CN107730084A - Repair of Transformer decision-making technique based on gray prediction and risk assessment - Google Patents

Repair of Transformer decision-making technique based on gray prediction and risk assessment Download PDF

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CN107730084A
CN107730084A CN201710842962.5A CN201710842962A CN107730084A CN 107730084 A CN107730084 A CN 107730084A CN 201710842962 A CN201710842962 A CN 201710842962A CN 107730084 A CN107730084 A CN 107730084A
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罗静
张玄武
蔡彪
蔡一彪
吴芳基
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Hangzhou Safety Intelligent Technology Co Ltd
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Abstract

The invention discloses a kind of Repair of Transformer decision-making technique based on gray prediction and risk assessment, including：Transformer state variable, the historical data of transformer fault type and current measured data are obtained respectively, and transformer is obtained in t according to the method for gray predictionkThe predicted value of the state variable at moment；Using the Fuzzy C-Means Cluster Algorithm based on comentropy, with reference to transformer in tkThe data of the state variable at moment, calculating transformer tkThe probability of happening of moment each failure；Determine tkThe expense of each Strategies of Maintenance of moment transformer it is expected, and compared with the expectation of transformer fault overall risk expense, determines transformer optimal repair time and optimal Strategies of Maintenance.The present invention takes into full account the difference of individual transformer and integrated transformers probability of malfunction in each fault rate of calculating transformer, and the comentropy for characterizing sample distribution situation is substituted into fuzzy clustering algorithm as priori, improve the accuracy of transformer fault probability calculation.

Description

Repair of Transformer decision-making technique based on gray prediction and risk assessment
Technical field
The present invention relates to Repair of Transformer decision-making technique, more particularly to a kind of change based on gray prediction and risk assessment Depressor maintenance decision method.
Background technology
Transformer is the crucial hub device of power system, and its running status is directly connected to the safety of whole power system With stably, therefore formulating rational Strategies of Maintenance to transformer and being even more important.Repair of Transformer decision-making technique is normally based on wind What is established assessed in danger, and maintenance decision is made by carrying out analysis to failure serious consequence and probability of malfunction.Repair of Transformer at present Experience of the decision-making technique when assessing transformer fault serious consequence mostly according to user or expert is given a mark, and this way is led to It is not strong with property；Use Weibull distribution calculating transformer probability of malfunction mostly when analyzing transformer fault probability, but it is individual The overall transformer probability of malfunction that transformer fault probability characterizes with Weibull distribution differs greatly, and can be drawn using Weibull distribution Enter very big error.In addition, existing Repair of Transformer decision-making technique only gives the suggestion of inspection operation mostly at present, for The specific repair time does not refer to.
The content of the invention
The present invention is directed to the shortcomings of existing computational methods versatility in the prior art is not strong and error is big, there is provided a kind of Repair of Transformer decision-making technique based on gray prediction and risk assessment.
In order to solve the above-mentioned technical problem, the present invention is addressed by following technical proposals：
A kind of Repair of Transformer decision-making technique based on gray prediction and risk assessment, including：
Acquisition transformer state variable, the historical data of transformer fault type and current measured data respectively, and according to The method of gray prediction obtains transformer in tkThe predicted value of the state variable at moment；
Using the Fuzzy C-Means Cluster Algorithm based on comentropy, with reference to transformer in tkThe data of the state variable at moment, Calculating transformer tkThe probability of happening of moment each failure；
The consequence of each failure of the transformer is quantified, and with reference to the transformer tkThe moment generation of each failure is general Rate, calculate tkMoment transformer fault overall risk expense it is expected；
Determine tkThe expense of each Strategies of Maintenance of moment transformer it is expected, and it is expected to carry out with transformer fault overall risk expense Compare, determine transformer optimal repair time and optimal Strategies of Maintenance.
As a kind of embodiment, the state variable of the transformer includes oil dissolved gas H2、CH4、C2H2、 C2H4、C2H6、CO、CO2, furfural content in micro- water, winding absorptance, iron core grounding current and oil in oil, be designated as x respectively1- x11；
The transformer fault type includes winding failure, sleeve pipe failure, dielectric failure, iron core failure, tap and opened Failure, lead wire fault and oil leakage fault are closed, is designated as y respectively1-y7；
Every kind of fault type is labeled as two kinds of fault modes, two kinds of fault modes include cause transformer stoppage in transit pattern and Cause transformer efficiency reduction mode, be designated as y respectively1,1、y1,2、y2,1、y2,2…y7,1、y7,2。
It is described that transformer is obtained in t according to the method for gray prediction as a kind of embodimentkThe state variable at moment The specific practice of predicted value include：
Use and transformer state variable is predicted with the gray model based on GM (1,1) model, with transformer shape State variable xi, i=1,2 ..., 11 historical data and current measured dataBased on be modeled, GM (1,1) was modeled Journey is as follows：
Wherein, t1,t2,...,tn-1For at intervals ofHistorical data at equal intervals collection the moment, tnFor current time,For tkMoment transformer state variable xiData, to current measured dataAdded up to obtainUse with Lower formula represents：
Wherein,
It is rightAlbinism differential equation is established, formula represents as follows：
Wherein, a is development coefficient, and b is grey actuating quantity, and by albinism differential equation discretization, formula represents as follows：
Wherein, k represents k-th of data acquisition moment,Data sampling interval is represented, is taken The albinism differential equation substituted into after discretization obtains：
Below equation is obtained by transposition：
K=1,2 ..., n-1 are made, the formula after transposition is converted into following result：
Now, following symbol is introduced：
Then have：
Order
Then have：
Further conversion obtains：
Make x(1)(t0)=x(0)(t1), obtain it is discrete after albinism differential equation solution, formula is as follows：
Wherein,For cumulative dataIn+1 sampling instant t of kthk+1Predicted value, now, Data convert is obtained into below equation：
Wherein,As transformer state variable xiIn k-th of sampling instant tkPredicted value xi(tk), k=1, 2,...；
By the current measured data of each state variable of the transformer and the transformer state variable prediction value xi(tk), k =1,2 ... recorded, record formula represents as follows：
Wherein, tnFor current time,Respectively equally spaced prediction time, x (t) represent t transformer shape State variable x value.
