CN105846437A - Interaction correlation-based choleskey decomposition half invariant flow calculating method - Google Patents

Abstract

The invention provides an interaction correlation-based choleskey decomposition half invariant flow calculating method which comprises the following steps: basic data, node injection power random distribution parameters and response participation factors are input; interaction response node injection power probability distribution and a response correlation coefficient matrix Cres are calculated; Newton-Raphson method flow calculation is conducted at new reference operating points, output variable node voltage X0 and a branch flow Z0 and sensitivity matrixes S0 and T0 are obtained; a choleskey decomposition method is adopted, and node injection power random variables in response correlation are converted into mutually independent random variables; all order half invariants of flow output variables are obtained with response correlation taken into consideration; probability distribution of the output variables is calculated via use of Gram-Charlier series. The method provided in the invention is suitable for analyzing actual operation condition of a power grid after a flexible load used as a high-quality responsive resource participates in system scheduling, and capacity of the power grid in admitting new energy can be further improved.

Description

The cholesky of a kind of meter and interactive dependency decomposes cumulant tidal current computing method

Technical field

The present invention relates to power system computation field, the cholesky being specifically related to a kind of meter and interactive dependency decomposes half Invariant tidal current computing method.

Background technology

The fast development of intermittent energy proposes new significant challenge to Power Systems balanced capacity, and flexible load is adjusted Spend and become the focus of concern both at home and abroad as supplementing of conventional electric power generation scheduling.The uncertainty of flexible load self response makes The uncertain factor of lotus both sides, source strengthens, and how utilizing probabilistic load flow to analyze the problems referred to above is a difficult problem.Probabilistic loadflow Being one of important method analyzing electrical network uncertain factor, how processing the dependency between stochastic variable is to affect Load flow calculation knot The key factor of fruit.

If one or more stochastic variables are the reasons causing its dependent variable to produce randomness, then stochastic variable and sound Response dependency should be there is between measuring.Such as the randomness in order to tackle wind power, ring usually through scheduling unit and flexible load Answer the randomness of wind-powered electricity generation and realize the equilibrium,transient of supply and demand, owing to wind power has certain randomness so that normal power supplies or The scheduling quantum of flexible load also has certain uncertainty, is defined as the interactive response dependency of stochastic variable here.

Current power flow algorithm cannot be simulated and calculated load participates in the randomness of scheduling and solves interactive response dependency, Under strong uncertain environment, electric network swim analysis and computing capability are more weak, promote the electrical network receiving ability aspect effect to new forms of energy The best.

Summary of the invention

For overcoming above-mentioned the deficiencies in the prior art, the present invention provides the cholesky of a kind of meter and interactive dependency to decompose half Invariant tidal current computing method, on the basis of analyzing stochastic variable interactive response dependency, in conjunction with responsive node and random note Enter correlation matrix computational methods between source node, it is proposed that the cumulant Probabilistic Load Flow modeling side decomposed based on cholesky Method and calculation process.

Realizing the solution that above-mentioned purpose used is:

The cholesky of a kind of meter and interactive dependency decomposes cumulant tidal current computing method, described computational methods bag Include:

(1) input basic data, node injecting power random distribution parameter and response participation factors；

(2) probability distribution and the response correlation matrix C of interactive response node injecting power are calculated_res；

(3) carry out Newton-Laphson method Load flow calculation at new benchmark operating point, obtain output variable node voltage X₀With Road power Z₀And sensitivity matrix S₀And T₀；

(4) utilize cholesky decomposition method, the node injecting power stochastic variable with response dependency is converted to phase The most independent stochastic variable；

(5) ask for meter and each rank cumulant of response dependency trend output variable, utilize Gram-Charlier progression Calculate the probability distribution of output variable.

Preferably, in described step (1), branch parameters, generating needed for described basic data comprising determining that property Load flow calculation Machine injecting power and load injecting power.

Preferably, in described step (1), described response participation factors such as following formula:

$K_j=\left\{\begin{array}{c}0,j\∉{\mathrm{\Ω}}_R\\ \frac{{\mathrm{\ΔP}}_{jmax}}{{\mathrm{\Σ\ΔP}}_{jmax}},j\∈{\mathrm{\Ω}}_R\end{array}\right.$

In formula: Δ P_jmaxAdj sp for node；Ω_RFor participating in the node set of interactive response, if certain node is connected to many When platform unit or multiple flexible load, taking it and for the equivalent participation factors of this node, all node participation factors sums are 1, j For jth node.

