CN111753252A - Nataf transformation-based random variable sample generation method and system - Google Patents

Abstract

The invention relates to a random variable sample generation method and system based on Nataf transformation, wherein the method comprises the following steps: establishing a conversion relation between a correlation coefficient matrix of the random variable in the original distribution domain and a correlation coefficient matrix of the random variable in the standard Gaussian distribution domain according to Nataf transformation; determining a correlation coefficient matrix of the random variable in the standard Gaussian distribution domain by adopting a radial basis network based on the conversion relation between the correlation coefficient matrix of the random variable in the original distribution domain and the correlation coefficient matrix of the random variable in the standard Gaussian distribution domain; and generating sample data of the random variable in the original distribution domain by using a correlation coefficient matrix of the random variable in the standard Gaussian distribution domain. According to the technical scheme provided by the invention, the conversion relation of the correlation coefficient of the random variable in the original distribution domain and the standard Gaussian distribution domain in the Nataf transformation is simplified by using the radial basis function neural network, the solving speed of the correlation coefficient of the random variable in the standard Gaussian distribution domain in the Nataf transformation is accelerated, and the generation speed of the random variable sample is accelerated.

Description

Nataf transformation-based random variable sample generation method and system

Technical Field

The invention relates to the technical field of new energy grid connection technology and optimal power flow analysis, in particular to a random variable sample generation method and system based on Nataf transformation.

Background

With the development of new energy technology, more and more new energy power plants are incorporated into the power network. For different kinds of new energy power plants, most of primary energy sources of the new energy power plants have strong random uncertainty, such as wind speed of a wind power plant, solar radiation intensity of a photovoltaic power plant, tidal flow rate of a tidal power plant and the like. In the case of large-scale grid connection of new energy, great uncertainty is brought to a power system by the uncertain sources.

Probabilistic power flow and probabilistic optimal power flow have been proposed as basic analysis methods capable of effectively handling power systems with probabilistic uncertainty sources. For a general probabilistic power flow and probabilistic optimal power flow algorithm, the algorithm mainly comprises three steps of probability modeling, probabilistic (optimal) power flow calculation, analysis of probability calculation results and application. As the scale of the power grid increases, the number of uncertain sources contained in the power system increases, and the probability analysis of the whole network will be a very time-consuming process (under the condition of ensuring reasonable analysis accuracy). The time of probability analysis is mainly consumed in probability modeling and probability (optimal) power flow calculation, and the time of probability modeling is often ignored by people.

Generally, probabilistic modeling is divided into two blocks — edge distribution modeling and correlation modeling. Edge distribution modeling is generally based on actually measured data, and probability density functions obeyed by variables are estimated through a mathematical statistics method, and the process needs a larger sample length. For correlation modeling, it is common to describe the correlation between random variables using pearson correlation coefficients, i.e., linear correlation coefficients. Similarly, the linear correlation coefficient is calculated from measured data corresponding to the variable.

However, in probabilistic (optimal) power flow analysis, in order to perform a probabilistic calculation, it is first necessary to generate a sample that satisfies a probabilistic model. Practice has shown, however, that a considerable number of random variables are subject to edge distribution types that are not gaussian, whereas non-gaussian samples with correlations cannot be generated directly. Therefore, when dealing with random variables with linear correlations, other methods are needed to generate appropriate samples, such as the Nataf transform.

On the premise of knowing the cumulative distribution function of the random variable or the inverse function thereof, the Nataf transformation effectively connects the standard Gaussian distribution domain and the original distribution domain of the variable by an equal probability method. Thus, the generation of the original domain samples translates into the problem of generating standard gaussian distributed samples with a specific correlation, which can be easily implemented. However, the problem is that the Nataf transform is a monotonous nonlinear transformation process, and the linear correlation coefficient values of two variables are changed before and after the nonlinear transformation based on the properties of the linear correlation coefficient. Therefore, it is an important step to calculate the correlation coefficient of the corresponding variable in the standard gaussian domain according to the distribution information of the variable in the original distribution domain, such as the edge distribution and the correlation coefficient. From the knowledge of probability theory, it can be found that the solving process is an implicit solving problem of a nonlinear equation containing double integral.

For the problem of solving the correlation coefficient of the original domain in the Nataf transformation, the traditional Simpson method and the dichotomy method can be used for calculating more accurately, but the calculation speed is very time-consuming. Practice has shown that this method takes up to minutes for the calculation of a correlation coefficient. The correlation coefficient calculations involved may take as long as several days for a large scale system that may contain hundreds or thousands of random variables. Therefore, in a computing scenario with high real-time requirements, such a method is very undesirable. Later through algorithm development, a calculation method based on a Gauss-Hermite product method and an interpolation method is developed, the method has considerable calculation advantages in speed, however, the numerical integration and the interpolation calculation both introduce extra errors, and improper parameter selection can have great influence on calculation results. Therefore, based on such a conventional correlation coefficient calculation method, the practical applicability of the method may not be high in the application of real-time probabilistic optimal power flow calculation of a large power grid containing high-dimensional random sources.

Disclosure of Invention

Aiming at the defects of the prior art, the invention aims to provide a random variable sample generation method based on Nataf transformation, which utilizes a radial basis function neural network to simplify the conversion relation of correlation coefficients of random variables in an original distribution domain and a standard Gaussian distribution domain in the Nataf transformation, accelerates the solving speed of the correlation coefficients of the random variables in the standard Gaussian distribution domain in the Nataf transformation, and further accelerates the generation speed of the random variable sample.

The purpose of the invention is realized by adopting the following technical scheme:

the invention provides a random variable sample generation method based on Nataf transformation, and the improvement is that the method comprises the following steps:

establishing a conversion relation between a correlation coefficient matrix of the random variable in the original distribution domain and a correlation coefficient matrix of the random variable in the standard Gaussian distribution domain according to Nataf transformation;

determining a correlation coefficient matrix of the random variable in the standard Gaussian distribution domain by adopting a radial basis network based on the conversion relation between the correlation coefficient matrix of the random variable in the original distribution domain and the correlation coefficient matrix of the random variable in the standard Gaussian distribution domain;

and generating sample data of the random variable in the original distribution domain by using a correlation coefficient matrix of the random variable in the standard Gaussian distribution domain.

The invention provides a random variable sample generation system based on Nataf transformation, and the improvement is that the system comprises:

the establishing module is used for establishing a conversion relation between a correlation coefficient matrix of the random variable in the original distribution domain and a correlation coefficient matrix of the random variable in the standard Gaussian distribution domain according to Nataf transformation;

the determining module is used for determining the correlation coefficient matrix of the random variable in the standard Gaussian distribution domain by adopting a radial basis network based on the conversion relation between the correlation coefficient matrix of the random variable in the original distribution domain and the correlation coefficient matrix of the random variable in the standard Gaussian distribution domain;

and the generating module is used for generating sample data of the random variable in the original distribution domain by using the correlation coefficient matrix of the random variable in the standard Gaussian distribution domain.

Compared with the closest prior art, the invention has the following beneficial effects:

according to the technical scheme provided by the invention, the conversion relation between the correlation coefficient matrix of the random variable in the original distribution domain and the correlation coefficient matrix of the random variable in the standard Gaussian distribution domain is established according to Nataf transformation; determining a correlation coefficient matrix of the random variable in the standard Gaussian distribution domain by adopting a radial basis network based on the conversion relation between the correlation coefficient matrix of the random variable in the original distribution domain and the correlation coefficient matrix of the random variable in the standard Gaussian distribution domain; and generating sample data of the random variable in the original distribution domain by using a correlation coefficient matrix of the random variable in the standard Gaussian distribution domain. The method simplifies the conversion relation of the correlation coefficients of the random variables in the original distribution domain and the standard Gaussian distribution domain in the Nataf transformation by using the radial basis function neural network, accelerates the solving speed of the correlation coefficients of the random variables in the standard Gaussian distribution domain in the Nataf transformation, and further accelerates the generation speed of random variable samples.

