CN112417768B - Wind power correlation condition sampling method based on vine structure Pair-Copula - Google Patents

Wind power correlation condition sampling method based on vine structure Pair-Copula Download PDF

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CN112417768B
CN112417768B CN202011437994.5A CN202011437994A CN112417768B CN 112417768 B CN112417768 B CN 112417768B CN 202011437994 A CN202011437994 A CN 202011437994A CN 112417768 B CN112417768 B CN 112417768B
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朱想
李新宸
师浩琪
郭力
李霞林
刘一欣
王中冠
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Abstract

The invention discloses a wind power correlation condition sampling method based on a rattan structure Pair-Copula, which comprises the following steps of 1, obtaining a true value of wind power output and edge probability distribution of a predicted value; step 2, constructing an optimal Pair-Copula function of the real value and the predicted value of the wind power output; and 3, sampling the vine structure Pair-Copula under the condition of giving a predicted value of the wind power output to obtain a sample considering time correlation and condition correlation. Compared with the prior art, the method simultaneously considers the time correlation of the wind power output prediction error and the condition correlation of the wind power output prediction error and the predicted value, and more accurately describes the distribution characteristics of the wind power output prediction error under different predicted values; the obtained sampling points can reasonably describe the prediction error distribution condition under the condition of a given predicted value, and can be well applied to the random optimization or robust optimization of a power grid.

Description

Wind power correlation condition sampling method based on vine structure Pair-Copula
Technical Field
The invention belongs to the technical field of correlation modeling and power grids, and particularly relates to a wind power correlation condition sampling method based on a vine structure Pair-Copula.
Background
With the exhaustion of non-renewable fossil fuels and the continuous increase of environmental pollution, renewable energy sources such as photovoltaic and wind power play an increasingly important role, and the proportion of the renewable energy sources in the power grid output is continuously increased.
However, renewable energy is affected by various factors, with uncertainty, which puts new demands on the stable, economical operation of the grid. Therefore, in a power grid containing renewable energy, uncertainty of the renewable energy is often considered, so that a random or robust optimization modeling method is adopted. However, random optimization often requires a large amount of sample data to obtain an accurate probability distribution, and robust optimization also requires constructing a reasonable uncertainty set as much as possible to balance the degree of conservation and economy.
Researches find that strong time correlation exists in the prediction error of renewable energy power generation, which has great influence on the optimization result of the power grid, and the size of the prediction error is in conditional correlation with the predicted value. The current correlation modeling method has the following problems:
1) the method for determining the uncertain set or performing random optimization by using the historical data of the prediction error at each moment can consider the time correlation, but does not consider the condition correlation with the predicted value, and can cause over-high or over-low estimation on the uncertainty when the predicted value is very high or very low;
2) the condition correlation can be embodied by a prediction error modeling method for the partition of the predicted value, but the number of partitions influences the quality of the model and is difficult to expand to multiple dimensions;
3) the Pair-Copula model based on the rattan structure can well model the time correlation, has strong adaptability to the statistical characteristics of data, but cannot model the condition correlation.
Disclosure of Invention
The invention aims to solve the problem of simultaneous consideration of time correlation of renewable energy power generation prediction errors represented by wind power and condition correlation of the renewable energy power generation prediction errors and predicted values, and provides a wind power correlation condition sampling method based on a rattan structure Pair-Copula.
