CN110084433A - Wind power prediction error piecewise fitting method based on gauss hybrid models - Google Patents
Wind power prediction error piecewise fitting method based on gauss hybrid models Download PDFInfo
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Abstract
The invention discloses the wind power prediction error piecewise fitting method based on gauss hybrid models, specifically: it is worth on the basis of wind energy turbine set installed capacity, the wind power prediction value and wind power prediction error of wind power plant historical data is normalized；It makes wind power prediction value and predicts the distribution scatter plot of error；According to each section difference of distribution scatter plot, wind power prediction value is divided into several sections from small to large, makes the frequency histogram of wind power prediction error in each section respectively；Using gauss hybrid models, the frequency histogram of wind power prediction error in each section is fitted respectively, parameter Estimation is carried out to the gauss hybrid models in each section respectively using expectation-maximization algorithm, obtains mean value, standard deviation, the weight of each sub- Gaussian Profile in the gauss hybrid models in each section；The present invention can accurately reflect the true distribution situation of wind power prediction error comprehensively, have considerable flexibility and fitting precision, and computational efficiency is higher.
Description
Technical field
The present invention relates to the Distribution estimation fields of wind power prediction error, and in particular to is based on gauss hybrid models
Wind power prediction error piecewise fitting method.
Background technique
As the concentration on a large scale of wind-powered electricity generation is grid-connected, wind-powered electricity generation permeability is greatly improved, and causes the uncertainty of electric system aobvious
It writes and increases, bring huge challenge to the traffic control of electric system.Due to influencing the influence factor of Wind turbines output power
It is sufficiently complex, the prior art be difficult to completely grasp Wind turbines output power changing rule, cause wind power prediction value with
It is inevitably present error between actual value, therefore needs the characteristic of meticulous depiction wind power prediction error, obtains wind-powered electricity generation function
The uncertainty models of rate prediction error concentrate wind-powered electricity generation so as to formulate reasonable power generation dispatching plan etc. on a large scale
Grid-connected safety and economy has certain practical significance.
Wang Chengfu, Wang Zhaoqing, Sun Hongbin wait to consider the built-up pattern containing Wind turbines of prediction error timing distribution characteristic
[J] Proceedings of the CSEE, 2016,36 (15): 4081-4090. uses t location-scale probability Distribution Model pair
The probability distribution of wind power prediction error is fitted, special for the probability distribution of wind power prediction error spikes and thick tail
Property all has preferable fitting effect；
Zhao Shuqiang, Wang Yang, Xu Yan dispatch [J] based on the fire storage joint Dependent-chance Programming of wind-powered electricity generation prediction error randomness
Proceedings of the CSEE, 2014,34 (S): 9-16. are divided using the probability of Cauchy's distributed model fitting wind power prediction error
Cloth, and wind power prediction value is divided into several sections by size, wind power prediction error in each section of piecewise fitting
Probability distribution；
Ye Lin, Zhang Yali, huge clouds wait for the Gaussian Mixture mould of the Probabilistic Load calculating containing wind power plant
Type [J] Proceedings of the CSEE, 2017,37 (15): 4379-4387. proposes to be fitted wind power using gauss hybrid models
The probability distribution for predicting error, achieves preferable fitting effect, but do not consider that wind power prediction value is located at different section models
The difference of probability of error distribution is predicted when enclosing；
Above-mentioned the mentioned method of document is primarily present two problems: first is that mostly using single probability Distribution Model to wind-powered electricity generation function greatly
The probability distribution of rate prediction error is fitted, and cannot take into account asymmetric wind power prediction error, spike, thick tail well very
To the probability density characteristics of multimodal；Second is that not considering when the probability density characteristics to wind power prediction error are analyzed
Wind power prediction value predicts the difference of probability of error distribution in different interval ranges.
