Substation full life cycle carbon emission evaluation method based on semi-invariant method

Abstract

The invention relates to a substation full-life-cycle carbon emission evaluation method based on a semi-invariant method, and belongs to the field of power system carbon emission evaluation, and the method comprises the steps: S1, building a substation full-life-cycle carbon emission uncertainty quantitative model: building a substation full-life-cycle carbon emission data quality evaluation matrix, establishing an uncertainty quantification model by using a Beta function; s2, calculating the statistical characteristics of the carbon emission of the transformer substation based on a semi-invariant method: solving each stage of semi-invariant of the total activity consumption of each stage, and converting each stage of semi-invariant of the total activity consumption of each stage into each stage of semi-invariant of the carbon emission according to the property of the semi-invariant; and S3, transformer substation carbon emission distribution function fitting: calculating an expansion coefficient of a Gram-Charlier expansion series, and performing scale transformation to obtain a probability density function of an original variable, thereby realizing improvement of the accuracy of carbon emission evaluation, improvement of the efficiency of uncertainty analysis and a systematic uncertainty quantification process.

Description

Technical Field

[0001] The present invention belongs to the technical field of carbon emission assessment of power systems, and relates to a substation full life cycle carbon emission assessment method based on a semi-invariant method, and in particular to a substation full life cycle carbon emission assessment method that integrates the semi-invariant method with a Gram-Charlier series expansion. Background Art

[0002] Substations, as a critical component of the power system, emit carbon throughout their lifecycle, encompassing multiple stages including construction, material production and transportation, operation, and demolition, significantly impacting the power industry's overall carbon footprint. However, existing substation carbon emission assessment methods lack data integrity, model accuracy, and uncertainty analysis, making them difficult to meet the needs of refined management.

[0003] While traditional life cycle assessment (LCA) methods can quantify substation carbon emissions on a phased basis, they primarily focus on deterministic accounting and fail to fully consider the impact of factors such as missing data, biased model assumptions, and parameter uncertainty on the results. For example, energy consumption data during the building materials production phase may contain errors due to differences in production processes or statistical calibers, and equipment energy consumption parameters during the operation phase may also experience performance degradation over time, making it difficult for traditional accounting results to truly reflect actual carbon emission levels. Existing studies have mostly used Monte Carlo simulation to address carbon emission uncertainties, but this method relies on large amounts of repeated sampling, is computationally inefficient, and can only provide discrete probability results, failing to intuitively reveal the higher-order statistical characteristics of carbon emission distributions. Furthermore, qualitative descriptions of data quality are difficult to directly translate into quantitative analysis inputs, resulting in a lack of systematicity in the quantification of sources of uncertainty.

[0004] Therefore, those skilled in the art are in urgent need of a method that is applicable to substations and can achieve a comprehensive and accurate assessment of carbon emissions. Summary of the Invention

[0005] In view of this, in order to solve the problem that the existing substation carbon emission evaluation method has shortcomings in data integrity, model accuracy and uncertainty analysis, and is difficult to meet the refined management needs under the "carbon peak and carbon neutrality" goals, the present invention provides a substation full life cycle carbon emission evaluation method based on the semi-invariant method.

[0006] In order to achieve the above object, the present invention provides the following technical solutions:

[0007] A method for evaluating carbon emissions over the entire life cycle of a substation based on a semi-invariant method includes the following steps:

[0008] S1. Construction of uncertainty quantification model for substation carbon emissions throughout its life cycle: Establish a data quality evaluation matrix for substation carbon emissions throughout its life cycle. After establishing the data quality evaluation matrix, use the Beta function to establish an uncertainty quantification model.

[0009] S2. Calculation of statistical characteristics of substation carbon emissions based on the semi-invariant method: Based on the uncertainty quantification model in step S1, solve the origin moment of the random variable. Based on the relationship between the semi-invariant and the origin moment, solve the various-order semi-invariants of the total activity consumption in each stage. Based on the properties of the semi-invariant, convert the various-order semi-invariants of the total activity consumption in each stage into various-order semi-invariants of carbon emissions.

[0010] S3. Fitting the carbon emission distribution function of the substation: Select the Gram-Charlier series expansion method, and perform Gram-Charlier series expansion based on the semi-invariants of each order of carbon emissions in step S2 to obtain the expansion of the probability density function and the cumulative distribution function. Then, perform a scale transformation to obtain the probability density function of the original variable.