It is described to use the Fuzzy C-Means Cluster Algorithm based on comentropy as a kind of embodiment, exist with reference to transformer tkThe data of the state variable at moment, calculating transformer tkThe probability of happening of moment each failure, specific steps include：
History, current and prediction data to the transformer state variable are normalized, and formula is as follows：
Wherein, t1~tn-1Moment, t are gathered for historical datanFor current time,It is pre- for transformer state variable Survey moment, xi(tk)*For t after normalizedkThe state variable x of moment transformeriData；
Transformer state variable historical data is arranged to set S=(z1,z2,...,zj,...,zn-1), wherein, zj= (x1(tj)*,x2(tj)*,...,x11(tj)*), j=1,2 ..., n-1 is j-th of data acquisition moment tjSample state variable is returned Value after one change, the classification number of the set S and corresponding initial cluster center of all categories are determined using max-min distance means, The classification each sample in the set S being included into respectively where closest cluster centre, and determine that fuzzy clustering is calculated Method other specification；
Further determined that using Fuzzy C-Means Cluster Algorithm is improved in new cluster corresponding to each classification of the set S The heart, formula are as follows：
Wherein, dijFor j-th of sample zjTo ith cluster center viEuclidean distance, c be classification number, ωjFor sample This zjDistribution of weights, ρijFor sample zjThe fuzzy membership of corresponding i-th of classification, l are the number of sample in each classification, m =2 be Fuzzy Exponential, and J is object function, wherein, fuzzy membership ρijObtained by below equation：
Wherein, dijFor j-th of sample zjTo ith cluster center viEuclidean distance, c be classification number；
The cluster centre of all categories is updated by the distribution of weights of training sample, formula is as follows：
Wherein, ωjFor sample zjDistribution of weights, ρijFor sample zjCorresponding classification i fuzzy membership, l is each classification The number of middle sample, zjRepresent j-th of sample, viRepresent new cluster centre；Repeat step is up to new cluster centre viIt is corresponding The difference of object function J and previous round object function be less than threshold epsilon, then algorithm terminates and determines c cluster centre；
T is calculated by the c cluster centre determinedk, k=n, n+1 ..., n+n1Moment transformer state variable number According to probability of all categories is belonged to, formula is as follows：
Wherein, ρiFor tkMoment transformer state variable data corresponds to classification i fuzzy membership, that is, belongs to the general of classification i Rate, diFor tkMoment transformer state variable data is to ith cluster center viEuclidean distance, c be classification number；
Calculating transformer tk, k=n, n+1 ..., n+n1The probability of happening of moment each failure, formula are as follows：
Wherein, ρjFor transformer t failure j probability of happening, ρiBelong to class for t transformer state variable data Other i probability,To count in the obtained set S jth kind fault type proportion, n in classification iijFor classification i The number of samples of middle jth kind fault type, niFor classification i number of samples；
Fuzzy clustering algorithm calculating transformer probability of malfunction based on comentropy described in the use, and by the transformer The record for the probability of happening that current measured data and prediction data correspond to each failure is as follows：
Wherein, tnFor current time,Respectively equally spaced prediction time, p (y, t) indication transformer t Failure y probability of happening.
As a kind of embodiment, the consequence to each failure of the transformer quantifies, according to the tkWhen The probability of happening of each failure of transformer is carved, calculates tkMoment transformer fault overall risk expense it is expected that specific steps include：
Transformer different faults pattern is analyzed, obtains different faults pattern consequence, by the different faults pattern Consequence is quantified as currency；
According to quantitative currency, t is determinedk, k=n, n+1 ..., n+n1The risk cost of each failure of moment transformer, it is public Formula is as follows：
Wherein, E (yi,j) for transformer occur fault of stop when risk cost, E*(yi,j) occur for transformer under efficiency Risk cost daily during failure, t dropnFor current time；
According to the tk, k=n, n+1 ..., n+n1The risk cost and the probability of happening of each failure of moment transformer Record determine tk, k=n, n+1 ..., n+n1Moment transformer fault overall risk expense it is expected that formula is as follows：
Wherein, cos t (yi,j,tk) represent tkMoment transformer breaks down yi,jWhen risk cost, p (yi,j,tk) table Show transformer tkMoment failure yi,jProbability of happening record.
As a kind of embodiment, Repair of Transformer strategy include interruption maintenance, maintenance of giving priority in arranging for, monitoring run, Periodic inspection and maintenance of delaying.