Summary of the invention
In order to accurately reflect that the true distribution situation of wind power prediction error, the present invention provide mixed based on Gauss comprehensively
The wind power prediction error piecewise fitting method of molding type, specific steps include:
Step 1: data prediction is worth on the basis of wind energy turbine set installed capacity, to the wind power of wind power plant historical data
Predicted value and wind power prediction error are normalized respectively；
Step 2: for the wind power prediction value and wind power prediction error information after normalized, with wind-powered electricity generation
Power prediction value makes wind power prediction value and wind as ordinate as abscissa, corresponding wind power prediction error
The distribution scatter plot of electrical power prediction error；
Step 3: being divided into several sections for wind power prediction value from small to large, makes wind-powered electricity generation function in each section respectively
The frequency histogram of rate prediction error；
Step 4: apply gauss hybrid models, respectively to the frequency histogram of wind power prediction error in each section into
Row fitting carries out parameter Estimation to the gauss hybrid models in each section respectively using expectation-maximization algorithm, obtains in each section
Mean value, standard deviation, the weight of each sub- Gaussian Profile in the gauss hybrid models of wind power prediction error.
Its step 1 specifically:
Regard the practical power output of wind-powered electricity generation as the sum of wind power prediction value and wind power prediction error, i.e. wind power prediction
Error is equal to the difference of wind-powered electricity generation practical power output and wind power prediction value, and wind power prediction error may be expressed as:
In formula, Δ PWFor wind power prediction error；PWFor the practical power output of wind-powered electricity generation；For wind power prediction value；
It is worth on the basis of wind energy turbine set installed capacity, wind power prediction value and wind power prediction error is returned respectively
One change processing, expression formula difference are as follows:
In formula, PWRFor wind energy turbine set installed capacity；ΔP′WIt is the wind power prediction value and wind after normalizing respectively
Electrical power predicts error.
Step 3 specifically:
3.1 according to the wind power prediction value of wind power plant historical data and the distribution scatterplot of corresponding wind power prediction error
Figure analyzes the difference of wind power predicted value distribution of wind power prediction error in different sections in distribution scatter plot；
3.2 difference according to the distribution of wind power prediction error in different sections of wind power prediction value, by wind-powered electricity generation
Power prediction value is divided into several sections from small to large；
3.3 count the frequency of wind power prediction error in each section respectively, and it is pre- to make wind power in each section respectively
Survey the frequency histogram of error.
Step 4 specifically:
4.1 apply gauss hybrid models (Gaussian mixture model, GMM), respectively to wind-powered electricity generation function in each section
The frequency histogram of rate prediction error is fitted:
The probability density function of the gauss hybrid models in certain section are as follows:
In formula, n is the number for the sub- Gaussian Profile that the gauss hybrid models in the section include；For comprising
The probability density function of k-th of sub- Gaussian Profile；X is that the wind power prediction error in the section observes any sample in data set
This value；μk、σk、ωkMean value, standard deviation, the weight of respectively k-th sub- Gaussian Profile；Wherein, weights omegakMeet:
4.2 use expectation-maximization algorithm (expectation maximization, EM), respectively to the Gauss in each section
Mixed model carries out parameter Estimation:
Using expectation-maximization algorithm, the max log possibility predication of the gauss hybrid models parameter vector in certain section is solved
Vector, the max log possibility predication vector of the gauss hybrid models in certain section are as follows:
Wherein,
θ=[ω1,μ1,σ1,…,ωn,μn,σn] (7)
Wherein, the parameter vector of θ gauss hybrid models thus；θ*For max log possibility predication vector, i.e. log-likelihood
The parameter vector of gauss hybrid models when function reaches maximum value；xiData set is observed for the wind power prediction error in the section
In i-th of sample value；N is that the wind power prediction error in the section observes the sample size of data set；
Define implicit variable
To i-th of sample in the wind power prediction error observation data set in Mr. Yu section, defining implicit variable is γik,
γik=1 indicates that the sample is generated by k-th in gauss hybrid models Gaussian Profile, γik=0 indicates the sample not by Gauss
K-th of sub- Gaussian Profile generates in mixed model, i.e., implicit variable γikMeet
γik∈{0,1} (8)
E step (Expectation step) calculates the expectation formula of implicit variable
If parameter vector θ=[ω of the gauss hybrid models of current interval1,μ1,σ1,…,ωn,μn,σn], utilize pattra leaves
The posterior probability that i-th of sample is generated by k-th in gauss hybrid models Gaussian Profile in this theorem calculating observation data set,
Calculation formula are as follows:
In formula,Indicate that i-th of sample is sub by k-th in gauss hybrid models in observation data set under "current" model parameter
The posterior probability that Gaussian Profile generates, responsiveness of the referred to as k-th sub- Gaussian Profile to i-th of sample in observation data set；
M step (Maximization step), i.e. the revaluation formula of computation model distribution parameter
According to the posterior probability values of the implicit variable of current each sample, pair that observation data set is complete data can be obtained
Number likelihood function；Then by seeking partial derivative of the log-likelihood function about each parameter, the partial derivative of each parameter is enabled respectively etc.