Preferably, in described step (2), described calculating includes:

(2-1) meter and stochastic source inject node new forms of energy randomness and load prediction randomness, obtain system imbalance merit Rate；

(2-2) obtained the sample of each random node injecting power by Monte Carlo sampling method, use correlation coefficient ρ_ijRetouch State the response randomness Δ P ' of interactive response node j_resjWith the randomness Δ P that stochastic source injects node i_iLinear correlation degree；

(2-3) obtain n stochastic source and inject node i, the correlation matrix C of m interactive response node j randomness_res, Matrix is that (m+n) ties up symmetrical matrix.

Further, in described step (2-1), described system imbalance power such as following formula:

P_unb=P_unb0+ΔP_unb

${\mathrm{\ΔP}}_{unb}=\underset{i\∈{\mathrm{\Ω}}_I}{\Σ}{\mathrm{\ΔP}}_i=\underset{i\∈{\mathrm{\Ω}}_I}{\Σ}({\mathrm{\ΔP}}_{wi}+{\mathrm{\ΔP}}_{li})$

In formula: P_unbFor system imbalance power；P_unb0For the definitiveness part in imbalance power；ΔP_unbFor randomness Part；Ω_IRepresent that stochastic source injects the set of node；ΔP_iFor the randomness of node i injecting power, this node connected new energy Source randomness Δ P_wiWith load randomness Δ P_liJointly cause.

Further, in described step (2-2), described correlation coefficient ρ_ijSuch as following formula:

${\mathrm{\ρ}}_{ij}=\frac{\mathrm{cov}({\mathrm{\ΔP}}_i,{\mathrm{\ΔP}}_{resj}^{\′})}{\sqrt{D\left({\mathrm{\ΔP}}_{resj}^{\′}\right)\·D\left({\mathrm{\ΔP}}_i\right)}}$

Wherein, cov () is covariance, and D () is variance.

Preferably, in described step (3), system load flow equation matrix form is as follows:

$\left\{\begin{array}{c}X=X_0+\mathrm{\Δ}x=X_0+S_0\mathrm{\Δ}w\\ Z=Z_0+\mathrm{\Δ}z=Z_0+T_0\mathrm{\Δ}w\end{array}\right.$

Wherein, X, Z represent that node voltage and branch power, subscript 0 represent benchmark running status respectively；Δ x, Δ z are respectively Represent node voltage and the change at random amount of branch power；Δ w represents the change at random amount of injecting power；S₀With T₀Represent respectively The sensitivity that injecting power is changed by node voltage and branch power.

Preferably, in described step (4), described decomposition such as following formula:

C_res=GG^T

In formula: G is lower triangular matrix, G^TTransposition for G.

Preferably, in described step (5), described each rank cumulant such as following formula:

$\left\{\begin{array}{c}\mathrm{\Δ}{x}^{\left(k\right)}=S_0^{\′\left(k\right)}\mathrm{\Δ}{w}^{\′\left(k\right)}+{\left(S_0^{\′\′}AG\right)}^{\left(k\right)}\mathrm{\Δ}{Y}^{\left(k\right)}\\ {\mathrm{\Δz}}^{\left(k\right)}=T_0^{\′\left(k\right)}{\mathrm{\Δw}}^{\′\left(k\right)}+{\left(T_0^{\′\′}AG\right)}^{\left(k\right)}{\mathrm{\ΔY}}^{\left(k\right)}\end{array}\right.$

In formula: (*)^(k)Represent k rank cumulant；After considering response dependency, Δ w carrying out piecemeal, Δ w ' is the most solely Vertical input variable, Δ w is " for having the input variable of response dependency；G is correlation matrix C_resDivided by cholesky Lower triangular matrix after solution；A is diagonal matrix, and diagonal element is the standard deviation of node injecting power relevant variable；Δ Y be standard just State be distributed, its single order cumulant is 0, and second order cumulant is 1, three rank and above be 0；S′₀Represent separate node The sensitivity matrix block that injecting power is changed by voltage, S "₀Represent that the node voltage with response dependency is to injecting power Sensitivity matrix block, T '₀Represent the separate node branch power sensitivity matrix block to injecting power, T "₀Expression has The node branch power of the response dependency sensitivity matrix block to injecting power.

