CN105182186A - Radial network fault branch identification method based on voltage distribution along line and full coverage of travelling wave information - Google Patents

Abstract

The invention relates to a radial network fault branch identification method based on voltage distribution along a line and full coverage of travelling wave information, and the method belongs to the technical field of relay protection of power systems. According to the invention, when a single-phase earth fault happens in a multi-branch radial distribution network, PCA-SVM is used to identify a cut set in which a fault branch of the multi-branch radial distribution network is. In the cut set, a line-model travelling wave is calculated from fault travelling wave data obtained from a measurement terminal of a trunk 1. According to the Bergeron line transmission equation, the obtained line-model travelling wave is used to calculate voltage travelling wave and current travelling wave distribution along a line, a voltage travelling wave along the line and a current travelling wave along the line are subjected to travelling wave decomposition along the direction of the line, and a direction travelling wave distributed along the line is obtained. A distance measurement function is constructed by multiplying a forward travelling wave with an opposite travelling wave and calculating integration in two successive time windows separately, and the length of each time window is L1/(2v). Finally, the fault branch in the multi-branch radial distribution network is identified according to the travelling wave distribution of the distance measurement function in a line length L1 range. Theoretical analysis and simulation results indicate that the method has a good effect.

Description

A kind of radiation network Fault branch identification method based on voltage's distribiuting along the line and all standing of row ripple information

Technical field

The present invention relates to a kind of radiation network Fault branch identification method based on voltage's distribiuting along the line and all standing of row ripple information, belong to Relay Protection Technology in Power System field.

Background technology

Distribution line failure, especially singlephase earth fault quick, accurately locate, to ensure the safety and stability of whole electric system and economical operation very important.Power distribution network generally adopts radial structure, and its lines branch is more, how accurately to identify that the fault branch of radiation network is a large difficult point of distribution line failure location.S injection method a kind ofly adds letter diagnostic method by what detect that the path of Injection Signal and feature realize the identification of fault branch and localization of fault, but its Injection Signal is limited, and larger by arrangement of conductors capacitive effect.Along with the successful Application of transmission line travelling wave range finding, the fault traveling wave location of existing scholar to power distribution network expands research, proposes to determine fault section by identifying from the reflection wave of trouble spot and point of discontinuity.But because in multiple-limb radiation network, the wave attenuation of fault initial row is large, the factors such as branch line length great disparity causes follow-up wave head disorderly and unsystematic affect, trouble spot reflection wave is not easily comformed identification in multiple reflection ripple, thus makes dependence time domain wavefront carry out the method for fault section differentiation and location by impact to a certain extent.Therefore proposition is badly in need of a kind of reliably not by the radiation network Fault branch identification method that the effective identification of fault traveling wave wave head affects.

Summary of the invention

The object of the invention is to overcome the not high limitation of existing radiation network Fault branch identification method reliability, a kind of radiation network Fault branch identification method based on voltage's distribiuting along the line and all standing of row ripple information is provided.

Technical scheme of the present invention is: a kind of radiation network Fault branch identification method based on voltage's distribiuting along the line and all standing of row ripple information, when multiple-limb radial distribution generation singlephase earth fault, PCA-SVM is adopted to identify the fault branch place cut set of the radial distribution of multiple-limb; Then under fault cut set, line line ripple is asked for the fault traveling wave data that trunk 1 measuring end obtains, the line line ripple that application obtains also calculates voltage traveling wave along the line and current traveling wave distribution according to Bei Jielong circuit equation of transfer, voltage traveling wave along the line and current traveling wave along the line are carried out direction along the line row Wave Decomposition, obtain the direction row ripple of distribution along the line, recycle its direct wave be multiplied by backward-travelling wave and respectively at two L in succession ₁carry out integration to construct range function in/(2v) window length, finally according to range function along line length L ₁scope expert ripple sudden change distribution carrys out the fault branch of identification multiple-limb radiation network.

Concrete steps are:

(1), during multiple-limb radial distribution generation singlephase earth fault, application PCA-SVM differentiates the fault cut set of this radial distribution.

(2) under sampling rate 1MHz, voltage, the electric current that trunk 1 measuring end obtains is sampled, obtains phase current sampling value sequence i _m,a(k), i _m,b(k), i _m,c(k), phase voltage sampled value sequence u _m,a(k), u _m,b(k), u _m,c(k), wherein k represents sampled point, k=1,2, M represents measuring end.

(3) the discrete series i of wire finishing die electric current and line mode voltage is asked respectively according to formula (1) and formula (2) _m,s(k) and u _m,s(k):

i _M,s(k)＝i _M,a(k)-i _M,b(k)(1)

u _M,s(k)＝u _M,a(k)-u _M,b(k)(2)

(4) calculating of distribution along the line: utilize formula (3) and formula (4) to calculate the voltage's distribiuting along the line of measuring end place feeder line trunk 1 and distribution of current along the line respectively.