In zero, so that it may calculate the weights omega of each sub- Gaussian Profile in the gauss hybrid modelsk, mean μk, varianceTheir calculating is public
Formula is as follows:
The E step that iterates and M step, until log-likelihood function value reaches maximum value；
After 4.3 carry out parameter Estimation to gauss hybrid models using EM algorithm, the wind power prediction in each section can be obtained
Mean value, standard deviation, the weight of each sub- Gaussian Profile in the gauss hybrid models of error.
The present invention obtains beneficial technical effect are as follows: the present invention can accurately reflect the true of wind power prediction error comprehensively
Real distribution situation has considerable flexibility and fitting precision, and computational efficiency is higher.
Detailed description of the invention
Fig. 1 is flow chart of the invention.
Fig. 2 be embodiment wind power prediction value with it is corresponding predict error distribution scatter plot and interval division.
Fig. 3 is the frequency histogram of wind power prediction error in the overall wind power prediction error and each section of embodiment
Figure.
Fig. 4 is that the gauss hybrid models of wind power prediction probability of error Density Distribution in each section of embodiment are fitted song
Line.
Fig. 5 is the different gauss hybrid models of embodiment [0,0.3) probability of wind power prediction error is close in section
Spend matched curve.
Fig. 6 be embodiment different gauss hybrid models [0.3,0.6) probability of wind power prediction error in section
Density matched curve.
Fig. 7 is that the probability of different gauss hybrid models wind power prediction error in [0.6,1] section of embodiment is close
Spend matched curve.
Specific embodiment
Detailed illustrate is carried out to technology contents of the invention below.
The process of wind power prediction error piecewise fitting method based on gauss hybrid models is as shown in Figure 1, specific step
Suddenly include:
Step 1: data prediction is worth on the basis of wind energy turbine set installed capacity, to the wind power of wind power plant historical data
Predicted value and wind power prediction error are normalized respectively；
Regard the practical power output of wind-powered electricity generation as the sum of wind power prediction value and wind power prediction error, i.e. wind power prediction
Error is equal to the difference of wind-powered electricity generation practical power output and wind power prediction value, and wind power prediction error may be expressed as:
In formula, Δ PWFor wind power prediction error；PWFor the practical power output of wind-powered electricity generation；For wind power prediction value；
If directlying adopt the absolute error of (1) formula, the variation range of the wind power prediction error amount counted is very big,
And the fluctuation situation that can not characterize wind power prediction error needs to be standardized place to wind power prediction error
Reason；Therefore, it need to be worth on the basis of wind energy turbine set installed capacity, wind power prediction value and wind power prediction error are carried out respectively
Normalized, expression formula difference are as follows:
In formula, PWRFor wind energy turbine set installed capacity；ΔP′WIt is the wind power prediction value and wind-powered electricity generation after normalizing respectively
Power prediction error.
