CN104834994A - Small sample relay protection reliability parameter estimation method based on SVM (Support Vector Machine) - Google Patents

Abstract

The invention relates to a small sample relay protection reliability parameter estimation method based on a SVM (Support Vector Machine). The method comprises the following steps: 1, using an empirical formula to calculate the empirical reliability of an original failure sample and performing linear processing; 2, using a SVM algorithm to perform regression of failure data and performing prediction to generate a new enlarged sample; 3, performing linear fitting of the original failure data to obtain a parameter estimation value as a new iteration initial value; and 4, performing least square fitting of the enlarged new sample to obtain a final reliability parameter estimation result. Compared with the prior art, the method provided by the invention can effectively solve problems that priori knowledge and statistical samples are extremely in shortage as the operation time of an intelligent transformer station is short, and is beneficial for objective analysis and evaluation of reliability of a protection system.

Description

Based on the small sample reliability of relay protection method for parameter estimation of SVM

Technical field

The present invention relates to a kind of reliability of relay protection method for parameter estimation, especially relate to a kind of small sample reliability of relay protection method for parameter estimation based on SVM.

Background technology

In intelligent substation, the change of sampling and wavy trajectory, causes the structure of protective device to there occurs great changes [1].In order to the reliability of accurate evaluation protection system, need the reliability underlying parameter obtaining these new devices.At present; reliability of relay protection parameter estimation many employings the least square estimation method; as document [2] utilizes least-squares estimation to obtain the existence degree function of imperfect censored data; document [3] utilizes the distribution function in least square method and Mean rank order estimation protective relaying device life-span, the parameter of document [4] application Least Square Method Weibull distribution.But because protection system belongs to highly reliable system, the inefficacy sample size that can obtain in practice seldom [5].Intelligent Substation System time of putting into operation is not long, and the problem that sample size is little is more outstanding.Under condition of small sample, least-squares estimation can produce multiple correlation, is easy to accuracy and the stability [6] of affecting parameters estimation.Therefore, setting up one and be enclosed within the method for protective device dependability parameter being carried out under condition of small sample to science estimation, is the prerequisite of intelligent substation protection system reliability being carried out to Correct Analysis and objective evaluation.

In order to improve the precision of small sample parameter estimation, a kind of resolving ideas utilizes priori.Document [7] proposes a kind of Bayes method, effectively in conjunction with the prior imformation of source multiple under small sample, various ways, can obtain more complete posterior information, do not need very large increment also can obtain good probabilistic estimated value.But due to the singularity of relay protected operation, its prior imformation lacks equally.Because in intelligent substation, the structural form of protective device there occurs great changes, it is made to be difficult to utilize the prior imformation of GPF (General Protection False.If prior distribution is inaccurate, by direct affecting parameters estimated result, this is the subject matter of the method.Another kind of common method manages enlarged sample capacity.Document [8] utilizes Bootstrap method small sample problem to be converted into large sample problem to estimate the APPROXIMATE DISTRIBUTION of load model parameters.But the method is too relied on original sample by the average estimating parameter, easily produces parameter shift, be unfavorable for the robustness of parameter estimation, very large deviation may be brought as document [9] describes Bootstrap method emulation under System in Small Sample Situation by example.

Its Literature [1]: Zhang Peichao, Gao Xiang. System Architecture of Digitized Substation [J]. electric power network technique, 2006,24:73-77.

Document [2]: Rausand M, Hoyland A.System reliability theory:models, statisticalmethods, and applications [M] .New York:John Wiley Sons, 2004.

Document [3]: Xu Yan, Bai Jing, Dai Zhihui. a kind of new method [J] of the protective relaying device fail-safe analysis based on Weibull distribution. North China Electric Power University's journal (natural science edition), 2012,04:15-19.

Document [4]: Dai Zhihui, Wang Zengping, Jiao Yanjun, etc. the protective device reliability evaluation based on defect analysis studies [J]. protecting electrical power system and control, 2013,12:54-59.

Document [5]: Dai Zhihui. reliability of relay protection and risk assessment study [D] thereof. North China Electric Power University, 2012.

Document [6]: Zhang Hengxi, Guo Jilian, Zhu Jiayuan. small sample multivariate data analysis method and application [M]. publishing house of Northwestern Polytechnical University, 2002.