Drawings

FIG. 1 is a flow chart of a method for generating random variable samples based on Nataf transformation;

FIG. 2 is a probability density diagram of the cost of power generation in an embodiment of the invention;

fig. 3 is a block diagram of a random variable sample generation system based on a Nataf transform.

Detailed Description

The following describes embodiments of the present invention in further detail with reference to the accompanying drawings.

In order to make the objects, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are some, but not all, embodiments of the present invention. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.

The invention provides a random variable sample generation method based on Nataf transformation, which comprises the following steps of:

step 101, establishing a conversion relation between a correlation coefficient matrix of a random variable in an original distribution domain and a correlation coefficient matrix of a random variable in a standard Gaussian distribution domain according to Nataf transformation;

102, determining a correlation coefficient matrix of the random variable in the standard Gaussian distribution domain by adopting a radial basis network based on the conversion relation between the correlation coefficient matrix of the random variable in the original distribution domain and the correlation coefficient matrix of the random variable in the standard Gaussian distribution domain;

and 103, generating sample data of the random variable in the original distribution domain by using the correlation coefficient matrix of the random variable in the standard Gaussian distribution domain.

Specifically, the step 101 includes:

step 101-1, forming an N-dimensional vector Z by using N random variables which obey standard Gaussian distribution, and determining a joint density function between a random variable i and a random variable j in the vector Z;

step 101-2, determining correlation coefficients ρ of the random variable i and the random variable j in the correlation coefficient matrix of the random variables in the original distribution domain based on the principle that the cumulative distribution function value of the random variables in the original distribution domain is equal to the cumulative distribution function value of the random variables in the gaussian distribution domain (Nataf transform) and the joint density function between the random variable i and the random variable j in the vector Z_XijAnd correlation coefficient rho of random variable i and random variable j in correlation coefficient matrix of random variable in standard Gaussian distribution domain_ZijThe conversion relationship between them;

in the preferred embodiment of the present invention, the matrix C of correlation coefficients of random variables in the original distribution domain_XMatrix C of correlation coefficients with random variables in the standard Gaussian distribution domain_ZAre all N × order N matrices, and are all strictly symmetric real number square matrices (rho)_Xij＝ρ_Xji、ρ_Zij＝ρ_Zji)，C_XAnd C_ZAll main diagonals ofThe elements are all 1, and correlation coefficient calculation formulas of the random variable i and the random variable j in the original distribution domain are determined by using a Pearson correlation coefficient method.

Wherein i, j ∈ (1-N), N is the number of random variables in the power system, and the joint density function between the random variable i and the random variable j in the vector Z

z_iIs an independent variable, z, of a random variable i in a standard Gaussian distribution domain_jIs an argument of a random variable j in the domain of a standard gaussian distribution.

In a preferred embodiment of the invention, the random variables of the power system may include: wind speed of a wind power plant, illumination intensity of a photovoltaic power plant, load, tidal energy speed of a tidal power plant, branch parameters of a power system and the like.

Further, the step 101-2 specifically includes:

determining the correlation coefficient rho of the random variable i and the random variable j in the correlation coefficient matrix of the random variable in the original distribution domain according to the following formula_XijCorrelation coefficient rho of random variable i and random variable j in correlation coefficient matrix with random variable in standard Gaussian distribution domain_ZijThe conversion relationship between:

in the formula (I), the compound is shown in the specification,

to be phi (z)_i) Substituting into the value obtained in the inverse function of the cumulative distribution function of the random variable i in the original distribution domain,

to be phi (z)_j) Substituting into the inverse function of the cumulative distribution function of the random variable j in the original distribution domain to obtain a value, Φ (z)_i) The argument for the random variable i in the normalized Gaussian distribution domain is z_iCumulative score of random variable i in time-corresponding standard Gaussian distribution domainValue of the distribution function, Φ (z)_j) The argument for the random variable j in the normalized Gaussian distribution domain is z_jThe value of the cumulative distribution function of the random variable j in the standard Gaussian distribution domain, mu_i·Is the mean value of the random variable i after the synchronous sampling of N random variables of the power system in the original distribution domain_jIs the mean value, sigma, of a random variable j after synchronously sampling N random variables of the power system in an original distribution domain_iFor the variance, σ, of the random variable i after the N random variables of the power system are synchronously sampled in the original distribution domain_jIs the variance of a random variable j after synchronously sampling N random variables of the power system in the original distribution domain.

In a specific embodiment of the present invention, the correlation coefficients ρ of the random variable i and the random variable j in the correlation coefficient matrix of the random variable in the original distribution domain_XijAnd correlation coefficient rho of random variable i and random variable j in correlation coefficient matrix of random variable in standard Gaussian distribution domain_ZijThe conversion relation between the random variables is based on the joint density function between the random variable i and the random variable j in the vector Z

Conversion relation of random variable i in original distribution domain and random variable i in standard Gaussian distribution domain

Conversion relation between random variable j in original distribution domain and random variable j in standard Gaussian distribution domain

And the correlation coefficients of the random variable i and the random variable j in the original distribution domain

μ_ijTo be the mean value of the product of the random variable i and the random variable j after synchronously sampling the N random variables of the power system in the original distribution domain,

as a function of the cumulative distribution of the random variable i in the standard gaussian distribution domain,

as a function of the cumulative distribution of the random variable i in the original distribution domain,

as a function of the cumulative distribution of the random variable j in the standard gaussian distribution domain,

is a cumulative distribution function of the random variable j in the original distribution domain.

In the preferred embodiment of the present invention, the process of obtaining the cumulative distribution function of the random variable i to the power system in the original distribution domain may be:

analyzing sampling data of a random variable i of the power system in the original distribution domain by adopting a mathematical statistics method, and acquiring a probability distribution function of the random variable i of the power system in the original distribution domain; and determining the cumulative distribution function of the random variable i of the power system in the original distribution domain according to the probability distribution function of the random variable i of the power system in the original distribution domain.

Solving the correlation coefficient matrix C of random variables in the standard Gaussian distribution domain in the invention_ZI.e. solving for all off-diagonal element values therein. According to the symmetry of the correlation coefficient matrix, the elements that need to be calculated are only the upper triangular elements (or the lower triangular elements) and do not include the main diagonal elements. On the premise of N random variables, the number of elements to be calculated is 0.5N (N-1), and correlation coefficients rho of the random variables i and j in a correlation coefficient matrix of the power system in a standard Gaussian distribution domain are calculated_ZijMay be illustrated by step 102, wherein step 102 comprises:

step 102-1, synchronously sampling N for random variables i and j of the power system in a standard Gaussian distribution domain_sThen, obtain N_sVector Z formed by sampling data of random variable i and random variable j in sub-sampling_ij,SAnd will move towardsQuantity Z_ij,SIn the formula

Obtaining a vector Z_ij,SCorresponding vector RBF_ij,s；

In the preferred embodiment of the invention, a radial basis function neural network is used to approximate Z_iAnd Z_jFor input, take X_iAnd X_jThe product of (a) is the transfer relationship of the output, formulated as

Sampling random variables i and j in a standard Gaussian distribution domain in a two-dimensional space formed by the random variables i and j to obtain N_sSamples by a vector Z_ij,SIs represented by the formula_ij,SRespectively substituting the medium elements into the above formula to obtain a vector RBF_ij,s。

Step 102-2, vector Z_ij,SSum vector RBF_ij,sRespectively as radial basis function neural networks W_ijInput data and output data of

As a radial basis neural network W_ijSetting function of the kth neuron of the hidden layer

