A Substation Life Cycle Carbon Emission Assessment Method Based on Semi-Invariant Method

Abstract

This invention relates to a substation life-cycle carbon emission assessment method based on the semi-invariant method, belonging to the field of power system carbon emission assessment. The method includes: S1, Construction of a substation life-cycle carbon emission uncertainty quantification model: Establishing a substation life-cycle carbon emission data quality evaluation matrix and using the Beta function to establish an uncertainty quantification model; S2, Calculation of substation carbon emission statistical characteristics based on the semi-invariant method: Solving for the semi-invariants of total activity consumption at each stage, and based on the properties of semi-invariants, transforming the semi-invariants of total activity consumption at each stage into semi-invariants of carbon emissions; S3, Fitting the substation carbon emission distribution function: Calculating the expansion coefficients of the Gram-Charlier series, performing scaling transformation to obtain the probability density function of the original variables, thereby improving the accuracy of carbon emission assessment, enhancing the efficiency of uncertainty analysis, and implementing a systematic uncertainty quantification process.

Description

Technical Field

[0001] This invention belongs to the field of power system carbon emission assessment technology, and relates to a substation full life cycle carbon emission assessment method based on semi-invariant method, and more particularly to a substation full life cycle carbon emission assessment method that integrates semi-invariant method and Gram-Charlier series expansion. Background Technology

[0002] As a critical component of the power system, substations' carbon emissions throughout their entire lifecycle encompass multiple stages, including construction, building material production and transportation, operation, and demolition, significantly impacting the overall carbon footprint of the power industry. However, existing substation carbon emission assessment methods suffer from shortcomings in data completeness, model accuracy, and uncertainty analysis, making it difficult to meet the needs of refined management.

[0003] While traditional Life Cycle Assessment (LCA) can quantify substation carbon emissions in stages, it primarily focuses on deterministic accounting and fails to adequately consider the impact of factors such as missing data, model assumption biases, and parameter uncertainties. For example, energy consumption data during the building materials production stage may contain errors due to differences in production processes or statistical methods, and equipment energy consumption parameters during operation may also degrade over time, making it difficult for traditional accounting results to accurately reflect actual carbon emission levels. Existing studies often use Monte Carlo simulation to handle carbon emission uncertainties, but this method relies on extensive repeated sampling, resulting in low computational efficiency and providing only discrete probability results, failing to intuitively reveal the higher-order statistical characteristics of carbon emission distribution. Furthermore, qualitative descriptions of data quality are difficult to directly translate into quantitative analysis inputs, leading to a lack of systematic approach in quantifying the sources of uncertainty.

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

[0005] In view of this, in order to address the shortcomings of existing substation carbon emission assessment methods in terms of data integrity, model accuracy, and uncertainty analysis, which make it difficult to meet the refined management needs under the goal of "carbon peaking and carbon neutrality", this invention provides a substation full life cycle carbon emission assessment method based on the semi-invariant method.

[0006] To achieve the above objectives, the present invention provides the following technical solution:

[0007] A method for assessing the carbon emissions of substations throughout their entire life cycle based on a semi-invariant method includes the following steps:

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

[0009] 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, solve the raw moments of random variables. According to the relationship between semi-invariants and raw moments, solve the semi-invariants of total activity consumption at each stage. According to the properties of semi-invariants, convert the semi-invariants of total activity consumption at each stage into semi-invariants of carbon emissions.

[0010] S3. Substation carbon emission distribution function fitting: The Gram-Charlier series expansion method is selected. Based on the semi-invariants of carbon emissions in step S2, the Gram-Charlier series expansion is performed to obtain the expansion of the probability density function and the cumulative distribution function. Then, the scaling transformation is performed 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 carbon emission data throughout the entire life cycle of substations: Using the data quality index evaluation method, score the carbon emission inventory data throughout the entire life cycle of substations from five dimensions: reliability, completeness, technical relevance, regional relevance, and time relevance, and establish a data quality evaluation matrix; Through the data quality evaluation matrix, conduct a comprehensive qualitative and quantitative assessment of the carbon emission data at each stage of the entire life cycle of substations, qualitatively analyze the causes of uncertainty, and quantitatively quantify 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, the comprehensive data quality score is transformed into the shape parameters and endpoint parameters of the quantification model.