Document [7]: Wang Chengshan, Xie Yinghua. based on the evaluating reliability of distribution network [J] of double-deck isomorphism Bayesian network model. electric power network technique, 2005,07:41-46.

Document [8]: Han Dong, Ma Jin, He Renmu. based on the actual measurement load model parameters preferred [J] of Bootstrap. electrotechnics journal, 2012,27 (8): 141-146.

Document [9]: Duan Xiaojun, Wang Zhengming. the Bootstrap method [J] under System in Small Sample Situation. trajectory journal, 2003,15 (3): 1-5.

Summary of the invention

Object of the present invention be exactly in order to overcome above-mentioned prior art exist defect and a kind of small sample reliability of relay protection method for parameter estimation based on SVM is provided; effectively can overcome because intelligent substation puts into operation the problem that time long, priori and statistical sample very lack, contribute to carrying out objective A+E to protection system reliability.

Object of the present invention can be achieved through the following technical solutions:

Based on a small sample reliability of relay protection method for parameter estimation of SVM, it is characterized in that, comprise the following steps:

1) utilize experimental formula, calculate the experience fiduciary level of original failure sample, and do linearization process;

2) adopt SVM algorithm, fail data is returned, and prediction generates new enlarged sample;

3) carry out linear fit to original failure data, the estimates of parameters of acquisition is as new iterative initial value;

4) to the new samples after expansion, do least square fitting, obtain final dependability parameter estimated result.

Weibull distribution is very important a kind of distributed model in reliability engineering ^], because it can describe out the tub curve that equipment changes in whole life cycle internal fault rate, become one of most popular model in fail-safe analysis equipment life in recent years.The reliability distribution utilizing Two-parameter Weibull distribution to describe protective device has obtained the support of numerous research.Described utilizes experimental formula, and experience fiduciary level R (t) calculating original failure sample is specific as follows:

$R\left(t\right)=e^{-{\left(\frac{t}{\mathrm{\λ}}\right)}^{\mathrm{\α}}}---\left(1\right)$

In formula: α is form parameter, can distinguish the different failure type of product according to the numerical values recited of form parameter, as α < 1, crash rate tapers off earlier failure period of distribution, general corresponding proterctive equipment over time; When α=1, crash rate is constant, and Two-parameter Weibull distribution is reduced to exponential distribution, and equipment operates in random failure period; As α >1, crash rate is in increasing progressively distribution, and equipment operates in loss failure period.λ is scale parameter, plays the effect zooming in or out coordinate scale.

Present invention utilizes two outstanding advantages of SVM algorithm.First, SVM adopts structural risk minimization to carry out training study machine, and uses VC dimension theory to carry out metrology structure risk.This algorithm is finally summed up as and solves convex quadratic programming problem, can ensure to obtain globally optimal solution, overcomes the local extremum problem of neural net method theoretically.Secondly, SVM is the Corpus--based Method theories of learning, and this makes SVM show distinctive advantage in the identification of solution small sample parameter.Described SVM algorithm is specific as follows:

If the training dataset of linear SVM regression problem is S={ (x _i, y _i), i=1,2 ..., l}, wherein x _i∈ R ⁿi-th input amendment, y _i∈ R is corresponding to x _idesired value, l is training sample number;

Object finds a real-valued function y=f (x) according to given training data, makes this function can represent the dependence of y and x, to infer the functional value y corresponding to any x with this function, defining linear ε insensitive loss function is

${|y-f\left(x\right)|}_{\mathrm{\&epsiv;}}=\left\{\begin{array}{cc}0,& |y-f\left(x\right)|\&le;\mathrm{\&epsiv;}\\ |y-f\left(x\right)|-\mathrm{\&epsiv;},& |y-f\left(x\right)|>\mathrm{\&epsiv;}\end{array}\right.---\left(2\right)$

If namely desired value y and the difference between the value f (x) of the regression estimates function of learn configuration are less than ε, then loss equals 0;

If there is a lineoid

f(x)＝ω ^Tx+b＝0 (3)

Wherein, ω ∈ R ⁿ, b ∈ R, makes

|y _i-f(x _i)|≤ε (4)