As a kind of embodiment, the determination tkThe expense of each Strategies of Maintenance of moment transformer it is expected, and and transformation Device failure overall risk expense it is expected to be compared, and determines transformer optimal repair time and optimal Strategies of Maintenance, specific steps bag Include：
Determine tk, k=n, n+1 ..., n+n1The expense of moment each Strategies of Maintenance it is expected that calculation formula is as follows：
Ei(tk)=fixedCost (i)+dayCosti(tk)
Wherein, fixedCost (i) is that the fixation cost of overhaul of i-th kind of scheme is used, dayCosti(tk) it is tkI-th kind of inspection of moment Repair the extra charge of strategy；
More each moment transformer fault overall risk expense it is expected R (tk) with the expense of each Strategies of Maintenance it is expected Ei (tk), optimal repair time point is found, optimal repair time point is in R (tk) ＜ Ei(tk), i=1,2 ..., 5 critical point, this When min (Ei(tk)) corresponding to strategy be optimal Strategies of Maintenance.
The present invention has significant technique effect as a result of above technical scheme：
The Repair of Transformer decision-making technique based on gray prediction and risk assessment of the present invention is in each failure of calculating transformer The difference of individual transformer and integrated transformers probability of malfunction is taken into full account during probability of happening, using Fuzzy C-Means Cluster Algorithm The probability of malfunction of calculating transformer, and the comentropy for characterizing sample distribution situation is substituted into fuzzy clustering as priori and calculated Method, improve the accuracy of transformer fault probability calculation；
After the Repair of Transformer decision-making technique based on gray prediction and risk assessment of the present invention has quantified transformer fault Fruit, each failure effect of transformer is characterized with the mode of monetary loss, is easily understood, it is versatile；
The Repair of Transformer decision-making technique based on gray prediction and risk assessment of the present invention proposes comprehensive transformer event Hinder the expectation of overall risk expense and the desired Repair of Transformer decision-making technique of each Strategies of Maintenance expense, can meet to become with the cost of minimum The reliability requirement of depressor；
The present invention's integrates gray prediction and risk based on the Repair of Transformer decision-making technique of gray prediction and risk assessment Appraisal procedure can not only propose optimal Strategies of Maintenance, moreover it is possible to recommend optimal maintenance time；
The present invention when carrying out maintenance decision, from economy and the aspect of reliability two, proposes rational transformation simultaneously Device Strategies of Maintenance and repair time suggest, contribute to the development of Repair of Transformer work.
Brief description of the drawings
In order to illustrate more clearly about the embodiment of the present invention or technical scheme of the prior art, below will be to embodiment or existing There is the required accompanying drawing used in technology description to be briefly described, it should be apparent that, drawings in the following description are only this Some embodiments of invention, for those of ordinary skill in the art, without having to pay creative labor, may be used also To obtain other accompanying drawings according to these accompanying drawings.
Fig. 1 is the flow chart of the present invention；
Fig. 2 is the schematic diagram of each fault rate of calculating transformer t in the present invention.
Embodiment
With reference to embodiment, the present invention is described in further detail, following examples be explanation of the invention and The invention is not limited in following examples.
Embodiment 1：
A kind of Repair of Transformer decision-making technique based on gray prediction and risk assessment, as shown in figure 1, including：
S1, transformer state variable, the historical data of transformer fault type and current measured data, and root are obtained respectively Transformer is obtained in t according to the method for gray predictionkThe predicted value of the state variable at moment；
S2, using the Fuzzy C-Means Cluster Algorithm based on comentropy, with reference to transformer in tkThe number of the state variable at moment According to calculating transformer tkThe probability of happening of moment each failure；
S3, the consequence to each failure of the transformer quantify, and with reference to the transformer tkThe hair of moment each failure Raw probability, calculates tkMoment transformer fault overall risk expense it is expected；
S4, determine tkThe expense of each Strategies of Maintenance of moment transformer it is expected, and it is expected with transformer fault overall risk expense It is compared, determines transformer optimal repair time and optimal Strategies of Maintenance.
In step sl, the state variable of the transformer includes oil dissolved gas H2、CH4、C2H2、C2H4、C2H6、CO、 CO2, furfural content in micro- water, winding absorptance, iron core grounding current and oil in oil, be designated as x respectively1-x11；
The transformer fault type includes winding failure, sleeve pipe failure, dielectric failure, iron core failure, tap and opened Failure, lead wire fault and oil leakage fault are closed, is designated as y respectively1-y7；
Every kind of fault type is labeled as two kinds of fault modes, two kinds of fault modes include cause transformer stoppage in transit pattern and Cause transformer efficiency reduction mode, be designated as y respectively1,1、y1,2、y2,1、y2,2…y7,1、y7,2。
Further, in step sl, it is described that state change of the transformer in t is obtained according to the method for gray prediction The specific practice of the predicted value of amount includes：
Use and transformer state variable is predicted with the gray model based on GM (1,1) model, with transformer shape State variable xi, i=1,2 ..., 11 historical data and current measured dataBased on be modeled, GM (1,1) was modeled Journey is as follows：
Wherein, t1,t2,...,tn-1For at intervals ofHistorical data at equal intervals collection the moment, tnFor current time,For tkMoment transformer state variable xiData, to current measured dataAdded up to obtainUse with Lower formula represents：
Wherein,
It is rightAlbinism differential equation is established, formula represents as follows：
Wherein, a is development coefficient, and b is grey actuating quantity, and by albinism differential equation discretization, formula represents as follows：
Wherein, k represents k-th of data acquisition moment,Data sampling interval is represented, is taken The albinism differential equation substituted into after discretization obtains：
Below equation is obtained by transposition：
K=1,2 ..., n-1 are made, the formula after transposition is converted into following result：
Now, following symbol is introduced：
Then have：
Order
Then have：
Further conversion obtains：
Make x(1)(t0)=x(0)(t1), obtain it is discrete after albinism differential equation solution, formula is as follows：
Wherein,For cumulative dataIn+1 sampling instant t of kthk+1Predicted value, now, Data convert is obtained into below equation：
Wherein,As transformer state variable xiIn k-th of sampling instant tkPredicted value xi(tk), k=1, 2,...；
By the current measured data of each state variable of the transformer and the transformer state variable prediction value xi(tk), k =1,2 ... recorded, record formula represents as follows：
Wherein, tnFor current time,Respectively equally spaced prediction time, x (t) represent t transformer shape State variable x value.