As a radial basis neural network W_ijObtaining a setting function of an output layer to obtain a radial basis function (Wn)_ijN of the hidden layer_sAn output weight matrix ω of each neuron;

in the preferred embodiment of the present invention, RBF (Z)_ij) Is the transfer function of the radial basis function neural network, which represents when a given input quantity is (Z)_i,Z_j) The approximate output value of the radial basis function neural network after the specific sample value is RBF (Z)_ij) The radial basis function neural network structure is a three-layer network structure including an input layer, a hidden layer (radial basis layer), and an output layer, the input layer including two neurons respectively corresponding to an input quantity Z_iAnd Z_j(ii) a The hidden layer (radial base layer) comprises N_sEach neuron is a radial basis function, and the radial basis function of the kth neuron is phi_RBF,k(Z_ij) With a function width parameter of 1, the radial basis function of the kth neuron is expanded to

(Z_i,k,Z_j,k) The number of neurons in the output layer is 1, which is the center of the kth neuron. The whole radial basis function neural network is fully connected between direct layers, and the weight omega is set in the process of transmitting the kth neuron of the hidden layer to the output layer_kThus, the transmission expression of the whole radial basis function neural network is

Or RBF (Z)_ij)＝Φ_RBF(Z_ij) ω; with Z_ij,SInputting data for radial basis function neural network, using RBF_ij,sAnd as the output data of the radial basis function neural network, the weight between each neuron in the hidden layer of the radial basis function neural network and the output layer of the radial basis function neural network can be obtained by utilizing the transfer expression of the radial basis function neural network.

102-3, based on the correlation coefficient rho of the random variable i and the random variable j in the correlation coefficient matrix of the random variable in the original distribution domain_XijAnd correlation coefficient rho of random variable i and random variable j in correlation coefficient matrix of random variable in standard Gaussian distribution domain_ZijThe conversion relation between them and the radial basis function neural network W_ijN of the hidden layer_sDetermining correlation coefficient rho of random variable i and random variable j in correlation coefficient matrix of random variable in standard Gaussian distribution domain by output weight matrix omega of each neuron_Zij；

Wherein the content of the first and second substances,

Z_i,hthe value Z of the random variable i is the value of the random variable i when the nth sampling of the N random variables of the power system is carried out in the standard Gaussian distribution domain_j,hFor N random numbers of power systems in a standard Gaussian distribution domainThe value of the random variable j at the h-th sampling of the machine variable,

ω_kis a radial basis network W_ijThe output weight of the kth neuron of the hidden layer,

is represented by (Z)_i,h,Z_j,h) As a radial basis network W_ijThe output data of the network, RBF (Z)_ij) Is represented by (Z)_i,Z_j) As a radial basis network W_ijThe input data of (c) is the output data of the network output layer, (Z)_i,k,Z_j,k) Is a radial basis network W_ijCenter of the kth neuron of the hidden layer, Φ_RBF,k(Z_ij) Is a radial basis network W_ijHidden layer output matrix of (1) h, k ∈ (1-N)_s)，

N_sIs the total number of sample points that are,

is represented by (Z)_i,Z_j) As a radial basis network W_ijWhen the data is input, the output data of the kth neuron of the network hidden layer, i, j ∈ (1-N), N is the number of random variables in the power system, z is_iIs an independent variable, z, of a random variable i in a standard Gaussian distribution domain_jIs an argument of a random variable j in the domain of a standard gaussian distribution.

Further, the step 102-3 includes:

step A: based on formula

And RBF (Z)_ij)＝Φ_RBF(Z_ij) ω the correlation coefficient ρ of the random variable i and the random variable j in the correlation coefficient matrix of the random variables in the original distribution domain_XijRandom variable in matrix of correlation coefficient with random variable in standard Gaussian distribution domaini and the correlation coefficient ρ of the random variable j_ZijThe conversion relationship between them is transformed into:

and B: order to

And the correlation coefficient rho of the random variable i and the random variable j in the correlation coefficient matrix of the random variable in the original distribution domain_XijCorrelation coefficient rho of random variable i and random variable j in correlation coefficient matrix with random variable in standard Gaussian distribution domain_ZijThe conversion relationship of (a) is transformed into:

and C: the correlation coefficient rho of the random variable i and the random variable j in the correlation coefficient matrix of the random variable in the original distribution domain_XijCorrelation coefficient rho of random variable i and random variable j in correlation coefficient matrix with random variable in standard Gaussian distribution domain_ZijThe conversion relationship between them is transformed into:

in the preferred embodiment of the invention, p is calculated considering that the radial basis functions are all gaussian functions with absolute multiplications_XijAnd rho_ZijOf the conversion type

The double integral part of (a) is denoted by I,

in I to

Transformation into formula

And the two norms thereof are expanded, I can be transformed into:

and then to

Wherein

Step D: order to

And initializing correlation coefficients rho of random variable i and random variable j in the correlation coefficient matrix of the random variable in the standard Gaussian distribution domain_Zij,0＝ρ_XijThe current iteration number L is 1;

in the preferred embodiment of the invention, simplified Newton's method is used, with p_XijAs ρ_ZijThe initial value of (c), the fixed Jacobian matrix element (i.e., the partial derivative function G' (ρ) in each iteration_Zij) Constant 1) and using ρ_Zij,L＝ρ_Zij,L-1+G(ρ_Zij,L-1) Iterate when G (ρ)_Zij,L-1) Satisfies the iteration termination condition, | G (ρ)_Zij,L-1) If is | <, the correlation coefficient rho of the random variable i and the random variable j in the correlation coefficient matrix of the random variable in the standard Gaussian distribution domain at the L-th iteration is determined_Zij,LAnd the correlation coefficients of the random variable i and the random variable j in the correlation coefficient matrix serving as the random variable in the standard Gaussian distribution domain.

General formula

The medium independent variable is rho_ZijSetting ρ at initial iteration_Zij,0＝ρ_Xij

Step (ii) ofE: will rho_Zij,L-1Substitution into

In (1), calculating G (ρ)_Zij,L-1) And G (ρ)_Zij,L-1) Substitution into the iterative equation ρ_Zij,L＝ρ_Zij,L-1+G(ρ_Zij,L-1) In the method, the correlation coefficient rho of the random variable i and the random variable j in the correlation coefficient matrix of the random variable in the standard Gaussian distribution domain during the L-th iteration is obtained_Zij,L；

Step F: when | G (ρ)_Zij,L-1) When | is less than | then let ρ_Zij,LThe correlation coefficients of the random variable i and the random variable j in the correlation coefficient matrix of the random variable in the standard Gaussian distribution domain are obtained, otherwise, the correlation coefficient is made L +1, and the step E is returned;

wherein, mu_i·Is the mean value of the random variable i after the synchronous sampling of N random variables of the power system in the original distribution domain_jIs the mean value, sigma, of a random variable j after synchronously sampling N random variables of the power system in an original distribution domain_iFor the variance, σ, of the random variable i after the N random variables of the power system are synchronously sampled in the original distribution domain_jTo synchronize the variance of the random variable j after sampling the N random variables of the power system in the original distribution domain,

to be phi (z)_i) Substituting into the value obtained in the inverse function of the cumulative distribution function of the random variable i in the original distribution domain,

to be phi (z)_j) Substituting into the inverse of the cumulative distribution function of the random variable j in the original distribution domain to obtain a value, U_iIndependent variables, U, for the standard Gaussian distribution-compliant and independent random variable i in the Gaussian distribution domain_jIs independent variable of independent random variable j in Gaussian distribution domain, which obeys standard Gaussian distribution, A is a first parameter, B_kIs a second parameter, E_kIs the third parameter, D is the fourth parameter, G (ρ)_Zij) Is the fifth parameter, is the end of iterationThe threshold value of the threshold value is set,

Z_i,kthe method is the value of a random variable i, Z in the k-th sampling of N random variables of the power system in a standard Gaussian distribution domain_j,kThe method is the value of the random variable j in the standard Gaussian distribution domain when the kth sampling is carried out on the N random variables of the power system.