Then sample set S is claimed to be ε-linear-apporximation, f (x)=ω ^tx+b is the linear regression estimation function to sample;

Sample point (x _i, y _i) ∈ S is to lineoid f (x)=ω ^tthe distance d of x+b=0 _ifor:

$d_i=\frac{|{\mathrm{\&omega;}}^T{x}_i+b-y_i|}{\sqrt{1+{\left|\right|\mathrm{\&omega;}\left|\right|}^2}}\&le;\frac{\mathrm{\&epsiv;}}{\sqrt{1+{\left|\right|\mathrm{\&omega;}\left|\right|}^2}}---\left(5\right)$

Namely be the upper bound of the point in S to lineoid distance, maximizing the lineoid that this upper bound obtains is best fit approximation lineoid, should make for this reason || ω || reach minimum;

Introduce Nonlinear Mapping φ (x) and the sample in the input space is mapped to high-dimensional feature space, then construct linear optimal hyperlane at high-dimensional feature space, because the inner product operation in higher dimensional space is very consuming time, SVM introduces kernel function K (x _i, x _j) replace in higher dimensional space inner product operation, select RBF kernel function as the kernel function of SVM, its form is as follows:

K(x _i,x _j)＝exp(-γ·||x _i-x _j|| ²) (6)

In formula: γ is the width of kernel function,

When constraint condition (4) can not realize, then need softening requirement of adjusting the distance, allow the sample not meeting constraint condition (4) to exist, for nonlinear regression problem, two slack variable ξ can be introduced _i, optimization problem objective function becomes:

$\min \frac{1}{2}{\left|\right|\mathrm{\&omega;}\left|\right|}^2+\frac{C}{l}\underset{i=1}{\overset{l}{\mathrm{\&Sigma;}}}({\mathrm{\&xi;}}_i+{\mathrm{\&xi;}}_i^{*})---\left(7\right)$

Constraint condition is:

$\begin{array}{c}{\mathrm{\&omega;}}^T\mathrm{\&phi;}\left(x_i\right)+b-y_i\&le;\mathrm{\&epsiv;}+{\mathrm{\&xi;}}_i\\ y_i-{\mathrm{\&omega;}}^T\mathrm{\&phi;}\left(x_i\right)-b\&le;\mathrm{\&epsiv;}+{\mathrm{\&xi;}}_i^{*}\end{array}---\left(8\right)$

In formula (7): C is penalty factor, the larger expression of C is larger to the sample punishment that error is large; Otherwise then large to error schedule of samples reveals higher tolerance, adjustment C can change the generalization ability of SVM; Slack variable ξ _i, represent the sample error allowed; L is training sample number; B is optimum interphase parameter.

Described linear fit carried out to original failure data be specially:

First twice logarithm is got continuously to formula (1):

$\ln \ln \frac{1}{R\left(t\right)}=\mathrm{\&alpha;}\ln t-\mathrm{\&alpha;}\ln \mathrm{\&lambda;}---\left(9\right)$

Make variable to replace, order

$y=\ln \ln \frac{1}{R\left(t\right)},x=\ln t---\left(10\right)$

a＝α,b＝-αlnλ

Then have

y＝ax+b (11)

Wherein a, b are linear fit coefficient, and x, y are respectively dependent variable and dependent variable, and t is the time that protection runs, and α is form parameter, and λ is scale parameter, utilizes Least Square Method parameter alpha, λ;

For checking the correlativity of fitting result and raw sample data, can the correlation coefficient ρ shown in introduction-type (12);

$\mathrm{\&rho;}=\frac{\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}(x_i-\stackrel{\&OverBar;}{x})(y_i-\stackrel{\&OverBar;}{y})}{\sqrt{\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{(x_i-\stackrel{\&OverBar;}{x})}^2}\sqrt{\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{(y_i-\stackrel{\&OverBar;}{y})}^2}}---\left(12\right)$

In formula: n is sample number; ρ more close to 1, then show fitting result and raw sample data correlativity closer.

Compared with prior art, the present invention has the following advantages:

(1) the SVM method Corpus--based Method theories of learning, can well solve the regression problem under condition of small sample; Carry out by grid search the results of learning that parameter optimization can improve SVM, adopt cross validation effectively can avoid problem concerning study.

