CN102721941B - Method for fusing and diagnosing fault information of circuit of electric meter on basis of SOM (self-organized mapping) and D-S (Dempster-Shafer) theories - Google Patents

Abstract

A method for fusing and diagnosing fault information of a circuit of an electric meter on the basis of SOM (self-organized mapping) and D-S (Dempster-Shafer) theories includes firstly, creating a fault mode set for the faulty circuit according to circuit analysis of the faulty circuit of the electric energy meter and fault modes specified by GJB299C; secondly, selecting to-be-observed fault signal points corresponding to fault modes in the set according to the fault mode set created in the first step and using the to-be-observed fault signal points as test points for functions and states of the circuit; thirdly, preprocessing fault signals acquired at the fault signal points selected in the second step; fourthly, fusing fault information by the aid of the SOM theory, outputting fault conclusions, selecting 70% of data for training and selecting 30% of the data for testing; and fifthly, fusing the fault conclusions by the aid of the D-S theory and making a decision for faults. By the aid of the method, confidence degree of a fault diagnostic result is further increased, integral uncertainty caused by errors is reduced, accuracy of fault diagnosis is greatly improved, and the method is an extremely important means in the field of information fusion.

Description

A kind of meter circuit failure message based on SOM and D-S theory merges and diagnostic method

(1) technical field:

The invention provides a kind of electric energy meter circuit failure message merges and faulty circuit diagnostic method, relate in particular to that a kind of electric energy meter circuit failure message based on Self-Organizing Feature Maps (based on SOM) merges and (D-S theory is first to be proposed in 1967 by Dempster based on D-S theory, a kind of inexact reasoning theory being further developed in 1976 by his student shafer, also referred to as evidence theory or D-S evidence theory) the method for electric energy meter circuit fault diagnosis.It belongs to fault information fusion and diagnostic field.

(2) background technology:

Nearly ten years, typical case as system failure emulation technology in circuit design field applies---and circuit performance, reliability and testability Comprehensive Analysis Technique based on fault simulation have been obtained very fast development, in this technical research, utilize eda tool to carry out malfunction emulation to circuit, obtain the fault quantitative analysis results of objective circuit, utilize the each node failure data of circuit of gained, by given analytical algorithm, estimate circuit test parameter, provide the predicted value of fault detect rate/isolation rate.This technology can realize robotization and the intellectuality of performance evaluation, fail-safe analysis and testability analysis process, is the important technology development of electronic product comprehensive Design and analytical technology.

But this technology Shortcomings still at present, this technical research and application also mainly concentrate on circuit board level at present, and for research and the application of Circuits System level, because components and parts fault mode is too many, simulation time is long and scale of model is excessive etc., reason also cannot, by the sorted generalization that affects of fault, cannot be delivered to upper strata circuit.On the other hand, in circuit fault diagnosis field, the development of failure prediction technology is also immature.Require to obtain the status information such as operation characteristic parameter, environmental parameter of real-time watch device, thereby the data of the current ruuning situation of institute's extraction equipment are carried out to parameter analysis and failure prediction, and these parameters of electronic system are difficult to obtain.

Meanwhile, information fusion technology becomes the focus of current research, for pattern-recognition and failure prediction have been opened up new approach.Multi-level, many-sided processing such as information fusion refers to automatically detecting with data from the information of single or multiple sensors (or information source), associated, relevant, estimation and combination, to obtain, accurate target location is estimated and complete target identities is estimated, and battlefield situation, threat and significance level thereof are carried out to appropriate estimation.It utilizes the redundancy of information and complementarity can expand space-time hunting zone, improves target detectivity, improves detection performance; The resolution in raising time or space, the dimension of increase target signature vector, the uncertainty of reduction information, the degree of confidence of improvement information; Strengthen system survivability and adaptive ability; The thing followed is to reduce the fog-level of reasoning, has improved decision-making capability, thereby the performance of whole system is improved greatly.

In this patent, characteristic layer merges by SOM input data is classified, and calculates the trusting degree of classification results, but, when evidence quantity increases or when classification results has conflict, only relies on SOM just can not draw compellent diagnostic result.Therefore carrying out D-S evidence fusion in decision-making level, the trusting degree of the result to fault diagnosis is further increased, reduced because the entirety uncertainty that error is brought, greatly improved the accuracy of fault diagnosis, is the very important means in information fusion field.

(3) summary of the invention:

1, object: a kind of method that the object of this invention is to provide fusion of the failure message based on Self-Organizing Feature Maps and the electric energy meter circuit fault diagnosis based on D-S evidence theory.

2, technical scheme: the present invention is achieved by the following technical solutions:

The invention provides a kind of meter circuit failure message based on SOM and D-S evidence theory and merge and diagnostic method, the method concrete steps are as follows:

Step 1: according to the circuit analysis to electrical energy meter fault circuit, and the fault mode specifying according to GJB299C, faulty circuit is set up to a fault mode set;

Step 2: according to the fault mode set of step 1 choose with gather in the corresponding fault-signal point (voltage of certain trouble spot, electric current etc.) to be observed of fault mode, as the test point of circuit function and state;

Step 3: the fault-signal that the fault-signal point place choosing in step 2 is obtained carries out pre-service;

The feature adopting due to the present invention comprises temporal characteristics, frequecy characteristic and statistical nature, so carrying out in Signal Pretreatment process also adopting the processing mode of going forward one by one, as described below:

(1) first to the fault-signal interpolation of obtaining, thereby unified signal granularity obtains the change procedure curve of fault-signal, i.e. temporal characteristics;

(2) on the basis of (1), temporal characteristics is normalized, carries out discrete Fourier transformation, picked up signal frequency spectrum and correlated characteristic;

(3) on the basis of (1), calculate the statistical property of signal, comprise second order distance, quadravalence distance, maximal values etc., set up eigenvectors matrix;

According to the feature of circuit signal, the present invention adopts the time, and the information of frequency and three dimensions of statistical nature is carried out signal fused, and its schematic diagram as shown in Figure 2;

Step 4: use SOM to carry out failure message fusion, output fault conclusion; The data of selection 70% are trained, and the data of selection 30% are tested; In order to keep three stack features data unified, SOM has identical basic setup; SOM is provided with 1 input layer; In network layer, only comprise 1 output layer, without hidden layer, 9 neurons are set, use and connect distance function calculating input vector to neuronic distance, transition function is made as competitive type compet, and topological structure is hexagonal network etale topology function hextop; The initialization function of input weight vector (inputWeights) is just value initialization of mid point, and learning function is Self-organizing Maps weights learning function (learnsom); Iterations is 80;