Compared with prior art, the method have the advantages that

The present invention devises the cholesky of a kind of meter and interactive dependency and decomposes cumulant tidal current computing method, the party Method can be counted and the uncertainty of flexible load interactive response behavior, and simulation calculated load participate in the randomness of scheduling, and utilize Cholesky decomposition method solves interactive response dependency.The present invention be conducive to improving under strong uncertain environment electric network swim analysis and Computing capability, is particularly suited for flexible load and can participate in the actual fortune of system call post analysis electrical network by resource response as a kind of high-quality Market condition, promotes the electrical network receiving ability to new forms of energy further.

Accompanying drawing explanation

Fig. 1 is meter and the cholesky decomposition cumulant tidal current computing method flow chart of interactive dependency of the present invention.

Fig. 2 is interactive response dependency diagram of the present invention.

Detailed description of the invention

Below in conjunction with the accompanying drawings the detailed description of the invention of the present invention is described in further detail.

1, input basis flow data, probability distribution parameters and response regulation coefficient.Basic data includes definitiveness trend Calculate required branch parameters, electromotor and load injecting power etc., additionally need input node injecting power random distribution parameter, Response participation factors etc..

When in system, some bus nodes accesses certain adjustable unit or flexible load, its scheduling quantum can be as balance The interactive response amount of system imbalance power, these bus nodes are referred to as interactive response node.Can be according to the connected unit of node Or the regulations speed (or variable capacity etc.) of flexible load determines its participation factors, with unit (or flexible load) creep speed As a example by being directly proportional, the response participation factors of node j is represented by:

$K_j=\left\{\begin{array}{c}0,j\∉{\mathrm{\Ω}}_R\\ \frac{{\mathrm{\ΔP}}_{jmax}}{{\mathrm{\Σ\ΔP}}_{jmax}},j\∈{\mathrm{\Ω}}_R\end{array}\right.---\left(1\right)$

In formula: Δ P_jmaxAdj sp for node；Ω_RFor participating in the node set of interactive response, if certain node is connected to many During platform unit (or multiple flexible load), taking it and for the equivalent participation factors of this node, all node participation factors sums are 1。

2, according to the imbalance power of system, use and calculate interactive response node injecting power based on monte carlo method Probability distribution and response correlation matrix C_res。

For certain electrical network, after counting and injecting node new forms of energy randomness and load prediction randomness, system imbalance power Also it is a stochastic variable, is represented by:

P_unb=P_unb0+ΔP_unb (2)

${\mathrm{\ΔP}}_{unb}=\underset{i\∈{\mathrm{\Ω}}_I}{\Σ}{\mathrm{\ΔP}}_i=\underset{i\∈{\mathrm{\Ω}}_I}{\Σ}({\mathrm{\ΔP}}_{wi}+{\mathrm{\ΔP}}_{li})---\left(3\right)$

In formula: P_unbFor system imbalance power；P_unb0For the definitiveness part in imbalance power；ΔP_unbFor randomness Part；Ω_IRepresent that stochastic source injects the set of node；ΔP_iRandomness for node i injecting power；New energy is connected by this node Source randomness Δ P_wiWith load randomness Δ P_liJointly cause.

Then system imbalance power expectation part and randomness part Δ P_unbAll according to participation factors K on node j_jEnter Row distribution, can be obtained the random partial Δ P of node j interactive response amount by formula (3)_resjIt is represented by:

${\mathrm{\ΔP}}_{resj}=K_j{\mathrm{\ΔP}}_{unb}=K_j\underset{j\∈{\mathrm{\Ω}}_I}{\Σ}{\mathrm{\ΔP}}_i---\left(4\right)$

Assume interactive node response quautity also Normal Distribution, the then random distribution of system imbalance power and interactive node The random distribution relation of response quautity is as shown in schematic diagram 2.It is to say, when system imbalance power is Δ P₁Time, interactive node j Response quautity expected value be K_jΔP₁, owing to considering that response self randomness then response quautity is obeyedNormal distribution, Consider that interactive response self randomness posterior nodal point interactive response amount is denoted as Δ P '_resj。

Additionally, according to formula (4) it is seen that, the response randomness Δ P ' of interactive node j_resjNode is injected with stochastic source The randomness Δ P of i_iThere is obvious dependency relation.The sample of each random node injecting power is obtained by Monte Carlo sampling method This, use correlation coefficient ρ on this basis_ijLinear correlation degree between the two described:

${\mathrm{\ρ}}_{ij}=\frac{\mathrm{cov}({\mathrm{\ΔP}}_i,{\mathrm{\ΔP}}_{resj}^{\′})}{\sqrt{D\left({\mathrm{\ΔP}}_{resj}^{\′}\right)\·D\left({\mathrm{\ΔP}}_i\right)}}---\left(5\right)$

In formula: cov () is covariance, D () is variance.