(2) utilize SVM regression model enlarged sample capacity, effectively can solve the multiple correlation between the variable that causes because sample size is not enough, thus improve accuracy and the stability of least-squares algorithm.

(3) new method is without the need to priori, has good anti-outlier ability, has comparatively hyposensitivity to sample size.Under condition of small sample, These characteristics is particularly important.

Accompanying drawing explanation

Fig. 1 be the present invention with existing algorithm compare process flow diagram;

Fig. 2 is the sample distribution figure of specific embodiment;

Fig. 3 is the fiduciary level change curve in time of specific embodiment;

Fig. 4 is process flow diagram of the present invention.

Embodiment

Below in conjunction with the drawings and specific embodiments, the present invention is described in detail.

Embodiment

As shown in Figure 4, based on the small sample reliability of relay protection method for parameter estimation of SVM, comprise the following steps:

S100, utilize experimental formula, calculate the experience fiduciary level of original failure sample, and do linearization process;

S200, employing SVM algorithm, return, and prediction generates new enlarged sample to fail data;

S300, carry out linear fit to original failure data, the estimates of parameters of acquisition is as new iterative initial value;

S400, to expand after new samples, do least square fitting, obtain final dependability parameter estimated result.

Sample size deficiency can cause or aggravate the multiple correlation between variable, thus has a strong impact on accuracy and the stability of least-squares algorithm ^[6].Utilize the outstanding advantage of SVM algorithm in small sample recurrence herein, propose the dependability parameter method of estimation under a kind of novel condition of small sample, as shown in Figure 1.In figure, step MLS1 ~ MLS5 is the common dependability parameter estimation procedure based on the principle of least square, step SVM1 ~ the SVM3 on right side is then the flow process that new method is newly-increased, and namely the entire flow of new method is: MLS1 ~ MLS2-SVM1 ~ SVM3-MLS3 ~ MLS5.

In new method, first gather original failure sample data, calculate experience fiduciary level by experimental formula.Then, first carry out SVM parameter optimization, then carry out SVM regression training and generate regression model, utilize the sample that this model generation is new, thus enlarged sample capacity.Finally, recycle the principle of least square and carry out parameter fitting.

During owing to adopting solution by iterative method least square fitting problem in a computer, need for target component gives initial value.For this reason, herein first by step MLS1 ~ MLS5, ordinary least square fitting is utilized to try to achieve roughness parameter estimated result, then when utilizing new method, iterative initial value during using this result as least square fitting.

Be provided with the logout sample data t of n platform protective relaying device ₁, t ₂..., t _n, fail data is t ₁≤ t ₂≤ ...≤t _r(r < n), and sample size r≤10.It should be noted that, because protective relaying device has higher reliability, during observing, therefore only have minority device to lose efficacy, most of device is until observation terminates still to be in normal operating conditions, and the sample data namely recorded has right truncation characteristic.In this case, experimental formula can be adopted to calculate the experience fiduciary level of sample.Different experimental formulas is used for the parameter estimation under different occasion, and the present invention selects mathematical expectation formula when studying ^]:

$R\left(t_i\right)=1-\frac{i}{n+1}$

Obtain the experience fiduciary level R (t that original small sample out-of-service time data are corresponding _i) after, it can be used as the training sample that SVM returns.

SVM parameter optimization and regression training

Performance during SVM training is subject to many factors, especially following two factors: 1) penalty factor, control wrong point sample proportion, namely in the proper subspace determined, regulate Learning machine fiducial range and empiric risk ratio, make the generalization ability of learner best; 2) selection of kernel function type and parameter thereof.Select the RBF kernel function comparatively commonly used herein, its kernel functional parameter is γ.Need to determine these two parameters of C and γ when training like this.The Selecting parameter of SVM is also an optimization problem, and the normal method adopted has experience back-and-forth method, gradient descent method, genetic algorithm, particle swarm optimization algorithm etc.

