CN117313304B - Gaussian mixture model method for analyzing overall sensitivity of power flow of power distribution network - Google Patents

Abstract

The invention discloses a Gaussian mixture model method for global sensitivity analysis of distribution network power flow, and relates to the field of distribution network. The method includes the following steps: Step 1. Establish the Gaussian joint probability density function of the distribution network power flow input variables and output variables. Hybrid model; Step 2, obtain the analytical formula of the probability density function of the distribution network power flow output variable, and calculate its variance; Step 3, obtain the analytical formula of the conditional probability density function and conditional variance of the distribution network power flow output variable with respect to the input variable , and calculate the expected value of the conditional variance; step 4, calculate the distribution network power flow global sensitivity index based on the results obtained in steps 2 and 3, and complete the distribution network power flow global sensitivity analysis. The present invention only requires fewer times of power flow calculations, takes a short time, avoids the time-consuming agent model construction and Monte Carlo simulation calculation processes, and significantly improves the calculation efficiency of the global sensitivity analysis of the distribution network power flow.

Description

Gaussian Mixture Model Method for Global Sensitivity Analysis of Distribution Network Power Flow

Technical field

The invention relates to the field of distribution network, specifically a Gaussian mixture model method for global sensitivity analysis of power flow in distribution network.

Background technique

With the acceleration of the construction of new distribution networks, a large number of highly uncertain new sources/loads are connected, resulting in outstanding uncertainty in the operating status of the distribution network. Global sensitivity analysis is a key technology for quantifying uncertainty propagation. Through global sensitivity analysis of distribution network power flow, the importance quantification results of each uncertainty source/load can be obtained, and the key sources/loads that cause uncertainty in the operating status of the distribution network can be identified. , thereby proactively managing the uncertainty of key sources/charges in a targeted manner. The uncertain sources/loads in the distribution network have complex probability distribution characteristics and significant correlations. At the same time, the distribution network power flow model has significant nonlinear characteristics. The global sensitivity analysis of the distribution network power flow generally adopts Monte Carlo simulation method. The Monte Carlo simulation method has a heavy computational burden. The Monte Carlo simulation method based on the agent model can reduce the time-consuming calculation and simulation of distribution network power flow, but the overall calculation time is still long.

Contents of the invention

The purpose of the present invention is to provide a Gaussian mixture model method for global sensitivity analysis of distribution network power flow, which greatly improves the calculation efficiency of global sensitivity analysis of distribution network power flow, so as to solve the problems raised in the above background technology.

In order to achieve the above objects, the present invention provides the following technical solutions:

The Gaussian mixture model method for global sensitivity analysis of distribution network power flow in the present invention includes the following steps:

Step 1: Establish a Gaussian mixture model of the joint probability density function of the distribution network power flow input variables and output variables;

Step 2: Obtain the analytical formula of the probability density function of the power flow output variable of the distribution network and calculate its variance;

Step 3: Obtain the analytical expression of the conditional probability density function and conditional variance of the distribution network power flow output variable with respect to the input variable, and calculate the expected value of the conditional variance;

Step 4: Calculate the distribution network power flow global sensitivity index based on the results obtained in steps 2 and 3, and complete the distribution network power flow global sensitivity analysis.

As a further solution of the present invention, the step 1 is specifically:

First, a Gaussian mixture model of joint probability density function of distribution network power flow input variables is established;

The distribution network power flow input variable vector is x, and its n historical samples are X ₁ , X ₂ ,..., X _n . The non-parametric kernel density estimation method is used to establish the joint probability density function of x:

In the formula: N(·) represents the Gaussian kernel function, H _k represents the bandwidth matrix corresponding to the k-th sample X _k . The probability density function based on non-parametric kernel density estimation is the basic Gaussian mixture model, and its Gaussian component K _b is equal n, the weight coefficient of the kth Gaussian component Equal to 1/n, mean vector/> Equal to X _k , covariance matrix/> equal to H _k ;

The density-preserving hierarchical expectation maximization algorithm is used to reduce the Gaussian components of the basic Gaussian mixture model shown in Equation (1), and the simplified Gaussian mixture model is obtained as:

In the formula: ω _k , μ _x,k , Σ _x,k respectively represent the weight coefficient, mean vector and covariance matrix of the kth Gaussian component, and K represents the number of reduced Gaussian components;

Secondly, based on equation (2), a piecewise linearized distribution network power flow model is established;

The compact form of the nonlinear distribution network power flow model is:

y＝Γ(x) (3)

In the formula: Γ(·) represents the nonlinear power flow equation, and y represents the output variable vector of the distribution network power flow;

Given the kth Gaussian component N(x|μ _x,k ,Σ _x,k ) of the Gaussian mixture model of input variables shown in equation (2), for _the nonlinear power flow model shown in equation (3) in Linearize at , and get the kth linearized model:

y＝T _k x+[Γ(μ _x,k )-T _k μ _x,k ] (4)

In the formula: T _k is the inverse Jacobian matrix of the nonlinear power flow equation at _μ model, that is, a piecewise linear power flow model containing K linear segments is established;

Then, based on equations (2) and (4), the joint probability density function of the distribution network power flow input variables and output variables is established;

For the kth Gaussian component N(x|μ _x,k ,Σ _x,k ) of the Gaussian mixture model given the input variables shown in equation (2), the distribution network power flow input variable vector x is expressed as:

x＝L _k u+μ _x,k (5)

In the formula: L _k is the lower triangular matrix obtained by Σ _{x, k} Choleski decomposition, u is a variable vector that satisfies the standard normal distribution;

According to equations (4) and (5), for the kth Gaussian component N(x|μ _x,k ,Σ _x,k ) of the Gaussian mixture model given the input variables shown in equation (2), the distribution network power flow The joint variable vector of input variables and output variables is:

According to equation (6) and the Choleski decomposition principle, for the kth Gaussian component N(x|μ _x,k ,Σ _x,k ) of the Gaussian mixture model given the input variables shown in equation (2), the power distribution The joint probability density function of the grid power flow input variables and output variables is:

In the formula: Θ _k represents the event of the kth Gaussian component N(x|μ _x,k ,Σ _x,k ) of the Gaussian mixture model given the input variables shown in equation (2), and its probability is P(Θ _k ) = _ωk ;

According to equation (7) and the full probability formula, the joint probability density function of the distribution network power flow input variables and output variables is finally obtained:

As a further solution of the present invention, the step 2 is specifically:

According to the analytical formula of the joint probability density function of the distribution network power flow input variables and output variables shown in Equation (8), the probability distribution of the l-th output variable y _l of the distribution network power flow is the marginal probability distribution of Equation (8), The analytical formula of the probability density function to obtain y _l is:

In the formula: μ _l,k =μ _y,k (l), Σ _ll,k = Σ _y,k (l,l), (l), (l,l) represent the index of vector and matrix elements respectively;

According to equation (9), the M-order origin moment of y _l is expressed as:

In the formula: E[N(y _l |μ _l,k ,Σ _ll,k ) ^M ] represents the M-order origin moment of Gaussian distribution N(y _l |μ _l,k ,Σ _ll,k );

According to equation (10), the variance of y _l is expressed as:

The variance of the power flow output variable of the distribution network is calculated according to Equation (11).

As a further solution of the present invention, the step 3 is specifically:

First, divide the distribution network power flow input variable vector x into complementary subsets x _c and x _d , that is, x = {x _c , x _d };

Secondly, calculate the expected value of the conditional variance of the output variable y _l with respect to the input variable x _c ;

Taking x _c as the input variable to be analyzed, the joint variable vector composed of x _c and the l-th output variable y _l is expressed as w, that is:

The probability distribution of w is the marginal distribution of equation (8), and the analytical formula of the joint probability density function of w is:

In the formula: μ _c,k =μ _x,k (c), Σ _cc,k = Σ _x,k (c,c), Σ _cl,k = Σ _xy,k (c,l), Σ _lc,k =Σ _yx,k (l,c), c represents the index of x _c in x, l represents the index of y _l in y;

According to equation (13) and Bayes’ theorem, the analytical formula for obtaining the conditional probability density function of y _l with respect to x _c is:

In the formula:

According to equations (14)-(17), the analytical formula of the conditional variance of y _l with respect to x _c is obtained:

Calculate the expected value E[Var(y _l |x _c )] of the conditional variance of y _l with respect to x _c according to equation (18);