Thus, available stochastic source injects node i (n altogether), the phase relation of interactive response node j (m altogether) randomness Matrix number C_res, matrix is (m+n) dimension symmetrical matrix:

3, carry out Newton-Laphson method Load flow calculation at new benchmark operating point, obtain output variable node voltage X and branch road Trend Z and sensitivity matrix S₀And T₀.When the expected value of system imbalance power is shared by responsive node according to participation factors After obtain the benchmark operating point that system is new, then system load flow equation matrix form is as follows:

$\left\{\begin{array}{c}X=X_0+\mathrm{\Δ}x=X_0+S_0\mathrm{\Δ}w\\ Z=Z_0+\mathrm{\Δ}z=Z_0+T_0\mathrm{\Δ}w\end{array}\right.---\left(7\right)$

Wherein, X, Z represent that node voltage and branch power, subscript 0 represent benchmark running status respectively；Δ x, Δ z are respectively Represent node voltage and the change at random amount of branch power；Δ w represents the change at random amount of injecting power；S₀With T₀Represent respectively The sensitivity that injecting power is changed by node voltage and branch power.

4, utilize Cholesky decomposition method, the node injecting power stochastic variable with response dependency is converted to mutually Independent stochastic variable.

Correlation matrix C_resGenerally positive definite matrix, then can carry out cholesky decomposition to this matrix:

C_res=GG^T (8)

In formula: G is lower triangular matrix, its element is represented by:

$\left\{\begin{array}{c}{g}_{kk}={({\mathrm{\ρ}}_{kk}-\underset{t=1}{\overset{k-1}{\Σ}}{g}_{kt}^2)}^2,k=1,2,\mathrm{...}m+n\\ g_{lk}=\frac{{\mathrm{\ρ}}_{lk}-\underset{t=1}{\overset{k-1}{\Σ}}{g}_{lt}{g}_{kt}}{g_{kk}},l=k+1,k+2,\mathrm{...}m+n\end{array}\right.---\left(9\right)$

In formula: ρ_kkAnd ρ_lkIt is respectively correlation matrix C_resIn correlation coefficient；K is to have relevant stochastic variable Number, is m+n herein.If random node and the injecting power of interactive response nodeClothes From normal distribution, orderX is the stochastic variable of one group of obedience standard normal distribution, and its correlation matrix is still C_res；A is diagonal matrix, and diagonal element is the standard deviation of node injecting power relevant variable；μ is its expectation.

Correlation matrix C is understood by formula (9)_resFor symmetrical matrix, then there is an orthogonal matrix B, can will have dependency Stochastic variable X be converted into the stochastic variable Y of incoherent obedience standard normal distribution:

Y=BX (10)

In formula: Y=[y₁,y₂,…,y_n+m]^TIt is one group of separate stochastic variable obeying standard normal distribution, then The correlation matrix C of Y_YFor unit matrix I, thus can obtain:

$\begin{array}{c}{C}_Y=\mathrm{\ρ}(Y,Y^T)=\mathrm{\ρ}(BX,X^T{B}^T)=B\mathrm{\ρ}(X,X^T)B^T\\ ={\mathrm{BC}}_{\mathrm{Re}s}{B}^T={\mathrm{BGG}}^T{B}^T=\left(BG\right){\left(BG\right)}^T=I\end{array}---\left(11\right)$

Take B=G^-1, i.e. Y=G^-1X, can have one group of stochastic variable of dependencyIt is expressed as incoherent obedience to mark The expression formula of quasi normal distribution stochastic variable Y:

$\mathrm{\Δ}{\tilde{P}}^{\′}=AGY+\mathrm{\μ}---\left(12\right)$

5, ask for meter and each rank cumulant of response dependency trend output variable, utilize Gram-Charlier progression Calculate the probability distribution of output variable.After system node injecting power random partial after meter and response, calculate system load flow defeated The each rank cumulant going out variable is represented by:

$\left\{\begin{array}{c}{\mathrm{\Δx}}^{\left(k\right)}=S_0^{\left(k\right)}{\mathrm{\Δw}}^{\left(k\right)}\\ {\mathrm{\Δz}}^{\left(k\right)}=T_0^{\left(k\right)}{\mathrm{\Δw}}^{\left(k\right)}\end{array}\right.---\left(13\right)$