In view of handled sample size is very little, present invention incorporates cross-validation method and grid data service carries out parameter optimization.First based on gridding method, C ∈ [C is made ₁, C ₂], change step is C _s; γ ∈ [γ ₁, γ ₂], change step is γ _s.Training for often organizing parameter (C, γ), utilizing cross validation to calculate root-mean-square error, getting one group of minimum parameter of error as model parameter.If there is many group parameters to there is identical training effect, then one group of parameter that prioritizing selection penalty factor is less, to avoid occurring study, thus improve the generalization ability of model.

After obtaining optimum SVM parameter, regression training can be carried out to sample, and with obtain model generation expand new samples collection.New samples collection has similar distribution to original sample collection, but larger sample size can improve accuracy and the stability of least-squares algorithm.

Simulation analysis

1.1 original sample

The microcomputer protecting device of certain model totally 60, under same working environment, puts into operation simultaneously, the fault-time of record protection device, and logout is as shown on the left of table 1.2013-3-98:00:00 fault-time selecting No. 52 devices is observation cut-off time, and namely total observation time is 17640h.

Table 1

The upper table record operation conditionss of 60 protective devices in about 2 years observation times, the device wherein broken down only has 8, and sample has right truncation characteristic.Employing formula (9) calculates experience fiduciary level corresponding to out-of-service time point, the results are shown in table 1 right column.The training sample that out-of-service time and corresponding experience fiduciary level return as SVM.

2.2 dependability parameter estimated results

Adopt grid data service find optimized parameter, hunting zone and step-length as follows:

C∈[2 ^-8,2 ⁸],log ₂C _S＝1

γ∈[2 ^-8,2 ⁸],log ₂γ _S＝1

After completing SVM regression training, utilize regression model that sample size is expanded 10 times herein, namely new samples number amounts to 80.Original sample and new samples are drawn in Fig. 2 in the lump.As seen from the figure, former sample set and new samples collection all show that ln (ln (1/R (t))) and lnt have the relation of approximately linear, but the latter is more level and smooth in sample distribution.

Least square fitting is carried out to the new samples generated, the reliability distribution parameter of this model protective device can be obtained.SVM parameter optimization result and reliability estimation of distribution parameters value one are listed in table 2.

Table 2

Penalty factor	Kernel functional parameter γ	Form parameter α	Scale parameter λ	Correlation coefficient ρ
					1	0.25	1.6441	54854	0.9981

1. the contrast of distinct methods

Below by middle to method in this paper (referred to as SVM method) and document [2-4] the common least square method (referred to as MLS method) used, and the Weibull++ software (referred to as Weibull++ method) of ReliaSoft company of U.S. exploitation contrasts.At present, Weibull++ software has become the standard software that global numerous enterprises carries out lifetime data analysis.

A. parameter estimation result compares

Utilize identical original failure sample set, the dependability parameter estimated result that each method obtains is as shown in table 3.

Table 3

According to form parameter and scale parameter, the fiduciary level change curve in time under three kinds of methods can be drawn, as shown in Figure 3.Give the point reliability utilizing experimental formula to calculate in figure simultaneously.

Point reliability corresponding to original failure time is as shown in table 4:

Table 4

Time/h	MLS method	Weibull++ method	SVM method
				6576	0.9839	0.9688	0.9699
7344	0.9790	0.9627	0.9640
				8808	0.9676	0.9500	0.9518
9144	0.9645	0.9469	0.9488
				12936	0.9199	0.9080	0.9112
13848	0.9063	0.8977	0.9012
				14160	0.9014	0.8941	0.8977
17640	0.8382	0.8516	0.8565

Observe above result and can obtain some conclusion following:

(1) three kind of form parameter that method obtains all is greater than 1, illustrates that the protection of this model is in the loss failure period that crash rate raises in time.

(2) from the correlation coefficient ρ of table 3, under condition of small sample, MLS method has been difficult to obtain parameter estimation result accurately, and Weibull++ method and SVM method in this paper still can obtain good estimated result, and parameter is close.

(3) as seen from Figure 3, the not enough problem of obvious matching is there is in MLS method to the 8th sample point, and the estimated result of Weibull++ has slight left avertence, this related coefficient from Weibull++ method is not so good as SVM method (see table 3) and also can be verified.