Step 5: use D-S evidence theory to merge fault conclusion, do the decision-making of being out of order; Under certain fault mode, failure message, through the information fusion of characteristic layer SOM, has comprised correct conclusion in acquired results, but the unsteady scope of SOM recognition result is larger, also need to dwindle its scope by decision-making level's information fusion, obtain more accurate fault diagnosis conclusion; According to the result of fault simulation, set up the identification framework of diagnosing information fusion fault;

Wherein, described in step 1, faulty circuit is being set up to fault mode set according to GJB299C, the method of its foundation is as follows: the fault mode comprising according to GJB299C, and the fault mode of each trouble spot to circuit under test is numbered and list, and concrete form can be as shown in Table 1;

Form 1 fault mode set

Sequence number	Fault mode	Sequence number	Fault mode
				M1	C1-parameter drift+5%	M8	R1-parameter drift-5%
M2	The Q1-open circuit c utmost point	M9	R1-open circuit
				M3	The Q1-open circuit b utmost point	M10	C1-parameter drift-5%
M4	The Q1-open circuit e utmost point	M11	C2-parameter drift+5%
				M5	The Q1-short circuit cb utmost point	M12	C2-parameter drift-5%

M6	The Q1-short circuit be utmost point	M13	C2-open circuit
				M7	R1-parameter drift+5%	M14	C2-short circuit

Wherein, described in step 3, the fault-signal obtaining being carried out to pre-service, is to adopt the mode of normalization and interpolation to carry out pre-service to signal; Normalization is that data larger amplitude range are mapped on another interval by certain rule change, as ([0,1] or [1,1]), conventionally the input data of model have different dimensions, represent different meanings, by normalization to reach minimizing data volume, the object of computation complexity and unified dimension; To a series of data point (x on [a, b] interval ₀, y ₀) (x ₁, y ₁) (x ₂, y ₂) ... (x _n, y _n), main a normalized mapping rule is as follows:

$y=\frac{(y_{\max }-y_{\min })(x-x_{\min })}{x_{\max }-x_{\min }}+y_{\min }---\left(1\right)$

$\stackrel{\‾}{x}=\frac{1}{N}\underset{i=1}{\overset{N}{\mathrm{\Σ}}}{x}_i$

$S^2=\frac{1}{N}\underset{i=1}{\overset{N}{\mathrm{\Σ}}}{(x_i-\stackrel{\‾}{x})}^2$

$y_i=\frac{x_i-\stackrel{\‾}{x}}{S}---\left(2\right)$

In formula, y _min, y _maxfor the minimum and maximum value of y, x _min, x _maxit is the minimum and maximum value of x.

The data of processing according to formula (1) can limit the border of mapping, belong to a kind of linear mapping; And according to the processing of formula (2) standardization that is otherwise known as, the data mean value that obtains is 0, variance is 1;

Interpolation has been generally the method that solves the functional value of middle unknown point between function region and adopt; The method of method of interpolation is:

If there is following data point on [a, b] interval:

(x ₀,y ₀)(x ₁,y ₁)(x ₂,y ₂)……(x _n,y _n)

Wherein y _i=f (x _i), i=0,1 ... n, x ₀, x ₁x _nbe called node; Value according to f (x) at node, constructs a smooth enough and fairly simple function be called interpolating function, as the approximate expression of f (x), then calculate in the functional value of interval [a, b] (being called interpolation section) upper any point x, the approximate value as original f (x) at this point; The general algebraic polynomial that adopts is as interpolating function;

In the time of n=1, Algebraic interpolation problem becomes, through two point (x ₀, y ₀) and (x ₁, y ₁), ask 1 interpolation polynomial p of function y=f (x) ₁(x),, according to cartesian geometry knowledge, can obtain

$p_1\left(x\right)=\frac{x-x_1}{x_0-x_1}{y}_0+\frac{x-x_0}{x_1-x_0}{y}_1=l_0\left(x\right)y_0+l_1\left(x\right)y_1---\left(4\right)$

Wherein, l ₀, l ₁being called Interpolation-Radix-Function one time, is all an order polynomial, p ₁(x) be a lagrange polynomial; Can verify, they meet

l ₀(x)+l ₁(x)=1 （5）

L ₀(x)=1, l ₁(x)=0 or l ₀(x)=0, l ₁(x)=1 (6)

At x, under the certain condition of y, given unified interpolation moment point, can carry out interpolation successively according to above-mentioned formula, obtains the unified time series of granularity;

Wherein, at the statistical nature of the signal described in step 3, computing method are as follows:

Suppose that the fault-signal obtaining after pre-service is x ₁, x ₂..., x _n, sampling time interval is Δ t, the statistical property of employing has secondary distance, and four squares, maximum value and maximum amplitude point, formula is as follows:

Second order distance $x_1={\left(\frac{1}{m}\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{y}_i^2\right)}^{\frac{1}{2}}---\left(7\right)$

Quadravalence distance $x_2={\left(\frac{1}{m}\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{y}_i^4\right)}^{\frac{1}{4}}---\left(8\right)$

Maximum value $x_3=\underset{i}{\max }\left|y_i\right|---\left(9\right)$

Maximum amplitude point x ₄if=k Δ t $y_k=\underset{i=1}{\overset{m}{\max }}{y}_i---\left(10\right)$

Wherein, carry out failure message fusion at the use self-organized mapping network described in step 4, the learning algorithm of the self-organized mapping network of use is as follows:

(1) initialization; The each weight vector of output layer is given little random number and is normalized, obtain random manifold after treatment j=1,2 ..., m, sets up initial winning neighborhood N _j. (0), learning rate is composed initial value;

(2) input learning sample; From training set, choose an input pattern and be normalized, obtaining the input vector after normalized p ∈ 1,2 ..., p}

(3) find triumph node; Calculate with dot product, j=1,2,, m, therefrom selects the triumph node j of dot product maximum ^*; If input pattern is without normalized, should be according to calculate Euclidean distance, find out the minimum triumph node of distance;

(4) define winning neighborhood N _j. (t) with j ^*centered by determine that the weights in t moment adjust territory, general initial neighborhood N _j. (0) is larger, N in training process _j. shrink in time gradually (0);