[0011] Furthermore, step S1 specifically includes the following steps:

[0012] S11. Establish a data quality evaluation matrix for substation carbon emissions throughout their life cycle: Use a data quality index evaluation method to score substation carbon emissions inventory data across its life cycle from five dimensions: reliability, completeness, technical relevance, regional relevance, and temporal relevance, and establish a data quality evaluation matrix. This data quality evaluation matrix will be used to conduct a qualitative and quantitative comprehensive assessment of carbon emissions data at each stage of the substation's life cycle, qualitatively analyzing the causes of uncertainty and quantitatively quantifying the deviations caused by uncertainty.

[0013] S12. Establish an uncertainty quantification model: After establishing the data quality evaluation matrix, use the Beta function to establish an uncertainty quantification model and link it with the data quality score in step S11, that is, convert the comprehensive data quality score into the shape parameters and endpoint parameters of the quantitative model.

[0014] Furthermore, the uncertainty quantification model in step S12 is constructed as follows:

[0015] (1)

[0016] Where: is the Gamma function; and is the shape parameter; and is the endpoint parameter;

[0017] The comprehensive data quality score is converted into shape parameters and endpoint parameters through empirical formulas. The conversion formula is as follows:

[0018] (2)

[0019] Where: Indicates conversion to an integer; is the parameter value of each indicator in the data list; Provide a comprehensive score for data quality; convert the comprehensive score for data quality into shape parameters and endpoint parameters through empirical formulas, thereby quantifying the uncertainty of carbon emissions.

[0020] Furthermore, in step S11, the substation life cycle carbon emission inventory data is scored using a [1,5] point scoring system, where 1 point represents the worst data quality, i.e., the greatest uncertainty, and 5 points represents the best data quality, i.e., the least uncertainty.

[0021] Furthermore, step S2 specifically includes the following steps:

[0022] S21. The relationship between the semi-invariant of a random variable and the origin moment:

[0023] (3)

[0024] Where: represents the nth-order semi-invariant of a random variable; represents the n-order origin moment of the random variable, where the first-order origin moment is the expectation of the random variable, For Take different elements The number of combinations of elements; the semi-invariant has the following two properties: the order of the semi-invariant of the sum of independent random variables is equal to the sum of the order of the semi-invariant of the variable and the random variable times the k-order semi-invariant is equal to the k-order semi-invariant of the variable times; these two important properties are the key to simplifying calculations;

[0025] S22. Obtaining the various order semi-invariants of carbon emissions: Based on the uncertainty quantification model obtained in step S12, use formula 3 to solve the various order semi-invariants of the total activity consumption in each stage of the substation's entire life cycle; according to the properties of the semi-invariants, convert the various order semi-invariants of the total activity consumption in each stage into various order semi-invariants of carbon emissions, providing a basis for subsequent distribution function fitting.

[0026] Furthermore, step S3 specifically includes the following steps:

[0027] S31. Select the series expansion method for the best approximate distribution function: For the problem of calculating the distribution function value of an unknown random variable, select the Gram-Charlier series expansion method from the Gram-Charlier series expansion, Cornish-Fisher series expansion, Edgeworth series expansion, and maximum entropy principle. This method achieves an approximate expression of the target distribution function by systematically utilizing high-order moment information.

[0028] S32. Obtain expansion coefficients of the Gram-Charlier expansion series: The Gram-Charlier expansion series includes the expansion of the probability density function and the cumulative distribution function. The expansion formula is as follows:

[0029] (4)

[0030] Where: is the symbol of the random variable after the original variable is standardized; is the probability density function of the standard normal distribution; is the cumulative distribution function of the standard normal distribution; is the expansion coefficient;

[0031] If the central moment and semi-invariant of the original variable are known, they can be obtained by the following formula 5, showing the first six order expansion coefficients:

[0032] (5)

[0033] Where: represents the k-order central moment of the original variable;

[0034] S33. Obtain the probability density function of the original variable: the density function fitted by Gram-Charlier series expansion It is the density function of the random variable Z. After scaling, we get the probability density function of the original variable. The scaling formula is as follows:

[0035] (6)

[0036] Where: Represents the original variable The density function of .