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

[0015] (1)

[0016] In the formula: It is the Gamma function; and For shape parameters; and For endpoint parameters;

[0017] The overall data quality score is transformed into shape parameters and endpoint parameters using empirical formulas, as follows:

[0018] (2)

[0019] In the formula: Indicates conversion to integer; These are the parameter values ​​for each indicator in the data list; A comprehensive data quality score is generated; the comprehensive data quality score is then converted into shape parameters and endpoint parameters using empirical formulas, thereby quantifying the uncertainty of carbon emissions.

[0020] Furthermore, in step S11, the substation's full life-cycle carbon emission inventory data is scored using a [1,5] score system, where 1 point represents the worst data quality, i.e., the greatest uncertainty, and 5 points represent 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-invariants of a random variable and its raw moments:

[0023] (3)

[0024] In the formula: Represents the nth-order semi-invariant of a random variable; Let n represent the nth raw moment of a random variable, where the first raw moment is the expectation of the random variable. From Take from different elements The number of combinations of elements; semi-invariants have the following two properties: the semi-invariants of the sum of independent random variables are equal to the sum of the semi-invariants of that variable and the random variable. A k-th order semi-invariant multiplied by itself is equal to the k-th order semi-invariant of that variable. These two important properties are key to simplifying calculations;

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

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

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

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

[0029] (4)

[0030] In the formula: This is the notation for the standardized random variable of the original variable; is the probability density function of the standard normal distribution; The cumulative distribution function of the standard normal distribution; These are the expansion coefficients;

[0031] If the central moments and semi-invariants of the original variables are known, they can be obtained using Equation 5, which shows the expansion coefficients of the first six orders:

[0032] (5)

[0033] In the formula: The k-th 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. Let Z be the probability density function of the random variable. After scaling Z, we obtain the probability density function of the original variable. The scaling formula is as follows:

[0035] (6)

[0036] In the formula: Represents the original variable The density function.

[0037] The beneficial effects of this invention are as follows:

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

[0039] 1. Improved the accuracy of carbon emission assessment; by calculating the statistical characteristics of carbon emissions at each stage of the substation's entire life cycle using the semi-invariant method, it can effectively handle uncertainty issues. Semi-invariants possess the property that the sum of all semi-invariants of independent random variables equals the sum of all semi-invariants of that variable, and that random variables... A k-th order semi-invariant multiplied by itself is equal to the k-th order semi-invariant of that variable. This multiple property makes the calculation process simpler and more accurate in reflecting the statistical characteristics of carbon emissions.

[0040] 2. This method improves the efficiency and systematic nature of uncertainty analysis. By constructing a quality evaluation matrix for carbon emission data throughout the entire life cycle of substations, uncertainty is quantified, and the carbon emission distribution function is fitted using Gram-Charlier series expansion, which can intuitively display the probability distribution characteristics of carbon emissions. This method not only improves the efficiency of uncertainty analysis but also compensates for the shortcomings of existing methods through a systematic uncertainty quantification process.

[0041] 3. The uncertainty quantification process was systematized. By constructing a data quality evaluation matrix for the entire life cycle of substation carbon emissions, the inventory data was scored from five dimensions: reliability, completeness, technical relevance, regional relevance, and temporal relevance, establishing quantitative evaluation standards. This systematic uncertainty quantification method can transform the qualitative description of data quality into quantitative analysis input, providing a reliable foundation for subsequent statistical characteristic calculations and distribution function fitting.

[0042] 4. Provides a reliable basis for carbon emission reduction strategies; by accurately calculating the carbon emissions at each stage of the substation's entire life cycle and quantifying its uncertainties, it can provide power industry practitioners with more accurate carbon emission data and more reliable uncertainty analysis results. This helps to formulate more scientific and effective carbon emission reduction strategies, optimize the design, construction, and operation processes of substations, and thus achieve the low-carbon transformation of the power industry.