Specifically, the step 103 includes:

step 103-1, generating N_POPFSet independent and standard Gaussian distribution compliant sample data, and using N_POPFThe group of independent sample data which obeys the standard Gaussian distribution forms N rows N_POPFSample matrix U of columns_POPF；

103-2, according to the correlation coefficient matrix C of the random variables in the standard Gaussian distribution domain_ZSum matrix U_POPFDetermining a sample matrix Z of random variables in a standard Gaussian distribution domain_POPF；

Step 103-3, a sample matrix Z of random variables in the standard Gaussian distribution domain_POPFSample matrix X converted to random variables in the original distribution domain_POPF；

Wherein each group of sample data comprises N elements,

X_POPF,ifis a matrix X_POPFRow i and column f of_POPF,ifIs a matrix Z_POPFRow i and column f of_POPF＝LN_POPFL is a correlation coefficient matrix C for random variables in a standard Gaussian distribution domain_ZThe resulting lower triangular matrix, i.e. C, using Cholesky decomposition_Z＝L·L^TT is a transposed symbol, i ∈ (1-N), f ∈ (1-N)_POPF) N is the total number of random variables of the power system, N_POPFThe number of generated sample sets.

In the preferred embodiment of the present inventionIn the method, a sample matrix X of random variables in an original distribution domain can be obtained by using a Monte Carlo method_POPFAnd performing probability optimal power flow calculation or power flow calculation, and counting output results of the probability optimal power flow calculation or power flow calculation so as to determine the probability distribution condition of random variables in the power system.

In the best embodiment of the invention, an example is constructed to verify the validity of the algorithm. The method is based on standard IEEE118 node calculation and adopts the following expansion mode: and fans are additionally arranged at part of nodes, and the specific expansion condition is shown in table 1.

TABLE 1

For the wind speeds of all the wind turbines, a Weibull distribution (Weibull distribution) with a scale parameter of 10.7 and a shape parameter of 3.97 is adopted, and a correlation coefficient matrix among the wind speeds is shown in Table 2.

TABLE 2

In addition, all active loads (99 in total) are subjected to normal distribution with the average value of the original value of the standard calculation example and the standard deviation of 0.015 times of the average value. In each calculation, the power factor is kept the same as the standard calculation example, so that the value of the reactive load is determined. In the node order, the first 50 loads were set as the first group loads, the remaining 49 loads were the second group loads, the correlation coefficient between the internal loads of each group was 0.8, the correlation coefficient between the inter-group loads was 0, and the correlation coefficient between the loads and the wind speed was 0. The conversion formula for converting the wind speed into the active power output of the corresponding fan is as follows:

wherein v is_windRepresenting wind speed in m/s; p_TThe active output of a single fan is shown,unit MW; the reactive power consumed by each fan is a constant value of-0.0002 MVar.

According to the setting, the correlation coefficient conversion method selects a Simpson method to obtain integral, and a dichotomy method is used for solving a result obtained by a nonlinear equation to serve as reference.

The correlation conversion accuracy and speed results are shown in table 3. The Simpson method and the Newton method are used as reference methods, the Gauss-Hermite method and the interpolation method are more advanced solving methods at present, and the radial basis function neural network and the simplified Newton method are used as methods provided by the invention.

TABLE 3

Based on a reference correlation coefficient conversion method and a method based on a radial basis function neural network provided by the invention, a Monte Carlo method (10000 simple random samples) is combined to calculate a probability optimal power flow, the probability distribution condition of an objective function is considered, the obtained result is shown in FIG. 2, FIG. 2 shows that the similarity of the probability densities of the power generation cost calculated by the reference method and the technical scheme provided by the invention is 0.9686, namely representing the proportion of two probability density coincidence parts in the whole distribution condition, and the larger the similarity (close to 1) is, the higher the similarity of the two probability densities is.

The invention provides a random variable sample generation system based on Nataf transformation, as shown in FIG. 3, the system comprises:

the establishing module is used for establishing a conversion relation between a correlation coefficient matrix of the random variable in the original distribution domain and a correlation coefficient matrix of the random variable in the standard Gaussian distribution domain according to Nataf transformation;

the determining module is used for determining the correlation coefficient matrix of the random variable in the standard Gaussian distribution domain by adopting a radial basis network based on the conversion relation between the correlation coefficient matrix of the random variable in the original distribution domain and the correlation coefficient matrix of the random variable in the standard Gaussian distribution domain;

and the generating module is used for generating sample data of the random variable in the original distribution domain by using the correlation coefficient matrix of the random variable in the standard Gaussian distribution domain.

Specifically, the establishing module includes:

the first determining unit is used for forming an N-dimensional vector Z by using N random variables which obey standard Gaussian distribution, and determining a joint density function between a random variable i and a random variable j in the vector Z;

a second determining unit, configured to determine a correlation coefficient ρ of the random variable i and the random variable j in the correlation coefficient matrix of the random variables in the original distribution domain according to a principle that a cumulative distribution function value of the random variables in the original distribution domain is equal to a cumulative distribution function value of the random variables in the gaussian distribution domain and a joint density function between the random variable i and the random variable j in the vector Z_XijCorrelation coefficient rho of random variable i and random variable j in correlation coefficient matrix with random variable in standard Gaussian distribution domain_ZijThe conversion relationship between them;

wherein i, j ∈ (1-N), N is the number of random variables in the power system, and the joint density function between the random variable i and the random variable j in the vector Z

z_iIs an independent variable, z, of a random variable i in a standard Gaussian distribution domain_jIs an argument of a random variable j in the domain of a standard gaussian distribution.

The second determining unit is specifically configured to: further, the third determining unit is specifically configured to:

determining the correlation coefficient rho of the random variable i and the random variable j in the correlation coefficient matrix of the random variable in the original distribution domain according to the following formula_XijCorrelation coefficient rho of random variable i and random variable j in correlation coefficient matrix with random variable in standard Gaussian distribution domain_ZijThe conversion relationship between:

in the formula (I), the compound is shown in the specification,

to be phi (z)_i) Substituting into the value obtained in the inverse function of the cumulative distribution function of the random variable i in the original distribution domain,

to be phi (z)_j) Substituting into the inverse function of the cumulative distribution function of the random variable j in the original distribution domain to obtain a value, Φ (z)_i) The argument for the random variable i in the normalized Gaussian distribution domain is z_iThe value of the cumulative distribution function of the random variable i in the standard Gaussian distribution domain, phi (z), corresponding to the time_j) The argument for the random variable j in the normalized Gaussian distribution domain is z_jThe value of the cumulative distribution function of the random variable j in the standard Gaussian distribution domain, mu_iIs the mean value, μ, of the random variable i after the synchronous sampling of the N random variables of the power system in the original distribution domain_jIs the mean value, sigma, of a random variable j after synchronously sampling N random variables of the power system in an original distribution domain_iFor the variance, σ, of the random variable i after the N random variables of the power system are synchronously sampled in the original distribution domain_jIs the variance of a random variable j after synchronously sampling N random variables of the power system in the original distribution domain.

Further, the third determining unit is specifically configured to:

determining the correlation coefficient rho of the random variable i and the random variable j in the correlation coefficient matrix of the random variable in the original distribution domain according to the following formula_XijAnd correlation coefficient rho of random variable i and random variable j in correlation coefficient matrix of random variable in standard Gaussian distribution domain_ZijThe conversion relationship between:

in the formula (I), the compound is shown in the specification,

to be phi (z)_i) Substituting into the value obtained in the inverse function of the cumulative distribution function of the random variable i in the original distribution domain,

to be phi (z)_j) Substitution into original distribution Domain randomnessThe value obtained in the inverse of the cumulative distribution function of the variable j.