[0037] The beneficial effects of the present invention are:

[0038] The substation life cycle carbon emission assessment method based on the semi-invariant method disclosed in the present invention has the following advantages:

[0039] 1. Improved the accuracy of carbon emission evaluation; the statistical characteristics of carbon emissions at all stages of the substation's life cycle are calculated using the semi-invariant method, which can effectively handle uncertainty issues. The semi-invariant has the property that the sum of the semi-invariants of each order of the sum of independent random variables is equal to the sum of the semi-invariants of each order of the variable, and the random variable times the k-order semi-invariant is equal to the k-order semi-invariant of the variable This makes the calculation process simpler and can accurately reflect the statistical characteristics of carbon emissions.

[0040] 2. Improved the efficiency and systematicity of uncertainty analysis. By constructing a data quality evaluation matrix for substation carbon emissions throughout their lifecycle, quantifying uncertainty, and fitting the carbon emissions distribution function using a Gram-Charlier series expansion, this method can intuitively demonstrate the probabilistic distribution characteristics of carbon emissions. This method not only improves the efficiency of uncertainty analysis but also addresses the shortcomings of existing methods through a systematic uncertainty quantification process.

[0041] 3. The uncertainty quantification process has been systematized. By constructing a quality evaluation matrix for substation carbon emissions data throughout its lifecycle, inventory data is scored based on five dimensions: reliability, completeness, technical relevance, regional relevance, and temporal relevance, establishing a quantitative evaluation standard. This systematic uncertainty quantification approach transforms qualitative descriptions of data quality into quantitative analysis inputs, providing a reliable foundation for subsequent statistical property calculations and distribution function fitting.

[0042] 4. Providing a reliable basis for carbon reduction strategies: By accurately calculating carbon emissions at each stage of a substation's lifecycle and quantifying its uncertainty, the system can provide power industry practitioners with more accurate carbon emissions data and more reliable uncertainty analysis results. This will help develop more scientific and effective carbon reduction strategies, optimize substation design, construction, and operation processes, and ultimately achieve a low-carbon transformation for the power industry.

[0043] Other advantages, objects, and features of the present invention will be described in part in the following description and, in part, will be apparent to those skilled in the art upon examination of the following description or may be learned from practice of the present invention. The objects and other advantages of the present invention may be realized and obtained through the following description. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] In order to make the purpose, technical solutions and advantages of the present invention more clear, the present invention will be described in detail below with reference to the accompanying drawings, in which:

[0045] Figure 1 This is a flowchart for solving the carbon emission evaluation of the substation throughout its life cycle based on the semi-invariant method in this embodiment. DETAILED DESCRIPTION

[0046] The following describes the embodiments of the present invention by means of specific examples, and those skilled in the art can easily understand other advantages and effects of the present invention from the contents disclosed in this specification. The present invention can also be implemented or applied through other different specific embodiments, and the details in this specification can also be modified or changed in various ways based on different viewpoints and applications without departing from the spirit of the present invention. It should be noted that the illustrations provided in the following embodiments are only schematic illustrations of the basic concept of the present invention, and the following embodiments and features in the embodiments can be combined with each other without conflict.

[0047] This paper proposes a substation life cycle carbon emissions evaluation method based on the semi-invariant method. With the goal of deterministic calculation and uncertainty analysis of carbon emissions at each stage of the substation life cycle, a data quality evaluation matrix for substation life cycle carbon emissions is constructed to quantify uncertainty. The semi-invariant method is used to solve the statistical characteristics, and the carbon emission distribution function is fitted through the Gram-Charlier series expansion. Finally, a 110kV substation is used as an example for case verification, and a multi-stage data collection and calculation process is set up to verify the effectiveness of the method in accurately calculating carbon emissions and quantifying uncertainty.

[0048] The evaluation method specifically includes the following steps:

[0049] S1. Construction of uncertainty quantification model for carbon emissions throughout the life cycle of substations: Establish a data quality evaluation matrix for carbon emissions throughout the life cycle of substations. After establishing the data quality evaluation matrix, use the Beta function to establish an uncertainty quantification model.

[0050] Specifically:

[0051] S11. Establish a quality evaluation matrix for carbon emission data of substations throughout their life cycle: In the process of quantifying uncertainty, the present invention implements a qualitative and quantitative comprehensive evaluation of the carbon emission inventory data of substations throughout their life cycle, wherein the causes of uncertainty are qualitatively analyzed and the deviations caused by uncertainty are quantitatively quantified. In view of the problem that it is difficult to obtain complete carbon emission inventory data due to differences in system boundaries, activity processes and life cycle stages of different research objects, a data quality index evaluation method is selected to carry out subsequent research, and a quality evaluation matrix for carbon emission data of substations throughout their life cycle is constructed, which includes five evaluation indicators: reliability, integrity, technical relevance, regional relevance, and time relevance. A [1,5] point scoring system is adopted (1 point represents the worst data quality, i.e., the maximum uncertainty, and 5 points represents the best data quality, i.e., the minimum uncertainty). The specific scoring criteria are shown in Table 1.

