CN104050604A - Electric power system static safety assessment method based on probabilistic tide - Google Patents

Abstract

The invention discloses an electric power system static safety assessment method based on a probabilistic tide. Statistical information of the random quantity in a system is obtained according to historical data; mutual conversion between a relevant non-normal random variable and an independent normal distribution random variable is achieved by improving Nataf conversion. By means of the electric power system static safety assessment method based on the probabilistic tide, arbitrary probability distribution of many random factors in the electric power system can be coped with, the correlation among the random factors is considered, the processing difficulty of the correlation is reduced, and the electric power system static safety assessment method is suitable for assessing the influence of all kinds of uncertain factors in the electric power system on the static safety of the electric power system.

Description

Power system static safety evaluation method based on Probabilistic Load Flow

Technical field

The invention belongs to power network safety operation technical field, particularly a kind of power system static safety evaluation method based on Probabilistic Load Flow.

Background technology

Regenerative resource is grid-connected on a large scale, to operation of power networks, many uncertain factors have been brought, now need to pass through probabilistic load flow, obtain system operation characteristic amount (as node voltage amplitude and phase angle, circuit is meritorious and idle etc.) statistical information, and then the weak link of discovery system operation.Probabilistic Load Flow has been successfully applied to power system dynamic stability analysis, the aspects such as fail-safe analysis.

But there is many limitation in existing Probabilistic Load Flow, Monte Carlo method precision is the highest, but the most consuming time, generally using its result as the standard of evaluating other computing method performances.Analytic calculation is rapid, is applicable to process a large amount of input variables, but need to do linearization to original system model, easily causes certain error when random quantity fluctuation range is larger.Method of approximation, calculates consuming time less, but computational accuracy is not high, can only obtain output low order probability square, cannot obtain High Order Moment.Thereby need to build one can be efficiently and have a Probabilistic Load Flow algorithm of higher computational accuracy.In electric system, the correlativity of random quantity also causes scholar's concern in recent years, and research shows that its static security on system has impact.But in the method for existing processing correlativity, traditional Nataf transformation calculations is complicated, the statistical information that polynomial transformation technology cannot the original stochastic variable of complete reservation in the process of conversion, thereby need to further improve Correlation treatment method.

Summary of the invention

The object of the invention is to overcome above-mentioned the deficiencies in the prior art, a kind of power system static safety evaluation method based on Probabilistic Load Flow is provided, for assessment of the impact of all kinds of uncertain factors on power system static safety in each electric system.

Technical solution of the present invention is as follows:

A power system static safety evaluation method based on Probabilistic Load Flow,, comprise the steps:

Step 1: build the probabilistic load flow model that is shown below,

In formula the clean meritorious injecting power that represents i node of electric system; P _ijrepresent that node j flows to the meritorious trend of circuit of node i; the clean idle injecting power that represents i node of electric system; V _ithe voltage magnitude that represents i node of electric system; V _jthe voltage magnitude that represents j node of electric system; θ _ijthe phase difference of voltage that represents i node of electric system and j node; Y _ijrepresent to connect the line admittance amplitude of ij node; represent to connect the line admittance phase angle of ij node; G _ijthe electricity that represents electric system connection ij node line is led; B _ijrepresent that electric system connects the susceptance of ij node line; N is electric system interstitial content.

Wherein,

$\left\{\begin{array}{c}{P}_i^{\mathrm{inj}}=P_{w,i}+P_{\mathrm{pv},i}-P_{L,i}\\ Q_i^{\mathrm{inj}}=Q_{w,i}-Q_{L,i}\end{array}\right.$

In formula, P _w,ifor the wind-power electricity generation that i Nodes of electric system injects is gained merit, P _{pv, i}for the photovoltaic generation that i Nodes of electric system injects is gained merit, P _l,ifor gaining merit that each type load of i Nodes of electric system consumes; Q _w,ifor the wind-power electricity generation that i Nodes of electric system injects idle, Q _l,ifor each type load of i Nodes of electric system consume idle.In system, the marginal probability distribution of each Uncertainty can obtain experience distribution according to historical data.