Specifically, the determining module includes:

a first acquisition unit for synchronously sampling N random variables of the power system in a standard Gaussian distribution domain_sThen, obtain N_sVector Z formed by sampling data of random variable i and random variable j in sub-sampling_ij,SAnd will vector Z_ij,SIn the formula

Obtaining a vector Z_ij,SCorresponding vector RBF_ij,s；

A second acquisition unit for acquiring the vector Z_ij,SSum vector RBF_ij,sRespectively as radial basis function neural networks W_ijInput data and output data of

As a radial basis neural network W_ijSetting function of the kth neuron of the hidden layer

As a radial basis neural network W_ijObtaining a setting function of an output layer to obtain a radial basis function (Wn)_ijN of the hidden layer_sAn output weight matrix ω of each neuron;

a fourth determining unit, configured to determine a correlation coefficient ρ of the random variable i and the random variable j based on the correlation coefficient matrix of the random variables in the original distribution domain_XijAnd correlation coefficient rho of random variable i and random variable j in correlation coefficient matrix of random variable in standard Gaussian distribution domain_ZijThe conversion relation between them and the radial basis function neural network W_ijN of the hidden layer_sDetermining correlation coefficient rho of random variable i and random variable j in correlation coefficient matrix of random variable in standard Gaussian distribution domain by output weight matrix omega of each neuron_Zij；

Wherein the content of the first and second substances,

Z_i,hthe value Z of the random variable i is the value of the random variable i when the nth sampling of the N random variables of the power system is carried out in the standard Gaussian distribution domain_j,hIs the value of a random variable j when the nth sampling of the N random variables of the power system is carried out in a standard Gaussian distribution domain,

ω_kis a radial basis network W_ijThe output weight of the kth neuron of the hidden layer,

is represented by (Z)_i,h,Z_j,h) As a radial basis network W_ijThe output data of the network, RBF (Z)_ij) Is represented by (Z)_i,Z_j) As a radial basis network W_ijThe input data of (c) is the output data of the network output layer, (Z)_i,k,Z_j,k) Is a radial basis network W_ijCenter of the kth neuron of the hidden layer, Φ_RBF,k(Z_ij) Is a radial basis network W_ijHidden layer output matrix of (1) h, k ∈ (1-N)_s)，

N_sIs the total number of sample points that are,

is represented by (Z)_i,Z_j) As a radial basis network W_ijWhen the data is input, the output data of the kth neuron of the network hidden layer, i, j ∈ (1-N), N is the number of random variables in the power system, z is_iIs an independent variable, z, of a random variable i in a standard Gaussian distribution domain_jIs an argument of a random variable j in the domain of a standard gaussian distribution.

Further, the fourth determining unit includes:

first transformation sub-sheetElement for based on

And RBF (Z)_ij)＝Φ_RBF(Z_ij) ω the correlation coefficient ρ of the random variable i and the random variable j in the correlation coefficient matrix of the random variables in the original distribution domain_XijCorrelation coefficient rho of random variable i and random variable j in correlation coefficient matrix with random variable in standard Gaussian distribution domain_ZijThe conversion relationship between them is transformed into:

a second conversion subunit for converting

And the correlation coefficient rho of the random variable i and the random variable j in the correlation coefficient matrix of the random variable in the original distribution domain_XijCorrelation coefficient rho of random variable i and random variable j in correlation coefficient matrix with random variable in standard Gaussian distribution domain_ZijThe conversion relationship of (a) is transformed into:

a third transformation subunit, configured to transform correlation coefficients ρ of the random variable i and the random variable j in a correlation coefficient matrix of the random variable in the original distribution domain_XijCorrelation coefficient rho of random variable i and random variable j in correlation coefficient matrix with random variable in standard Gaussian distribution domain_ZijThe conversion relationship between them is transformed into:

an initialization subunit for ordering

And initializing the phases of random variables in a standard Gaussian distribution domainCorrelation coefficient rho of random variable i and random variable j in relational number matrix_Zij,0＝ρ_XijThe current iteration number L is 1;

an iteration subunit for dividing p_Zij,L-1Substitution into

In (1), calculating G (ρ)_Zij,L-1) And G (ρ)_Zij,L-1) Substitution into the iterative equation ρ_Zij,L＝ρ_Zij,L-1+G(ρ_Zij,L-1) In the method, the correlation coefficient rho of the random variable i and the random variable j in the correlation coefficient matrix of the random variable in the standard Gaussian distribution domain during the L-th iteration is obtained_Zij,L；

A judgment subunit for judging when | G (ρ)_Zij,L-1) When | is less than | then let ρ_Zij,LThe correlation coefficients of the random variable i and the random variable j in the correlation coefficient matrix of the random variable in the standard Gaussian distribution domain are obtained, otherwise, the correlation coefficient is made L +1, and the step E is returned;

wherein, mu_i·Is the mean value of the random variable i after the synchronous sampling of N random variables of the power system in the original distribution domain_jIs the mean value, sigma, of a random variable j after synchronously sampling N random variables of the power system in an original distribution domain_iFor the variance, σ, of the random variable i after the N random variables of the power system are synchronously sampled in the original distribution domain_jTo synchronize the variance of the random variable j after sampling the N random variables of the power system in the original distribution domain,

to be phi (z)_i) Substituting into the value obtained in the inverse function of the cumulative distribution function of the random variable i in the original distribution domain,

to be phi (z)_j) Substituting into the inverse of the cumulative distribution function of the random variable j in the original distribution domain to obtain a value, U_iIndependent variables, U, for the standard Gaussian distribution-compliant and independent random variable i in the Gaussian distribution domain_jFor complying with standard height in Gaussian distribution domainIndependent variables of a random variable j which is distributed in a gaussian manner, A is a first parameter, B_kIs a second parameter, E_kIs the third parameter, D is the fourth parameter, G (ρ)_Zij) For the fifth parameter, for the iteration termination threshold,

Z_i,kthe method is the value of a random variable i, Z in the k-th sampling of N random variables of the power system in a standard Gaussian distribution domain_j,kThe method is the value of the random variable j in the standard Gaussian distribution domain when the kth sampling is carried out on the N random variables of the power system.

Specifically, the generating module includes:

a sample construction unit for generating N_POPFSet independent and standard Gaussian distribution compliant sample data, and using N_POPFThe group of independent sample data which obeys the standard Gaussian distribution forms N rows N_POPFSample matrix U of columns_POPF；

A fifth determining unit for determining a correlation coefficient matrix C based on the random variables in the standard Gaussian distribution domain_ZSum matrix U_POPFDetermining a sample matrix Z of random variables in a standard Gaussian distribution domain_POPF；

A conversion unit for converting a sample matrix Z of random variables in a standard Gaussian distribution domain_POPFSample matrix X converted to random variables in the original distribution domain_POPF；

Wherein each group of sample data comprises N elements,

X_POPF,ifis a matrix X_POPFRow i and column f of_POPF,ifIs a matrix Z_POPFRow i and column f of_POPF＝LN_POPFL is a correlation coefficient matrix C for random variables in a standard Gaussian distribution domain_ZThe resulting lower triangular matrix, i.e. C, using Cholesky decomposition_Z＝L·L^TT is a transposed symbol, i ∈ (1-N), f ∈ (1-N)_POPF) N is the total number of random variables of the power system, N_POPFThe number of generated sample sets.

In the best embodiment of the invention, the transfer process of the product of the random variable in the standard Gaussian distribution domain and the random variable in the original distribution domain is approximated through the radial basis function neural network, and an analytical expression of the transfer process is given; analyzing the analytical expression of the transmission process, and changing the correlation coefficient conversion relation of the random variables in the original distribution domain and the Gaussian distribution domain in the integral form into a summation form, thereby simplifying the conversion relation of the correlation coefficients of the random variables in the original distribution domain and the standard Gaussian distribution domain, improving the calculation speed of the correlation coefficient of the random variables in the standard Gaussian distribution domain, and improving the sample generation speed of the random variables in the original distribution domain.