Step 2: for the wind power prediction value and wind power prediction error information after normalized, with wind-powered electricity generation
Power prediction value makes wind power prediction value and wind as ordinate as abscissa, corresponding wind power prediction error
The distribution scatter plot of electrical power prediction error；
Step 3: being divided into several sections for wind power prediction value from small to large, makes wind-powered electricity generation function in each section respectively
The frequency histogram of rate prediction error；
3.1 according to the wind power prediction value of wind power plant historical data and the distribution scatterplot of corresponding wind power prediction error
Figure analyzes the difference of wind power predicted value distribution of wind power prediction error in different sections in distribution scatter plot；
3.2 difference according to the distribution of wind power prediction error in different sections of wind power prediction value, by wind-powered electricity generation
Power prediction value is divided into several sections from small to large；
3.3 count the frequency of wind power prediction error in each section respectively, and it is pre- to make wind power in each section respectively
Survey the frequency histogram of error.
Step 4: apply gauss hybrid models, respectively to the frequency histogram of wind power prediction error in each section into
Row fitting carries out parameter Estimation to the gauss hybrid models in each section respectively using expectation-maximization algorithm, obtains in each section
Mean value, standard deviation, the weight of each sub- Gaussian Profile in the gauss hybrid models of wind power prediction error.
4.1 apply gauss hybrid models (Gaussian mixture model, GMM), respectively to wind-powered electricity generation function in each section
The frequency histogram of rate prediction error is fitted:
The probability density function of the gauss hybrid models in certain section are as follows:
In formula, n is the number for the sub- Gaussian Profile that the gauss hybrid models in the section include；For comprising
K-th of sub- Gaussian Profile probability density function；X is any in the wind power prediction error observation data set in the section
Sample value；μk、σk、ωkMean value, standard deviation, the weight of respectively k-th sub- Gaussian Profile；Wherein, weights omegakMeet:
4.2 use expectation-maximization algorithm (expectation maximization, EM), respectively to the Gauss in each section
Mixed model carries out parameter Estimation:
Using expectation-maximization algorithm, the max log possibility predication of the gauss hybrid models parameter vector in certain section is solved
Vector, the max log possibility predication vector of the gauss hybrid models in certain section are as follows:
Wherein,
θ=[ω1,μ1,σ1,…,ωn,μn,σn] (7)
Wherein, the parameter vector of θ gauss hybrid models thus；θ*For max log possibility predication vector, i.e. log-likelihood
The parameter vector of gauss hybrid models when function reaches maximum value；xiData set is observed for the wind power prediction error in the section
In i-th of sample value；N is that the wind power prediction error in the section observes the sample size of data set；
Using expectation maximization (expectation maximization, EM) algorithm, the Gaussian Mixture in certain section is solved
The max log possibility predication vector of model parameter, it is necessary first to determine implicit variable, after defining implicit variable, be calculated using EM
Calculate the revaluation formula of the expectation formula and computation model distribution parameter of implicit variable in method, respectively referred to as EM algorithm
Expectation step and Maximization step, individually below referred to as E step and M step；Iterate E step and M
Step, until log-likelihood function value reaches maximum value；
The calculation process of EM algorithm are as follows:
(a) implicit variable is defined
To i-th of sample in the wind power prediction error observation data set in Mr. Yu section, defining implicit variable is γik,
γik=1 indicates that the sample is generated by k-th in gauss hybrid models Gaussian Profile, γik=0 indicates the sample not by Gauss
K-th of sub- Gaussian Profile generates in mixed model, i.e., implicit variable γikMeet
γik∈{0,1} (8)
(b) the parameter vector θ of the gauss hybrid models in certain section is randomly initialized；
(c) primary iteration number t=0 is set.
(d) E step
If parameter vector θ=[ω of the gauss hybrid models of current interval1,μ1,σ1,…,ωn,μn,σn], utilize pattra leaves
The posterior probability that i-th of sample is generated by k-th in gauss hybrid models Gaussian Profile in this theorem calculating observation data set,
Calculation formula are as follows:
In formula,Indicate current interval gauss hybrid models parameter under observe data set in the i-th sample by Gaussian Mixture
The posterior probability that k-th of sub- Gaussian Profile generates in model, referred to as k-th sub- Gaussian Profile is to i-th of sample in observation data set
This responsiveness；
(e) M step
According to the posterior probability values of the implicit variable of current each sample, pair that observation data set is complete data can be obtained
Number likelihood function；Then by seeking partial derivative of the log-likelihood function about each parameter, the partial derivative of each parameter is enabled respectively etc.