$\begin{array}{c}{u}_{M,x,s}(x,k)=\frac{1}{2}{\left(\frac{Z_{c,s}+r_{s}x/4}{Z_{c,s}}\right)}^2\[u_{M,s}(k+\frac{x}{v_s})-i_{M,s}(k+\frac{x}{v_s})(Z_{c,s}+\frac{r_{s}x}{4})\]\\ +\frac{1}{2}{\left(\frac{Z_{c,s}-r_{s}x/4}{Z_{c,s}}\right)}^2\[u_{M,s}(k-\frac{x}{v_s})+i_{M,s}(k-\frac{x}{v_s})(Z_{c,s}-r_{s}x)\]\\ -{\left(\frac{r_{s}x/4}{Z_{c,s}}\right)}^2{u}_{M,s}\left(k\right)-\frac{r_{s}x}{4}\left(\frac{Z_{c,s}+r_{s}x/4}{Z_{c,s}}\right)\left(\frac{Z_{c,s}-r_{s}x/4}{Z_{c,s}}\right)i_{M,s}\left(k\right)\end{array}---\left(3\right)$

$\begin{array}{c}{i}_{M,x,s}(x,k)=\frac{1}{2Z_{c,s}}\left(\frac{Z_{c,s}+r_{s}x/4}{Z_{c,s}}\right)\[u_{M,s}(k+x/v_s)-i_{M,s}(k+x/v_s)\·(Z_{c,s}+r_{s}x/4)\]\\ -\frac{1}{2Z_{c,s}}\left(\frac{Z_{c,s}-r_{s}x/4}{Z_{c,s}}\right)\[u_{M,s}(k-x/v_s)+i_{M,s}(k-x/v_s)\·(Z_{c,s}-r_{s}x/4)\]\\ -\frac{1}{2Z_{c,s}}\·\frac{r_{s}x}{2Z_{c,s}}\[u_{M,s}\left(k\right)-i_{M,s}\left(k\right)(r_{s}x/4)\]\end{array}---\left(4\right)$

In formula, s is Aerial mode component; X is the distance of any point along the line to measuring end; V is the wave velocity of circuit; Z _c,sfor the characteristic impedance of circuit; r _sfor circuit resistance per unit length; u _m,sk line mode voltage row ripple that () is measuring end; i _m,sk line mould current traveling wave that () is measuring end; u _{m, x, s}(x, k) is for the k moment is apart from the voltage at measuring end x place; i _{m, x, s}(x, k) is for the k moment is apart from the electric current at measuring end x place.

(5) direct wave and the backward-travelling wave of distribution along the line is calculated: calculate the capable ripple of forward voltage of trunk 1 distribution along the line, the capable ripple of reverse voltage of distribution along the line respectively according to formula (5) and formula (6), namely

u ⁺ _M,x,s＝(u _M,x,s+Z _c,si _M,x,s)/2(5)

u ^- _M,x,s＝(u _M,x,s-Z _c,si _M,x,s)/2(6)

(6) the direct wave gradient of distribution along the line and the calculating of backward-travelling wave gradient: the forward voltage gradient utilizing the difference structure distribution along the line of adjacent two sampled values of the capable ripple of forward voltage of distribution along the line, namely

c ⁺ _M,dif—u(k)＝u ⁺ _k,x,s(k)-u ⁺ _k,x,s(k-1)(7)

Utilize the reverse voltage gradient of the difference structure distribution along the line of adjacent two sampled values of the capable ripple of reverse voltage of distribution along the line, namely

c ^- _M,dif—u(k)＝u ^- _k,x,s(k)-u ^- _k,x,s(k-1)(8)

(7) direct wave sudden change and the backward-travelling wave sudden change of distribution along the line is calculated: the capable ripple sudden change of forward voltage of extracting trunk 1 distribution along the line according to formula (9), namely

${S^{+}}_{M,2u}(x,k)=\underset{n=k-R+1}{\overset{k}{\Σ}}{\[{c^{+}}_{M,dif\_u}\left(k\right)\]}^3---\left(9\right)$

The capable ripple sudden change of reverse voltage of trunk 1 distribution is along the line extracted, namely according to formula (10)

${S^{-}}_{M,2u}(x,k)=\underset{n=k-R+1}{\overset{k}{\Σ}}{\[{c^{-}}_{M,dif\_u}\left(k\right)\]}^3---\left(10\right)$

In formula, R is taken as 3.

(8) structure of range function: adopt formula (11) and formula (12), window [k when direct wave step (6) obtained sudden change is multiplied with backward-travelling wave sudden change and observes respectively at row ripple ₀, k ₀+ L ₁/ (2v)] and [k ₀+ L ₁/ (2v), k ₀+ L ₁/ v] in carry out integration, obtain range function f _uI(x) and f _uIIthe row ripple sudden change along the line of (x).

$\begin{array}{cc}{f}_{uI}\left(x\right)={\∫}_{k_0}^{k_0+L_1/2v}{S}_{M,2u}^{+}(x,k)\×S_{M,2u}^{-}(x,k)dk& x\∈\[0,L_1/2\]\end{array}---\left(11\right)$

$\begin{array}{cc}{f}_{uII}\left(x\right)={\∫}_{k_0+L_1/2v}^{k_0+L_1/v}{S}_{M,2u}^{+}(x,k)\×S_{M,2u}^{-}(x,k)dk& x\∈\[L_1/2,L_1\]\end{array}---\left(12\right)$

In formula, k ₀represent the fault initial row ripple due in that measuring end M detects; L ₁for the line length of trunk 1.