As will be appreciated by one skilled in the art, embodiments of the present application may be provided as a method, system, or computer program product. Accordingly, the present application may take the form of an entirely hardware embodiment, an entirely software embodiment or an embodiment combining software and hardware aspects. Furthermore, the present application may take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, and the like) having computer-usable program code embodied therein.

The present application is described with reference to flowchart illustrations and/or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the application. It will be understood that each flow and/or block of the flow diagrams and/or block diagrams, and combinations of flows and/or blocks in the flow diagrams and/or block diagrams, can be implemented by computer program instructions. These computer program instructions may be provided to a processor of a general purpose computer, special purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.

These computer program instructions may also be stored in a computer-readable memory that can direct a computer or other programmable data processing apparatus to function in a particular manner, such that the instructions stored in the computer-readable memory produce an article of manufacture including instruction means which implement the function specified in the flowchart flow or flows and/or block diagram block or blocks.

These computer program instructions may also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer implemented process such that the instructions which execute on the computer or other programmable apparatus provide steps for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.

Finally, it should be noted that: the above embodiments are only for illustrating the technical solutions of the present invention and not for limiting the same, and although the present invention is described in detail with reference to the above embodiments, those of ordinary skill in the art should understand that: modifications and equivalents may be made to the embodiments of the invention without departing from the spirit and scope of the invention, which is to be covered by the claims.

Claims (12)

1. A method for generating random variable samples based on a Nataf transform, the method comprising:

establishing a conversion relation between a correlation coefficient matrix of the random variable in the original distribution domain and a correlation coefficient matrix of the random variable in the standard Gaussian distribution domain according to Nataf transformation;

determining a correlation coefficient matrix of the random variable in the standard Gaussian distribution domain by adopting a radial basis network based on the conversion relation between the correlation coefficient matrix of the random variable in the original distribution domain and the correlation coefficient matrix of the random variable in the standard Gaussian distribution domain;

and generating sample data of the random variable in the original distribution domain by using a correlation coefficient matrix of the random variable in the standard Gaussian distribution domain.

2. The method according to claim 1, wherein the establishing a conversion relationship between the correlation coefficient matrix of the random variable in the original distribution domain and the correlation coefficient matrix of the random variable in the standard gaussian distribution domain according to the Nataf transform comprises:

forming an N-dimensional vector Z by using N random variables which obey standard Gaussian distribution, and determining a joint density function between a random variable i and a random variable j in the vector Z;

determining the correlation coefficient rho of the random variable i and the random variable j in the correlation coefficient matrix of the random variables in the original distribution domain according to the principle that the cumulative distribution function value of the random variables in the original distribution domain is equal to the cumulative distribution function value of the random variables in the Gaussian distribution domain and the joint density function between the random variable i and the random variable j in the vector Z_XijCorrelation coefficient rho of random variable i and random variable j in correlation coefficient matrix with random variable in standard Gaussian distribution domain_ZijThe conversion relationship between them;

wherein i, j ∈ (1-N), N is the number of random variables in the power system, and the joint density function between the random variable i and the random variable j in the vector Z

z_iIs an independent variable, z, of a random variable i in a standard Gaussian distribution domain_jIs an argument of a random variable j in the domain of a standard gaussian distribution.

3. The method according to claim 2, wherein the correlation coefficient ρ of the random variable i and the random variable j in the correlation coefficient matrix of the random variables in the original distribution domain is determined according to a rule that the cumulative distribution function value of the random variables in the original distribution domain is equal to the cumulative distribution function value of the random variables in the gaussian distribution domain and a joint density function between the random variable i and the random variable j in the vector Z_XijCorrelation coefficient rho of random variable i and random variable j in correlation coefficient matrix with random variable in standard Gaussian distribution domain_ZijThe conversion relationship between the two, including:

determining the correlation coefficient rho of the random variable i and the random variable j in the correlation coefficient matrix of the random variable in the original distribution domain according to the following formula_XijCorrelation coefficient rho of random variable i and random variable j in correlation coefficient matrix with random variable in standard Gaussian distribution domain_ZijThe conversion relationship between:

in the formula (I), the compound is shown in the specification,

to be phi (z)_i) Substituting into the value obtained in the inverse function of the cumulative distribution function of the random variable i in the original distribution domain,

to be phi (z)_j) Substituting into the inverse function of the cumulative distribution function of the random variable j in the original distribution domain to obtain a value, Φ (z)_i) The argument for the random variable i in the normalized Gaussian distribution domain is z_iThe value of the cumulative distribution function of the random variable i in the standard Gaussian distribution domain, phi (z), corresponding to the time_j) The argument for the random variable j in the normalized Gaussian distribution domain is z_jThe value of the cumulative distribution function of the random variable j in the standard Gaussian distribution domain, mu_i·Is the mean value of the random variable i after the synchronous sampling of N random variables of the power system in the original distribution domain_jIs the mean value, sigma, of a random variable j after synchronously sampling N random variables of the power system in an original distribution domain_iFor the variance, σ, of the random variable i after the N random variables of the power system are synchronously sampled in the original distribution domain_jIs the variance of a random variable j after synchronously sampling N random variables of the power system in the original distribution domain.

4. The method of claim 1, wherein determining the correlation coefficient matrix of the random variable in the standard gaussian distribution domain using the radial basis network based on a transformation relationship between the correlation coefficient matrix of the random variable in the original distribution domain and the correlation coefficient matrix of the random variable in the standard gaussian distribution domain comprises:

synchronous sampling N for N random variables of power system in standard Gaussian distribution domain_sThen, obtain N_sVector Z formed by sampling data of random variable i and random variable j in sub-sampling_ij,SAnd will vector Z_ij,SIn the formula

Obtaining a vector Z_ij,SCorresponding vector RBF_ij,s；

Will vector Z_ij,SSum vector RBF_ij,sRespectively as radial basis function neural networks W_ijInput data and output data of

As a radial basis neural network W_ijSetting function of the kth neuron of the hidden layer

As a radial basis neural network W_ijObtaining a setting function of an output layer to obtain a radial basis function (Wn)_ijN of the hidden layer_sAn output weight matrix ω of each neuron;

correlation coefficient rho of random variable i and random variable j in correlation coefficient matrix based on random variable in original distribution domain_XijAnd correlation coefficient rho of random variable i and random variable j in correlation coefficient matrix of random variable in standard Gaussian distribution domain_ZijThe conversion relation between them and the radial basis function neural network W_ijN of the hidden layer_sDetermining the output weight matrix omega of each neuron, and determining the correlation coefficient rho of the random variable i and the random variable j in the correlation coefficient matrix of the random variables in the standard Gaussian distribution domain_Zij；

Wherein the content of the first and second substances,

Z_i,hthe value Z of the random variable i is the value of the random variable i when the nth sampling of the N random variables of the power system is carried out in the standard Gaussian distribution domain_j,hIs the value of a random variable j when the nth sampling of the N random variables of the power system is carried out in a standard Gaussian distribution domain,

ω_kis a radial basis network W_ijThe output weight of the kth neuron of the hidden layer,

is represented by (Z)_i,h,Z_j,h) As a radial basis network W_ijThe output data of the network, RBF (Z)_ij) Is represented by (Z)_i,Z_j) As a radial basis network W_ijThe input data of (c) is the output data of the network output layer, (Z)_i,k,Z_j,k) Is a radial basis network W_ijCenter of the kth neuron of the hidden layer, Φ_RBF,k(Z_ij) Is a radial basis network W_ijHidden layer output matrix of (1) h, k ∈ (1-N)_s)，

N_sIs the total number of sample points that are,

is represented by (Z)_i,Z_j) As a radial basis network W_ijWhen the data is input, the output data of the kth neuron of the network hidden layer, i, j ∈ (1-N), N is the number of random variables in the power system, z is_iIs an independent variable, z, of a random variable i in a standard Gaussian distribution domain_jIs an argument of a random variable j in the domain of a standard gaussian distribution.