Step 2: random quantity is normal distribution and when separate in electric system, marginal probability distribution and power flow equation according to random quantity, adopt multiple integration method rapid solving to obtain each rank statistic of node voltage amplitude, phase angle and the Line Flow of electric system, its calculating formula is as follows:

The expectation of note output y is m _y, each rank centre distance is μ _i, input x=[x ₁, x ₂..., x _n], f (x ₁, x ₂..., x _n) be associating normal probability density function.The m based on Stroud integral formula _yand μ _icalculating formula as follows:

$\begin{array}{c}{m}_y={\∫}_{-\∞}^{+\∞}...{\∫}_{-\∞}^{+\∞}y(x_1,x_2,...,x_n)f(x_1,x_2,...,x_n)\·{\mathrm{dx}}_1...{\mathrm{dx}}_n\\ =\underset{j=1}{\overset{M}{\mathrm{\Σ}}}{A}_{j}y(p_{j1},p_{j2},...,p_{\mathrm{jn}})\end{array}$

$\begin{array}{c}{\mathrm{\μ}}_i={\∫}_{-\∞}^{+\∞}...{\∫}_{-\∞}^{+\∞}{(y(x_1,x_2,...,x_n)-m_y)}^{i}f(x_1,x_2,...,x_n)\·{\mathrm{dx}}_1...{\mathrm{dx}}_n\\ =\underset{j=1}{\overset{M}{\mathrm{\Σ}}}{A}_j{(y(p_{j1},p_{j2},...,p_{\mathrm{jn}})-m_y)}^i\end{array}$

In formula, weight function is f, and integrand is y or (y-m _y) ⁱ, A and p _j=[p _j1, p _j2..., p _jn] for the weight coefficient in integral formula with join a little, M is for joining a number, the weight coefficient of different accuracy formula and join some the works < < Approximate calculation of multiple integrals > > referring to A.H Stroud.According to each, join point (p- _j1, p _j2..., p _jn) (j=1,2 ..., M) carry out deterministic trend calculating, can obtain exporting accordingly y (p- _j1, p _j2..., p _jn), and then utilize above formula weighted sum, can obtain exporting each rank statistic of y.

In electric system, random quantity is skewed distribution or while having correlativity, adopt and improve Nataf conversion process, be converted into independent normally distributed random variable, thereby available multiple integration method is processed correlativity random quantity, calculating probability trend, its calculation procedure is as follows:

The input v=[v distributing arbitrarily ₁, v ₂..., v _n] ^t, v _iaverage is μ _vi, standard deviation is σ _vi, edge PDF is f (v _i), edge accumulated probability distribution function (cumulative distribution function, CDF) is F (v _i).V is converted to relevant criterion normal distribution vector z=[z ₁, z ₂..., z _n] ^t, i.e. normal transformation

Φ(z _i)＝F(v _i)

Φ (z in formula _i) be z _icDF.Note ρ _ijfor v _iand v _jrelated coefficient, after conversion, related coefficient changes, and can adopt the TNPT based on moments method to calculate the equivalent correlation coefficient ρ after conversion _1ij.

In TNPT method, with standardized normal distribution stochastic variable z _ithree rank polynomial expressions represent v _i,

$v_i=a_{0,i}+a_{1,i}{z}_i+a_{2,i}{z}_i^2+a_{3,i}{z}_i^3$

By v _istandardization, v _si=(v _i-μ _vi)/σ _vi, above formula is transformed to

$v_{\mathrm{si}}=b_{0,i}+b_{1,i}{z}_i+b_{2,i}{z}_i^2+b_{3,i}{z}_i^3$

Coefficient has following relation:

$\left\{\begin{array}{cc}{a}_{0,i}=b_{0,i}{\mathrm{\&sigma;}}_{\mathrm{vi}}+{\mathrm{\&mu;}}_{\mathrm{vi}};& \\ a_{r,i}=b_{r,i}{\mathrm{\&sigma;}}_{\mathrm{vi}}& r=\mathrm{1,2,3}\end{array}\right.$

According to moments method principle, one of equation the right and left equates to quadravalence moment of the orign, that is:

$E\left(v_i^r\right)=E\left[{(a_{0,i}+a_{1,i}{z}_i+a_{2,i}{z}_i^2+a_{3,i}{z}_i^3)}^r\right],r=\mathrm{1,2,3,4}$

By solving equation shown in following formula, can obtain coefficient.In formula, γ and κ are respectively v _sithe coefficient of skewness and kurtosis.