In step s 2, specific steps also include：
History, current and prediction data to the transformer state variable are normalized, and formula is as follows：
Wherein, t1~tn-1Moment, t are gathered for historical datanFor current time,It is pre- for transformer state variable Survey moment, xi(tk)*For t after normalizedkThe state variable x of moment transformeriData；
Transformer state variable historical data is arranged to set S=(z1,z2,...,zj,...,zn-1), wherein zj=(x1 (tj)*,x2(tj)*,...,x11(tj)*), j=1,2 ..., n-1 is j-th of data acquisition moment tjSample state variable normalizing Value after change, the classification number of the set S and corresponding initial cluster center of all categories are determined using max-min distance means, will Each sample in the set S is included into the classification where closest cluster centre respectively, and determines fuzzy clustering algorithm Other specification；
Further determined that using Fuzzy C-Means Cluster Algorithm is improved in new cluster corresponding to each classification of the set S The heart, formula are as follows：
Wherein, dijFor j-th of sample zjTo ith cluster center viEuclidean distance, c be classification number, ωjFor sample This zjDistribution of weights, ρijFor sample zjThe fuzzy membership of corresponding i-th of classification, l are the number of sample in each classification, m =2 be Fuzzy Exponential, and J is object function, wherein, fuzzy membership ρijObtained by below equation：
Wherein, dijFor j-th of sample zjTo ith cluster center viEuclidean distance, c be classification number；
The cluster centre of all categories is updated by the distribution of weights of training sample, formula is as follows：
Wherein, ωjFor sample zjDistribution of weights, ρijFor sample zjCorresponding classification i fuzzy membership, l is each classification The number of middle sample, zjRepresent j-th of sample, viRepresent new cluster centre；Repeat step is up to new cluster centre viIt is corresponding The difference of object function J and previous round object function be less than threshold epsilon, then algorithm terminates and determines c cluster centre；
T is calculated by the c cluster centre determinedk, k=n, n+1 ..., n+n1Moment transformer state variable number According to probability of all categories is belonged to, formula is as follows：
Wherein, ρiFor tkMoment transformer state variable data corresponds to classification i fuzzy membership, that is, belongs to the general of classification i Rate, diFor tkMoment transformer state variable data is to ith cluster center viEuclidean distance, c be classification number；
Calculating transformer tk, k=n, n+1 ..., n+n1The probability of happening of moment each failure, formula are as follows：
Wherein, ρjFor transformer t failure j probability of happening, ρiBelong to class for t transformer state variable data Other i probability,To count in the obtained set S jth kind fault type proportion, n in classification iijFor classification i The number of samples of middle jth kind fault type, niFor classification i number of samples；
Fuzzy clustering algorithm calculating transformer probability of malfunction based on comentropy described in the use, and by the transformer The record for the probability of happening that current measured data and prediction data correspond to each failure is as follows：
Wherein, tnFor current time,Respectively equally spaced prediction time, p (y, t) indication transformer t Failure y probability of happening.
In step s3, specific steps also include：
Transformer different faults pattern is analyzed, obtains different faults pattern consequence, by the different faults pattern Consequence is quantified as currency, as follows：
The risk cost of each failure of the transformer of table 1
Referring to the risk cost of each failure of the transformer of table 1, according to quantitative currency, t is determinedk, k=n, n+1 ..., n+ n1The failure risk expense of moment transformer, formula are as follows：
Wherein, E (yi,j) for transformer occur fault of stop when risk cost, E*(yi,j) occur for transformer under efficiency Risk cost daily during failure, t dropnFor current time；
According to the tk, k=n, n+1 ..., n+n1The risk cost and the probability of happening of each failure of moment transformer Record determine tk, k=n, n+1 ..., n+n1Moment transformer fault overall risk expense it is expected that formula is as follows：
Wherein, cos t (yi,j,tk) represent tkMoment transformer breaks down yi,jWhen risk cost, p (yi,j,tk) table Show transformer tkMoment failure yi,jProbability of happening record.
Further, Repair of Transformer strategy includes interruption maintenance, maintenance of giving priority in arranging for, monitoring operation, periodic inspection And maintenance of delaying.