5. The method according to claim 4, wherein the correlation coefficient p of the random variable i and the random variable j in the correlation coefficient matrix based on the random variables in the original distribution domain_XijAnd correlation coefficient rho of random variable i and random variable j in correlation coefficient matrix of random variable in standard Gaussian distribution domain_ZijThe conversion relation between them and the radial basis function neural network W_ijN of the hidden layer_sDetermining correlation coefficient rho of random variable i and random variable j in correlation coefficient matrix of random variable in standard Gaussian distribution domain by output weight matrix omega of each neuron_ZijThe method comprises the following steps:

step A: based on formula

And RBF (Z)_ij)＝Φ_RBF(Z_ij) ω the correlation coefficient ρ of the random variable i and the random variable j in the correlation coefficient matrix of the random variables in the original distribution domain_XijCorrelation coefficient rho of random variable i and random variable j in correlation coefficient matrix with random variable in standard Gaussian distribution domain_ZijThe conversion relationship between them is transformed into:

and B: order to

And the correlation coefficient rho of the random variable i and the random variable j in the correlation coefficient matrix of the random variable in the original distribution domain_XijCorrelation coefficient rho of random variable i and random variable j in correlation coefficient matrix with random variable in standard Gaussian distribution domain_ZijThe conversion relationship of (a) is transformed into:

and C: correlating coefficient moment of random variable in original distribution domainCorrelation coefficient rho of random variable i and random variable j in array_XijCorrelation coefficient rho of random variable i and random variable j in correlation coefficient matrix with random variable in standard Gaussian distribution domain_ZijThe conversion relationship between them is transformed into:

step D: order to

And initializing correlation coefficients rho of random variable i and random variable j in the correlation coefficient matrix of the random variable in the standard Gaussian distribution domain_Zij,0＝ρ_XijThe current iteration number L is 1;

step E: will rho_Zij,L-1Substitution into

In (1), calculating G (ρ)_Zij,L-1) And G (ρ)_Zij,L-1) Substitution into the iterative equation ρ_Zij,L＝ρ_Zij,L-1+G(ρ_Zij,L-1) In the method, the correlation coefficient rho of the random variable i and the random variable j in the correlation coefficient matrix of the random variable in the standard Gaussian distribution domain during the L-th iteration is obtained_Zij,L；

Step F: when | G (ρ)_Zij,L-1) When | is less than | then let ρ_Zij,LThe correlation coefficients of the random variable i and the random variable j in the correlation coefficient matrix of the random variable in the standard Gaussian distribution domain are obtained, otherwise, the correlation coefficient is made L +1, and the step E is returned;

wherein, mu_i·Is the mean value of the random variable i after the synchronous sampling of N random variables of the power system in the original distribution domain_jIs the mean value, sigma, of a random variable j after synchronously sampling N random variables of the power system in an original distribution domain_iFor the variance, σ, of the random variable i after the N random variables of the power system are synchronously sampled in the original distribution domain_jTo synchronize the variance of the random variable j after sampling the N random variables of the power system in the original distribution domain,

to be phi (z)_i) Substituting into the value obtained in the inverse function of the cumulative distribution function of the random variable i in the original distribution domain,

to be phi (z)_j) Substituting into the inverse of the cumulative distribution function of the random variable j in the original distribution domain to obtain a value, U_iIndependent variables, U, for the standard Gaussian distribution-compliant and independent random variable i in the Gaussian distribution domain_jIs independent variable of independent random variable j in Gaussian distribution domain, which obeys standard Gaussian distribution, A is a first parameter, B_kIs a second parameter, E_kIs the third parameter, D is the fourth parameter, G (ρ)_Zij) For the fifth parameter, for the iteration termination threshold,

Z_i,kthe method is the value of a random variable i, Z in the k-th sampling of N random variables of the power system in a standard Gaussian distribution domain_j,kThe method is the value of the random variable j in the standard Gaussian distribution domain when the kth sampling is carried out on the N random variables of the power system.

6. The method of claim 1, wherein generating sample data of random variables in an original distribution domain using a matrix of correlation coefficients of the random variables in a standard gaussian distribution domain comprises:

generating N_POPFSet independent and standard Gaussian distribution compliant sample data, and using N_POPFThe group of independent sample data which obeys the standard Gaussian distribution forms N rows N_POPFSample matrix U of columns_POPF；

Correlation coefficient matrix C based on random variables in standard Gaussian distribution domain_ZSum matrix U_POPFDetermining a sample matrix Z of random variables in a standard Gaussian distribution domain_POPF；

Sample matrix Z of random variables in standard Gaussian distribution domain_POPFSample matrix X converted to random variables in the original distribution domain_POPF；

Wherein each group of sample data comprises N elements,

X_POPF,ifis a matrix X_POPFRow i and column f of_POPF,ifIs a matrix Z_POPFRow i and column f of_POPF＝LN_POPFL is a correlation coefficient matrix C for random variables in a standard Gaussian distribution domain_ZThe resulting lower triangular matrix, i.e. C, using Cholesky decomposition_Z＝L·L^TT is a transposed symbol, i ∈ (1-N), f ∈ (1-N)_POPF) N is the total number of random variables of the power system, N_POPFThe number of generated sample sets.

7. A Nataf transform-based random variable sample generation system, the system comprising:

the establishing module is used for establishing a conversion relation between a correlation coefficient matrix of the random variable in the original distribution domain and a correlation coefficient matrix of the random variable in the standard Gaussian distribution domain according to Nataf transformation;

the determining module is used for determining the correlation coefficient matrix of the random variable in the standard Gaussian distribution domain by adopting a radial basis network based on the conversion relation between the correlation coefficient matrix of the random variable in the original distribution domain and the correlation coefficient matrix of the random variable in the standard Gaussian distribution domain;

and the generating module is used for generating sample data of the random variable in the original distribution domain by using the correlation coefficient matrix of the random variable in the standard Gaussian distribution domain.

8. The system of claim 7, wherein the setup module comprises:

the first determining unit is used for forming an N-dimensional vector Z by using N random variables which obey standard Gaussian distribution, and determining a joint density function between a random variable i and a random variable j in the vector Z;

a second determining unit, configured to determine a correlation coefficient ρ of the random variable i and the random variable j in the correlation coefficient matrix of the random variables in the original distribution domain according to a principle that a cumulative distribution function value of the random variables in the original distribution domain is equal to a cumulative distribution function value of the random variables in the gaussian distribution domain and a joint density function between the random variable i and the random variable j in the vector Z_XijCorrelation coefficient rho of random variable i and random variable j in correlation coefficient matrix with random variable in standard Gaussian distribution domain_ZijThe conversion relationship between them;

wherein i, j ∈ (1-N), N is the number of random variables in the power system, and the joint density function between the random variable i and the random variable j in the vector Z

z_iIs an independent variable, z, of a random variable i in a standard Gaussian distribution domain_jIs an argument of a random variable j in the domain of a standard gaussian distribution.

9. The system of claim 8, wherein the second determining unit is specifically configured to:

determining the correlation coefficient rho of the random variable i and the random variable j in the correlation coefficient matrix of the random variable in the original distribution domain according to the following formula_XijCorrelation coefficient rho of random variable i and random variable j in correlation coefficient matrix with random variable in standard Gaussian distribution domain_ZijThe conversion relationship between:

in the formula (I), the compound is shown in the specification,

to be phi (z)_i) Accumulation of random variable i substituted into original distribution domainThe values obtained in the inverse of the distribution function,

to be phi (z)_j) Substituting into the inverse function of the cumulative distribution function of the random variable j in the original distribution domain to obtain a value, Φ (z)_i) The argument for the random variable i in the normalized Gaussian distribution domain is z_iThe value of the cumulative distribution function of the random variable i in the standard Gaussian distribution domain, phi (z), corresponding to the time_j) The argument for the random variable j in the normalized Gaussian distribution domain is z_jThe value of the cumulative distribution function of the random variable j in the standard Gaussian distribution domain, mu_i·Is the mean value of the random variable i after the synchronous sampling of N random variables of the power system in the original distribution domain_jIs the mean value, sigma, of a random variable j after synchronously sampling N random variables of the power system in an original distribution domain_iFor the variance, σ, of the random variable i after the N random variables of the power system are synchronously sampled in the original distribution domain_jIs the variance of a random variable j after synchronously sampling N random variables of the power system in the original distribution domain.