$\left\{\begin{array}{c}{b}_{1,i}^2+6b_{1,i}{b}_{3,i}+2b_{2,i}^2+15b_{3,i}^2=1\\ 2b_{2,i}(b_{1,i}^2+24b_{1,i}{b}_{3,i}+105b_{3,i}^2+2)=\mathrm{\&gamma;}\\ 24[(b_{1,i}{b}_{3,i}+b_{2,i}^2(1+b_{1,i}^2+28b_{1,i}{b}_{3,i})+b_{3,i}^2\&times;\\ (12+48b_{1,i}{b}_{3,i}+141b_{2,i}^2+225b_{3,i}^2)]=\mathrm{\&kappa;}-3\\ b_{0,i}+b_{2,i}=0\end{array}\right.$

Further, solve following formula and obtain equivalent correlation coefficient ρ _1ij,

$\begin{array}{c}6a_{3,i}{a}_{3,j}{\mathrm{\&rho;}}_{1\mathrm{ij}}^3+2a_{2,i}{a}_{2,j}{\mathrm{\&rho;}}_{1\mathrm{ij}}^2+(a_{1,i}+3a_{3,i})\&times;(a_{1,j}+3a_{3,j}){\mathrm{\&rho;}}_{1\mathrm{ij}}+[(a_{0,i}+a_{2,i})\\ \&times;(a_{0,j}+a_{2,j})-{\mathrm{\&rho;}}_{\mathrm{ij}}{\mathrm{\&sigma;}}_{\mathrm{vi}}{\mathrm{\&sigma;}}_{\mathrm{vj}}-{\mathrm{\&mu;}}_{\mathrm{vi}}{\mathrm{\&mu;}}_{\mathrm{vj}}]=0\end{array}$

ρ _1ijmeet

-1≤ρ _1ij≤1，ρ _1ijρ _ij≥0

The correlation matrix of note z is C _z, further utilize Cholesky to decompose C _z=BB ^t, have

In formula, e is independent normal distribution vector.And then can be by a p that joins for original Stroud formula under normal distribution _jbe converted to the p that joins under actual input variable probability space _j'; By p _jbring above formula into,

$P_j^{\text{'}}=\left[\begin{array}{c}{p}_{j1}^{\text{'}}\\ p_{j2}^{\text{'}}\\ .\\ .\\ .\\ p_{\mathrm{jn}}^{\text{'}}\end{array}\right]=\left[\begin{array}{c}{F}^{-1}\left(\mathrm{\&Phi;}\left(z_1\right)\right)\\ F^{-1}\left(\mathrm{\&Phi;}\left(z_2\right)\right)\\ .\\ .\\ .\\ F^{-1}\left(\mathrm{\&Phi;}\left(z_n\right)\right)\end{array}\right]$

F-in formula ¹() is F (v _i) inverse function.To p' _jcarry out the calculating of determinacy trend, and then each rank square that utilizes the weighted sum of Stroud formula to export.

Step 3, utilize limited rank Cornish-fisher series expansion to obtain the probability distribution of node voltage amplitude, phase angle and the Line Flow of electric system, and then obtain the out-of-limit probability of system according to setting threshold, appraisal procedure is as follows:

According to step 2, to each rank square of output, each rank cumulant γ of note output is

γ ₁＝m _y

γ ₂＝μ ₂

γ ₃＝μ ₃

${\mathrm{\&gamma;}}_4={\mathrm{\&mu;}}_4-3{\mathrm{\&mu;}}_2^2$

Stochastic variable canonical form probability density function be expressed as:

In formula probability density function for normal distribution.According to above formula, obtain edge accumulated probability distribution function F (y) (cumulative distribution function, CDF), be shown below.

$F\left(y\right)={\&Integral;}_{-\&infin;}^{(y-m_y)/\sqrt{{\mathrm{\&mu;}}_2}}f\left(y_s\right){\mathrm{dy}}_s$

According to setting threshold P _limitthe not out-of-limit probability of the system that obtains, brings above formula F (P into _limit).