[0052] Table 1: Substation life cycle carbon emission data quality evaluation matrix

[0053]

[0054] S12. Establish uncertainty quantification model: After establishing the data quality evaluation matrix, use the Beta function to establish an uncertainty quantification model and link it with the data quality score in step S11, that is, convert the comprehensive data quality score into the shape parameters and endpoint parameters of the quantitative model.

[0055] The uncertainty quantification model is as follows:

[0056] (1)

[0057] Where: is the Gamma function; and is the shape parameter; and is the endpoint parameter;

[0058] The comprehensive data quality score is converted into shape parameters and endpoint parameters through empirical formulas. The conversion formula is as follows:

[0059] (2)

[0060] Where: Indicates conversion to an integer; is the parameter value of each indicator in the data list; Provide a comprehensive score for data quality; convert the comprehensive score for data quality into shape parameters and endpoint parameters through empirical formulas, thereby quantifying the uncertainty of carbon emissions.

[0061] S2. Calculation of statistical characteristics of carbon emissions from substations based on the semi-invariant method: Based on the uncertainty quantification model in step S1, the origin moment of the random variable is solved. According to the relationship between the semi-invariant and the origin moment, the semi-invariants of each order of the total activity consumption in each stage are solved. According to the properties of the semi-invariant, the semi-invariants of each order of the total activity consumption in each stage are converted into semi-invariants of each order of carbon emissions.

[0062] Specifically:

[0063] S21. The moments of a random variable are its numerical characteristics. Semi-invariants are also a kind of numerical characteristics of random variables. They can be obtained from moments of the corresponding order or less. The relationship between the semi-invariants of random variables and the moments at the origin is:

[0064] (3)

[0065] Where: represents the nth-order semi-invariant of a random variable; represents the n-order origin moment of the random variable, where the first-order origin moment is the expectation of the random variable, For Take different elements The number of combinations of elements.

[0066] The semi-invariant has the following two properties: the order of the semi-invariant of the sum of independent random variables is equal to the sum of the order of the semi-invariant of the variable and the random variable times the k-order semi-invariant is equal to the k-order semi-invariant of the variable times; these two important properties are the key to simplifying calculations.

[0067] S22: Obtaining the various order semi-invariants of carbon emissions: Based on the uncertainty quantification model obtained in step S12, use formula 3 to solve the various order semi-invariants of the total activity consumption in each stage of the substation's entire life cycle; according to the properties of the semi-invariants, the various order semi-invariants of the total activity consumption in each stage are converted into various order semi-invariants of carbon emissions, providing a basis for subsequent distribution function fitting.

[0068] S3. Fitting the carbon emission distribution function of the substation: Select the Gram-Charlier series expansion method, and perform Gram-Charlier series expansion based on the semi-invariants of each order of carbon emissions in step S2 to obtain the expansion of the probability density function and the cumulative distribution function. Then, perform a scale transformation to obtain the probability density function of the original variable.

[0069] Specifically:

[0070] S31. Series Expansion Method for Selecting the Optimal Approximate Distribution Function: For calculating the distribution function value of an unknown random variable, various methods based on moments or semi-invariants exist, including the Gram-Charlier series expansion, the Cornish-Fisher series expansion, the Edgeworth series expansion, and the maximum entropy principle. The present invention employs the Gram-Charlier series expansion method for specific calculations. This method systematically utilizes high-order moment information to achieve an approximate expression of the target distribution function.

[0071] S32. Obtain expansion coefficients of the Gram-Charlier expansion series: The Gram-Charlier expansion series includes the expansion of the probability density function and the cumulative distribution function. The expansion formula is as follows:

[0072] (4)

[0073] Where: is the symbol of the random variable after the original variable is standardized; is the probability density function of the standard normal distribution; is the cumulative distribution function of the standard normal distribution; is the expansion coefficient;

[0074] If the central moment and semi-invariant of the original variable are known, they can be obtained by the following formula 5, showing the first six order expansion coefficients:

[0075] (5)

[0076] Where: Represents the k-order central moment of the original variable.