In step s 4, specific steps also include：
Determine tk, k=n, n+1 ..., n+n1The expense of each Strategies of Maintenance of moment transformer it is expected that calculation formula is as follows：
Ei(tk)=fixedCost (i)+dayCosti(tk)
Wherein, fixedCost (i) is that the fixation cost of overhaul of i-th kind of scheme is used, dayCosti(tk) it is tkI-th kind of inspection of moment Repair the extra charge of strategy；
Each Strategies of Maintenance expense of transformer is given in table 2 it is expected：
Each Strategies of Maintenance expense of the transformer of table 2 it is expected
More each moment transformer fault overall risk expense it is expected R (tk) with the expense of each Strategies of Maintenance it is expected Ei (tk), optimal repair time point is in R (tk) ＜ Ei(tk), i=1,2 ..., 5 critical point, now min (Ei(tk)) corresponding to plan Slightly optimal Strategies of Maintenance, transformer fault overall risk expense it is expected it is expected as shown in table 3 with each Strategies of Maintenance expense：
t R(t) E1(t) E2(t) E3(t) E4(t) E5(t)
tn R(tn) E1(tn) E2(tn) E3(tn) E3(tn) E3(tn)
tn+1 R(tn+1) E1(tn+1) E2(tn+1) E3(tn+1) E3(tn+1) E3(tn+1)
… … … … … … …
tk R(tk) E1(tk) E2(tk) E3(tk) E3(tk) E3(tk)
… … … … … … …
tn+n1 R(tn+n1) E1(tn+n1) E2(tn+n1) E3(tn+n1) E3(tn+n1) E3(tn+n1)
The transformer fault overall risk expense of table 3 it is expected it is expected with each Strategies of Maintenance expense
Furthermore, it is necessary to illustrate, the specific embodiment described in this specification, the shape of its parts and components, it is named Title etc. can be different.The equivalent or simple change that all construction, feature and principles according to described in inventional idea of the present invention are done, is wrapped Include in the protection domain of patent of the present invention.Those skilled in the art can be to described specific implementation Example is made various modifications or supplement or substituted using similar mode, structure without departing from the present invention or surmounts this Scope as defined in the claims, protection scope of the present invention all should be belonged to.

Claims (7)

  1. A kind of 1. Repair of Transformer decision-making technique based on gray prediction and risk assessment, it is characterised in that including：
    Transformer state variable, the historical data of transformer fault type and current measured data are obtained respectively, and according to grey The method of prediction obtains transformer in tkThe predicted value of the state variable at moment；
    Using the Fuzzy C-Means Cluster Algorithm based on comentropy, with reference to transformer in tkThe data of the state variable at moment, calculate Transformer tkThe probability of happening of moment each failure；
    The consequence of each failure of the transformer is quantified, and with reference to the transformer tkThe probability of happening of moment each failure, meter Calculate tkMoment transformer fault overall risk expense it is expected；
    Determine tkThe expense expectation of each Strategies of Maintenance of moment transformer, and compared with the expectation of transformer fault overall risk expense, Determine transformer optimal repair time and optimal Strategies of Maintenance.
  2. 2. the Repair of Transformer decision-making technique based on gray prediction and risk assessment according to claim 1, it is characterised in that The state variable of the transformer includes oil dissolved gas H2、CH4、C2H2、C2H4、C2H6、CO、CO2, oil in micro- water, winding inhale Furfural content in receipts ratio, iron core grounding current and oil, is designated as x respectively1-x11；
    The transformer fault type includes winding failure, sleeve pipe failure, dielectric failure, iron core failure, shunting switch event Barrier, lead wire fault and oil leakage fault, are designated as y respectively1-y7；
    Every kind of fault type is labeled as two kinds of fault modes, two kinds of fault modes, which include, to be caused transformer stoppage in transit pattern and cause Transformer efficiency reduction mode, is designated as y respectively1,1、y1,2、y2,1、y2,2…y7,1、y7,2。
  3. 3. the Repair of Transformer decision-making technique based on gray prediction and risk assessment according to claim 1, it is characterised in that It is described that transformer is obtained in t according to the method for gray predictionkThe specific practice of the predicted value of the state variable at moment includes：
    Use and transformer state variable is predicted with the gray model based on GM (1,1) model, become with transformer state Measure xi, i=1,2 ..., 11 historical data and current measured dataBased on be modeled, GM (1,1) modeling process is such as Under：
    <mrow> <msubsup> <mi>X</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <mrow> <mo>(</mo> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </msubsup> <mo>(</mo> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>)</mo> <mo>,</mo> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </msubsup> <mo>(</mo> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>)</mo> <mo>,</mo> <mn>...</mn> <mo>,</mo> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </msubsup> <mo>(</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>)</mo> <mo>,</mo> <mn>...</mn> <mo>,</mo> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </msubsup> <mo>(</mo> <msub> <mi>t</mi> <mi>n</mi> </msub> <mo>)</mo> <mo>)</mo> </mrow> </mrow>
    Wherein, t1,t2,...,tn-1For at intervals ofHistorical data at equal intervals collection the moment, tnFor current time,For tkMoment transformer state variable xiData, to current measured dataAdded up to obtainUsing below equation table Show：
    <mrow> <msubsup> <mi>X</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <mrow> <mo>(</mo> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mo>(</mo> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>)</mo> <mo>,</mo> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mo>(</mo> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>)</mo> <mo>,</mo> <mn>...</mn> <mo>,</mo> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mo>(</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>)</mo> <mo>,</mo> <mn>...</mn> <mo>,</mo> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </msubsup> <mo>(</mo> <msub> <mi>t</mi> <mi>n</mi> </msub> <mo>)</mo> <mo>)</mo> </mrow> </mrow>
    Wherein,
    It is rightAlbinism differential equation is established, formula represents as follows：
    <mrow> <mfrac> <mrow> <msubsup> <mi>dX</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> <mo>+</mo> <msubsup> <mi>aX</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <mi>b</mi> </mrow>
    Wherein, a is development coefficient, and b is grey actuating quantity, and by albinism differential equation discretization, formula represents as follows：
    <mrow> <msubsup> <mi>dX</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mo>-</mo> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow>
    Wherein, k represents k-th of data acquisition moment,Data sampling interval is represented, is taken The albinism differential equation substituted into after discretization obtains：
    <mrow> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>a</mi> <mrow> <mo>(</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>(</mo> <mrow> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mo>=</mo> <mi>b</mi> </mrow>
    Below equation is obtained by transposition：