[0043] Other advantages, objectives, and features of the invention will be set forth in part in the description which follows, and in part will be apparent to those skilled in the art from the following examination, or may be learned from practice of the invention. The objectives and other advantages of the invention can be realized and obtained through the following description. Attached Figure Description

[0044] To make the objectives, technical solutions, and advantages of the present invention clearer, the preferred embodiments of the present invention will be described in detail below with reference to the accompanying drawings, wherein:

[0045] Figure 1 This is a flowchart of the substation life-cycle carbon emission assessment solution based on the semi-invariant method in this embodiment. Detailed Implementation

[0046] The following specific examples illustrate the implementation of the present invention. Those skilled in the art can easily understand other advantages and effects of the present invention from the content disclosed in this specification. The present invention can also be implemented or applied through other different specific embodiments, and various details in this specification can be modified or changed 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 representations of the basic concept of the present invention. Unless otherwise specified, the following embodiments and features can be combined with each other.

[0047] This invention proposes a substation lifecycle carbon emission assessment method based on the semi-invariant method. The method aims to determine the carbon emissions at each stage of the substation's lifecycle and analyze the uncertainty. It constructs a data quality evaluation matrix for the substation's lifecycle carbon emissions to quantify uncertainty, uses the semi-invariant method to solve statistical characteristics, and fits the carbon emission distribution function using Gram-Charlier series expansion. Finally, a 110kV substation is used as an example for verification. A multi-stage data acquisition 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 the uncertainty quantification model for carbon emissions throughout the entire life cycle of substations: Establish a data quality evaluation matrix for carbon emissions throughout the entire 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. Establishing a Substation Life Cycle Carbon Emission Data Quality Evaluation Matrix: In the process of quantifying uncertainty, this invention conducts a comprehensive qualitative and quantitative evaluation of the substation life cycle carbon emission inventory data. The qualitative analysis addresses the causes of uncertainty, while the quantitative analysis quantifies the deviations caused by uncertainty. Given the difficulty in obtaining complete carbon emission inventory data due to differences in system boundaries, activity processes, and life cycle stages among different research objects, a data quality index evaluation method is selected for subsequent research. A substation life cycle carbon emission data quality evaluation matrix is ​​constructed, which includes five evaluation indicators: reliability, completeness, technical relevance, regional relevance, and time relevance. A [1,5] scoring system is adopted (1 point represents the worst data quality, i.e., the greatest uncertainty, and 5 points represent the best data quality, i.e., the least 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 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, the comprehensive data quality score is transformed into the shape parameters and endpoint parameters of the quantification model.

[0055] The uncertainty quantification model is as follows:

[0056] (1)

[0057] In the formula: It is the Gamma function; and For shape parameters; and For endpoint parameters;

[0058] The overall data quality score is transformed into shape parameters and endpoint parameters using empirical formulas, as follows:

[0059] (2)

[0060] In the formula: Indicates conversion to integer; These are the parameter values ​​for each indicator in the data list; A comprehensive data quality score is generated; the comprehensive data quality score is then converted into shape parameters and endpoint parameters using empirical formulas, thereby quantifying the uncertainty of carbon emissions.

[0061] S2. Calculation of substation carbon emission statistical characteristics based on semi-invariant method: Based on the uncertainty quantification model in step S1, solve the raw moments of the random variables. According to the relationship between semi-invariants and raw moments, solve the semi-invariants of the total activity consumption in each stage. According to the properties of semi-invariants, transform the semi-invariants of the total activity consumption in each stage into the semi-invariants of carbon emissions.

[0062] Specifically:

[0063] S21. The moments of a random variable are its numerical characteristics, and semi-invariants are also a type of numerical characteristic of a random variable. They can be obtained from moments of orders no higher than the corresponding orders. The relationship between the semi-invariants of a random variable and the raw moments is as follows:

[0064] (3)

[0065] In the formula: Represents the nth-order semi-invariant of a random variable; Let n represent the nth raw moment of a random variable, where the first raw moment is the expectation of the random variable. From Take from different elements The number of combinations of elements.