Compared with prior art, the invention has the beneficial effects as follows:

The arbitrariness probability distributing that can tackle many enchancement factors in electric system distributes, consider the correlativity between enchancement factor and simplified correlativity intractability, use the statistical information of the acquisition system character of multiple integration method efficiently and accurately, further can be to the out-of-limit probability of system after obtaining its probability distribution, be that static system is assessed safely, be applicable to assess the impact of all kinds of uncertain factors on power system static safety in each electric system.

Accompanying drawing explanation

Fig. 1 is the implementing procedure figure of the power system static safety assessment based on Probabilistic Load Flow.

Fig. 2 is the improvement Nataf transform method schematic diagram in the power system static safety assessment based on Probabilistic Load Flow.

Embodiment

Below in conjunction with accompanying drawing, illustrate the inventive method, but should not limit guard method of the present invention with this.

Fig. 1 is the implementing procedure figure of the power system static safety assessment based on Probabilistic Load Flow, and as shown in the figure, a kind of power system static safety evaluation method based on Probabilistic Load Flow, comprises the steps:

Step 1: build the power flow algorithm that is shown below,

In formula the clean meritorious injecting power that represents i node of electric system; the clean idle injecting power that represents i node of electric system; V _ithe voltage magnitude that represents i node of electric system; V _jthe voltage magnitude that represents j node of electric system; P _ijrepresent that node j flows to the meritorious trend of circuit of node i; θ _ijthe phase difference of voltage that represents i node of electric system and j node; Y _ijrepresent to connect the line admittance amplitude of ij node; represent to connect the line admittance phase angle of ij node; G _ijthe electricity that represents electric system connection ij node line is led; B _ijrepresent that electric system connects the susceptance of ij node line; N is electric system interstitial content.

Wherein,

$\left\{\begin{array}{c}{P}_i^{\mathrm{inj}}=P_{w,i}+P_{\mathrm{pv},i}-P_{L,i}\\ Q_i^{\mathrm{inj}}=Q_{w,i}-Q_{L,i}\end{array}\right.$

In formula, P _w,ifor the wind-power electricity generation that i Nodes of electric system injects is gained merit, P _{pv, i}for the photovoltaic generation that i Nodes of electric system injects is gained merit, P _l,ifor gaining merit that each type load of i Nodes of electric system consumes; Q _w,ifor the wind-power electricity generation that i Nodes of electric system injects idle, Q _l,ifor each type load of i Nodes of electric system consume idle.In system, the marginal probability distribution of each Uncertainty can obtain experience distribution according to historical data.

Step 2: random quantity is normal distribution and when separate in electric system, marginal probability distribution and power flow equation according to random quantity, adopt multiple integration method rapid solving to obtain each rank statistic of node voltage amplitude, phase angle and the Line Flow of electric system, its calculating formula is as follows:

The expectation of note output y is m _y, each rank centre distance is μ _i, input x=[x ₁, x ₂..., x _n], f (x ₁, x ₂..., x _n) be associating normal probability density function.M based on Stroud integral formula _yand μ _icalculating formula as follows:

$\begin{array}{c}{m}_y={\&Integral;}_{-\&infin;}^{+\&infin;}...{\&Integral;}_{-\&infin;}^{+\&infin;}y(x_1,x_2,...,x_n)f(x_1,x_2,...,x_n)\&CenterDot;{\mathrm{dx}}_1...{\mathrm{dx}}_n\\ =\underset{j=1}{\overset{M}{\mathrm{\&Sigma;}}}{A}_{j}y(p_{j1},p_{j2},...,p_{\mathrm{jn}})\end{array}$

$\begin{array}{c}{\mathrm{\&mu;}}_i={\&Integral;}_{-\&infin;}^{+\&infin;}...{\&Integral;}_{-\&infin;}^{+\&infin;}{(y(x_1,x_2,...,x_n)-m_y)}^{i}f(x_1,x_2,...,x_n)\&CenterDot;{\mathrm{dx}}_1...{\mathrm{dx}}_n\\ =\underset{j=1}{\overset{M}{\mathrm{\&Sigma;}}}{A}_j{(y(p_{j1},p_{j2},...,p_{\mathrm{jn}})-m_y)}^i\end{array}$

In formula, weight function is f, and integrand is y or (y-m _y) ⁱ, A and p _j=[p _j1, p _j2..., p _jn] for the weight coefficient in integral formula with join a little, M is for joining a number, the weight coefficient of different accuracy formula and join some the works < < Approximate calculation of multiple integrals > > referring to A.H Stroud.According to each, join point (p- _j1, p _j2..., p _jn) (j=1,2 ..., M) carry out deterministic trend calculating, can obtain exporting accordingly y (p- _j1, p _j2..., p _jn), and then utilize above formula weighted sum, can obtain exporting each rank statistic of y.