The calculation formula of E[Var(y _l |x _c )] is:

E[Var(y _l |x _c )]=∫Var(y _l |x _c )f(x _c )dx _c (19)

In the formula:

The numerical integration method is used to calculate equation (19), specifically:

In the formula: _{X c, 1 ,}

Then, calculate the expected value of the conditional variance of the output variable y _l with respect to the input variable x _d ;

Taking x _d as the input variable to be analyzed, the joint variable vector composed of x _d and the l-th output variable y _l is expressed as v, that is:

The probability distribution of v is the marginal distribution of equation (8), and the analytical formula of the joint probability density function of v is obtained:

In the formula: μ _d,k =μ _x,k (d), Σ _dd,k = Σ _x,k (d,d), Σ _dl,k = Σ _xy,k (d,l), Σ _ld,k =Σ _yx,k (l,d), d represents the index of x _d in x, l represents the index of y _l in y;

According to equation (23) and Bayes’ theorem, the analytical formula for obtaining the conditional probability density function of y _l with respect to x _d is:

In the formula:

According to equations (24)-(27), the analytical formula of the conditional variance of y _l with respect to x _d is obtained:

Calculate the expected value E[Var(y _l |x _d )] of the conditional variance of y _l with respect to x _d according to Equation (28);

The calculation formula of E[Var(y _l |x _d )] is:

E[Var(y _l |x _d )]=∫Var(y _l |x _d )f(x _d )dx _d (29)

In the formula:

The numerical integration method is used to calculate equation (29), specifically:

In the _formula : X _{d, 1 ,}

As a further solution of the present invention, the step 4 is specifically:

According to Var(y _l ) obtained in step 2 and E[Var(y _l |x _c )] obtained in step 3, calculate the main effect global sensitivity index of the output variable y _l with respect to the input variable x _c :

According to Var(y _l ) obtained in step 2 and E[Var(y _l |x _d )] obtained in step 3, calculate the global sensitivity index of the total effect of the output variable y _l on the input variable x _c :

In summary, the global sensitivity analysis of distribution network power flow is completed.

Compared with the prior art, the beneficial effects of the present invention are:

1) Based on the Gaussian mixture model and piecewise linearized power flow model, the analytical formula of the joint probability density function of the distribution network power flow input variables and output variables can be obtained. The joint probability density function includes the complete probabilities of the input variables and the output variables. Information and piecewise linearized distribution network power flow model information, the construction process only requires a smaller number of power flow calculations, avoiding the time-consuming proxy model construction process;

2) Based on the analytical formula of the joint probability density function of the power flow input variables and output variables of the distribution network, the analytical formula of the variance of the output variable and the analytical formula of the conditional variance of the output variable with respect to the input variable are obtained, avoiding the time-consuming construction of the distribution network Monte Carlo simulation calculation process of power flow;

Thanks to the above characteristics, the solution of the present invention realizes the analytical method of global sensitivity analysis of distribution network power flow, and significantly improves the calculation efficiency of global sensitivity analysis of distribution network power flow.

Description of drawings

Figure 1 is a flow chart of the method of the present invention.

Detailed ways

The technical solutions in the embodiments of the present invention are described clearly and completely below. Obviously, the described embodiments are only some of the embodiments of the present invention, rather than all the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts fall within the scope of protection of the present invention.

In the embodiment of the present invention, the Gaussian mixture model method for global sensitivity analysis of distribution network power flow includes the following steps:

Step 1: Establish a Gaussian mixture model of the joint probability density function of the distribution network power flow input variables and output variables.

The detailed process is as follows:

First, a Gaussian mixture model of joint probability density function of distribution network power flow input variables is established.

The distribution network power flow input variable vector is x, and its n historical samples are X ₁ , X ₂ ,..., X _n . The non-parametric kernel density estimation method is used to establish the joint probability density function of x:

In the formula: N(·) represents the Gaussian kernel function, and H _k represents the bandwidth matrix corresponding to the k-th sample X _k . The probability density function based on non-parametric kernel density estimation is a basic Gaussian mixture model, whose number of Gaussian components K _b is equal to n, and the weight coefficient of the kth Gaussian component Equal to 1/n, mean vector/> Equal to X _k , covariance matrix/> Equal to H _k .