[0066] Semi-invariants possess the following two properties: the sum of the semi-invariants of independent random variables equals the sum of the semi-invariants of that variable plus the sum of the random variables. A k-th order semi-invariant multiplied by itself is equal to the k-th order semi-invariant of that variable. These two important properties are key to simplifying calculations.

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

[0068] S3. Substation carbon emission distribution function fitting: The Gram-Charlier series expansion method is selected. Based on the semi-invariants of carbon emissions in step S2, the Gram-Charlier series expansion is performed to obtain the expansion of the probability density function and the cumulative distribution function. Then, the scaling transformation is performed to obtain the probability density function of the original variable.

[0069] Specifically:

[0070] S31. Selection of the Best Approximate Distribution Function Series Expansion Method: For the problem of calculating the distribution function value of an unknown random variable, there are various existing methods based on moments or semi-invariants, including Gram-Charlier series expansion, Cornish-Fisher series expansion, Edgeworth series expansion, and the maximum entropy principle. This invention employs the Gram-Charlier series expansion method for specific calculations. This method achieves an approximate expression of the target distribution function by systematically utilizing information from higher-order moments.

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

[0072] (4)

[0073] In the formula: This is the notation for the standardized random variable of the original variable; is the probability density function of the standard normal distribution; The cumulative distribution function of the standard normal distribution; These are the expansion coefficients;

[0074] If the central moments and semi-invariants of the original variables are known, they can be obtained using Equation 5, which shows the expansion coefficients of the first six orders:

[0075] (5)

[0076] In the formula: It represents the k-th 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. Let Z be the probability density function of the random variable. After scaling Z, we obtain the probability density function of the original variable. The scaling formula is as follows:

[0078] (6)

[0079] In the formula: Represents the original variable The density function.

[0080] Example

[0081] Taking a typical 110kV substation as an example, the main transformer capacity is 2*50MVA; the general design scheme of 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, and 24 outgoing lines in the current phase and 36 in the future.

[0082] The calculation process is as follows Figure 1 As shown, Figure 1 The left side shows the workflow based on the semi-invariant method, with Monte Carlo simulation serving as the evaluation criterion for the semi-invariant method. The right side shows the algorithm flow under Monte Carlo simulation. Methods for estimating the probability density distribution of a dataset are divided into parametric estimation and non-parametric estimation, the difference being whether prior knowledge is incorporated. To fit a more accurate model, this invention uses non-parametric estimation, that is, fitting the distribution through the characteristics and properties of the data itself; specifically, kernel density estimation is chosen.

[0083] Specific implementation steps:

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

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

[0086] S3. Substitute the semi-invariants 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] Table 2 shows the 95% confidence intervals for carbon emissions under 6000 Monte Carlo simulations, 10000 Monte Carlo simulations, and semi-infinite method for each stage (construction stage, building material production and transportation stage, operation stage, demolition stage, and overall stage) of the 110kV substation.

[0089] Table 2: 95% Confidence Intervals for Carbon Emissions at Different Stages and Using Different Methods

[0090]

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

[0092] Table 3: Relative width error of confidence interval under different stages and methods

[0093]

[0094] The above calculation results show that the 95% confidence interval for carbon emissions obtained by the method of this invention is more accurate at all stages compared to the 95% confidence intervals for carbon emissions under 6000 Monte Carlo simulations and 10000 Monte Carlo simulations. The relative width error of the confidence interval obtained by the method of this invention is smaller than that of the 6000 Monte Carlo confidence interval, indicating that the semi-invariant method is more accurate in describing the uncertainty of carbon emissions at each stage, and the statistical information on carbon emissions under the semi-invariant method has higher reliability.

[0095] This evaluation method quantifies uncertainty by constructing a quality evaluation matrix for carbon emission data throughout the entire life cycle of a substation, solves for statistical characteristics using the semi-invariant method, and fits the carbon emission distribution function using Gram-Charlier series expansion. A 110kV substation is used as an example for verification. The results show that the confidence intervals of the semi-invariant method at each stage and throughout the entire life cycle are in high agreement with high-order Monte Carlo simulations, and the relative width error is significantly lower, verifying the effectiveness of this method for accurate carbon emission calculation and uncertainty quantification.