Figure 2 shows that the improvement Nataf mapping algorithm schematic diagram in the power system static safety assessment based on Probabilistic Load Flow.In electric system, random quantity is skewed distribution or while having correlativity, adopt and improve Nataf conversion process, be converted into independent normally distributed random variable, thereby available multiple integration method is processed correlativity random quantity, calculating probability trend, its calculation procedure is as follows:

The input v=[v distributing arbitrarily ₁, v ₂..., v _n] ^t, v _iaverage is μ _vi, standard deviation is σ _vi, edge PDF is f (v _i), edge accumulated probability distribution function (cumulative distribution function, CDF) is F (v _i).V is converted to relevant criterion normal distribution vector z=[z ₁, z ₂..., z _n] ^t, i.e. normal transformation

Φ(z _i)＝F(v _i)

Φ (z in formula _i) be z _icDF.Note ρ _ijfor v _iand v _jrelated coefficient, after conversion, related coefficient changes, and can adopt the polynomial transformation based on moments method to calculate the equivalent correlation coefficient ρ after conversion _1ij.

In polynomial transformation method, with standardized normal distribution stochastic variable z _ithree rank polynomial expressions represent v _i,

$v_i=a_{0,i}+a_{1,i}{z}_i+a_{2,i}{z}_i^2+a_{3,i}{z}_i^3$

By v _istandardization, v _si=(v _i-μ _vi)/σ _vi, above formula is transformed to

$v_{\mathrm{si}}=b_{0,i}+b_{1,i}{z}_i+b_{2,i}{z}_i^2+b_{3,i}{z}_i^3$

Coefficient has following relation:

$\left\{\begin{array}{cc}{a}_{0,i}=b_{0,i}{\mathrm{\&sigma;}}_{\mathrm{vi}}+{\mathrm{\&mu;}}_{\mathrm{vi}};& \\ a_{r,i}=b_{r,i}{\mathrm{\&sigma;}}_{\mathrm{vi}}& r=\mathrm{1,2,3}\end{array}\right.$

According to moments method principle, one of equation the right and left equates to quadravalence moment of the orign, that is:

$E\left(v_i^r\right)=E\left[{(a_{0,i}+a_{1,i}{z}_i+a_{2,i}{z}_i^2+a_{3,i}{z}_i^3)}^r\right],r=\mathrm{1,2,3,4}$

By solving equation shown in following formula, can obtain coefficient.In formula, γ and κ are respectively v _sithe coefficient of skewness and kurtosis.

$\left\{\begin{array}{c}{b}_{1,i}^2+6b_{1,i}{b}_{3,i}+2b_{2,i}^2+15b_{3,i}^2=1\\ 2b_{2,i}(b_{1,i}^2+24b_{1,i}{b}_{3,i}+105b_{3,i}^2+2)=\mathrm{\&gamma;}\\ 24[(b_{1,i}{b}_{3,i}+b_{2,i}^2(1+b_{1,i}^2+28b_{1,i}{b}_{3,i})+b_{3,i}^2\&times;\\ (12+48b_{1,i}{b}_{3,i}+141b_{2,i}^2+225b_{3,i}^2)]=\mathrm{\&kappa;}-3\\ b_{0,i}+b_{2,i}=0\end{array}\right.$

Further, solve following formula and obtain equivalent correlation coefficient ρ _1ij,