(5) adjust weights, for winning neighborhood N _j. all knot adjustment weights in (0):

$W_{\mathrm{ij}}(t+1)=W_{\mathrm{ij}}\left(t\right)+\mathrm{\η}(t,N)[x_i^p-W_{\mathrm{ij}}\left(t\right)];i=\mathrm{1,2},\·\·\·,n,j\∈N_j.\left(0\right)$

Wherein, η (t, N) is the function of the topology distance N between interior j the neuron of training time t and neighborhood and triumph neuron;

(6) finish to check, training finish be taking learning rate whether decay to 0 or certain predetermined positive decimal as condition, do not satisfy condition and get back to step 2,3,4;

Wherein, the step at the use D-S evidence theory described in step 5, fault conclusion being merged is as follows:

(1) taking the output p (Fi) of feature fusion as basis, calculate fault elementary probability according to following formula (16) and count mT, mF and mS, as the evidence group mass (F) of decision-making level's information fusion, the error E n that wherein uncertain probability m (θ) is SOM;

(2), after adopting D-S evidence combined rule to carry out decision-making level's information fusion, obtain the comprehensive evidence group of fault;

(3) the reliability interval of calculating fault; If the fault hypothesis set in Fault Identification framework is all singleton, belief function Bel (F)=mass (F) so, likelihood function Pls (F)=Bel (F)+m (θ);

(4), according to decision rule, draw fault diagnosis conclusion;

According to the network structure of SOM, following basic probability assignment function is proposed

$d_i=\sqrt{\underset{j=1}{\overset{N}{\mathrm{\Σ}}}{(y\left(j\right)-P_{F_i}\left(j\right))}^2}---\left(11\right)$

$K_i=\frac{1}{d_i}---\left(12\right)$

${\mathrm{Nd}}_i=\sqrt{\underset{j}{\mathrm{\Σ}}{(P_{F_0}\left(j\right)-P_{F_i}\left(j\right))}^2}---\left(13\right)$

$\mathrm{Err}=\frac{|{\mathrm{Nd}}_i-d_i|}{\mathrm{\Σ}|{\mathrm{Nd}}_i-d_i|}---\left(14\right)$

$E_n=\sqrt{\frac{\mathrm{\Σ}{\mathrm{Err}}^2}{L}}---\left(15\right)$

$m\left(F_i\right)=\frac{K_i}{\mathrm{\Σ}{K}_i}(1-E_n)---\left(16\right)$

m(θ)=E _n (17)

Wherein, d _ifor desired output y is to each neuronic Euclidean distance, the less expression matching degree of distance is higher, and corresponding elementary probability assignment should be larger, therefore to d _iget inverse and obtain K _t.Nd _ifor reality output to each neuronic Euclidean distance, the length that L is weight vector, E _nfor desired output distance and reality are exported the poor of distance, then be normalized, also can represent the uncertainty degree of evidence body; And basic probability function is exactly by K _iand E _njointly calculate;

By basic probability assignment function, can calculate belief function Bel (F) and plausibility function Pls (F), the reliability interval that in DS evidence fusion, criterion can be made up of Bel (F) and Pls (F) represents; F (0,1): explanation cannot determine whether fault F occurs; Bel (F)=0, illustrates that it is 0 that fault F occurs as genuine degree of belief; illustrate that it is also 0 that fault F does not occur as genuine degree of belief, that is to say and cannot whether occur by failure judgement F; F (0,0): illustrate that fault F does not occur one and is decided to be very; F (1,1): illustrate that fault F occurs one and is decided to be very;

In Circuit Fault Simulation, it is generally acknowledged that fault is all independent generation, does not exist common factor, each fault F between each fault _jonly there is common factor with self and indeterminate θ.

The invention provides that a kind of electric energy meter circuit failure message based on Self-Organizing Feature Maps merges and the method for electric energy meter circuit fault diagnosis based on D-S evidence theory, its advantage and effect have:

In this patent, characteristic layer fusion is classified to input data by SOM network, and calculate the trusting degree of classification results, but, when evidence body quantity increases or when classification results has conflict, only relies on SOM just can not draw compellent diagnostic result.Therefore carrying out D-S evidence fusion in decision-making level, the trusting degree of the result to fault diagnosis is further increased, reduced because the entirety uncertainty that error is brought, greatly improved the accuracy of fault diagnosis, is the very important means in information fusion field.

(4) brief description of the drawings:

The process flow diagram of Fig. 1 the method for the invention

Fig. 2 feature extraction and fusion process schematic diagram

Fig. 3 (a) SOM neural network classification result

Fig. 3 (b) SOM neural network classification result

(5) embodiment:

The process chart of the method for the invention as shown in Figure 1.The invention provides a kind of electric energy meter circuit failure message fusion method based on Self-Organizing Feature Maps and the electric energy meter circuit method for diagnosing faults based on D-S evidence theory, its step is as follows:

Step 1: electrical energy meter fault circuit is carried out to circuit analysis, and the fault mode comprising with reference to GJB299C is set up fault mode set.

Step 2: according to the fault mode set of step 1 choose with gather in the corresponding fault-signal point to be observed of fault mode, as the test point of circuit function and state.

Step 3: the fault-signal that the fault-signal point place choosing in step 2 is obtained carries out pre-service.This patent adopts the mode of normalization and interpolation to carry out pre-service to signal.Normalization is that data larger amplitude range are mapped on another interval by certain rule change, as ([0,1] or [1,1]), conventionally the input data of model have different dimensions, represent different meanings, by normalization to reach minimizing data volume, the object of computation complexity and unified dimension.Main a normalized mapping rule is as follows:

$y=\frac{(y_{\max }-y_{\min })(x-x_{\min })}{x_{\max }-x_{\min }}+y_{\min }---\left(18\right)$

$\stackrel{\‾}{x}=\frac{1}{N}\underset{i=1}{\overset{N}{\mathrm{\Σ}}}{x}_i$

$S^2=\frac{1}{N}\underset{i=1}{\overset{N}{\mathrm{\Σ}}}{(x_i-\stackrel{\‾}{x})}^2$

$y_i=\frac{x_i-\stackrel{\‾}{x}}{S}---\left(19\right)$

In formula, ymin, ymax is the minimum and maximum value of y, xmin, xmax is the minimum and maximum value of x.