The density-preserving hierarchical expectation maximization algorithm is used to reduce the Gaussian components of the basic Gaussian mixture model shown in Equation (1), and the simplified Gaussian mixture model is obtained as:

In the formula: ω _k , μ _x,k , Σ _x,k respectively represent the weight coefficient, mean vector and covariance matrix of the kth Gaussian component, and K represents the number of reduced Gaussian components.

Secondly, based on equation (2), a piecewise linearized distribution network power flow model is established.

The compact form of the nonlinear distribution network power flow model is:

y＝Γ(x) (3)

In the formula: Γ(·) represents the nonlinear power flow equation, and y represents the output variable vector of the distribution network power flow.

Given the kth Gaussian component N(x|μ _x,k ,Σ _x,k ) of the Gaussian mixture model of input variables shown in equation (2), for _the nonlinear power flow model shown in equation (3) in Linearize at , and get the kth linearized model:

y＝T _k x+[Γ(μ _x,k )-T _k μ _x,k ] (4)

In the formula: T _k is the inverse Jacobian matrix of the nonlinear power flow equation at μ _x,k . Each Gaussian component of the Gaussian mixture model of the joint probability density function of the input variables shown in Equation (2) will correspond to a linearized power flow model, that is, a piecewise linear power flow model containing K linear segments is established.

Then, based on equations (2) and (4), the joint probability density function of the distribution network power flow input variables and output variables is established.

For the kth Gaussian component N(x|μ _x,k ,Σ _x,k ) of the Gaussian mixture model given the input variables shown in equation (2), the distribution network power flow input variable vector x is expressed as:

x＝L _k u+μ _x,k (5)

In the formula: L _k is the lower triangular matrix obtained by Σ _{x, k} Choleski decomposition, and u is a variable vector that satisfies the standard normal distribution.

According to equations (4) and (5), for the kth Gaussian component N(x|μ _x,k ,Σ _x,k ) of the Gaussian mixture model given the input variables shown in equation (2), the distribution network power flow The joint variable vector of input variables and output variables is:

According to equation (6) and the Choleski decomposition principle, for the kth Gaussian component N(x|μ _x,k ,Σ _x,k ) of the Gaussian mixture model given the input variables shown in equation (2), the power distribution The joint probability density function of the grid power flow input variables and output variables is:

In the formula: Θ _k represents the event of the kth Gaussian component N(x|μ _x,k ,Σ _x,k ) of the Gaussian mixture model given the input variables shown in equation (2), and its probability is P(Θ _k ) =ω _k .

According to equation (7) and the full probability formula, the joint probability density function of the distribution network power flow input variables and output variables is finally obtained:

Step 2: Obtain the analytical formula of the probability density function of the power flow output variable of the distribution network and calculate its variance.

The detailed process is as follows:

According to the analytical formula of the joint probability density function of the distribution network power flow input variables and output variables shown in Equation (8), the probability distribution of the l-th output variable y _l of the distribution network power flow is the marginal probability distribution of Equation (8), The analytical formula of the probability density function to obtain y _l is:

In the formula: μ _l,k =μ _y,k (l), Σ _ll,k = Σ _y,k (l,l), (l), (l,l) represent the index of vector and matrix elements respectively.

According to equation (9), the M-order origin moment of y _l is expressed as:

In the formula: E[N(y _l |μ _l,k ,Σ _ll,k ) ^M ] represents the M-order origin moment of Gaussian distribution N(y _l |μ _l,k ,Σ _ll,k ).

According to equation (10), the variance of y _l is expressed as:

The variance of the power flow output variable of the distribution network is calculated according to Equation (11).

Step 3: Obtain the analytical expression of the conditional probability density function and conditional variance of the distribution network power flow output variable with respect to the input variable, and calculate the expected value of the conditional variance.

The detailed process is as follows:

First, the distribution network power flow input variable vector x is divided into complementary subsets x _c and x _d , that is, x = {x _c , x _d }.