In zero, so that it may calculate the weights omega of each sub- Gaussian Profile in the gauss hybrid modelsk, mean μk, varianceTheir calculating is public
Formula is as follows:
(f) gauss hybrid models are calculated for the log-likelihood function of observation data set according to formula (6).
(g) judge whether log-likelihood function value restrains, or whether reach maximum number of iterations.If so, stop iteration,
And go to step (h)；If it is not, then enabling t ← t+1, and goes to step (d) and continue to iterate to calculate.
(h) mean μ of each sub- Gaussian Profile in the gauss hybrid models in the section is returnedk, varianceWeights omegak；
After 4.3 carry out parameter Estimation to sub- Gaussian Profile each in gauss hybrid models using EM algorithm, each area can be obtained
Between wind power prediction error gauss hybrid models in each sub- Gaussian Profile mean value, standard deviation, weight；
4.4 construction fitting precision evaluation indexes
In order to verify the accuracy of gauss hybrid models fitting result, using mean absolute error (Mean Absolute
Error, MAE), root-mean-square error (Root Mean Square Error, RMSE), cosine angle transform (Icos) 3 refer to
Mark, to evaluate the fitting effect of gauss hybrid models；
Note X, Y be respectively in the frequency histogram of wind power prediction error the value of each histogram center with it is corresponding
The sequence of empirical probability density functional value composition, i.e.,
X=[x1,x2,…,xm] (14)
Y=[y1,y2,…,ym] (15)
Wherein, m is the histogram quantity that frequency histogram includes.
After being fitted using probability distribution of the gauss hybrid models to wind power prediction error, Gaussian Mixture mould is calculated
Functional value of the type in sequence X at each element, is denoted asIt is represented by
The calculation formula of mean absolute error (MAE) is
The calculation formula of root-mean-square error (RMSE) is
Cosine angle transform (Icos) calculation formula be
Above-mentioned 3 kinds of fingers target value is smaller, shows that the fitting precision of probability Distribution Model is higher.
Embodiment
Using the historical statistical data in certain practical wind power plant on June 30th, 1 day 1 January in 2017 as analysis object,
Predicted time scale predicted that temporal resolution is 1 hour for 24 hours a few days ago.It is worth on the basis of wind energy turbine set installed capacity, to wind
Electrical power predicted value and prediction error are normalized.Wind power prediction value is made to dissipate with the distribution of corresponding prediction error
Point diagram, as shown in Figure 2.
From Figure 2 it can be seen that wind power prediction value [0,0.3), [0.3,0.6), [0.6,1] this 3 sections when, it is corresponding
Prediction error shows different probability density characteristics, therefore wind power prediction value is divided into 3 sections, with this 3 sections
Carry out the probability distribution of piecewise fitting wind power prediction error.
Respectively statistics wind power prediction value [0,0.3), [0.3,0.6), the frequency of [0.6,1] section interior prediction error
Number, each section interior prediction error after the frequency histogram of macro-forecast error when then making unsegmented respectively, and segmentation
Frequency histogram, as shown in Figure 3.
By taking the gauss hybrid models that 3 sub- Gauss distributing lines are composed as an example, to the general of wind power prediction error
Rate distribution is fitted, and obtains the fitting result and use piecewise fitting method that the macro-forecast probability of error is distributed when unsegmented
The fitting result of each section interior prediction probability of error distribution afterwards, as shown in Figure 4.
By Fig. 3 and Fig. 4 as it can be seen that being equally fitted using gauss hybrid models, the probability point of 3 section interior prediction errors
Cloth has significant difference:
1) wind power prediction value be located at [0,0.3) in interval range when, predict the Probability Distribution Fitting curve tool of error
There is apparent spike behavior；
2) wind power prediction value be located at [0.3,0.6) in interval range when, predict the Probability Distribution Fitting curve of error
Symmetry is presented, and there is significant thick tail characteristic；
3) when wind power prediction value is located in [0.6,1] interval range, predict that the Probability Distribution Fitting curve of error is in
Reveal asymmetrical distribution character, there is apparent left avertence characteristic.