10. The system of claim 7, wherein the determination module comprises:

a first acquisition unit for synchronously sampling N random variables of the power system in a standard Gaussian distribution domain_sThen, obtain N_sVector Z formed by sampling data of random variable i and random variable j in sub-sampling_ij,SAnd will vector Z_ij,SIn the formula

Obtaining a vector Z_ij,SCorresponding vector RBF_ij,s；

A second acquisition unit for acquiring the vector Z_ij,SSum vector RBF_ij,sRespectively as radial basis function neural networks W_ijInput data and output data of

As a radial basis neural network W_ijSetting function of the kth neuron of the hidden layer

As a radial basis neural network W_ijObtaining a setting function of an output layer to obtain a radial basis function (Wn)_ijN of the hidden layer_sAn output weight matrix ω of each neuron;

a fourth determining unit, configured to determine a correlation coefficient ρ of the random variable i and the random variable j based on the correlation coefficient matrix of the random variables in the original distribution domain_XijAnd correlation coefficient rho of random variable i and random variable j in correlation coefficient matrix of random variable in standard Gaussian distribution domain_ZijThe conversion relation between them and the radial basis function neural network W_ijN of the hidden layer_sDetermining correlation coefficient rho of random variable i and random variable j in correlation coefficient matrix of random variable in standard Gaussian distribution domain by output weight matrix omega of each neuron_Zij；

Wherein the content of the first and second substances,

Z_i,hthe value Z of the random variable i is the value of the random variable i when the nth sampling of the N random variables of the power system is carried out in the standard Gaussian distribution domain_j,hIs the value of a random variable j when the nth sampling of the N random variables of the power system is carried out in a standard Gaussian distribution domain,

ω_kis a radial basis network W_ijThe output weight of the kth neuron of the hidden layer,

is represented by (Z)_i,h,Z_j,h) As a radial basis network W_ijThe output data of the network, RBF (Z)_ij) Is represented by (Z)_i,Z_j) As a radial basis network W_ijThe input data of (c) is the output data of the network output layer, (Z)_i,k,Z_j,k) Is a radial basis network W_ijCenter of the kth neuron of the hidden layer, Φ_RBF,k(Z_ij) Is a radial basis network W_ijHidden layer output matrix of (1) h, k ∈ (1-N)_s)，

N_sIs the total number of sample points that are,

is represented by (Z)_i,Z_j) As a radial basis network W_ijWhen the data is input, the output data of the kth neuron of the network hidden layer, i, j ∈ (1-N), N is the number of random variables in the power system, z is_iIs an independent variable, z, of a random variable i in a standard Gaussian distribution domain_jIs an argument of a random variable j in the domain of a standard gaussian distribution.

11. The system of claim 10, wherein the fourth determination unit comprises:

a first transformation subunit for being based on

And RBF (Z)_ij)＝Φ_RBF(Z_ij) ω the correlation coefficient ρ of the random variable i and the random variable j in the correlation coefficient matrix of the random variables in the original distribution domain_XijCorrelation coefficient rho of random variable i and random variable j in correlation coefficient matrix with random variable in standard Gaussian distribution domain_ZijThe conversion relationship between them is transformed into:

a second transformation sub-unit for performing a second transformation,for making

And the correlation coefficient rho of the random variable i and the random variable j in the correlation coefficient matrix of the random variable in the original distribution domain_XijCorrelation coefficient rho of random variable i and random variable j in correlation coefficient matrix with random variable in standard Gaussian distribution domain_ZijThe conversion relationship of (a) is transformed into:

a third transformation subunit, configured to transform correlation coefficients ρ of the random variable i and the random variable j in a correlation coefficient matrix of the random variable in the original distribution domain_XijCorrelation coefficient rho of random variable i and random variable j in correlation coefficient matrix with random variable in standard Gaussian distribution domain_ZijThe conversion relationship between them is transformed into:

an initialization subunit for ordering

And initializing correlation coefficients rho of random variable i and random variable j in the correlation coefficient matrix of the random variable in the standard Gaussian distribution domain_Zij,0＝ρ_XijThe current iteration number L is 1;

an iteration subunit for dividing p_Zij,L-1Substitution into

In (1), calculating G (ρ)_Zij,L-1) And G (ρ)_Zij,L-1) Substitution into the iterative equation ρ_Zij,L＝ρ_Zij,L-1+G(ρ_Zij,L-1) In the method, the correlation coefficient rho of the random variable i and the random variable j in the correlation coefficient matrix of the random variable in the standard Gaussian distribution domain during the L-th iteration is obtained_Zij,L；

A judgment subunit for judging when | G (ρ)_Zij,L-1) When | is less than | then let ρ_Zij,LThe correlation coefficients of the random variable i and the random variable j in the correlation coefficient matrix of the random variable in the standard Gaussian distribution domain are obtained, otherwise, the correlation coefficient is made L +1, and the step E is returned;

wherein, mu_i·Is the mean value of the random variable i after the synchronous sampling of N random variables of the power system in the original distribution domain_jIs the mean value, sigma, of a random variable j after synchronously sampling N random variables of the power system in an original distribution domain_iFor the variance, σ, of the random variable i after the N random variables of the power system are synchronously sampled in the original distribution domain_jTo synchronize the variance of the random variable j after sampling the N random variables of the power system in the original distribution domain,

to be phi (z)_i) Substituting into the value obtained in the inverse function of the cumulative distribution function of the random variable i in the original distribution domain,

to be phi (z)_j) Substituting into the inverse of the cumulative distribution function of the random variable j in the original distribution domain to obtain a value, U_iIndependent variables, U, for the standard Gaussian distribution-compliant and independent random variable i in the Gaussian distribution domain_jIs independent variable of independent random variable j in Gaussian distribution domain, which obeys standard Gaussian distribution, A is a first parameter, B_kIs a second parameter, E_kIs the third parameter, D is the fourth parameter, G (ρ)_Zij) For the fifth parameter, for the iteration termination threshold,

Z_i,kthe method is the value of a random variable i, Z in the k-th sampling of N random variables of the power system in a standard Gaussian distribution domain_j,kThe method is the value of the random variable j in the standard Gaussian distribution domain when the kth sampling is carried out on the N random variables of the power system.

12. The system of claim 7, wherein the generation module comprises:

a sample construction unit for generating N_POPFSet independent and standard Gaussian distribution compliant sample data, and using N_POPFThe group of independent sample data which obeys the standard Gaussian distribution forms N rows N_POPFSample matrix U of columns_POPF；

A fifth determining unit for determining a correlation coefficient matrix C based on the random variables in the standard Gaussian distribution domain_ZSum matrix U_POPFDetermining a sample matrix Z of random variables in a standard Gaussian distribution domain_POPF；

A conversion unit for converting a sample matrix Z of random variables in a standard Gaussian distribution domain_POPFSample matrix X converted to random variables in the original distribution domain_POPF；

Wherein each group of sample data comprises N elements,

X_POPF,ifis a matrix X_POPFRow i and column f of_POPF,ifIs a matrix Z_POPFRow i and column f of_POPF＝LN_POPFL is a correlation coefficient matrix C for random variables in a standard Gaussian distribution domain_ZThe resulting lower triangular matrix, i.e. C, using Cholesky decomposition_Z＝L·L^TT is a transposed symbol, i ∈ (1-N), f ∈ (1-N)_POPF) N is the total number of random variables of the power system, N_POPFThe number of generated sample sets.

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