Secondly, calculate the expected value of the conditional variance of the output variable y _l with respect to the input variable x _c ;

Taking x _c as the input variable to be analyzed, the joint variable vector composed of x _c and the l-th output variable y _l is expressed as w, that is:

The probability distribution of w is the marginal distribution of equation (8), and the analytical formula of the joint probability density function of w is:

In the formula: μ _c,k =μ _x,k (c), Σ _cc,k = Σ _x,k (c,c), Σ _cl,k = Σ _xy,k (c,l), Σ _lc,k =Σ _yx,k (l,c), c represents the index of x _c in x, and l represents the index of y _l in y.

According to equation (13) and Bayes’ theorem, the analytical formula for obtaining the conditional probability density function of y _l with respect to x _c is:

In the formula:

According to equations (14)-(17), the analytical formula of the conditional variance of y _l with respect to x _c is obtained:

Calculate the expected value E[Var(y _l |x _c )] of the conditional variance of y _l with respect to x _c according to equation (18).

The calculation formula of E[Var(y _l |x _c )] is:

E[Var(y _l |x _c )]=∫Var(y _l |x _c )f(x _c )dx _c (19)

In the formula:

The numerical integration method is used to calculate equation (19), specifically:

In the formula: X _c,1 , X _c,2 , ..., X _c,N are N samples of x _c generated according to formula (20).

Then, calculate the expected value of the conditional variance of the output variable y _l with respect to the input variable x _d ;

Taking x _d as the input variable to be analyzed, the joint variable vector composed of x _d and the l-th output variable y _l is expressed as v, that is:

The probability distribution of v is the marginal distribution of equation (8), and the analytical formula of the joint probability density function of v is obtained:

In the formula: μ _d,k =μ _x,k (d), Σ _dd,k = Σ _x,k (d,d), Σ _dl,k = Σ _xy,k (d,l), Σ _ld,k =Σ _yx,k (l,d), d represents the index of x _d in x, and l represents the index of y _l in y.

According to equation (23) and Bayes’ theorem, the analytical formula for obtaining the conditional probability density function of y _l with respect to x _d is:

In the formula:

According to equations (24)-(27), the analytical formula of the conditional variance of y _l with respect to x _d is obtained:

Calculate the expected value E[Var(y _l |x _d )] of the conditional variance of y _l with respect to x _d according to equation (28).

The calculation formula of E[Var(y _l |x _d )] is:

E[Var(y _l |x _d )]=∫Var(y _l |x _d )f(x _d )dx _d (29)

In the formula:

The numerical integration method is used to calculate equation (29), specifically:

In the formula: X _{d , 1 ,}

Step 4: Calculate the distribution network power flow global sensitivity index based on the results obtained in steps 2 and 3, and complete the distribution network power flow global sensitivity analysis.

The detailed process is as follows:

According to Var(y _l ) obtained in step 2 and E[Var(y _l |x _c )] obtained in step 3, calculate the main effect global sensitivity index of the output variable y _l with respect to the input variable x _c :

According to Var(y _l ) obtained in step 2 and E[Var(y _l |x _d )] obtained in step 3, calculate the global sensitivity index of the total effect of the output variable y _l on the input variable x _c :

In summary, the global sensitivity analysis of distribution network power flow is completed.

As a preferred embodiment of the present invention, the details are as follows:

Simulation case analysis

An IEEE 33-node system containing 4 distributed photovoltaics, 1 distributed wind power, and 1 electric vehicle charging station was used to verify the inventive solution. The results obtained by the traditional Monte Carlo simulation method were used as the benchmark, and the inventive solution was compared with the polynomial-based Comparing the Monte Carlo simulation methods of the agent model of Chaos Expansion (PCE) and Gaussian Random Process Regression (GPR), the Monte Carlo simulation method based on the agent model is the closest existing technical solution to the present invention, and PCE and GPR are commonly used 2 agency models. The rated active power of each distributed photovoltaic unit is 0.8MW, the rated active power of distributed wind power is 1.2MW, and the electric vehicle charging station is equipped with 20 charging piles (the rated power of each charging pile is 42kW). The output data of wind and solar distributed power sources are obtained by converting actual measured wind speed and irradiation intensity respectively. The load data of electric vehicle charging stations are collected and obtained by actual electric charging stations. The original load data of IEEE 33 nodes is used as the annual average and normalized The load curve is obtained from an actual system. The regular load in each hour time window satisfies the normal distribution, and its standard deviation is 10% of the mean. The number of Gaussian components of the solution of the present invention is 100, the number of samples for numerical integration is 1000, the number of training samples for agent model modeling is 100, and the number of samples for Monte Carlo simulation is 10 ⁶ . The simulation program is compiled and run on a computer configured with Intel i7-10510U 1.8GHz CPU and 16GB RAM.