    <mrow> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mi>a</mi> <mrow> <mo>(</mo> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>(</mo> <mrow> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mo>+</mo> <mi>b</mi> </mrow>
    K=1,2 ..., n-1 are made, the formula after transposition is converted into following result：
    <mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>...</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>a</mi> <mrow> <mo>(</mo> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>(</mo> <mrow> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mo>+</mo> <mi>b</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>a</mi> <mrow> <mo>(</mo> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>(</mo> <mrow> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mo>+</mo> <mi>b</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>...</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>a</mi> <mrow> <mo>(</mo> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>(</mo> <mrow> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mo>+</mo> <mi>b</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mi>a</mi> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mo>(</mo> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>)</mo> <mo>+</mo> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mo>(</mo> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mo>(</mo> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>)</mo> <mo>+</mo> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mo>(</mo> <msub> <mi>t</mi> <mn>3</mn> </msub> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>...</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mo>(</mo> <msub> <mi>t</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> <mo>+</mo> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mo>(</mo> <msub> <mi>t</mi> <mi>n</mi> </msub> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mi>b</mi> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>...</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> </mrow>
    Now, following symbol is introduced：
    <mrow> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>...</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <msup> <mi>Z</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msup> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mo>(</mo> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>)</mo> <mo>+</mo> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mo>(</mo> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mo>(</mo> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>)</mo> <mo>+</mo> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mo>(</mo> <msub> <mi>t</mi> <mn>3</mn> </msub> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>...</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mo>(</mo> <msub> <mi>t</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> <mo>+</mo> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mo>(</mo> <msub> <mi>t</mi> <mi>n</mi> </msub> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <mi>E</mi> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>...</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> </mrow>
    Then have：
    <mrow> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mo>=</mo> <msup> <mi>aZ</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mi>E</mi> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msup> <mi>Z</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msup> </mtd> <mtd> <mi>E</mi> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mi>a</mi> </mtd> </mtr> <mtr> <mtd> <mi>b</mi> </mtd> </mtr> </mtable> </mfenced> </mrow>
    Order
    <mrow> <mi>B</mi> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msup> <mi>Z</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msup> </mtd> <mtd> <mi>E</mi> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mo>(</mo> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>)</mo> <mo>+</mo> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mo>(</mo> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mo>(</mo> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>)</mo> <mo>+</mo> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mo>(</mo> <msub> <mi>t</mi> <mn>3</mn> </msub> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>...</mn> </mtd> <mtd> <mn>...</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mo>(</mo> <msub> <mi>t</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> <mo>+</mo> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mo>(</mo> <msub> <mi>t</mi> <mi>n</mi> </msub> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <mover> <mi>a</mi> <mo>^</mo> </mover> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mi>a</mi> </mtd> </mtr> <mtr> <mtd> <mi>b</mi> </mtd> </mtr> </mtable> </mfenced> </mrow>
    Then have：
    <mrow> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mo>=</mo> <mi>B</mi> <mover> <mi>a</mi> <mo>^</mo> </mover> </mrow>
    Further conversion obtains：
    <mrow> <mover> <mi>a</mi> <mo>^</mo> </mover> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mi>a</mi> </mtd> </mtr> <mtr> <mtd> <mi>b</mi> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>B</mi> <mi>T</mi> </msup> <mi>B</mi> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>B</mi> <mi>T</mi> </msup> <msub> <mi>Y</mi> <mi>n</mi> </msub> </mrow>
    Make x(1)(t0)=x(0)(t1), obtain it is discrete after albinism differential equation solution, formula is as follows：
    <mrow> <msubsup> <mover> <mi>x</mi> <mo>^</mo> </mover> <mi>i</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </msubsup> <mo>(</mo> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>)</mo> <mo>-</mo> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>a</mi> <mi>k</mi> </mrow> </msup> <mo>+</mo> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> </mrow>
    Wherein,For cumulative dataIn+1 sampling instant t of kthk+1Predicted value, now, by data Reduction obtains below equation：
    <mrow> <msubsup> <mover> <mi>x</mi> <mo>^</mo> </mover> <mi>i</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mover> <mi>x</mi> <mo>^</mo> </mover> <mi>i</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msubsup> <mover> <mi>x</mi> <mo>^</mo> </mover> <mi>i</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow>
    Wherein,As transformer state variable xiIn k-th of sampling instant tkPredicted value xi(tk), k=1,2 ...；
    By the current measured data of each state variable of the transformer and the transformer state variable prediction value xi(tk), k=1, 2 ... recorded, record formula represents as follows：
    <mrow> <mi>x</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msub> <mi>x</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mrow> <msub> <mi>x</mi> <mn>11</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msub> <mi>x</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mrow> <msub> <mi>x</mi> <mn>11</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>...</mn> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mn>...</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mrow> <mi>n</mi> <mo>+</mo> <msub> <mi>n</mi> <mn>1</mn> </msub> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msub> <mi>x</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mrow> <mi>n</mi> <mo>+</mo> <msub> <mi>n</mi> <mn>1</mn> </msub> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mrow> <msub> <mi>x</mi> <mn>11</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mrow> <mi>n</mi> <mo>+</mo> <msub> <mi>n</mi> <mn>1</mn> </msub> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow>
    Wherein, tnFor current time,Respectively equally spaced prediction time, x (t) represent that t transformer state becomes Measure x value.