The technical solution of the invention is as follows:
a wind power correlation condition sampling method based on a vine structure Pair-Copula specifically comprises the following steps:
1. a wind power correlation condition sampling method based on a vine structure Pair-Copula is characterized by comprising the following steps:
step 1, obtaining edge probability distribution of a real value and a predicted value of wind power output, namely obtaining the edge probability distribution of the real value and the predicted value of the wind power output by adopting a parameter estimation method or a non-parameter estimation method according to the distribution characteristics of wind power historical data;
step 2, constructing an optimal Pair-Copula function of a true value and a predicted value of wind power output, determining the optimal Pair-Copula function type and parameters of multidimensional variables in pairs layer by layer from the layer 1 of the rattan structure Pair-Copula, namely selecting one of the fitted Copula functions with the minimum Euclidean distance to the empirical Copula function as the optimal Pair-Copula function, and fitting to obtain a Copula function Cp(u, v) and empirical Copula function CnThe Euclidean distance between (u, v) is defined as:
Figure BDA0002821176410000021
the specific expression of the empirical Copula function is as follows:
Figure BDA0002821176410000031
in the formula (x)i,yi) Samples taken from variables (X, Y) that construct the Copula function, F (X)i) And G (y)i) Are each xiAnd yiThe distribution function of (a) is determined,
Figure BDA0002821176410000032
and
Figure BDA0002821176410000033
n is the number of samples for an indicative function;
step 3, sampling the vine structure Pair-Copula under the condition of giving a predicted value of wind power output;
giving a wind power output predicted value ytTo find out the probability distribution function ut＝Fy,t(yt) T is 1,2, …, the value of N and known u2,…,uNUnder the condition u1Conditional probability distribution F (u)1|u2,…,uN) (ii) a According to the formula of the vine structure Pair-Copula function, obtaining:
Figure BDA0002821176410000034
Figure BDA0002821176410000035
k＝N+1,N+2,…,2N；m＝1,2,…,2N-2；m+1≤k-1
ut＝Fy,t(yt) For wind power output prediction value ytDistribution function of ut+N＝Fx,t(xt) For the true value x of wind power outputtT 1,2, …, N; f (u)m|um+1,…,uk-1) And F (u)k|um+1,…,uk-1) Are each um+1,…,uk-1Under the condition umAnd ukConditional probability distribution of (1), Cm,k|m+1,…,k-1As a function of Copula, F (u)k|um,…,uk-1) Is um,…,uk-1Under the condition ukA conditional probability distribution of (a);
finally, according to the probability distribution function relation uk＝Fy,k(xk-N) Real value sample x for calculating wind power output in k-N time periodk-NThe expression is as follows:
Figure BDA0002821176410000036
Figure BDA0002821176410000037
actual value probability distribution function value u for representing wind power output in k-N time periodkThe inverse distribution function of (c).
Compared with the prior art, the invention has the following advantages:
(1) based on data driving, the accuracy of the correlation model is improved by using the real value and the historical data of the predicted value of the wind power output;
(2) meanwhile, the time correlation of the wind power output prediction error and the condition correlation of the wind power output prediction error and the predicted value are considered, and the distribution characteristics of the wind power output prediction error under different predicted values are more accurately described;
(3) the obtained sampling points can reasonably describe the prediction error distribution condition under the condition of a given predicted value, and can be well applied to the random optimization or robust optimization of a power grid.
Drawings
FIG. 1 is an overall flow chart of a wind power correlation condition sampling method based on a vine structure Pair-Copula;
FIG. 2 is a logic diagram of a D vine structure Pair-Copula;
FIG. 3 is a scatter plot of wind power prediction error versus predicted value for historical data in accordance with an embodiment of the present invention;
FIG. 4 is a daily power curve for a lower predicted power selected by an embodiment of the present invention;
FIG. 5 is a daily power curve for a higher predicted power selected by an embodiment of the present invention;
FIG. 6 is a graph of the temporal correlation of historical data for an embodiment of the present invention;
FIG. 7 is a graph of the time dependence of sample points for low predicted power days for an embodiment of the present invention;
FIG. 8 is a graph of the time dependence of sample points for high predicted power days for an embodiment of the present invention;
FIG. 9 is a graph of a lower power empirical distribution of historical data for an embodiment of the present invention;
FIG. 10 is a graph of an empirical distribution of higher power for historical data for an embodiment of the present invention;
FIG. 11 is an empirical distribution of sample points for low predicted power days of an embodiment of the present invention;
FIG. 12 is an empirical distribution of sample points for high predicted power days of an embodiment of the present invention.
Detailed Description
The technical solution of the present invention is further explained with reference to the drawings and the embodiments.