$\begin{array}{c}6a_{3,i}{a}_{3,j}{\mathrm{\&rho;}}_{1\mathrm{ij}}^3+2a_{2,i}{a}_{2,j}{\mathrm{\&rho;}}_{1\mathrm{ij}}^2+(a_{1,i}+3a_{3,i})\&times;(a_{1,j}+3a_{3,j}){\mathrm{\&rho;}}_{1\mathrm{ij}}+[(a_{0,i}+a_{2,i})\\ \&times;(a_{0,j}+a_{2,j})-{\mathrm{\&rho;}}_{\mathrm{ij}}{\mathrm{\&sigma;}}_{\mathrm{vi}}{\mathrm{\&sigma;}}_{\mathrm{vj}}-{\mathrm{\&mu;}}_{\mathrm{vi}}{\mathrm{\&mu;}}_{\mathrm{vj}}]=0\end{array}$

ρ _1ijmeet

-1≤ρ _1ij≤1，ρ _1ijρ _ij≥0

The correlation matrix of note z is C _z, further utilize Cholesky to decompose C _z=BB ^t, have

In formula, e is independent normal distribution vector.And then can be by a p that joins for original Stroud formula under normal distribution _jbe converted to the p that joins under actual input variable probability space _j'.By p _jbring above formula into,

$P_j^{\text{'}}=\left[\begin{array}{c}{p}_{j1}^{\text{'}}\\ p_{j2}^{\text{'}}\\ .\\ .\\ .\\ p_{\mathrm{jn}}^{\text{'}}\end{array}\right]=\left[\begin{array}{c}{F}^{-1}\left(\mathrm{\&Phi;}\left(z_1\right)\right)\\ F^{-1}\left(\mathrm{\&Phi;}\left(z_2\right)\right)\\ .\\ .\\ .\\ F^{-1}\left(\mathrm{\&Phi;}\left(z_n\right)\right)\end{array}\right]$

F in formula ^-1() is F (v _i) inverse function.To each p' _jcarry out the calculating of determinacy trend, and then each rank square that utilizes the weighted sum of Stroud formula to export.

Step 3: use limited rank Cornish-fisher series expansion to obtain the probability distribution of node voltage amplitude, phase angle and the Line Flow of electric system, and then obtain the out-of-limit probability of system according to setting threshold, appraisal procedure is as follows:

Each rank square of the output obtaining according to step 2, each rank cumulant γ of note output is

γ1＝m _y

γ ₂＝μ ₂

γ ₃＝μ ₃

${\mathrm{\&gamma;}}_4={\mathrm{\&mu;}}_4-3{\mathrm{\&mu;}}_2^2$

Stochastic variable canonical form probability density function be expressed as:

In formula probability density function for normal distribution.According to above formula, obtain edge accumulated probability distribution function F (y) (cumulative distribution function, CDF), be shown below.

$F\left(y\right)={\&Integral;}_{-\&infin;}^{(y-m_y)/\sqrt{{\mathrm{\&mu;}}_2}}f\left(y_s\right){\mathrm{dy}}_s$

According to setting threshold P _limitthe not out-of-limit probability of the system that obtains, brings above formula F (P into _limit).According to the horizontal Pro of predefined safe probability _limit(Pro for example _limit=0.9), as the not out-of-limit probability F of system (P _limit) >Pro _limittime, decision-making system safety, otherwise the static security level of system does not meet the demands, and has potential risk.

Claims (2)

1. the power system static safety evaluation method based on Probabilistic Load Flow, is characterized in that, comprises the steps:

Step 1, structure probabilistic load flow model, as shown in the formula:

In formula the clean meritorious injecting power that represents i node of electric system; P _ijrepresent that node j flows to the meritorious trend of circuit of node i; the clean idle injecting power that represents i node of electric system; V _ithe voltage magnitude that represents i node of electric system; V _jthe voltage magnitude that represents j node of electric system; θ _ijthe phase difference of voltage that represents i node of electric system and j node; Y _ijrepresent to connect the line admittance amplitude of ij node; represent to connect the line admittance phase angle of ij node; G _ijthe electricity that represents electric system connection ij node line is led; B _ijrepresent that electric system connects the susceptance of ij node line; N is electric system interstitial content;

Wherein,

$\left\{\begin{array}{c}{P}_i^{\mathrm{inj}}=P_{w,i}+P_{\mathrm{pv},i}-P_{L,i}\\ Q_i^{\mathrm{inj}}=Q_{w,i}-Q_{L,i}\end{array}\right.$