After considering response dependency, Δ w is carried out piecemeal, it may be assumed that

$\mathrm{\Δ}w=\left[\begin{array}{c}\mathrm{\Δ}{w}^{\′}\\ \mathrm{\Δ}{w}^{\′\′}\end{array}\right]---\left(14\right)$

" for there is the input variable of response dependency, on the other hand in formula: Δ w ' is separate input variable, Δ w The S answered₀, T₀Also piecemeal is:

$\left\{\begin{array}{c}{S}_0=\[S_0^{\′},S_0^{\′\′}\]\\ T_0=\[T_0^{\′},T_0^{\′\′}\]\end{array}\right.---\left(15\right)$

By Cholesky decomposition method, the input variable with response dependency is converted into separate standard normal Distribution, hereinI.e. Δ w "=AG Δ Y, shown under the form that formula (13) is final:

$\left\{\begin{array}{c}\mathrm{\Δ}{x}^{\left(k\right)}=S_0^{\′\left(k\right)}\mathrm{\Δ}{w}^{\′\left(k\right)}+{\left(S_0^{\′\′}AG\right)}^{\left(k\right)}\mathrm{\Δ}{Y}^{\left(k\right)}\\ {\mathrm{\Δz}}^{\left(k\right)}=T_0^{\′\left(k\right)}{\mathrm{\Δw}}^{\′\left(k\right)}+{\left(T_0^{\′\′}AG\right)}^{\left(k\right)}{\mathrm{\ΔY}}^{\left(k\right)}\end{array}\right.---\left(16\right)$

In formula: (*)^(k)Represent k rank cumulant；After considering response dependency, Δ w carrying out piecemeal, Δ w ' is the most solely Vertical input variable, Δ w is " for having the input variable of response dependency；G is correlation matrix C_resDivided by cholesky Lower triangular matrix after solution；A is diagonal matrix, and diagonal element is the standard deviation of node injecting power relevant variable；Δ Y be standard just State be distributed, its single order cumulant is 0, and second order cumulant is 1, three rank and above be 0；S′₀Represent separate node The sensitivity matrix block that injecting power is changed by voltage, S "₀Represent that the node voltage with response dependency is to injecting power Sensitivity matrix block, T '₀Represent the separate node branch power sensitivity matrix block to injecting power, T "₀Expression has The node branch power of the response dependency sensitivity matrix block to injecting power.

On this basis, Gram-Charlier progression is utilized to calculate output variable probability distribution.

Finally should be noted that: above example is merely to illustrate the technical scheme of the application rather than to its protection domain Restriction, although being described in detail the application with reference to above-described embodiment, those of ordinary skill in the field should Understand: those skilled in the art read the application after still can to application detailed description of the invention carry out all changes, amendment or Person's equivalent, but these changes, amendment or equivalent, all within the claims that application is awaited the reply.

Claims (9)

1. the cholesky of a meter and interactive dependency decomposes cumulant tidal current computing method, it is characterised in that described meter Calculation method includes:

(1) input basic data, node injecting power random distribution parameter and response participation factors；

(2) probability distribution and the response correlation matrix C of interactive response node injecting power are calculated_res；

(3) carry out Newton-Laphson method Load flow calculation at new benchmark operating point, obtain output variable node voltage X₀And branch power Z₀And sensitivity matrix S₀And T₀；

(4) utilize cholesky decomposition method, the node injecting power stochastic variable with response dependency is converted to the most solely Vertical stochastic variable；

(5) ask for meter and each rank cumulant of response dependency trend output variable, utilize Gram-Charlier progression to calculate The probability distribution of output variable.

2. computational methods as claimed in claim 1, it is characterised in that in described step (1), described basic data includes: really Branch parameters, electromotor injecting power and load injecting power needed for qualitative Load flow calculation.

3. computational methods as claimed in claim 1, it is characterised in that in described step (1), described response participation factors K_jAs Following formula:

$K_j=\left\{\begin{array}{c}0,j\∉{\mathrm{\Ω}}_R\\ \frac{{\mathrm{\ΔP}}_{jmax}}{{\mathrm{\Σ\ΔP}}_{jmax}},j\∈{\mathrm{\Ω}}_R\end{array}\right.$

In formula: Δ P_jmaxAdj sp for node；Ω_RFor participating in the node set of interactive response, if certain node is connected to multiple stage machine When group or multiple flexible load, taking it and for the equivalent participation factors of this node, all node participation factors sums are 1, and j is the J node.