The data of processing according to formula (1) can limit the border of mapping, belong to a kind of linear mapping.And according to the processing of formula (19) standardization that is otherwise known as, the data mean value that obtains is 0, variance is 1.

Interpolation has been generally the method that solves the functional value of middle unknown point between function region and adopt.The method of method of interpolation is:

If there is following data point on [a, b] interval:

(x ₀,y ₀)(x ₁,y ₁)(x ₂,y ₂)……(x _n,y _n)

Wherein y _i=f (x _i), i=0,1 ... n, x ₀, x ₁x _nbe called node.Value according to f (x) at node, constructs a smooth enough and fairly simple function be called interpolating function, as the approximate expression of f (x), then calculate in the functional value of interval [a, b] (being called interpolation section) upper any point x, the approximate value as original f (x) at this point.The general algebraic polynomial that adopts is as interpolating function.

In the time of n=1, Algebraic interpolation problem becomes, through two point (x ₀, y ₀) and (x ₁, y ₁), ask 1 interpolation polynomial p of function y=f (x) ₁(x),, according to cartesian geometry knowledge, can obtain

$p_1\left(x\right)=\frac{x-x_1}{x_0-x_1}{y}_0+\frac{x-x_0}{x_1-x_0}{y}_1=l_0\left(x\right)y_0+l_1\left(x\right)y_1---\left(21\right)$

Wherein, l ₀, l ₁being called Interpolation-Radix-Function one time, is all an order polynomial, p ₁(x) be a lagrange polynomial.Can verify, they meet

l ₀(x)+l ₁(x)=1 （22）

L ₀(x)=1, l ₁(x)=0 or l ₀(x)=0, l ₁(x)=1 (23)

At x, under the certain condition of y, given unified interpolation moment point, can carry out interpolation successively according to above-mentioned formula, obtains the unified time series of granularity.

The feature that this patent adopts comprises temporal characteristics, frequecy characteristic and statistical nature, and carrying out in Signal Pretreatment process also adopting the processing mode of going forward one by one.

(1) first to the fault-signal interpolation of obtaining, thereby unified signal granularity obtains the change procedure curve of fault-signal, i.e. temporal characteristics;

(2) on the basis of (1), temporal characteristics is normalized, carries out discrete Fourier transformation, picked up signal frequency spectrum and correlated characteristic;

(3) on the basis of (1), calculate the statistical property of signal, comprise second order distance, quadravalence distance, maximal values etc., set up eigenvectors matrix;

Adopt the time according to the feature this patent of circuit signal, the information of frequency and three dimensions of statistical nature is carried out signal fused, and its schematic diagram as shown in Figure 2.

1 temporal characteristics

Fault feature can be reflected in the procedure parameter correlation properties of circuit operation.The procedure parameter of circuit operation comprises magnitude of voltage, current value, and performance number etc., these procedure parameters are time dependent, therefore can say fault also reaction to some extent on time coordinate, are the temporal characteristics of fault.

In the time being input as the data of different faults pattern, its seasonal effect in time series length and sampling granularity are all inconsistent, therefore need step 3 to carry out pre-service to data, mainly reach stylistic unification by interpolation and difference, to meet the requirement of Intelligence Classifier to input data.

2 frequecy characteristics

Fault, except showing as certain temporal signatures, is also presented as the frequency change in simulation process.Different fault modes can produce different impacts to circuit signal frequency component.Aspect data processing, particularly signal process field Fourier transform (Fourier) has vital role, see that theoretically many common computings have good character (for example, difference quotient computing becomes multinomial operation, and convolution becomes common multiplication) under Fourier transform; Describe each periodic vibration from practice Fourier expansion from mathematical angle and be by the unifrequent simple harmonic oscillation of letter this physical phenomenon that superposes.Fourier transform has clearly disclosed the spectrum structure of data.This patent utilizes Fourier transform that time-domain signal is converted into frequency-region signal, thereby obtains its frequency domain character.Generally adopt Fast Fourier Transform (FFT) for discrete signal, its (contrary) transformation for mula is as follows

$X\left(k\right)=\underset{j=1}{\overset{N}{\mathrm{\Σ}}}x\left(j\right){\mathrm{\ω}}_N^{(j-1)(k-1)}---\left(24\right)$

$x\left(j\right)=\frac{1}{N}\underset{k=1}{\overset{N}{\mathrm{\Σ}}}X\left(k\right){\mathrm{\ω}}_N^{-(j-1)(k-1)}---\left(25\right)$

Wherein ω _n=e ^{(2 π i)/N}..When this patent carries out Fast Fourier Transform (FFT) to the fault-signal under different faults pattern, apply unified sample frequency, thereby the frequency-region signal obtaining has identical dimension.

3 statistical natures

In order to improve accuracy and the reliability of fusion, should adopt multiple statistical natures to merge.The concrete source of statistical nature can be single-sensor, can be also multiple sensors, and they are consistent in mathematical form.The meaning directly perceived that adopts multiple statistical natures is exactly to provide more fault status information from different sides, for the fusion of fault provides detailed as far as possible information, thereby improves syncretizing effect.

Suppose that the fault-signal obtaining after pre-service is x ₁, x ₂..., x _n, sampling time interval is Δ t, the statistical property of employing has secondary distance, and four squares, maximum value and maximum amplitude point, formula is as follows:

Second order distance $x_1={\left(\frac{1}{m}\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{y}_i^2\right)}^{\frac{1}{2}}---\left(26\right)$

Quadravalence distance $x_2={\left(\frac{1}{m}\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{y}_i^4\right)}^{\frac{1}{4}}---\left(27\right)$

Maximum value $x_3=\underset{i}{\max }\left|y_i\right|---\left(28\right)$

Maximum amplitude point x ₄if=k Δ t $y_k=\underset{i=1}{\overset{m}{\max }}{y}_i---\left(29\right)$

Fault-signal can be similar to the stack of regarding some simple harmonic waves as, is expressed as

$y=f\left(x\right)=a_0+\underset{n=1}{\overset{\∞}{\mathrm{\Σ}}}{a}_n\cos \frac{\mathrm{n\πx}}{L}+b_n\sin \frac{\mathrm{n\πx}}{L}---\left(30\right)$

Often its magnitude shift amount a ₀much larger than the amplitude a of vibration harmonics _n, b _n, thereby also not obvious from the vibration of macroscopic view signal, its secondary distance is very little apart from difference with four times, causes data field calibration not high.In order to address this problem, this patent is first normalized signal time sequence, is mapped to interval [0,1], then counting statistics feature, and its data distribute more satisfactory, are easy to distinguish different faults pattern.