1. Comparison of accuracy

Table 1 shows some global sensitivity indicators of the IEEE 33 node obtained by the scheme of the present invention. Four representative output variables are selected: node 33 voltage amplitude V ₃₃ , branch 31-32 current amplitude I _31-32 , branch 31-32 active power P _31-32 , network total active loss _PL , the input variable under investigation is the active power P _PV4 of the fourth photovoltaic power generation unit. S _KM (V ₃₃ ,P _PV4 ), S _KM (I _31-32 ,P _PV4 ), S _KM (P _31-32 ,P _PV4 ) and S _KM (P _L ,P _PV4 ) respectively represent the above four output variables. V ₃₃ , I _31-32 , P _31-32 , P _L main effect global sensitivity index on P _PV4 . The results show that the solution of the present invention can obtain accurate global sensitivity index of distribution network power flow, and the accuracy is significantly better than the existing technical solution.

Table 1 Relative error comparison of global sensitivity indicators

2. Comparison of efficiency

This simulation case includes a total of 98 output variables (including node voltage, branch current, branch active power, total active power loss, and total voltage deviation) and 38 input variables (32 conventional loads, 4 photovoltaics, 1 wind power, 1 electric vehicle charging stations), global sensitivity analysis requires calculating a total of 2×38×98 global sensitivity indicators. Table 2 shows the comparison of the calculation time of different methods. The Monte Carlo simulation method based on the surrogate model takes a long time in the construction of the surrogate model and the Monte Carlo simulation process, although the calculation time of the Monte Carlo simulation of the surrogate model method is significantly lower than that of the traditional Monte Carlo simulation method. , but due to the large number of indicators to be calculated, the total time is still long. Compared with the existing technical solutions, the technical solution of the present invention avoids the time-consuming construction of the agent model and the Monte Carlo simulation process, and is an analytical method. Therefore, the calculation time is significantly lower than that of the existing technical solutions, and the calculation efficiency is improved by an order of magnitude.

Table 2 Comparison of global sensitivity index calculation time

In summary, the solution of the present invention has the following advantages.

(1) The modeling process of the Gaussian mixture model of the joint probability density function of the distribution network power flow input variables and output variables only requires a small number of power flow calculations, takes a very short time, and avoids the time-consuming surrogate model construction process.

(2) The analytical formula of the variance of the distribution network power flow output variable and the analytical formula of the conditional variance of the distribution network power flow output variable with respect to the input variable are obtained, which makes the calculation of the global sensitivity index very short and avoids the time-consuming power distribution Monte Carlo simulation calculation process of network power flow.

To sum up, the solution of the present invention significantly improves the calculation efficiency of global sensitivity analysis of distribution network power flow.

It is obvious to those skilled in the art that the present invention is not limited to the details of the above-described exemplary embodiments, and the present invention can be implemented in other specific forms without departing from the spirit or essential characteristics of the present invention. Therefore, the embodiments should be regarded as illustrative and non-restrictive from any point of view, and the scope of the present invention is defined by the appended claims rather than the above description, and it is therefore intended that all claims falling within the claims All changes within the meaning and scope of equivalent elements are included in the present invention.

In addition, it should be understood that although this specification is described in terms of implementations, not each implementation only contains an independent technical solution. This description of the specification is only for the sake of clarity, and those skilled in the art should take the specification as a whole. , the technical solutions in each embodiment can also be appropriately combined to form other implementations that can be understood by those skilled in the art.