[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 intended to limit it. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the present invention without departing from the spirit and scope of the present invention, and all such modifications or substitutions should be covered within the scope of the claims of the present invention.

Claims

1. A method for assessing the carbon emissions of a substation throughout its entire life cycle based on the semi-invariant method, characterized in that, Includes the following steps: S1. Construction of the uncertainty quantification model for carbon emissions throughout the entire life cycle of substations: Establish a data quality evaluation matrix for carbon emissions throughout the entire life cycle of substations. After establishing the data quality evaluation matrix, use the Beta function to establish an uncertainty quantification model. S2. Calculation of substation carbon emission statistical characteristics based on semi-invariant method: Based on the uncertainty quantification model in step S1, solve the raw moments of random variables. According to the relationship between semi-invariants and raw moments, solve the semi-invariants of total activity consumption in each stage. According to the properties of semi-invariants, convert the semi-invariants of total activity consumption in each stage into semi-invariants of carbon emissions. S3. Substation carbon emission distribution function fitting: Select the Gram-Charlier series expansion method. Based on the semi-invariants of carbon emissions in step S2, perform Gram-Charlier series expansion to obtain the expansion of the probability density function and the cumulative distribution function. Then, perform scaling transformation to obtain the probability density function of the original variable. S1 specifically includes the following steps: S11. Establish a data quality evaluation matrix for carbon emission data throughout the entire life cycle of substations: Using the data quality index evaluation method, score the carbon emission inventory data throughout the entire life cycle of substations from 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, the comprehensive data quality score is transformed into the shape parameters and endpoint parameters of the quantification model. The uncertainty quantification model in step S12 is constructed as follows: （1） In the formula: It is the Gamma function; and For shape parameters; and For endpoint parameters; The overall data quality score is transformed into shape parameters and endpoint parameters using empirical formulas, as follows: （2） In the formula: Indicates conversion to integer; These are the parameter values ​​for each indicator in the data list; A comprehensive score for data quality.

2. The substation life-cycle carbon emission assessment method based on the semi-invariant method as described in claim 1, characterized in that, In step S11, the carbon emission inventory data of the substation throughout its entire life cycle is scored using a [1,5] score system, where 1 point represents the worst data quality, i.e., the greatest uncertainty, and 5 points represent the best data quality, i.e., the least uncertainty.

3. The substation life-cycle carbon emission assessment method based on the semi-invariant method as described in claim 2, characterized in that, Step S2 specifically includes the following steps: S21. The relationship between the semi-invariants of a random variable and its raw moments: （3） In the formula: Represents the nth-order semi-invariant of a random variable; Let n represent the nth raw moment of a random variable, where the first raw moment is the expectation of the random variable. From Take from different elements The number of combinations of elements. Semi-invariants possess the following two properties: the sum of the semi-invariants of independent random variables equals the sum of the semi-invariants of that variable plus the sum of the random variables. A k-th order semi-invariant multiplied by itself is equal to the k-th order semi-invariant of that variable. times; S22: Obtain semi-invariants of carbon emissions: Based on the uncertainty quantification model obtained in step S12, use formula 3 to solve for the semi-invariants of total activity consumption at each stage of the substation's entire life cycle; according to the properties of semi-invariants, transform the semi-invariants of total activity consumption at each stage into semi-invariants of carbon emissions, providing a basis for subsequent distribution function fitting.

4. The substation life-cycle carbon emission assessment method based on the semi-invariant method as described in claim 3, characterized in that, 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 unknown random variables, select the Gram-Charlier series expansion method from Gram-Charlier series expansion, Cornish-Fisher series expansion, Edgeworth series expansion and the maximum entropy principle. S32. Obtain the 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; if the central moments and semi-invariants of the original variables are known, find the expansion coefficients of the first six orders. S33. Obtain the probability density function of the original variable: the density function fitted by Gram-Charlier series expansion. It is the probability density function of the random variable Z. After scaling it, we get the probability density function of the original variable.

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