[0077] S33. Obtain the probability density function of the original variable: the density function fitted by Gram-Charlier series expansion It is the density function of the random variable Z. After scaling, we get the probability density function of the original variable. The scaling formula is as follows:

[0078] (6)

[0079] Where: Represents the original variable The density function of .

[0080] Example

[0081] Taking a typical 110kV substation as an example, the main transformer capacity is 2*50MVA; the general design scheme 110-A3-3 is adopted, and the voltage level is 110kV; there are 2 10kV incoming lines in the current phase and 3 in the future phase, and 24 outgoing lines in the current phase and 36 in the future phase.

[0082] The calculation process is as follows Figure 1 As shown, Figure 1 The left side of the figure shows the process based on the semi-invariant method, with Monte Carlo simulation used as the evaluation criteria for the semi-invariant method. The right side shows the algorithm process under Monte Carlo simulation. Methods for estimating the probability density distribution of a data set are divided into parametric estimation and non-parametric estimation, which differ in whether or not prior knowledge is incorporated. To fit a more accurate model, the present invention uses non-parametric estimation, which is to fit the distribution based on the characteristics and properties of the data itself. The specific method used is kernel density estimation.

[0083] Specific implementation steps:

[0084] S1. Based on the inventory data compiled in the life cycle assessment method, the quantitative model of carbon emission uncertainty is calculated by combining Formula 1 and Formula 2.

[0085] S2. Solve the various order semi-invariants of the total activity consumption in each stage based on formula 3; according to the two properties of the semi-invariants, convert the various order semi-invariants of the total activity consumption in each stage into the various order semi-invariants of carbon emissions.

[0086] S3. Substitute the semi-invariants of each order of carbon emissions into Formula 4 and combine them with Formula 6 to obtain the distribution function of carbon emissions.

[0087] The specific results are analyzed as follows:

[0088] The 95% confidence intervals of carbon emissions under 6000 Monte Carlo simulations, 10,000 Monte Carlo simulations, and the semi-quantitative method for the 110 kV substation at each stage (construction, building materials production and transportation, operation, demolition, and overall stage) are shown in Table 2.

[0089] Table 2: 95% confidence intervals for carbon emissions at different stages and using different methods

[0090]

[0091] Taking the confidence interval of 10,000 Monte Carlo carbon emission placements as the standard, the calculated relative width error of the confidence interval is shown in Table 3.

[0092] Table 3: Relative width error of confidence intervals at different stages and methods

[0093]

[0094] The above calculation results show that the 95% confidence intervals for carbon emissions using the method of the present invention are more accurate at all stages than the 95% confidence intervals for carbon emissions using the 6000 Monte Carlo and 10,000 Monte Carlo methods. The relative width error of the confidence intervals using the method of the present invention is smaller than the relative width error of the confidence intervals using the 6000 Monte Carlo method, indicating that the semi-invariant method is more accurate in describing the uncertainty of carbon emissions at each stage and that the carbon emission statistical information obtained using the semi-invariant method is more credible.

[0095] This evaluation method quantifies uncertainty by constructing a data quality evaluation matrix for substation carbon emissions throughout their lifecycle. It then uses a semi-invariant method to solve for statistical characteristics and, combined with a Gram-Charlier series expansion, fits the carbon emissions distribution function. This method is validated using a 110kV substation as an example. Results show that the confidence intervals of the semi-invariant method at each stage and throughout the lifecycle are highly consistent with those of high-order Monte Carlo simulations, with significantly lower relative width errors. This validates the method's effectiveness in accurately calculating carbon emissions and quantifying uncertainty.

[0096] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not limiting. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that the technical solutions of the present invention can be modified or replaced by equivalents without departing from the purpose and scope of the technical solutions, which should all be included in the scope of the claims of the present invention.

Claims

1. A method for evaluating carbon emissions over the entire life cycle of a substation based on a semi-invariant method, characterized in that: The following steps are involved: S1. Construction of uncertainty quantification model for substation carbon emissions throughout its life cycle: Establish a data quality evaluation matrix for substation carbon emissions throughout its life cycle. After establishing the data quality evaluation matrix, use the Beta function to establish an uncertainty quantification model. S2. Calculation of statistical characteristics of substation carbon emissions based on the semi-invariant method: Based on the uncertainty quantification model in step S1, solve the origin moment of the random variable. Based on the relationship between the semi-invariant and the origin moment, solve the semi-invariants of the total activity consumption in each stage. Based on the properties of the semi-invariant, convert the semi-invariants of the total activity consumption in each stage into the semi-invariants of the carbon emissions. S3. Fitting the carbon emission distribution function of the substation: Select the Gram-Charlier series expansion method, and perform a Gram-Charlier series expansion based on the semi-invariants of each order of carbon emissions in step S2 to obtain the expansion of the probability density function and the cumulative distribution function, and then perform a scale transformation to obtain the probability density function of the original variable.