Claims (2)

1. The Gaussian mixture model method for analyzing the overall sensitivity of the power flow of the power distribution network is characterized by comprising the following steps of:

step 1: establishing a Gaussian mixture model of a joint probability density function of a power flow input variable and an output variable of the power distribution network;

step 2: obtaining an analytic type of probability density function of a power flow output variable of the power distribution network, and calculating variance of the analytic type;

step 3: obtaining an analytic expression of a conditional probability density function and a conditional variance of a power flow output variable of the power distribution network relative to an input variable, and calculating an expected value of the conditional variance;

step 4: calculating a power flow global sensitivity index of the power distribution network according to the results obtained in the step 2 and the step 3, and completing power flow global sensitivity analysis of the power distribution network;

the step 1 specifically comprises the following steps:

firstly, establishing a Gaussian mixture model of a power flow input variable joint probability density function of a power distribution network;

the power flow input variable vector of the distribution network is X, and n historical samples of the power flow input variable vector are X ₁ 、X ₂ 、…、X _n Establishing a joint probability density function of x by adopting a non-parameter kernel density estimation method:

wherein: n (·) represents a Gaussian kernel function, H _k Represents the kth sample X _k A corresponding bandwidth matrix, the probability density function based on non-parametric kernel density estimation being a basic Gaussian mixture model with Gaussian component number K _b Weight coefficient equal to n, kth gaussian componentEqual to 1/n, mean vector->Equal to X _k Covariance matrix->Equal to H _k ；

And (3) adopting a layering expectation maximization algorithm of density retention to perform Gaussian component number reduction on the basic Gaussian mixture model shown in the formula (1), and obtaining a simplified Gaussian mixture model as follows:

wherein: omega _k 、μ _x,k 、Σ _x,k Respectively representing a weight coefficient, a mean vector and a covariance matrix of the kth Gaussian component, wherein K represents the reduced Gaussian component number;

secondly, on the basis of the formula (2), establishing a piecewise linearization power flow model of the power distribution network;

the compact form of the nonlinear power distribution network power flow model is:

y＝Γ(x) (3)

wherein: Γ (·) represents a nonlinear power flow equation, and y represents an output variable vector of power flow of the power distribution network;

the kth Gaussian component N (x|mu) of the Gaussian mixture model given the input variable of formula (2) _x,k ,Σ _x,k ) For the nonlinear tide model shown in (3), the nonlinear tide model is shown in mu _x,k Linearization is performed, and a kth linearization model is obtained:

y＝T _k x+pΓ(μ _x,k )-T _k μ _x,k ] (4)

wherein: t (T) _k Mu for nonlinear tide equation _x,k An inverse jacobian matrix at; each Gaussian component of the Gaussian mixture model of the input variable joint probability density function shown in the formula (2) corresponds to a linearized power flow model, namely, 1 piecewise linear power flow model comprising K linear segments is established;

then, on the basis of the formula (2) and the formula (4), establishing a joint probability density function of the power flow input variable and the power flow output variable of the power distribution network;

the kth Gaussian component N (x|mu) of the Gaussian mixture model for the input variable of the given formula (2) _x,k ,Σ _x,k ) The power flow input variable vector x of the power distribution network is expressed as:

x＝L _k u+μ _x,k (5)

wherein: l (L) _k Is sigma-delta _x,k The lower triangular matrix obtained by the George decomposition is a variable vector meeting standard normal distribution;

according to the formulas (4) and (5), for the kth gaussian component N (x|μ) of the gaussian mixture model given the input variable represented by the formula (2) _x,k ,Σ _x,k ) The combined variable vector of the power flow input variable and the power flow output variable of the power distribution network is as follows:

according to the formula (6) and the principle of the Georgi decomposition, the kth Gaussian component N (x|mu) of the Gaussian mixture model for the input variable represented by the given formula (2) _x,k ,Σ _x,k ) The joint probability density function of the power flow input variable and the power flow output variable of the power distribution network is as follows:

wherein: theta (theta) _k The kth Gaussian component N (x|mu) of the Gaussian mixture model representing the input variable of the given formula (2) _x,k ,Σ _x,k ) With a probability P (Θ) _k )＝ω _k ；