B. anti-outlier ability compares

When pen recorder failure event, due to the reason such as mistake of the registering capacity of monitoring arrangement, the restriction of record period or maintainer, there will be indivedual sample point and the larger problem of other deviations, this kind of data point is called as outlier or outlier.Under condition of small sample, the harm of outlier is more obvious.For the anti-outlier ability of comparative approach, by artificial for the out-of-service time of No. 31 devices left avertence 20%, namely the out-of-service time is changed into 10349h by 12936h.Parameter after reappraising is in table 5.

Contrast visible with table 3, the scale parameter result that three kinds of method for parameter estimation draw all has certain reduction, but the deviation of MLS and Weibull++ method has all exceeded 3%.By comparison, the parameter estimation result of SVM algorithm still has the highest related coefficient, illustrates that the method has good anti-outlier ability.

Table 5

C. the impact of sample size is compared

Under condition of small sample, the stability of parameter estimation will sharply be deteriorated.For comparative approach is to the susceptibility of sample size, be reduced to further 5 (shortening to 12936h by observation time) from 8 by original failure sample number, re-start parameter estimation, result is as shown in table 6:

Table 6

Contrast visible with table 3, now the related coefficient of MLS and Weibull++ method falls too low, and the deviation showing the estimated result of these two kinds of methods when sample size is not enough and original failure data very greatly, is difficult to truly reflect invalid cost model.But SVM method still can reach higher related coefficient, and the estimated bias of form parameter α, scale parameter λ is no more than 9.1%, 15.5% respectively, and this illustrates that SVM method is more insensitive to sample size.

Relay protection failure event has typical small sample feature.The present invention proposes a kind of based on reliability of relay protection method for parameter estimation under the condition of small sample of SVM; effectively can overcome because intelligent substation puts into operation the problem that time long, priori and statistical sample very lack, contribute to carrying out objective A+E to protection system reliability.Conclusion is as follows:

(1) the SVM method Corpus--based Method theories of learning, can well solve the regression problem under condition of small sample; Carry out by grid search the results of learning that parameter optimization can improve SVM, adopt cross validation effectively can avoid problem concerning study.

(2) utilize SVM regression model enlarged sample capacity, effectively can solve the multiple correlation between the variable that causes because sample size is not enough, thus improve accuracy and the stability of least-squares algorithm.

(3) new method is without the need to priori, has good anti-outlier ability, has comparatively hyposensitivity to sample size.Under condition of small sample, These characteristics is particularly important.

Claims (4)

1., based on a small sample reliability of relay protection method for parameter estimation of SVM, it is characterized in that, comprise the following steps:

1) utilize experimental formula, calculate the experience fiduciary level of original failure sample, and do linearization process;

2) adopt SVM algorithm, fail data is returned, and prediction generates new enlarged sample;

3) carry out linear fit to original failure data, the estimates of parameters of acquisition is as new iterative initial value;

4) to the new samples after expansion, do least square fitting, obtain final dependability parameter estimated result.

2. a kind of small sample reliability of relay protection method for parameter estimation based on SVM according to claim 1, it is characterized in that, described utilizes experimental formula, and experience fiduciary level R (t) calculating original failure sample is specific as follows:

$R\left(t\right)=e^{-{\left(\frac{t}{\mathrm{\&lambda;}}\right)}^{\mathrm{\&alpha;}}}---\left(1\right)$

In formula: α is form parameter, λ is scale parameter.

3. a kind of small sample reliability of relay protection method for parameter estimation based on SVM according to claim 2, it is characterized in that, described SVM algorithm is specific as follows:

If the training dataset of linear SVM regression problem is S={ (x _i, y _i), i=1,2 ..., l}, wherein x _i∈ R ⁿi-th input amendment, y _i∈ R is corresponding to x _idesired value, l is training sample number;

Object finds a real-valued function y=f (x) according to given training data, makes this function can represent the dependence of y and x, to infer the functional value y corresponding to any x with this function, defining linear ε insensitive loss function is

${|y-f\left(x\right)|}_{\mathrm{\&epsiv;}}=\left\{\begin{array}{cc}0,& |y-f\left(x\right)|\&le;\mathrm{\&epsiv;}\\ |y-f\left(x\right)|-\mathrm{\&epsiv;},& |y-f\left(x\right)|>\mathrm{\&epsiv;}\end{array}\right.---\left(2\right)$

