the scaled dot product attention for each head: Finally, the outputs of h heads are concatenated and a linear transformation is performed to obtain the final output: Concat is a function that concatenates h attention results along the feature dimension; The output projection matrix is ​​trainable and optimized through backpropagation.

10. A method for generating wind and solar power output scenes based on a diffusion model improved by multi-head attention, as described in claim 1, characterized in that... A day-ahead scene of wind and solar power was generated using a conditional denoising diffusion probability model, and the generated scene was compared with the measured values, including: During model training, historical power curves are sampled from historical wind and solar power output as input. For each sample, a time step t is randomly selected, and noise is added according to a noise schedule to obtain noisy data xt. The meteorological feature matrix is ​​used as a condition c and input into the noise prediction network. By minimizing prediction noise With actual noise The mean squared error between them is used for training; During scene generation, initial noise is sampled from a standard Gaussian distribution. The trained noise prediction network Substituting into the formula for the reverse diffusion process, the wind and solar power output scene x0 is generated through iterative denoising; Using the mean absolute error E MAE Root mean square error and relative entropy D KL The generated scene is evaluated as a performance metric, and the calculation formulas are as follows: Where x(i) is the measured value, Generate values ​​for the model. Let P be the average value, and Q be the distribution of the measured data and the distribution of the generated data, respectively. Let N be the total number of samples, i represent the i-th data point, p(x) be the probability density function of the measured data distribution, and q(x) be the probability density function of the generated data distribution.