A quantitative evaluation method for the output error probability distribution of a parallel local model of a power system

Abstract

This invention provides a quantitative evaluation method for the probability distribution of output error in a parallel local model of a power system, comprising the following steps: S1 using a Gram-Charlier series to approximate an unknown error probability density function, including unknown quantities g3 and g4; S2 modeling the probability density function of the output error of the parallel local model based on the statistical results of test sample errors, solving for unknown quantities g3 and g4, and obtaining the probability density function of the output error of the parallel local model. By combining the statistical results of test sample errors with the Gram-Charlier series, the probability density function of the output error of the parallel local model is modeled, and the accuracy of the modeling method is analyzed. It is found that the error probability density function established based on the Gram-Charlier series is more accurate than other methods.

Description

Technical Field

[0001] This invention belongs to the field of quantitative evaluation of output error probability distribution, and specifically relates to a method for quantitative evaluation of the output error probability distribution of a parallel local model of a power system. Background Technology

[0002] Since the parallel local model is based on sample data, errors are inevitable. Therefore, it is necessary to model the probability distribution of the output error of the parallel local model, and then quantify the impact of the output error to ensure the accuracy and reliability of the transient voltage safety assessment results.

[0003] Since the output error of a machine learning model is affected by many factors, including model structure, training samples and training algorithms, it is impossible to determine the form of the output error distribution in advance. Modeling can only be based on the statistical results of the test sample error. Summary of the Invention

[0004] This invention provides a quantitative evaluation method for the probability distribution of output error of a parallel local model of a power system, so as to reduce the output error of the parallel local model of the power system and ensure the accuracy and reliability of transient voltage safety evaluation results.

[0005] A method for quantitatively evaluating the probability distribution of output error in a parallel local model of a power system includes the following steps:

[0006] S1 uses Gram-Charlier series to approximate the unknown error probability density function, which includes the unknowns g3 and g4.

[0007] S2 models the probability density function of the output error of the parallel local model based on the statistical results of the test sample error, solves for the unknown quantities g3 and g4, and obtains the probability density function of the output error of the parallel local model.

[0008] Furthermore, S1 specifically includes:

[0009] S1.1 Assumption It is a random variable with mean μ and standard deviation σ;

[0010] For ease of analysis, Standardization process:

[0011]

[0012] Where x is a pair The standardized random variable has a mean of 0 and a standard deviation of 1.

[0013] The Gram-Charlier series expansion of the probability density function f(x) of the random variable x in S1.2 is:

[0014]

[0015] Among them, c i These are the coefficients of the Gram-Charlier series expansion, φ (i) φ(x) is the i-th derivative of the standard normal distribution φ(x); the expression for φ(x) is:

[0016]

[0017] Gram-Charlier series expansion coefficients c i The calculation can be performed based on the central moment of x, as shown in the following expression:

[0018]

[0019] Among them, g i Let x be the i-th central moment, defined as:

[0020]

[0021] Where f(x) is the probability density function of x, and μ is the mean of x;

[0022] Since the mean of x is 0, equation (5) can be simplified to:

[0023]

[0024] If the central moment is calculated based on the sampled data of x, the calculation expression is:

[0025]

[0026] Among them, g i It is the i-th central moment of x; x n These are the sampled data of x, with a total number of samples of N;

[0027] It can be observed that g1 is the mean of x, and g2 is the variance of x; furthermore, in the field of statistics, g3 is defined as the skewness of x, and g4 is defined as the kurtosis of x.

[0028] The i-th derivative φ(x) of the standard normal distribution (i) (x) can be calculated based on Hermitian polynomials; in probability theory, Hermitian polynomials are defined as follows:

[0029]

[0030] According to the definition, the expression for the Hermitian polynomial can be derived as follows:

[0031] H0(x)=1, H1(x)=x, H2(x)=x 2 -1, H3(x)=x 3 -3x, H4(x)=x 4 -6x 2 +3, ... (9)

[0032] From equations (4) and (9), we can see that as the order increases, the coefficients c of the Gram-Charlier series expansion increase. i Decrease, c i and φ (i) The computational complexity of (x) increases; if the first 5 terms in equation (2) are retained, that is, the probability density function f(x) of the random variable x is approximated by a 4th-order Gram-Charlier series, then the expression is:

[0033]

[0034] Among them, φ(x), H3(x) and H4(x) are fixed, and the unknowns are g3 and g4.

[0035] Furthermore, S2 specifically includes:

[0036] S2.1 Calculate the error output by the parallel local model on the test sample, expressed as:

[0037]

[0038] in, and These are the error, true value, and estimated value of the t-th component of the TVSI index for the n-th test sample, respectively.