According to the formula (7) and the full probability formula, the final obtaining of the joint probability density function of the power flow input variable and the power flow output variable of the power distribution network is as follows:

the step 2 specifically comprises the following steps:

according to the analysis of the joint probability density function of the input variable and the output variable of the power flow of the power distribution network shown in the formula (8), the first output variable y of the power flow of the power distribution network _l The probability distribution of (2) is the edge probability distribution of equation (8), and y is obtained _l The probability density function of (2) is as follows:

wherein: mu (mu) _l,k ＝μ _y,k (l)、Σ _ll,k ＝Σ _y,k (l, l), (l, l) represent the indices of the vector and matrix elements, respectively;

according to formula (9), y _l Is expressed as:

wherein: e [ N (y) _l |μ _l,k ,Σ _ll,k ) ^M ]Shows a Gaussian distribution N (y) _l |μ _l,k ,Σ _ll,k ) An M-order origin moment of (a);

according to formula (10), y _l The variance of (c) is expressed as:

calculating to obtain the variance of the power flow output variable of the power distribution network according to the formula (11);

the step 3 specifically comprises the following steps:

first, the power distribution network power flow input variable vector x is divided into complementary subsets x _c And x _d I.e. x= { x _c ,x _d }；

Next, the output variable y is calculated _l With respect to input variable x _c Is a desired value of conditional variance of (a);

will x _c As input variable to be analyzed, x _c And the first output variable y _l The combined variable vector is represented as w, namely:

the probability distribution of w is the edge distribution of the formula (8), and the analytical formula of the joint probability density function of w is:

wherein: mu (mu) _c,k ＝μ _x,k (c)、Σ _cc,k ＝Σ _x,k (c,c)、Σ _cl,k ＝Σ _xy,k (c,l)、Σ _lc,k ＝Σ _yx,k (l, c), c represents x _c Index in x, l denotes y _l An index in y;

obtaining y according to formula (13) and Bayes theorem _l In relation to x _c The analytical formula of the conditional probability density function is:

wherein:

according to formulas (14) - (17), y is obtained _l Concerning x _c The conditional variance analysis formula of (2) is:

calculating y according to formula (18) _l Concerning x _c The expected value E [ Var (y) _l |x _c )]；

E[Var(y _l |x _c )]The calculation formula of (2) is as follows:

E[Var(y _l |x _c )]＝∫Var(y _l |x _c )f(x _c )dx _c (19)

wherein:

the numerical integration method is adopted to calculate the formula (19), and the method is specifically as follows:

wherein: x is X _c,1 、X _c,2 、…、X _c,N For x generated according to formula (20) _c Is a sample of N samples;

then, calculate the output variable y _l With respect to input variable x _d Is a desired value of conditional variance of (a);

will x _d As input variable to be analyzed, x _d And the first output variable y _l The composed joint variable vector is denoted v, namely:

the probability distribution of v is the edge distribution of the formula (8), and the analytical formula of the joint probability density function for obtaining v is as follows:

wherein: mu (mu) _d,k ＝μ _x,k (d)、Σ _dd,k ＝Σ _x,k (d,d)、Σ _dl,k ＝Σ _xy,k (d,l)、Σ _ld,k ＝Σ _yx,k (l, d), d represents x _d Index in x, l denotes y _l An index in y;

obtaining y according to formula (23) and Bayes theorem _l Concerning x _d The analytical formula of the conditional probability density function is:

wherein:

according to formulae (24) - (27), y is obtained _l Concerning x _d The conditional variance analysis formula of (2) is:

calculating y according to formula (28) _l Concerning x _d The expected value E [ Var (y) _l |x _d )]；

E[Var(y _l |x _d )]The calculation formula of (2) is as follows:

E[Var(y _l |x _d )]＝∫Var(y _l |x _d )f(x _d )dx _d (29)

wherein:

the numerical integration method is adopted to calculate the formula (29), specifically:

wherein: x is X _d,1 、X _d,2 、…、X _d,N For x generated according to formula (30) _d Is a sample of the N samples.

2. The gaussian mixture model method for power distribution network power flow global sensitivity analysis according to claim 1, wherein said step 4 is specifically:

var (y) obtained according to step 2 _l ) And E [ Var (y) obtained in step 3 _l |x _c )]Calculating the output variable y _l With respect to input variable x _c Main effect global sensitivity index of (2):

var (y) obtained according to step 2 _l ) And E [ Var (y) obtained in step 3 _l |x _d )]Calculating the output variable y _l With respect to input variable x _c Global sensitivity index of the total effect of (a):

and comprehensively, completing the overall sensitivity analysis of the power flow of the power distribution network.