If namely desired value y and the difference between the value f (x) of the regression estimates function of learn configuration are less than ε, then loss equals 0;

If there is a lineoid

f(x)＝ω ^Tx+b＝0 (3)

Wherein, ω ∈ R ⁿ, b ∈ R, makes

|y _i-f(x _i)|≤ε (4)

Then sample set S is claimed to be ε-linear-apporximation, f (x)=ω ^tx+b is the linear regression estimation function to sample;

Sample point (x _i, y _i) ∈ S is to lineoid f (x)=ω ^tthe distance d of x+b=0 _ifor:

$d_i=\frac{|{\mathrm{\&omega;}}^T{x}_i+b-y_i|}{\sqrt{1+{\left|\right|\mathrm{\&omega;}\left|\right|}^2}}\&le;\frac{\mathrm{\&epsiv;}}{\sqrt{1+{\left|\right|\mathrm{\&omega;}\left|\right|}^2}}---\left(5\right)$

Namely be the upper bound of the point in S to lineoid distance, maximizing the lineoid that this upper bound obtains is best fit approximation lineoid, should make for this reason || ω || reach minimum;

Introduce Nonlinear Mapping φ (x) and the sample in the input space is mapped to high-dimensional feature space, then construct linear optimal hyperlane at high-dimensional feature space, because the inner product operation in higher dimensional space is very consuming time, SVM introduces kernel function K (x _i, x _j) replace in higher dimensional space inner product operation, select RBF kernel function as the kernel function of SVM, its form is as follows:

K(x _i,x _j)＝exp(-γ·||x _i-x _j|| ²) (6)

In formula: γ is the width of kernel function,

When constraint condition (4) can not realize, then need softening requirement of adjusting the distance, allow the sample not meeting constraint condition (4) to exist, for nonlinear regression problem, two slack variable ξ can be introduced _i, optimization problem objective function becomes:

$\min \frac{1}{2}{\left|\right|\mathrm{\&omega;}\left|\right|}^2+\frac{C}{l}\underset{i=1}{\overset{l}{\mathrm{\&Sigma;}}}({\mathrm{\&xi;}}_i+{\mathrm{\&xi;}}_i^{*})---\left(7\right)$

Constraint condition is:

ω ^Tφ(x _i)+b-y _i≤ε+ξ _i(8)

$y_i-{\mathrm{\&omega;}}^T\mathrm{\&phi;}\left(x_i\right)-b\&le;\mathrm{\&epsiv;}+{\mathrm{\&xi;}}_i^{*}$

In formula (7): C is penalty factor, the larger expression of C is larger to the sample punishment that error is large; Otherwise then large to error schedule of samples reveals higher tolerance; Slack variable ξ _i, represent the sample error allowed; L is training sample number; B is optimum interphase parameter.

4. a kind of small sample reliability of relay protection method for parameter estimation based on SVM according to claim 3, is characterized in that, described carry out linear fit to original failure data and is specially:

First twice logarithm is got continuously to formula (1):

$\ln \ln \frac{1}{R\left(t\right)}=\mathrm{\&alpha;}\ln t-\mathrm{\&alpha;}\ln \mathrm{\&lambda;}---\left(9\right)$

Make variable to replace, order

$y=\ln \ln \frac{1}{R\left(t\right)},x=\ln t---\left(10\right)$

a＝α,b＝-αlnλ

Then have

y＝ax+b (11)

Wherein a, b are linear fit coefficient, and x, y are respectively dependent variable and dependent variable, and t is the time that protection runs, and α is form parameter, and λ is scale parameter, utilizes Least Square Method parameter alpha, λ;

For checking the correlativity of fitting result and raw sample data, can the correlation coefficient ρ shown in introduction-type (12);

$\mathrm{\&rho;}=\frac{\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}(x_i-\stackrel{\&OverBar;}{x})(y_i-\stackrel{\&OverBar;}{y})}{\sqrt{\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{(x_i-\stackrel{\&OverBar;}{x})}^2}\sqrt{\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{(y_i-\stackrel{\&OverBar;}{y})}^2}}---\left(12\right)$

In formula: n is sample number; ρ more close to 1, then show fitting result and raw sample data correlativity closer.