Step 4: use self-organized mapping network to carry out failure message fusion, output fault conclusion.The data of selection 70% are trained, and the data of selection 30% are tested.In order to keep three stack features data unified, SOM network has identical basic setup.SOM is provided with 1 input layer; In network layer, only comprise 1 output layer, without hidden layer, 9 neurons are set, use and connect distance function calculating input vector to neuronic distance, transition function is made as competitive type compet, and topological structure is hexagonal network etale topology function hextop; The initialization function of input weight vector (inputWeights) is just value initialization of mid point, and learning function is Self-organizing Maps weights learning function (learnsom); Iterations is 80.

The learning algorithm of self-organized mapping network is as follows:

1 initialization; The each weight vector of output layer is given little random number and is normalized, obtain random manifold after treatment j=1,2 ..., m, sets up initial winning neighborhood N _j. (0), learning rate is composed initial value;

2 input learning samples; From training set, choose an input pattern and be normalized, obtaining the input vector after normalized p ∈ 1,2 ..., p}

3 find triumph node; Calculate with dot product, j=1,2,, m, therefrom selects the triumph node j of dot product maximum ^*; If input pattern is without normalized, should be according to calculate Euclidean distance, find out the minimum triumph node of distance;

The winning neighborhood N of 4 definition _j. (t) with j ^*centered by determine that the weights in t moment adjust territory, general initial neighborhood N _j. (0) is larger, N in training process _j. shrink in time gradually (0);

5 adjust weights, for winning neighborhood N _j. all knot adjustment weights in (0):

$W_{\mathrm{ij}}(t+1)=W_{\mathrm{ij}}\left(t\right)+\mathrm{\η}(t,N)[x_i^p-W_{\mathrm{ij}}\left(t\right)];i=\mathrm{1,2},\·\·\·,n,j\∈N_j.\left(0\right)$

Wherein, η (t, N) is the function of the topology distance N between interior j the neuron of training time t and neighborhood and triumph neuron;

6 finish to check, training finish be taking learning rate whether decay to 0 or certain predetermined positive decimal as condition, do not satisfy condition and get back to step 2,3,4;

The training of self-organized mapping network is not stable, and in the time training repeatedly, even the network of the identical setting of same group of data substitution, its result also might not be identical.The characteristic of this and data self has relation, also relevant with neural network self initialization function.Therefore, in training process, can suitably artificially adjust, to obtain satisfied classification results.

Step 5: use D-S evidence theory to merge fault conclusion, do the decision-making of being out of order.

Under certain fault mode, failure message, through the information fusion of characteristic layer SOM, has comprised correct conclusion in acquired results, but the unsteady scope of SOM recognition result is larger, also need to dwindle its scope by decision-making level's information fusion, obtain more accurate fault diagnosis conclusion.According to the result of fault simulation, set up the identification framework of diagnosing information fusion fault.

This patent adopts D-S evidence theory as information fusion method in decision-making level, and key step is as follows:

(1) taking the output p (Fi) of feature fusion as basis, calculate fault elementary probability according to formula (36) and count mT, mF and mS, as the evidence group mass (F) of decision-making level's information fusion, the error E n that wherein uncertain probability m (θ) is SOM;

(2), after adopting D-S evidence combined rule to carry out decision-making level's information fusion, obtain the comprehensive evidence group of fault;

(3) the reliability interval of calculating fault.If the fault hypothesis set in Fault Identification framework is all singleton, belief function Bel (F)=mass (F) so, likelihood function Pls (F)=Bel (F)+m (θ).

(4), according to decision rule, draw fault diagnosis conclusion.

According to the network structure of SOM, following basic probability assignment function is proposed

$d_i=\sqrt{\underset{j=1}{\overset{N}{\mathrm{\Σ}}}{(y\left(j\right)-P_{F_i}\left(j\right))}^2}---\left(31\right)$

$K_i=\frac{1}{d_i}---\left(32\right)$

${\mathrm{Nd}}_i=\sqrt{\underset{j}{\mathrm{\Σ}}{(P_{F_0}\left(j\right)-P_{F_i}\left(j\right))}^2}---\left(33\right)$

$\mathrm{Err}=\frac{|{\mathrm{Nd}}_i-d_i|}{\mathrm{\Σ}|{\mathrm{Nd}}_i-d_i|}---\left(34\right)$

$E_n=\sqrt{\frac{\mathrm{\Σ}{\mathrm{Err}}^2}{L}}---\left(35\right)$

$m\left(F_i\right)=\frac{K_i}{\mathrm{\Σ}{K}_i}(1-E_n)---\left(36\right)$

m(θ)=E _n (37)

Wherein, d _ifor desired output y is to each neuronic Euclidean distance, the less expression matching degree of distance is higher, and corresponding elementary probability assignment should be larger, therefore to d _iget inverse and obtain K _i.Nd _ifor reality output to each neuronic Euclidean distance, the length that L is weight vector, E _nfor desired output distance and reality are exported the poor of distance, then be normalized, also can represent the uncertainty degree of evidence body.And basic probability function is exactly by K _iand E _njointly calculate.

By above basic probability assignment function, can calculate belief function Bel (F) and plausibility function Pls (F), the reliability interval that in DS evidence fusion, criterion can be made up of Bel (F) and Pls (F) represents.F (0,1): explanation cannot determine whether fault F occurs.Bel (F)=0, illustrates that it is 0 that fault F occurs as genuine degree of belief; illustrate that it is also 0 that fault F does not occur as genuine degree of belief.That is to say and cannot whether occur by failure judgement F.F (0,0): illustrate that fault F does not occur one and is decided to be very.F (1,1): illustrate that fault F occurs one and is decided to be very.

In Circuit Fault Simulation, it is generally acknowledged that fault is all independent generation, between each fault, there is not common factor, each fault Fj only occurs simultaneously with self and indeterminate θ existence.According to the data after training, calculate elementary probability number and uncertain value according to above-mentioned formula, mT is the fault mode elementary probability number taking temporal characteristics as input, mF is the fault mode elementary probability number taking frequecy characteristic as input, and mS is the fault mode elementary probability number taking statistical nature as input.By D-S evidence combined rule, to mT, mF and mS merge between any two, calculate belief function Bel (F) and the likelihood function Pls (F) of fault.