As shown in fig. 1, the overall flow chart of the wind power correlation condition sampling method based on the rattan structure Pair-Copula of the present invention includes the following specific steps:
step 1, obtaining edge probability distribution of a real value and a predicted value of wind power output, namely obtaining the edge probability distribution of the real value and the predicted value of the wind power output by adopting a parameter estimation method or a non-parameter estimation method according to the distribution characteristics of wind power historical data; counting the total scheduling time period as N, xtAnd ytT is 1,2, …, and N is the real value and the predicted value of the wind power output in the period of t respectively; then can be composed of xtAnd ytRespectively estimating to obtain a distribution function F of a real value and a predicted value of wind power outputx,t(xt) And Fy,t(yt)；
Step 2, constructing an optimal Pair-Copula function of the real value and the predicted value of the wind power output:
let ui＝Fi(xi)，Fi(xi) Is a variable xiAccording to Copula probability distribution function C (-), dividing x into1,x2,…,xnIs expressed as F (x)1,x2,…,xn)＝C(u1,u2,…,un)；
The relationship between the Copula probability density function C (-) and the Copula probability distribution function C (-) is as follows:
Figure BDA0002821176410000051
taking the example of D rattan Pair-Copula, the logic diagram of the D rattan structure Pair-Copula is shown in FIG. 2.
Let ut＝Fy,t(yt)，uN+t＝Fx,t(xt)，t＝1,2,…,N；
Layer 1 ci,i+1Is ci,i+1(ui,ui+1) I-1, 2, …, abbreviation for n-1;
in the j-th layer:
ci,i+j|i+1,i+2,…,i+j-1is ci,i+j|i+1,i+2,…,i+j-1(F(ui|ui+1,ui+2,…,ui+j-1)，F(ui+j|ui+1,ui+2,…,ui+j-1) J ═ 2,3, …, n-1, i ═ 1,2, …, abbreviations for n-j. According to the property of the Copula function C and the property of the conditional distribution F (. |), the following vine structure Pair-Copula relation can be obtained, wherein
Figure BDA0002821176410000052
Represents the partial derivative:
Figure BDA0002821176410000053
Figure BDA0002821176410000054
Figure BDA0002821176410000055
Figure BDA0002821176410000061
x of final configuration1,…,xnIs combined with the probability density function f (x)1,…,xn) Comprises the following steps:
Figure BDA0002821176410000062
wherein, f (x)k) Is xkIs determined.
And determining the optimal Pair-Copula function type and parameters between every two multidimensional variables layer by layer from the layer 1 of the vine structure Pair-Copula. Commonly used Copula functions are of the elliptic Copula and archimedes Copula classes, including Gaussian Copula, t-Copula and Gumbel Copula, Clayton Copula, frank Copula, respectively. And respectively fitting by using each Copula function, and obtaining corresponding parameters by adopting maximum likelihood estimation. And selecting the one with the minimum Euclidean distance between the fitted Copula functions and the empirical Copula function as the optimal Pair-Copula function. Wherein the empirical Copula function is:
Figure BDA0002821176410000063
in the formula (x)i,yi) For samples taken from variables (X, Y) that construct the Copula function, u, v ∈ [0,1 ]]，F(xi) And G (y)i) Are each xiAnd yiThe distribution function of (a) is determined,
Figure BDA0002821176410000064
and
Figure BDA0002821176410000065
for demonstration letterNumber (subscript condition F (x)i) U or G (y)i) When v is satisfied, I is equal to 1, otherwise, I is equal to 0), and n is the number of samples.