  4. 4. the Repair of Transformer decision-making technique based on gray prediction and risk assessment according to claim 1, it is characterised in that It is described to use the Fuzzy C-Means Cluster Algorithm based on comentropy, with reference to transformer in tkThe data of the state variable at moment, calculate Transformer tkThe probability of happening of moment each failure, specific steps include：
    History, current and prediction data to the transformer state variable are normalized, and formula is as follows：
    <mrow> <msub> <mi>x</mi> <mi>i</mi> </msub> <msup> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>*</mo> </msup> <mo>=</mo> <mfrac> <mrow> <msub> <mi>x</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <munderover> <mi>max</mi> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>k</mi> <mo>=</mo> <mi>n</mi> <mo>+</mo> <msub> <mi>n</mi> <mn>1</mn> </msub> </mrow> </munderover> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>(</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mfrac> <mo>,</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mn>11</mn> </mrow>
    Wherein, t1~tn-1Moment, t are gathered for historical datanFor current time,For transformer state variable prediction when Carve, xi(tk)*For t after normalizedkThe state variable x of moment transformeriData；
    Transformer state variable historical data is arranged to set S=(z1,z2,...,zj,...,zn-1), wherein, zj=(x1 (tj)*,x2(tj)*,...,x11(tj)*), j=1,2 ..., n-1 is j-th of data acquisition moment tjSample state variable normalizing Value after change, the classification number of the set S and corresponding initial cluster center of all categories are determined using max-min distance means, will Each sample in the set S is included into the classification where closest cluster centre respectively, and determines fuzzy clustering algorithm Other specification；
    New cluster centre corresponding to each classification of the set S is further determined that using Fuzzy C-Means Cluster Algorithm is improved, it is public Formula is as follows：
    <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mi>J</mi> <mo>=</mo> <mstyle> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>1</mn> </munderover> </mstyle> <mstyle> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>c</mi> </munderover> </mstyle> <msubsup> <mi>&amp;omega;</mi> <mi>j</mi> <mi>m</mi> </msubsup> <mo>*</mo> <msubsup> <mi>&amp;rho;</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> <mi>m</mi> </msubsup> <mo>*</mo> <msubsup> <mi>d</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> <mn>2</mn> </msubsup> </mtd> </mtr> <mtr> <mtd> <mrow> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>c</mi> </munderover> <msub> <mi>&amp;rho;</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;rho;</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>&amp;Element;</mo> <mo>&amp;lsqb;</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>&amp;rsqb;</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&amp;omega;</mi> <mi>j</mi> </msub> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>c</mi> </munderover> <msubsup> <mi>d</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> <mn>2</mn> </msubsup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mtd> </mtr> </mtable> </mfenced>
    Wherein, dijFor j-th of sample zjTo ith cluster center viEuclidean distance, c be classification number, ωjFor sample zj Distribution of weights, ρijFor sample zjThe fuzzy membership of corresponding i-th of classification, l are the number of sample in each classification, and m=2 is Fuzzy Exponential, J are object function, wherein, fuzzy membership ρijObtained by below equation：
    <mrow> <msub> <mi>&amp;rho;</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mfrac> <msubsup> <mi>d</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> <mfrac> <mn>2</mn> <mrow> <mi>m</mi> <mo>-</mo> <mn>1</mn> </mrow> </mfrac> </msubsup> <mrow> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>c</mi> </munderover> <msubsup> <mi>d</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> <mfrac> <mn>2</mn> <mrow> <mi>m</mi> <mo>-</mo> <mn>1</mn> </mrow> </mfrac> </msubsup> </mrow> </mfrac> </mrow>
    Wherein, dijFor j-th of sample zjTo ith cluster center viEuclidean distance, c be classification number；
    The cluster centre of all categories is updated by the distribution of weights of training sample, formula is as follows：
    <mrow> <msub> <mi>v</mi> <mi>i</mi> </msub> <mo>=</mo> <mfrac> <mrow> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>l</mi> </munderover> <msubsup> <mi>&amp;omega;</mi> <mi>j</mi> <mi>m</mi> </msubsup> <msubsup> <mi>&amp;rho;</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> <mi>m</mi> </msubsup> <mo>*</mo> <msub> <mi>z</mi> <mi>j</mi> </msub> </mrow> <mrow> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msubsup> <mi>&amp;omega;</mi> <mi>j</mi> <mi>m</mi> </msubsup> <msubsup> <mi>&amp;rho;</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> <mi>m</mi> </msubsup> </mrow> </mfrac> </mrow>
    Wherein, ωjFor sample zjDistribution of weights, ρijFor sample zjCorresponding classification i fuzzy membership, l are sample in each classification This number, zjRepresent j-th of sample, viRepresent new cluster centre；Repeat step is up to new cluster centre viCorresponding mesh Scalar functions J and previous round object function difference are less than threshold epsilon, then algorithm terminates and determines c cluster centre；
    T is calculated by the c cluster centre determinedk, k=n, n+1 ..., n+n1Moment transformer state variable data category It is as follows in probability of all categories, formula：
    <mrow> <msub> <mi>&amp;rho;</mi> <mi>i</mi> </msub> <mo>=</mo> <mfrac> <msubsup> <mi>d</mi> <mi>i</mi> <mfrac> <mn>2</mn> <mrow> <mi>m</mi> <mo>-</mo> <mn>1</mn> </mrow> </mfrac> </msubsup> <mrow> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>c</mi> </munderover> <msubsup> <mi>d</mi> <mi>i</mi> <mfrac> <mn>2</mn> <mrow> <mi>m</mi> <mo>-</mo> <mn>1</mn> </mrow> </mfrac> </msubsup> </mrow> </mfrac> </mrow>