According to expertise, formulate decision rule, draw diagnosis.

Case study on implementation 1

Choose the metering circuit of certain model electric energy meter herein as case.The defective device of choosing has metalfilmresistor resistance R 1, solid tantalum electrochemical capacitor C1, and 1 class Leaded Ceramic Disc Capacitor C2, common double bipolar transistor Q1, sets up fault mode set according to GJB299C, and typical fault pattern in totally 14, as form 2.The voltage of selecting circuit output point is signaling point to be seen, using this as circuit function with the test point of state.

Form 2 fault mode definition

Sequence number	Fault mode	Sequence number	Fault mode
				M1	C1-parameter drift+5%	M8	R1-parameter drift-5%
M2	The Q1-open circuit c utmost point	M9	R1-open circuit
				M3	The Q1-open circuit b utmost point	M10	C1-parameter drift-5%
M4	The Q1-open circuit e utmost point	M11	C2-parameter drift+5%
				M5	The Q1-short circuit cb utmost point	M12	C2-parameter drift-5%
M6	The Q1-short circuit be utmost point	M13	C2-open circuit
				M7	R1-parameter drift+5%	M14	C2-short circuit

Select SOM network as Intelligence Classifier, carry out pattern classification.Always have the data of 14 kinds of fault modes, set up knowledge base through training.In order to keep three stack features data systems, SOM network has identical basic setup, and the training result of temporal characteristics is as shown in Fig. 3 (a), (b).Wherein Fig. 3 (a) transverse axis represents fault mode M1 ~ M14, and the longitudinal axis represents hit neuron sequence number, and the model representative that sequence number is identical is divided into same class; Fig. 3 (b) is neuronic classification situation, and from left to right, neuronic sequence number is 1 ~ 9 from top to bottom, corresponding with the value of the longitudinal axis in histogram.Can see in the network topology structure of 3*3,14 groups of inputs have been divided into 5 classes, the blue neuron for competition triumph, the input vector number that numeral wherein comprises.The classification results of frequecy characteristic and statistical nature is identical therewith, as form 3.

Form 3 neuron classification results

Neuron classification	Fault mode
		F1	M1，M7，M10，M11，M12
F2	M2，M3，M4
		F3	M5，M6，M9
F4	M8，M13

F5

M14

According to the data after training, calculate elementary probability number and uncertain value is as shown in table 4 according to above-mentioned formula, mT is the fault mode elementary probability number taking temporal characteristics as input, mF is the fault mode elementary probability number taking frequecy characteristic as input, and mS is the fault mode elementary probability number taking statistical nature as input.Calculate belief function Bel (F) and plausibility function Pls (F) with formula again, as shown in 1 ~ 3 row in form 5.

Form 4 decision-making level's fusion results

	mT	mF	mS	mT&mF	mT&mS	mF&mS	(mT&mF)&mS
								K				0.4699	0.6224	0.5912	0.6657
m(θ)	0.0144	0.1482	0.2236	0.0046	0.0052	0.0561	0.0015
								m(F 1)	0.0400	0.0656	0.0231	0.0202	0.0164	0.0332	0.0076
m(F 2)	0.0235	0.1375	0.0143	0.0185	0.0093	0.0589	0.0067
								m(F 3)	0.6388	0.4004	0.5338	0.7582	0.7898	0.6468	0.8664
m(F 4)	0.0396	0.0627	0.0229	0.0197	0.0162	0.0319	0.0074
								m(F 5)	0.2437	0.1856	0.1823	0.1788	0.1631	0.1731	0.1103

Adopt D-S evidence combined rule to mT, mF and mS merge between any two, calculate belief function Bel (F) and the likelihood function Pls (F) of fault, after merging, the elementary probability number of F5 is respectively 0.7582,0.7898 and 0.6468, before merging, be significantly increased.The value of uncertain probability m (θ) becomes 0.0046,0.0052 and 0.0561, before merging, significantly reduces.Experimental data shows, process information fusion, and the confidence level that target faults occurs improves, and the uncertainty of judgement weakens.

Further adopt D-S evidence combined rule to mT, between mF and mS three, merge, fusion results is as shown in the 7th row in table 4.The belief function Bel (F) of fault and likelihood function Pls (F), as shown in the 7th row in form 5.

Form 5 decision-making level's reliability intervals

	T	F	S	T&F	T&S	F&S	(T&F)&S
								m(θ)	0.0144	0.1482	0.2236	0.0046	0.0052	0.0561	0.0015

Pls(F1)	0.0544	0.2139	0.2467	0.0248	0.0216	0.0892	0.0092
								Bel(F1)	0.0400	0.0656	0.0231	0.0202	0.0164	0.0332	0.0076
Pls(F2)	0.0379	0.2857	0.2379	0.0231	0.0145	0.1150	0.0082
								Bel(F2)	0.0235	0.1375	0.0143	0.0185	0.0093	0.0589	0.0067
Pls(F3)	0.6533	0.5486	0.7574	0.7628	0.7950	0.7029	0.8679
								Bel(F3)	0.6388	0.4004	0.5338	0.7582	0.7898	0.6468	0.8664
Pls(F4)	0.0540	0.2109	0.2465	0.0243	0.0214	0.0879	0.0090

With reference to expertise, adopt following decision rule.

(1) the belief function value of target faults is the maximal value in all belief function values;

(2) the belief function value of target faults is at least 2 times of belief function value of other fault;

(3) belief function Bel (F) > 0.5;

(4) uncertain probability m (θ) <0.01.

Rule (1) illustrates in all fault types of evidence group to only have the fault of belief function value maximum to be only target faults; Rule (2) illustrates to only have the belief function value of target faults and other fault type to have enough difference, could confirm that target faults occurs; Rule (3) only illustrates that group has enough supports to fault on evidence, could confirm that fault occurs; Rule (4) illustrates by the judgement of evidence body have enough determinacy, could confirm that target faults occurs.

Can find by analysis, after fusion, the elementary probability number of fault F3 further improves, and becomes 0.8664, meets the decision-making judgment rule of " belief function Bel (F) >0.5 ".Meanwhile, the belief function value of F3 is the maximal value in all belief function values, and is at least 2 times of other fault belief function values.Illustrate that, through information fusion, the confidence level of " F3 generation " improves, and has reached decision rule requirement.The uncertain probability m (θ) of judgement " F3 generation " continues to reduce, and becomes 0.0015.