Copula function C obtained by fittingp(u, v) and empirical Copula function CnThe Euclidean distance between (u, v) is defined as:
Figure BDA0002821176410000066
step 3, under the condition of giving the wind power output predicted value, carrying out conditional sampling on Pair-Copula of the rattan structure to give the wind power output predicted value ytTo find out the probability distribution function ut＝Fy,t(yt) And t is 1,2, …, the value of N. According to the formula of the vine structure Pair-Copula function, the following results are obtained:
Figure BDA0002821176410000067
substitution of ut＝Fy,t(yt) And t is 1,2, …, N, and the known u is obtained2,…,uN-1Under the condition u1And uNConditional probability distribution F (u)1|u2,…,uN-1) And F (u)N|u2,…,uN-1) And Copula function C composed of them1,N|2,3,…,N-1And known u2,…,uNUnder the condition u1Conditional probability distribution F (u)1|u2,…,uN) A value of (d);
according to the formula of the vine structure Pair-Copula function, obtaining:
Figure BDA0002821176410000071
k＝N+1,N+2,…,2N；m＝1,2,…,2N-2；m+1≤k-1
ut＝Fy,t(yt) For wind power output prediction value ytDistribution function of ut+N＝Fx,t(xt) For the true value x of wind power outputtThe distribution function of (a) is determined,t＝1,2,…,N；F(um|um+1,…,uk-1) And F (u)k|um+1,…,uk-1) Are each um+1,…,uk-1Under the condition umAnd ukConditional probability distribution of (1), Cm,k|m+1,…,k-1For the Copula function composed of them, F (u)k|um,…,uk-1) Is um,…,uk-1Under the condition ukA conditional probability distribution of (a);
when m is 1, F (u) is obtained by sampling the independent uniform distributionk|u1,…,uk-1) When k is N +1, F (u)m|um+1,…,uk-1)＝F(u1|u2,…,uN) Thus, F (u) can be obtainedk|um+1,…,uk-1)＝F(uN+1|u2,…,uN) The value of (c). Similarly, F (u) can be finally obtained by traversing m ═ 1,2, …,2N-2 and k ═ N +1, N +2, …,2Nk|uk-1) And then according to the property of the Copula function:
Figure BDA0002821176410000072
Ck-1,kis uk-1,ukThe Copula function of (a) can solve the unary nonlinear equation to obtain uk。
Finally, according to the probability distribution function relation uk＝Fy,k(xk-N) Real value sample x for calculating wind power output in k-N time periodk-NThe expression is as follows:
Figure BDA0002821176410000073
Figure BDA0002821176410000074
actual value probability distribution function value u for representing wind power output in k-N time periodkThe inverse distribution function of (c).
The specific steps of the above sampling process are exemplified as follows:
obtaining independent and uniformly distributed samples zk＝F(uk|u1,…,uk-1) When k is N +1, N +2, …, and after 2N, k is N +1, the expression is followed
Figure BDA0002821176410000081
Solving for uN+1＝Fy,N+1(x1) The method comprises the following steps:
when m is 1, the formula is shown
Figure BDA0002821176410000082
Because of F (u)N+1|u1,…,uN) As sample point, F (u)1|u2,…,uN) Has already passed ut＝Fy,t(yt) Solving, therefore, a one-dimensional nonlinear equation can be solved to obtain F (u)N+1|u2,…,uN)。
When m is 2, the reason is that
Figure BDA0002821176410000083
Since F (u) is obtainedN+1|u2,…,uN) And the same principle can be defined by ut＝Fy,t(yt) Find F (u)2|u3,…,uN) I.e. by
Figure BDA0002821176410000084
Thus, a one-dimensional nonlinear equation can be solved to obtain F (u)N+1|u3,…,uN)。
Iterating until F (u) is obtainedN+1|uN) From
Figure BDA0002821176410000085
Can solve a unary nonlinear equation to obtain uN+1。
Fourthly, according to the probability distribution function uN+1＝Fy,N+1(x1) And the true value sample of the wind power output at 1 time interval can be obtained
Figure BDA0002821176410000086
Because of the pair zN+1＝F(uN+1|u1,…,uN) A large amount of samples are taken, so that a large amount of real value samples of wind power output can be obtained.
And fifthly, repeating the steps for k being N +2, … and 2N to finally obtain a large number of samples of the actual value of the wind power output in 1-N time intervals.
And obtaining a power sampling point which considers the time correlation of the wind power prediction error and the condition correlation of the wind power prediction error and the predicted value under the condition of giving the wind power output predicted value.
And (3) selecting wind power data of a certain wind power plant in 2019 years for example analysis, and testing the effect of the sampling point generated by the proposed method. The calculation example compares the similarity degree of original data and sampling data under two wind power prediction powers of low power and high power, and the more similar the statistical data, the better the sampling effect.
(1) And a scatter diagram of the wind power prediction error in the historical data about the predicted value and the selected wind power curve of two days.
As shown in fig. 3, a scatter plot of the predicted values versus the predicted values for the wind power prediction errors in the historical data is shown. It can be seen that the larger the wind power prediction value is, the more dispersed the distribution of the prediction error is relatively. As shown in fig. 4 and 5, daily power curves for two days with lower predicted power (mean 117kW) and higher predicted power (mean 465kW), respectively, were selected for the embodiments of the present invention.