    Wherein, ρiFor tkMoment transformer state variable data corresponds to classification i fuzzy membership, that is, belongs to classification i probability, diFor tkMoment transformer state variable data is to ith cluster center viEuclidean distance, c be classification number；
    Calculating transformer tk, k=n, n+1 ..., n+n1The probability of happening of moment each failure, formula are as follows：
    <mrow> <msub> <mi>&amp;rho;</mi> <mi>j</mi> </msub> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>c</mi> </munderover> <msub> <mi>&amp;rho;</mi> <mi>i</mi> </msub> <mo>&amp;times;</mo> <msub> <mi>f</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow>
    Wherein, ρjFor transformer t failure j probability of happening, ρiBelong to classification i's for t transformer state variable data Probability,To count in the obtained set S jth kind fault type proportion, n in classification iijFor jth in classification i The number of samples of kind fault type, niFor classification i number of samples；
    Fuzzy clustering algorithm calculating transformer probability of malfunction based on comentropy described in the use, and the transformer is current The record that measured data and prediction data correspond to the probability of happening of each failure is as follows：
    <mrow> <mi>p</mi> <mrow> <mo>(</mo> <mi>y</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>p</mi> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mrow> <mi>p</mi> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>p</mi> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mrow> <mi>p</mi> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>...</mn> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mn>...</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>p</mi> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow> <mi>n</mi> <mo>+</mo> <msub> <mi>n</mi> <mn>1</mn> </msub> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mrow> <mi>p</mi> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mrow> <mn>7</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow> <mi>n</mi> <mo>+</mo> <msub> <mi>n</mi> <mn>1</mn> </msub> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow>
    Wherein, tnFor current time,Respectively equally spaced prediction time, p (y, t) indication transformer t failure Y probability of happening.
  5. 5. the Repair of Transformer decision-making technique based on gray prediction and risk assessment according to claim 4, it is characterised in that The consequence to each failure of the transformer quantifies, according to the tkThe probability of happening of each failure of moment transformer, meter Calculate tkMoment transformer fault overall risk expense it is expected that specific steps include：
    Transformer different faults pattern is analyzed, obtains different faults pattern consequence, by the different faults pattern consequence It is quantified as currency；
    According to quantitative currency, t is determinedk, k=n, n+1 ..., n+n1The risk cost of each failure of moment transformer, formula is such as Under：
    Wherein, E (yi,j) for transformer occur fault of stop when risk cost, E*(yi,j) it is that efficiency decline event occurs for transformer Daily risk cost, t during barriernFor current time；
    According to the tk, k=n, n+1 ..., n+n1The risk cost of each failure of moment transformer and the note of the probability of happening Record to determine tk, k=n, n+1 ..., n+n1Moment transformer fault overall risk expense it is expected that formula is as follows：
    <mrow> <mi>R</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>7</mn> </munderover> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>2</mn> </munderover> <mi>cos</mi> <mi>t</mi> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>&amp;times;</mo> <mi>p</mi> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>(</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>)</mo> <mo>)</mo> </mrow> </mrow>
    Wherein, cost (yi,j,tk) represent tkMoment transformer breaks down yi,jWhen risk cost, p (yi,j,tk) represent transformation Device tkMoment failure yi,jProbability of happening record.
  6. 6. the Repair of Transformer decision-making technique based on gray prediction and risk assessment according to claim 1, it is characterised in that Repair of Transformer strategy includes interruption maintenance, maintenance of giving priority in arranging for, monitoring operation, periodic inspection and maintenance of delaying.
  7. 7. the Repair of Transformer decision-making technique based on gray prediction and risk assessment according to claim 6, it is characterised in that The determination tkThe expense of each Strategies of Maintenance of moment transformer it is expected, and it is expected to be compared with transformer fault overall risk expense Compared with determining that transformer optimal repair time and optimal Strategies of Maintenance, specific steps include：
    Determine tk, k=n, n+1 ..., n+n1The expense of moment each Strategies of Maintenance it is expected that calculation formula is as follows：
    Ei(tk)=fixedCost (i)+dayCosti(tk)
    Wherein, fixedCost (i) is that the fixation cost of overhaul of i-th kind of scheme is used, dayCosti(tk) it is tkI-th kind of maintenance plan of moment Extra charge slightly；
    More each moment transformer fault overall risk expense it is expected R (tk) with the expense of each Strategies of Maintenance it is expected Ei(tk), seek Optimal repair time point is looked for, optimal repair time point is in R (tk) ＜ Ei(tk), i=1,2 ..., 5 critical point, now min (Ei (tk)) corresponding to strategy be optimal Strategies of Maintenance.
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CN111539515A (en) * 2020-04-21 2020-08-14 中国电子科技集团公司第三十八研究所 Complex equipment maintenance decision method based on fault prediction
CN112115417A (en) * 2020-08-10 2020-12-22 贵州电网有限责任公司 Transformer real-time fault probability calculation method based on probability density clustering
CN112115417B (en) * 2020-08-10 2024-02-27 贵州电网有限责任公司 Transformer real-time fault probability calculation method based on probability density clustering
CN114077933A (en) * 2022-01-13 2022-02-22 常州艾肯智造科技有限公司 Real-time transmission system and method for steam on-line monitoring data

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