This patent is undertaken by SOM network in the fusion of characteristic layer, but, when evidence body quantity increases or when classification results has conflict, only relies on SOM just can not draw compellent diagnostic result.Therefore carrying out D-S evidence fusion in decision-making level, the trusting degree of the result to fault diagnosis is further increased, reduced because the entirety uncertainty that error is brought, greatly improved the accuracy of fault diagnosis, is the very important means in information fusion field.

Claims (6)

1. the meter circuit failure message based on SOM and D-S evidence theory merges and a diagnostic method, it is characterized in that: the method concrete steps are as follows:

Step 1: according to the circuit analysis to electrical energy meter fault circuit, and the fault mode specifying according to GJB299C, faulty circuit is set up to a fault mode set;

Step 2: according to the fault mode set of step 1 choose with gather in the corresponding fault-signal point to be observed of fault mode, as the test point of circuit function and state;

Step 3: the fault-signal that the fault-signal point place choosing in step 2 is obtained carries out pre-service; Because the feature adopting comprises temporal characteristics, frequecy characteristic and statistical nature, so carrying out in Signal Pretreatment process adopting the processing mode of going forward one by one, as described below:

(1) first to the fault-signal interpolation of obtaining, thereby unified signal granularity obtains the change procedure curve of fault-signal, i.e. temporal characteristics;

(2) on the basis of (1), temporal characteristics is normalized, carries out discrete Fourier transformation, picked up signal frequency spectrum and correlated characteristic;

(3) on the basis of (1), calculate the statistical property of signal, comprise second order distance, quadravalence distance and maximal value, set up eigenvectors matrix;

According to the feature of circuit signal, the information of employing time, frequency and three dimensions of statistical nature is carried out signal fused;

Step 4: use SOM to carry out failure message fusion, output fault conclusion; The data of selection 70% are trained, and the data of selection 30% are tested; In order to keep three stack features data unified, SOM has identical basic setup; SOM is provided with 1 input layer; In network layer, only comprise 1 output layer, without hidden layer, 9 neurons are set, use and connect distance function calculating input vector to neuronic distance, transition function is made as competitive type compet, and topological structure is hexagonal network etale topology function hextop; Input weight vector is that the initialization function of inputWeights is just value initialization of mid point, and learning function is that Self-organizing Maps weights learning function is learnsom; Iterations is 80;

Step 5: use D-S evidence theory to merge fault conclusion, do the decision-making of being out of order; Under certain fault mode, failure message, through the information fusion of characteristic layer SOM, has comprised correct conclusion in acquired results, but the unsteady scope of SOM recognition result is larger, also need to dwindle its scope by decision-making level's information fusion, obtain more accurate fault diagnosis conclusion; According to the result of fault simulation, set up the identification framework of diagnosing information fusion fault.

2. a kind of meter circuit failure message based on SOM and D-S evidence theory according to claim 1 merges and diagnostic method, it is characterized in that: described in step 1, faulty circuit is being set up to fault mode set according to GJB299C, the method of its foundation is as follows: the fault mode comprising according to GJB299C, the fault mode of each trouble spot to circuit under test is numbered and list, and concrete form as shown in Table 1.

Form 1 fault mode set

Sequence number Fault mode Sequence number Fault mode M1 C1-parameter drift+5% M8 R1-parameter drift-5% M2 The Q1-open circuit c utmost point M9 R1-open circuit M3 The Q1-open circuit b utmost point M10 C1-parameter drift-5% M4 The Q1-open circuit e utmost point M11 C2-parameter drift+5% M5 The Q1-short circuit cb utmost point M12 C2-parameter drift-5% M6 The Q1-short circuit be utmost point M13 C2-open circuit M7 R1-parameter drift+5% M14 C2-short circuit

3. a kind of meter circuit failure message based on SOM and D-S evidence theory according to claim 1 merges and diagnostic method, it is characterized in that: described in step 3, the fault-signal obtaining being carried out to pre-service, is to adopt the mode of normalization and interpolation to carry out pre-service to signal; Normalization is that data larger amplitude range are mapped on another interval by certain rule change, conventionally the input data of model have different dimensions, represent different meanings, by normalization to reach minimizing data volume, the object of computation complexity and unified dimension; To a series of data point (x on [a, b] interval ₀, y ₀) (x ₁, y ₁) (x ₂, y ₂) ... (x _n, y _n), main a normalized mapping rule is as follows:

$\begin{array}{c}y=\frac{(y_{\max }-y_{\min })(x-x_{\min })}{x_{\max }-x_{\min }}+y_{\min }\\ \stackrel{\&OverBar;}{x}=\frac{1}{N}\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}{x}_i\end{array}---\left(1\right)$

$\begin{array}{c}{S}^2=\frac{1}{N}\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}{(x_i-\stackrel{\&OverBar;}{x})}^2\\ y_i=\frac{x_i-x}{S}\end{array}---\left(2\right)$

In formula, y _min, y _maxfor minimum and the maximal value of y, x _min, x _maxminimum and the maximal value of x;

The data of processing according to formula (1) limit the border of mapping, belong to a kind of linear mapping; And according to the processing of formula (2) standardization that is otherwise known as, the data mean value that obtains is 0, variance is 1;

Interpolation has been generally the method that solves the functional value of middle unknown point between function region and adopt; The method of method of interpolation is:

If there is following data point on [a, b] interval:

(x ₀,y ₀)(x ₁,y ₁)(x ₂,y ₂)......(x _n,y _n)

Wherein y _i=f (x _i), i=0,1......n, x ₀, x ₁... x _nbe called node; Value according to f (x) at node, constructs a smooth enough and fairly simple function (x) be called interpolating function, as the approximate expression of f (x), then calculate (x) be called the functional value of any point x on interpolation section at interval [a, b], as originally

F (x) is in the approximate value of this point; The general algebraic polynomial that adopts is as interpolating function;

In the time of n=1, Algebraic interpolation problem becomes, through two point (x ₀, y ₀) and (x ₁, y ₁), ask 1 interpolation polynomial p of function y=f (x) ₁(x),, according to cartesian geometry knowledge, obtain;

$p_1\left(x\right)=\frac{x-x_1}{x_0-x_1}{y}_0+\frac{x-x_0}{x_1-x_0}{y}_1=l_0\left(x\right)y_0+l_1\left(x\right)y_1---\left(4\right)$

Wherein, l ₀, l ₁being called Interpolation-Radix-Function one time, is all an order polynomial, p ₁(x) be a lagrange polynomial; Empirical tests, they meet

l ₀(x)+l ₁(x)=1（5）

L ₀(x)=1, l ₁(x)=0 or l ₀(x)=0, l ₁(x)=1(6)

At x, under the certain condition of y, given unified interpolation moment point, carries out interpolation successively according to above-mentioned formula, obtains the unified time series of granularity.