2. The method for evaluating carbon emissions from a substation throughout its life cycle based on the semi-invariant method according to claim 1, wherein: The S1 specifically includes the following steps: S11. Establish a data quality evaluation matrix for the carbon emissions data of the entire life cycle of a substation: Use a data quality index evaluation method to score the carbon emissions inventory data of the entire life cycle of a substation based on five dimensions: reliability, completeness, technical relevance, regional relevance, and time relevance, and establish a data quality evaluation matrix; S12. Establish an uncertainty quantification model: After establishing the data quality evaluation matrix, use the Beta function to establish an uncertainty quantification model and link it with the data quality score in step S11, that is, convert the comprehensive data quality score into the shape parameters and endpoint parameters of the quantitative model.

3. The method for evaluating carbon emissions from a substation throughout its life cycle based on the semi-invariant method according to claim 2, wherein: In step S11, the substation life cycle carbon emission inventory data is scored using a [1,5] scoring system, where 1 point represents the worst data quality, i.e., the greatest uncertainty, and 5 points represents the best data quality, i.e., the least uncertainty.

4. The method for evaluating carbon emissions from a substation throughout its life cycle based on the semi-invariant method according to claim 2, wherein: The uncertainty quantification model in step S12 is constructed as follows: （1） Where: is the Gamma function; and is the shape parameter; and is the endpoint parameter; The comprehensive data quality score is converted into shape parameters and endpoint parameters through empirical formulas. The conversion formula is as follows: （2） Where: Indicates conversion to an integer; is the parameter value of each indicator in the data list; Provide an overall score for data quality.

5. The method for evaluating carbon emissions from a substation throughout its life cycle based on the semi-invariant method according to claim 2, wherein: The step S2 specifically includes the following steps: S21. The relationship between the semi-invariant of a random variable and the origin moment: （3） Where: represents the nth-order semi-invariant of a random variable; represents the n-order origin moment of the random variable, where the first-order origin moment is the expectation of the random variable, For Take different elements The number of combinations of elements; the semi-invariant has the following two properties: the order of the semi-invariant of the sum of independent random variables is equal to the sum of the order of the semi-invariant of the variable and the random variable times the k-order semi-invariant is equal to the k-order semi-invariant of the variable times; S22: Obtaining the various order semi-invariants of carbon emissions: Based on the uncertainty quantification model obtained in step S12, use formula 3 to solve the various order semi-invariants of the total activity consumption in each stage of the substation's entire life cycle; according to the properties of the semi-invariants, the various order semi-invariants of the total activity consumption in each stage are converted into various order semi-invariants of carbon emissions, providing a basis for subsequent distribution function fitting.

6. The method for evaluating carbon emissions from a substation throughout its life cycle based on the semi-invariant method according to claim 5, characterized in that: The step S3 specifically includes the following steps: S31. Select the best approximate distribution function series expansion method: For the problem of calculating the distribution function value of an unknown random variable, select the Gram-Charlier series expansion method from the Gram-Charlier series expansion, Cornish-Fisher series expansion, Edgeworth series expansion, and maximum entropy principle; S32. Obtain expansion coefficients of the Gram-Charlier expansion series: The Gram-Charlier expansion series includes the expansion of the probability density function and the cumulative distribution function. The expansion formula is as follows: （4） Where: is the symbol of the random variable after the original variable is standardized; is the probability density function of the standard normal distribution; is the cumulative distribution function of the standard normal distribution; is the expansion coefficient; If the central moment and semi-invariant of the original variable are known, they can be obtained by the following formula 5, showing the first six order expansion coefficients: （5） Where: represents the k-order central moment of the original variable; S33. Obtain the probability density function of the original variable: the density function fitted by Gram-Charlier series expansion It is the density function of the random variable Z. After scaling, we get the probability density function of the original variable. The scaling formula is as follows: （6） Where: Represents the original variable The density function of .

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