It is fitted accordingly, with respect to the probability distribution directly to macro-forecast error when not being segmented, using according to wind-powered electricity generation
The method of power prediction value piecewise fitting more comprehensively can accurately reflect the true distribution situation of wind power prediction error.
The fitting effect of gauss hybrid models depend on it includes sub- Gauss quantity, sub- Gauss quantity is more, fitting essence
Degree is higher, but also results in model information redundancy and complexity increase simultaneously, so that calculation amount becomes larger.Therefore it needs to imitate in fitting
Suitable sub- Gauss quantity is selected between fruit and model complexity.Application is composed of 2~5 sub- Gauss distributing lines respectively
Gauss hybrid models (being denoted as GMM-2, GMM-3, GMM-4 and GMM-5 respectively) to each wind power prediction value section interior prediction
The probability distribution of error is fitted.Fitting result is as shown in Fig. 5~7, and fitting precision evaluation index is as shown in table 1~3.
Table 1 [0,0.3) in section each gauss hybrid models fitting precision evaluation index
Table 2 [0.3,0.6) the fitting precision evaluation index of each gauss hybrid models in section
The fitting precision evaluation index of each gauss hybrid models in table 3 [0.6,1] section
The fitting result for comparing each gauss hybrid models can be seen that, the fitting effect of gauss hybrid models with it includes son
Gauss quantity is closely related.Further the fitting result of comparison GMM-3, GMM-4 and GMM-5 model can be seen that, GMM-4 and GMM-
The promotion of 5 pairs of fitting precisions has not been to show excessive son it is obvious that even partial fitting precision evaluation index is also increased
Gauss quantity will lead to model information redundancy, so that fitting effect is deteriorated instead.
Therefore, for selected wind power plant historical statistical data, in the fitting precision for comprehensively considering gauss hybrid models
After computational efficiency, it is fitted using probability distribution of the GMM-3 model to each wind power prediction value section interior prediction error,
The estimates of parameters of the GMM-3 model of wind power prediction error is as shown in table 4 in each section obtained by EM algorithm.
The estimates of parameters of the GMM-3 model of wind power prediction error in each section of table 4
The above result shows that be respectively adopted GMM-3 model to the probability distribution of the wind power prediction error in 3 sections into
Row fitting is that reasonably, have considerable flexibility and fitting precision, and computational efficiency is higher.
Claims (4)
1. the wind power prediction error piecewise fitting method based on gauss hybrid models, which is characterized in that specific steps include:
Step 1: data prediction is worth on the basis of wind energy turbine set installed capacity, to the wind power prediction of wind power plant historical data
Value and wind power prediction error are normalized respectively；
Step 2: for the wind power prediction value and wind power prediction error information after normalized, with wind power
Predicted value makes wind power prediction value and wind-powered electricity generation function as ordinate as abscissa, corresponding wind power prediction error
The distribution scatter plot of rate prediction error；
Step 3: being divided into several sections for wind power prediction value from small to large, and it is pre- to make wind power in each section respectively
Survey the frequency histogram of error；
Step 4: gauss hybrid models are applied, the frequency histogram of wind power prediction error in each section is intended respectively
It closes, parameter Estimation is carried out to the gauss hybrid models in each section respectively using expectation-maximization algorithm, obtains wind-powered electricity generation in each section
Mean value, standard deviation, the weight of each sub- Gaussian Profile in the gauss hybrid models of power prediction error.
2. the wind power prediction error piecewise fitting method according to claim 1 based on gauss hybrid models, special
Sign is, the step 1 specifically:
Regard the practical power output of wind-powered electricity generation as the sum of wind power prediction value and wind power prediction error, i.e. wind power prediction error
The difference of practical equal to wind-powered electricity generation power output and wind power prediction value, wind power prediction error may be expressed as:
In formula, Δ PWFor wind power prediction error；PWFor the practical power output of wind-powered electricity generation；For wind power prediction value；
It is worth on the basis of wind energy turbine set installed capacity, wind power prediction value and wind power prediction error is normalized respectively
Processing, expression formula difference are as follows:
In formula, PWRFor wind energy turbine set installed capacity；ΔP′WIt is the wind power prediction value and wind power after normalizing respectively
Predict error.