4. computational methods as claimed in claim 1, it is characterised in that in described step (2), described calculating includes:

(2-1) meter and stochastic source inject node new forms of energy randomness and load prediction randomness, obtain system imbalance power；

(2-2) obtained the sample of each random node injecting power by Monte Carlo sampling method, use correlation coefficient ρ_ijDescribe mutually The response randomness Δ P ' of dynamic response node j_resjWith the randomness Δ P that stochastic source injects node i_iLinear correlation degree；

(2-3) obtain n stochastic source and inject node i, the correlation matrix C of m interactive response node j randomness_res, matrix Symmetrical matrix is tieed up for (m+n).

5. computational methods as claimed in claim 4, it is characterised in that in described step (2-1), described system imbalance power P_unbSuch as following formula:

P_unb=P_unb0+ΔP_unb

${\mathrm{\ΔP}}_{unb}=\underset{i\∈{\mathrm{\Ω}}_I}{\Σ}{\mathrm{\ΔP}}_i=\underset{i\∈{\mathrm{\Ω}}_I}{\Σ}({\mathrm{\ΔP}}_{wi}+{\mathrm{\ΔP}}_{li})$

In formula: P_unb0For the definitiveness part in imbalance power；ΔP_unbFor randomness part；Ω_IRepresent that stochastic source injects joint The set of point；ΔP_iFor the randomness of node i injecting power, by this node connected new forms of energy randomness Δ P_wiWith load randomness ΔP_liJointly cause.

6. computational methods as claimed in claim 4, it is characterised in that in described step (2-2), described correlation coefficient ρ_ijAs follows Formula:

${\mathrm{\ρ}}_{ij}=\frac{\mathrm{cov}({\mathrm{\ΔP}}_i,{\mathrm{\ΔP}}_{resj}^{\′})}{\sqrt{D\left({\mathrm{\ΔP}}_{resj}^{\′}\right)\·D\left({\mathrm{\ΔP}}_i\right)}}$

Wherein, cov () is covariance, and D () is variance.

7. computational methods as claimed in claim 1, it is characterised in that in described step (3), system load flow equation matrix form As follows:

$\left\{\begin{array}{c}X=X_0+\mathrm{\Δ}x=X_0+S_0\mathrm{\Δ}w\\ Z=Z_0+\mathrm{\Δ}z=Z_0+T_0\mathrm{\Δ}w\end{array}\right.$

Wherein, X, Z represent that node voltage and branch power, subscript 0 represent benchmark running status respectively；Δ x, Δ z represent respectively The change at random amount of node voltage and branch power；Δ w represents the change at random amount of injecting power；S₀With T₀Represent node respectively The sensitivity that injecting power is changed by voltage and branch power.

8. computational methods as claimed in claim 1, it is characterised in that in described step (4), described decomposition such as following formula:

C_res=GG^T

In formula: G is lower triangular matrix, G^TTransposition for G.

9. computational methods as claimed in claim 1, it is characterised in that in described step (5), described each rank cumulant is as follows Formula:

$\left\{\begin{array}{c}{\mathrm{\Δx}}^{\left(k\right)}=S_0^{\′\left(k\right)}{\mathrm{\Δw}}^{\′\left(k\right)}+{\left(S_0^{\′\′}AG\right)}^{\left(k\right)}{\mathrm{\ΔY}}^{\left(k\right)}\\ {\mathrm{\Δz}}^{\left(k\right)}=T_0^{\′\left(k\right)}{\mathrm{\Δw}}^{\′\left(k\right)}+{\left(T_0^{\′\′}AG\right)}^{\left(k\right)}{\mathrm{\ΔY}}^{\left(k\right)}\end{array}\right.$

In formula: (*)^(k)Represent k rank cumulant；After considering response dependency, Δ w carrying out piecemeal, Δ w ' is for separate Input variable, Δ w is " for having the input variable of response dependency；G is correlation matrix C_resAfter being decomposed by cholesky Lower triangular matrix；A is diagonal matrix, and diagonal element is the standard deviation of node injecting power relevant variable；Δ Y is that standard normal is divided Cloth, its single order cumulant is 0, and second order cumulant is 1, three rank and above be 0；S′₀Represent separate node voltage Sensitivity matrix block to injecting power change, S "₀Represent sensitive to injecting power of node voltage with response dependency Degree matrix-block, T '₀Represent the separate node branch power sensitivity matrix block to injecting power, T "₀Represent that there is response The node branch power of the dependency sensitivity matrix block to injecting power.