[0039] TVSI Index Estimate It is the output of the parallel local model, the true value of the TVSI index. This is the output of the test sample;

[0040] S2.2 TVSI index error Standardization process:

[0041]

[0042] in, It is the error of the t-th component of the TVSI index of the nth test sample after standardization, N test It is the number of test samples; The mean is 0, and the variance and standard deviation are 1.

[0043] S2.3 Calculate the standardized TVSI index error ΔTVSI pred The central moment of is expressed as follows:

[0044]

[0045] Among them, g i,t It is the i-th order central moment of the error of the t-th component of the TVSI index;

[0046] For equation (10), it is only necessary to calculate the third and fourth order central moments;

[0047] S2.4 The central moment g calculated based on equation (13) i,t Substituting into equation (10), we can obtain the probability density function of the standardized TVSI index error, which is expressed as:

[0048]

[0049] Among them, f t ΔTVSI (x) is the probability density function of the error of the t-th component of the standardized TVSI index, g 3,t and g 4,t These are the third and fourth order central moments of the error of the t-th component of the standardized TVSI index, respectively.

[0050] Based on equations (14) and (12), the probability density function of the TVSI index before standardization can be further derived, which is also the probability density function of the output error of the parallel local model, and its expression is:

[0051]

[0052] in, It is the probability density function of the error of the t-th component of the TVSI index output by the parallel local model.

[0053] The beneficial effects of this invention are as follows:

[0054] The error probability density function established based on the Gram-Charlier series is more accurate than other methods. Detailed Implementation

[0055] Case Analysis

[0056] By combining the statistical results of the test sample error with the Gram-Charlier series, the probability density function of the output error of the parallel local model is modeled, and the accuracy of the modeling method is analyzed.

[0057] This example uses the IEEE 39-bus model. A three-phase N-1 fault is considered on line "Bus15-Bus16," with a fault duration of 250ms. Initial grid topology parameters and grid operating conditions are considered, and the initial number of operating scenarios is 1. The load model is a composite load consisting of induction motors and constant impedance loads, with induction motors accounting for 60%.

[0058] The output error of the parallel local model on the test sample set is modeled based on the fourth-order Gram-Charlier series, and the expression is as follows:

[0059]

[0060] in,

[0061] σ1=1.04×10 -3 σ² = 6.20 × 10 -3 σ3=4.86×10 -3 (17)

[0062] and σ1, σ2, and σ3 are the probability density functions of the errors of the three components of the TVSI index output by the parallel local model. σ1, σ2, and σ3 are the standard deviations of the error of the t-th component of the TVSI index on the test sample set. φ(x) is a standard normal distribution.

[0063] The following is based on the error probability density function Calculate the quantiles of the TVSI index error to quantify the assessment. The precision. For a random variable x, its τ quantile is defined as:

[0064] P(x≤x τ )=τ (18)

[0065] Where, x τ It is the τ quantile of x.

[0066] Quantiles are important statistical characteristics of random variables, and confidence intervals for random variables can be derived based on quantiles. For example, suppose the 2.5th quantile of x is x_2.5. 0.025 The 97.5th percentile of x is x 0.975 Then [x 0.025 ,x 0.975 [ ] represents the 95% confidence interval for x. The quantiles of a random variable can be calculated based on its probability density function, expressed as:

[0067]

[0068] Here, f(x) is the probability density function of x.

[0069] Three methods are considered for calculating the quantiles of the output error of a parallel local model: (1) based on the statistical results of a large number of test samples; (2) based on the fourth-order Gram-Charlier series; and (3) based on the normal distribution. Among them, the calculation results of the first method are used as the true values ​​of the quantiles; the second and third methods are based on fitting the probability density function of the output error with 500 test samples and calculating the quantiles of the output error based on Equation (19).

[0070] To evaluate the accuracy of the output error quantile calculation, the relative error index of the quantile is defined as follows:

[0071]

[0072] in, and These are the true and estimated values ​​of the τ quantile, respectively. σ is the standard deviation.

[0073] The quantiles of the TVSI index error calculated using the three methods are compared below, as shown in Tables 1-3. The 0.5% and 99.5% quantiles form a 99% confidence interval, and the 2.5% and 97.5% quantiles form a 95% confidence interval.