(2) Temporal correlation of historical data and temporal correlation of prediction error of sampled data at low power and high power.
Fig. 6 is a time dependency graph of historical data according to an embodiment of the present invention, and fig. 7 and 8 are time dependency graphs of sampling points obtained by using the method according to the present invention for low predicted power days and high predicted power days according to an embodiment of the present invention, respectively. The abscissa and ordinate represent the time periods, the different colors represent the magnitudes of the linear correlation coefficients, and the legend shows the corresponding numerical values. It can be seen that the correlations of the three figures are very similar.
(3) Distribution function (conditional dependence) of prediction error of historical data and sampled data at low power and high power.
As shown in FIGS. 9 and 10, the empirical distribution curves of lower power (50-150 kW daily) and higher power (400 kW daily) in the historical data of the embodiment of the present invention are shown respectively; fig. 11 and 12 are empirical distribution curves of sampling points obtained by the method of the present invention on low predicted power days and high predicted power days, respectively, according to the embodiment of the present invention. It can be seen that the historical data has conditional relevance, that is, when the wind power is higher, the prediction error distribution is more dispersed. The method provided by the invention can fit the distribution characteristics and obtain results similar to the original data.
In conclusion, under the condition of giving a predicted value of wind power, a large number of samples considering time correlation and condition correlation simultaneously can be obtained according to historical data of the predicted value and an actual value, the samples can well fit the distribution characteristics of original data, and the problem of insufficient sample amount of the original data is solved.

Claims (1)

1. A wind power correlation condition sampling method based on a vine structure Pair-Copula is characterized by comprising the following steps:
step 1, obtaining edge probability distribution of a real value and a predicted value of wind power output, namely obtaining the edge probability distribution of the real value and the predicted value of the wind power output by adopting a parameter estimation method or a non-parameter estimation method according to the distribution characteristics of wind power historical data;
step 2, constructing an optimal Pair-Copula function of a true value and a predicted value of wind power output, and determining optimal P between every two multidimensional variables layer by layer from the layer 1 of the Pair-Copula of the rattan structureSelecting the minimum Euclidean distance from the empirical Copula function as the optimal Pair-Copula function from the fitted Copula functions, and fitting to obtain a Copula function Cp(u, v) and empirical Copula function CnThe Euclidean distance between (u, v) is defined as:
Figure FDA0002821176400000011
the specific expression of the empirical Copula function is as follows:
Figure FDA0002821176400000012
in the formula (x)i,yi) Samples taken from variables (X, Y) that construct the Copula function, F (X)i) And G (y)i) Are each xiAnd yiThe distribution function of (a) is determined,
Figure FDA0002821176400000013
and
Figure FDA0002821176400000014
n is the number of samples for an indicative function;
step 3, sampling the vine structure Pair-Copula under the condition of giving a predicted value of wind power output;
giving a wind power output predicted value ytTo find out the probability distribution function ut＝Fy,t(yt) T is 1,2, …, the value of N, and known u2,…,uNUnder the condition u1Conditional probability distribution F (u)1|u2,…,uN) (ii) a According to the formula of the vine structure Pair-Copula function, obtaining:
Figure FDA0002821176400000015
Figure FDA0002821176400000021
k＝N+1,N+2,…,2N；m＝1,2,…,2N-2；m+1≤k-1
ut＝Fy,t(yt) For wind power output prediction value ytDistribution function of ut+N＝Fx,t(xt) For the true value x of wind power outputtT 1,2, …, N; f (u)m|um+1,…,uk-1) And F (u)k|um+1,…,uk-1) Are each um+1,…,uk-1Under the condition umAnd ukConditional probability distribution of (1), Cm,k|m+1,…,k-1As a function of Copula, F (u)k|um,…,uk-1) Is um,…,uk-1Under the condition ukA conditional probability distribution of (a);
finally, according to the probability distribution function relation uk＝Fy,k(xk-N) Real value sample x for calculating wind power output in k-N time periodk-NThe expression is as follows:
Figure FDA0002821176400000022
Figure FDA0002821176400000023
actual value probability distribution function value u for representing wind power output in k-N time periodkThe inverse distribution function of (c).
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