4. a kind of meter circuit failure message based on SOM and D-S evidence theory according to claim 1 merges and diagnostic method, it is characterized in that: at the statistical nature of the signal described in step 3, computing method are as follows:

Suppose that the fault-signal obtaining after pre-service is x ₁, x ₂..., x _n, sampling time interval is Δ t, the statistical property of employing has second order distance, and Fourth-order moment, maximum value and maximum amplitude point, formula is as follows:

Second order distance

Quadravalence distance $x_1={\left(\frac{1}{m}\underset{i=1}{\overset{m}{\mathrm{\&Sigma;}}}{y}_i^2\right)}^{\frac{1}{2}}---\left(7\right)$

Maximum value $x_3=\underset{i}{\max }\left|y_i\right|---\left(9\right)$

Maximum amplitude point x ₄if=k △ is t $y_k=\underset{i=1}{\overset{m}{\max }}{y}_i.---\left(10\right)$

5. a kind of meter circuit failure message based on SOM and D-S evidence theory according to claim 1 merges and diagnostic method, it is characterized in that: carry out failure message fusion at the use self-organized mapping network described in step 4, the learning algorithm of the self-organized mapping network of use is as follows:

(1) initialization; The each weight vector of output layer is given little random number and is normalized, obtain random manifold after treatment j=1,2 ..., m, sets up initial winning neighborhood N _j.(0), learning rate is composed initial value;

(2) input learning sample; From training set, choose an input pattern and be normalized, obtaining the input vector after normalized p ∈ 1,2 ..., p};

(3) find triumph node; Calculate with dot product, j=1,2,, m, therefrom selects the triumph node j of dot product maximum ^*; If input pattern is without normalized, should be according to calculate Euclidean distance, find out the minimum triumph node of distance;

(4) define winning neighborhood N _j.(t) with j ^*centered by determine that the weights in t moment adjust territory, general initial neighborhood N _j.(0) larger, N in training process _j.(0) shrink gradually in time;

(5) adjust weights, for winning neighborhood N _j.(0) all knot adjustment weights in:

W _ij（t+1）=W _ij（t）+η（t，N）[ -W _ij（t）]；i=1,2，…，n，j∈N _j.（0）

Wherein, η (t, N) is the function of the topology distance N between interior j the neuron of training time t and neighborhood and triumph neuron;

(6) finish to check, training finish be taking learning rate whether decay to 0 or certain predetermined positive decimal as condition, do not satisfy condition and get back to step (2), (3), (4).

6. a kind of meter circuit failure message based on SOM and D-S evidence theory according to claim 1 merges and diagnostic method, it is characterized in that: step fault conclusion being merged at the use D-S evidence theory described in step 5 is as follows:

(1) taking the output p (Fi) of feature fusion as basis, calculate fault elementary probability according to following formula (16) and count mT, mF and mS, as the evidence group mass (F) of decision-making level's information fusion, the error E n that wherein uncertain probability m (θ) is SOM;

(2), after adopting D-S evidence combined rule to carry out decision-making level's information fusion, obtain the comprehensive evidence group of fault;

(3) the reliability interval of calculating fault; If the fault hypothesis set in Fault Identification framework is all singleton, belief function Bel (F)=mass (F) so, likelihood function Pls (F)=Bel (F)+m (θ);

(4), according to decision rule, draw fault diagnosis conclusion;

According to the network structure of SOM, following basic probability assignment function is proposed

$d_i=\sqrt{\underset{j=1}{\overset{N}{\mathrm{\&Sigma;}}}}{(y\left(j\right)-P_{\mathrm{Fi}}\left(j\right))}^2---\left(11\right)$

$K_i=\frac{1}{d_i}---\left(12\right)$

${\mathrm{Nd}}_i=\sqrt{\underset{j}{\mathrm{\&Sigma;}}}{(P_{F0}\left(j\right)-P_{\mathrm{Fi}}\left(j\right))}^2---\left(13\right)$

$\mathrm{Err}=\frac{|{\mathrm{Nd}}_i-d_i|}{\mathrm{\&Sigma;}|{\mathrm{Nd}}_i-d_i|}---\left(14\right)$

$E_n\sqrt{\frac{\mathrm{\&Sigma;}{\mathrm{Err}}^2}{L}}---\left(15\right)$

$m\left(F_i\right)=\frac{K_i}{\mathrm{\&Sigma;}{K}_i}(1-E_n)---\left(16\right)$

Wherein, d _ifor desired output y is to each neuronic Euclidean distance, the less expression matching degree of distance is higher, and corresponding elementary probability assignment is larger, therefore to d _iget inverse and obtain K _i; Kd _ifor reality output to each neuronic Euclidean distance, the length that L is weight vector, E _nfor desired output distance and reality are exported the poor of distance, then be normalized, also represent the uncertainty degree of evidence body; And basic probability function is exactly by K _iand K _njointly calculate;

By basic probability assignment function, calculate belief function Bel (F) and plausibility function Pls (F), the reliability interval that in DS evidence fusion, criterion is made up of Bel (F) and Pls (F) represents; F (0,1): explanation cannot determine whether fault F occurs; Bel (F)=0, illustrates that it is 0 that fault F occurs as genuine degree of belief; Bel (P)=1-Pls (F)=0, illustrates that it is also 0 that fault F does not occur as genuine degree of belief, that is to say and cannot whether occur by failure judgement F; F (0,0): illustrate that fault F does not occur one and is decided to be very; F (1,1): illustrate that fault F occurs one and is decided to be very;

In Circuit Fault Simulation, it is generally acknowledged that fault is all independent generation, does not exist common factor, each fault F between each fault _jonly there is common factor with self and indeterminate θ.