[0074] Table 1 Comparison of TVSI1 error quantile calculation results

[0075]

[0076] Table 2 Comparison of TVSI2 error quantile calculation results

[0077]

[0078]

[0079] Table 3 Comparison of TVSI3 error quantile calculation results

[0080]

[0081] Analyzing the results in Tables 1-3, the average relative errors of the three error quantiles of the TVSI index calculated based on the 4th-order Gram-Charlier series are 12.93%, 16.06%, and 12.40%, respectively, while the average relative errors calculated based on the normal distribution are 54.12%, 58.01%, and 50.18%, respectively. In comparison, the errors of the TVSI index error quantiles calculated based on the Gram-Charlier series are smaller, decreasing by 76.11%, 72.32%, and 75.29% compared to those calculated based on the normal distribution, indicating that the error probability density function established based on the Gram-Charlier series is more accurate.

[0082] In practical applications, different orders can be selected according to the required computational accuracy, thus making the error probability density modeling method based on Gram-Charlier series more flexible. Here, a 4th-order Gram-Charlier series is used by default.

[0083] The above description is merely a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any equivalent substitutions or modifications made by those skilled in the art within the technical scope disclosed in the present invention, based on the technical solution and inventive concept of the present invention, should be covered within the scope of protection of the present invention.

Claims

1. A method for quantitatively evaluating the probability distribution of output error in a parallel local model of a power system, characterized in that, Includes the following steps: S1 uses Gram-Charlier series to approximate the unknown error probability density function, which contains the unknown quantity. and Specifically, this includes: based on Calculation of central moments of Gram-Charlier series expansion coefficients The expression is as follows: ， ， ， ， ， （4） in, yes The The first central moment; Approximating random variables using 4th-order Gram-Charlier series probability density function The expression is: （10） in, , and All are fixed, the quantity to be determined is and ; S2 models the probability density function of the output error of the parallel local model based on the statistical results of the test sample error, and solves for the unknown quantity. and The probability density function of the output error of the parallel local model is derived, specifically including: S2.1 According to the first TVSI index of the test sample The error, true value, and estimated value of each component are used to calculate the error of the parallel local model output on the test sample. S2.2 Error of TVSI index Standardization process is performed to obtain ; S2.3 Calculate the standardized TVSI index error The central moments g of the third and fourth orders 3,t and g 4,t ; S2.4 Calculate the g 3,t and g 4,t Substituting into equation (10), we obtain the probability density function of the standardized TVSI index error, which is expressed as: （14） in, It is the standardized TVSI index. The probability density function of the error component; based on and The probability density function of the output error of the parallel local model is obtained, and its expression is: （15） in, Calculate the standard deviation.

2. The method according to claim 1, characterized in that, S1 specifically includes: S1.1 Assumptions It is a mean Standard deviation is random variables; For ease of analysis, Standardization process: （1） in, Yes The standardized random variable has a mean of 0 and a standard deviation of 1. S1.2 Random Variables probability density function The Gram-Charlier series expansion is: （2） in, These are the coefficients of the Gram-Charlier series expansion. It is a standard normal distribution The First derivative; The expression is: （3） Gram-Charlier series expansion coefficients based on central moments Perform calculations. Defined as: （5） in, yes The probability density function, yes The mean; because The mean of is 0, and equation (5) simplifies to: （6） If based on The central moments are calculated from the sampled data, and the expression for the calculation is: （7） in, yes The The first central moment; yes The sampling data, the total number of samples is ; It can be observed that, yes The mean, yes The variance; furthermore, in the field of statistics, Defined as skewness, Defined as Peak value; Standard normal distribution The Derivative Calculations are performed based on Hermitian polynomials; in probability theory, a Hermitian polynomial is defined as follows: （8） According to the definition, the expression for the Hermitian polynomial is: ， ， ， ， ，… （9） Approximating random variables using 4th-order Gram-Charlier series The probability density function is obtained. .

3. The method according to claim 1, characterized in that, S2 specifically includes: S2.1 Calculate the error output by the parallel local model on the test sample, expressed as: （11） in, , and They are the first TVSI index of the test sample The error, true value, and estimated value of each component; TVSI Index Estimate It is the output of the parallel local model, the true value of the TVSI index. This is the output of the test sample; S2.2 Error of TVSI index Standardization process: （12） in, It is the standardized first TVSI index of the test sample The error of each component, It is the number of test samples; The mean is 0, and the variance and standard deviation are 1. S2.3 Calculate the standardized TVSI index error The central moment of is expressed as follows: （13） in, It is the TVSI index number The first component error The first central moment; For equation (10), only the third and fourth order central moments are calculated; S2.4 Calculate the g 3,t and g 4,t Substituting into equation (10), we obtain the probability density function of the standardized TVSI index error. ,based on and The probability density function of the output error of the parallel local model is obtained.

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