In formula, P _w,ifor the wind-power electricity generation that i Nodes of electric system injects is gained merit, P _{pv, i}for the photovoltaic generation that i Nodes of electric system injects is gained merit, P _l,ifor gaining merit that each type load of i Nodes of electric system consumes; Q _w,ifor the wind-power electricity generation that i Nodes of electric system injects idle, Q _l,ifor each type load of i Nodes of electric system consume idle;

Step 2, in electric system, random quantity is normal distribution and when separate, according to the marginal probability distribution of random quantity and power flow equation, adopt multiple integration method rapid solving to obtain each rank statistic of node voltage amplitude, phase angle and the Line Flow of electric system, its calculating formula is as follows:

The expectation of note output y is m _y, each rank centre distance is μ _i, input x=[x ₁, x ₂..., x _n], f (x ₁, x ₂..., x _n) be associating normal probability density function, the m based on Stroud integral formula _yand μ _icalculating formula as follows:

$\begin{array}{c}{m}_y={\&Integral;}_{-\&infin;}^{+\&infin;}...{\&Integral;}_{-\&infin;}^{+\&infin;}y(x_1,x_2,...,x_n)f(x_1,x_2,...,x_n)\&CenterDot;{\mathrm{dx}}_1...{\mathrm{dx}}_n\\ =\underset{j=1}{\overset{M}{\mathrm{\&Sigma;}}}{A}_{j}y(p_{j1},p_{j2},...,p_{\mathrm{jn}})\end{array}$

$\begin{array}{c}{\mathrm{\&mu;}}_i={\&Integral;}_{-\&infin;}^{+\&infin;}...{\&Integral;}_{-\&infin;}^{+\&infin;}{(y(x_1,x_2,...,x_n)-m_y)}^{i}f(x_1,x_2,...,x_n)\&CenterDot;{\mathrm{dx}}_1...{\mathrm{dx}}_n\\ =\underset{j=1}{\overset{M}{\mathrm{\&Sigma;}}}{A}_j{(y(p_{j1},p_{j2},...,p_{\mathrm{jn}})-m_y)}^i\end{array}$

In formula, weight function is f, and integrand is y or (y-m _y) ⁱ, A and p _j=[p _j1, p _j2..., p _jn] for the weight coefficient in integral formula with join a little, M is for joining a number;

According to each, join point (p _j1, p _j2..., p _jn) (j=1,2 ..., M) carry out deterministic trend calculating, obtain exporting accordingly y (p _j1, p _j2..., p _jn), and then utilize above formula weighted sum, obtain exporting each rank statistic of y;

In electric system, random quantity is skewed distribution or while having correlativity, adopts and improves Nataf conversion process, is converted into independent normally distributed random variable, with multiple integration method, processes correlativity random quantity, calculating probability trend, and its calculation procedure is as follows:

The input v=[v distributing arbitrarily ₁, v ₂..., v _n] ^t, v _iaverage is μ _vi, standard deviation is σ _vi, marginal probability density function is f (v _i), edge accumulated probability distribution function is F (v _i), v is converted to relevant criterion normal distribution vector z=[z ₁, z ₂..., z _n] ^t,

Φ(z _i)＝F(v _i)

Φ (z in formula _i) be z _iedge accumulated probability distribution function;

Note ρ _ijfor v _iand v _jrelated coefficient, adopts the TNPT based on moments method to calculate the equivalent correlation coefficient ρ after conversion _1ij,

The correlation matrix of note z is C _z, utilize Cholesky to decompose C _z=BB ^t, obtain transition matrix B,

z＝Be

In formula, e is independent normally distributed variable;

Utilize z=Be and Φ (z _i)=F (v _i), by a p that joins for original Stroud formula under normal distribution _jbe converted to joining a little under actual input variable probability space, and then according to Stroud formula, try to achieve each rank square of output y;

Step 3, utilize limited rank Cornish-fisher series expansion to obtain the probability distribution of node voltage amplitude, phase angle and the Line Flow of electric system, according to setting threshold, obtain the out-of-limit probability of system, appraisal procedure is as follows:

According to step 2, obtain each rank square of output, each rank cumulant γ of note output y is

γ ₁＝m _y

γ ₂＝μ ₂

γ ₃＝μ ₃

${\mathrm{\&gamma;}}_4={\mathrm{\&mu;}}_4-3{\mathrm{\&mu;}}_2^2$

Stochastic variable canonical form probability density function be expressed as:

In formula for the probability density function of normal distribution, obtain edge accumulated probability distribution function F (y), be shown below:

$F\left(y\right)={\&Integral;}_{-\&infin;}^{(y-m_y)/\sqrt{{\mathrm{\&mu;}}_2}}f\left(y_s\right){\mathrm{dy}}_s$

According to setting threshold P _limitthe not out-of-limit probability of the system that obtains, brings above formula F (P into _limit).

2. the power system static safety evaluation method based on Probabilistic Load Flow according to claim 1, is characterized in that, calculates equivalent correlation coefficient ρ _1ijconcrete steps are as follows:

In TNPT method, with standardized normal distribution stochastic variable z _ithree rank polynomial expressions represent v _i,

$v_i=a_{0,i}+a_{1,i}{z}_i+a_{2,i}{z}_i^2+a_{3,i}{z}_i^3$

By v _istandardization, v _si=(v _i-μ _vi)/σ _vi, above formula is transformed to

$v_{\mathrm{si}}=b_{0,i}+b_{1,i}{z}_i+b_{2,i}{z}_i^2+b_{3,i}{z}_i^3$

Coefficient has following relation:

$\left\{\begin{array}{cc}{a}_{0,i}=b_{0,i}{\mathrm{\&sigma;}}_{\mathrm{vi}}+{\mathrm{\&mu;}}_{\mathrm{vi}};& \\ a_{r,i}=b_{r,i}{\mathrm{\&sigma;}}_{\mathrm{vi}}& r=\mathrm{1,2,3}\end{array}\right.$

According to moments method principle, one of equation the right and left equates to quadravalence moment of the orign, that is:

$E\left(v_i^r\right)=E\left[{(a_{0,i}+a_{1,i}{z}_i+a_{2,i}{z}_i^2+a_{3,i}{z}_i^3)}^r\right],r=\mathrm{1,2,3,4}$

By solving equation shown in following formula, can obtain coefficient, in formula, γ and κ are respectively v _sithe coefficient of skewness and kurtosis,

$\left\{\begin{array}{c}{b}_{1,i}^2+6b_{1,i}{b}_{3,i}+2b_{2,i}^2+15b_{3,i}^2=1\\ 2b_{2,i}(b_{1,i}^2+24b_{1,i}{b}_{3,i}+105b_{3,i}^2+2)=\mathrm{\&gamma;}\\ 24[(b_{1,i}{b}_{3,i}+b_{2,i}^2(1+b_{1,i}^2+28b_{1,i}{b}_{3,i})+b_{3,i}^2\&times;\\ (12+48b_{1,i}{b}_{3,i}+141b_{2,i}^2+225b_{3,i}^2)]=\mathrm{\&kappa;}-3\\ b_{0,i}+b_{2,i}=0\end{array}\right.$

Solve following formula and obtain equivalent correlation coefficient ρ _1ij,

$\begin{array}{c}6a_{3,i}{a}_{3,j}{\mathrm{\&rho;}}_{1\mathrm{ij}}^3+2a_{2,i}{a}_{2,j}{\mathrm{\&rho;}}_{1\mathrm{ij}}^2+(a_{1,i}+3a_{3,i})\&times;(a_{1,j}+3a_{3,j}){\mathrm{\&rho;}}_{1\mathrm{ij}}+[(a_{0,i}+a_{2,i})\\ \&times;(a_{0,j}+a_{2,j})-{\mathrm{\&rho;}}_{\mathrm{ij}}{\mathrm{\&sigma;}}_{\mathrm{vi}}{\mathrm{\&sigma;}}_{\mathrm{vj}}-{\mathrm{\&mu;}}_{\mathrm{vi}}{\mathrm{\&mu;}}_{\mathrm{vj}}]=0\end{array}$

ρ _1ijmeet-1≤ρ _1ij≤ 1, ρ _1ijρ _ij>=0.