3. the wind power prediction error piecewise fitting method according to claim 1 based on gauss hybrid models, special
Sign is, the step 3 specifically:
3.1 according to the wind power prediction value of wind power plant historical data and the distribution scatter plot of corresponding wind power prediction error,
Analyze the difference of wind power predicted value distribution of wind power prediction error in different sections in distribution scatter plot；
3.2 difference according to the distribution of wind power prediction error in different sections of wind power prediction value, by wind power
Predicted value is divided into several sections from small to large；
3.3 count the frequency of wind power prediction error in each section respectively, make wind power prediction in each section respectively and miss
The frequency histogram of difference.
4. the wind power prediction error piecewise fitting method according to claim 1 based on gauss hybrid models, special
Sign is, the step 4 specifically:
4.1 apply gauss hybrid models, are fitted respectively to the frequency histogram of wind power prediction error in each section:
The probability density function of the gauss hybrid models in certain section are as follows:
In formula, n is the number for the sub- Gaussian Profile that the gauss hybrid models in the section include；For comprising kth
The probability density function of a sub- Gaussian Profile；X is that the wind power prediction error in the section observes the arbitrary sample in data set
Value；μk、σk、ωkMean value, standard deviation, the weight of respectively k-th sub- Gaussian Profile；Wherein, weights omegakMeet:
4.2 use expectation-maximization algorithm, carry out parameter Estimation to the gauss hybrid models in each section respectively:
Using expectation-maximization algorithm, solve the max log possibility predication of the gauss hybrid models parameter vector in certain section to
Amount, the max log possibility predication vector of the gauss hybrid models in certain section are as follows:
Wherein,
θ=[ω1,μ1,σ1,…,ωn,μn,σn] (7)
Wherein, the parameter vector of θ gauss hybrid models thus；θ*For max log possibility predication vector, i.e. log-likelihood function reaches
The parameter vector of gauss hybrid models when to maximum value；xiIt is observed i-th in data set for the wind power prediction error in the section
A sample value；N is that the wind power prediction error in the section observes the sample size of data set；
Define implicit variable
To i-th of sample in the wind power prediction error observation data set in Mr. Yu section, defining implicit variable is γik, γik=
1 indicates that the sample is generated by k-th in gauss hybrid models Gaussian Profile, γik=0 indicates the sample not by Gaussian Mixture mould
K-th of sub- Gaussian Profile generates in type, i.e., implicit variable γikMeet
γik∈{0,1} (8)
E step calculates the expectation formula of implicit variable
If parameter vector θ=[ω of the gauss hybrid models of current interval1,μ1,σ1,…,ωn,μn,σn], it is fixed using Bayes
The posterior probability that i-th of sample is generated by k-th in gauss hybrid models Gaussian Profile in calculating observation data set is managed, is calculated
Formula are as follows:
In formula,Indicate "current" model parameter under observe data set in i-th of sample by k-th in gauss hybrid models Gauss
It is distributed the posterior probability generated, responsiveness of the referred to as k-th sub- Gaussian Profile to i-th of sample in observation data set；
M step, i.e. the revaluation formula of computation model distribution parameter
According to the posterior probability values of the implicit variable of current each sample, can obtain observation data set be the logarithm of complete data seemingly
Right function；Then by seeking partial derivative of the log-likelihood function about each parameter, the partial derivative of each parameter is enabled to be respectively equal to zero,
The weights omega of each sub- Gaussian Profile in the gauss hybrid models can be calculatedk, mean μk, varianceTheir calculation formula is such as
Under:
The E step that iterates and M step, until log-likelihood function value reaches maximum value；
After 4.3 carry out parameter Estimation to gauss hybrid models using EM algorithm, the wind power prediction error in each section can be obtained
Gauss hybrid models in each sub- Gaussian Profile mean value, standard deviation, weight.
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