(9) structure of localization of fault criterion:

[k is calculated according to step (8) ₀, k ₀+ L ₁/ (2v)] and [k ₀+ L ₁/ (2v), k ₀+ L ₁/ v] two in succession time window in, range function f _uI(x) and f _uIIx the distribution catastrophe point along the line of (), its respective distances is designated as [x respectively _i1, x _i2,] and [x _iI1, x _iI2,].

If in step (1), SVM exports is 1, then failure judgement occurs on trunk 1;

If SVM exports in step (1) is 2, then failure judgement occurs in fault cut set 2 (that is: trunk 2 or branch 1), if now [x _i1, x _i2,] in mutation distance x ^* _i[x _iI1, x _iI2,] in mutation distance x ^* _iImeet

x ^* _I+x ^* _II＝L ₂(13)

And work as x ^* _icatastrophe point polarity be negative, x ^* _iIcatastrophe point polarity be timing, failure judgement is positioned within trunk 2 half line length; Work as x ^* _icatastrophe point polarity be just, x ^* _iIcatastrophe point polarity for time negative, failure judgement is positioned at outside trunk 2 half line length.In formula (13), L ₂for the line length of trunk 2.

If [x _i1, x _i2,] in mutation distance x ^* _i[x _iI1, x _iI2,] in mutation distance x ^* _iImeet

x ^* _I+x ^* _II＝L ₆(14)

And x ^* _iand x ^* _iIcatastrophe point polarity be negative polarity, then failure judgement occurs in branch 1.In formula (14), L ₆for the line length of branch 1.

If SVM exports in step (1) is 3, then failure judgement occurs in fault cut set 3 (that is: trunk 3 or branch 2), if now [x _i1, x _i2,] in mutation distance x ^* _i[x _iI1, x _iI2,] in mutation distance x ^* _iImeet constraint condition shown in formula (13), and x ^* _iand x ^* _iIcatastrophe point polarity be negative polarity, then failure judgement occurs in trunk 3.

If [x _i1, x _i2,] in mutation distance x ^* _i[x _iI1, x _iI2,] in mutation distance x ^* _iImeet constraint condition shown in formula (14), and x ^* _iand x ^* _iIcatastrophe point polarity be negative polarity, then failure judgement occurs in branch 1.

The invention has the beneficial effects as follows:

This method carries out localization of fault for transmission line of electricity, and its principle is simple, does not need to demarcate fault traveling wave wave-wave head, and not by the impact of the factor such as fault instantaneity, fault resistance change, range measurement accurately and reliably.

Accompanying drawing explanation

Fig. 1 is the radial Distributing network structure figure of multiple-limb of embodiment 1, embodiment 2;

Fig. 2 is under embodiment 1 trunk 2 fault, [k ₀, k ₀+ L ₁/ (2v)] time window in the sudden change distribution results of range function;

Fig. 3 is under embodiment 1 trunk 2 fault, [k ₀+ L ₁/ (2v), k ₀+ L ₁/ v] time window in the sudden change distribution results of range function;

Fig. 4 is under embodiment 2 trunk 3 fault, [k ₀, k ₀+ L ₁/ (2v)] time window in the sudden change distribution results of range function;

Fig. 5 is under embodiment 2 trunk 3 fault, [k ₀+ L ₁/ (2v), k ₀+ L ₁/ v] time window in the sudden change distribution results of range function.

Embodiment

Below in conjunction with the drawings and specific embodiments, the invention will be further described.

When multiple-limb radial distribution generation singlephase earth fault, PCA-SVM is adopted to identify the fault branch place cut set of the radial distribution of multiple-limb; Then under fault cut set, line line ripple is asked for the fault traveling wave data that trunk 1 measuring end obtains, the line line ripple that application obtains also calculates voltage traveling wave along the line and current traveling wave distribution according to Bei Jielong circuit equation of transfer, voltage traveling wave along the line and current traveling wave along the line are carried out direction along the line row Wave Decomposition, obtain the direction row ripple of distribution along the line, recycle its direct wave be multiplied by backward-travelling wave and respectively at two L in succession ₁carry out integration to construct range function in/(2v) window length, finally according to range function along line length L ₁scope expert ripple sudden change distribution carrys out the fault branch of identification multiple-limb radiation network.

Embodiment 1:

Adopt the multiple-limb radial power distribution network network topological structure as shown in Figure 1 with outgoing lines wiring, measuring end is positioned at head end M, perfects circuit L ₄=48km, L ₅=4km.Suppose that circuit MN is made up of trunk 1, trunk 2, trunk 3, spoke 1 and spoke 2, line length is respectively L ₁=20km, L ₂=15km, L ₃=15km, L ₆=10km, L ₇=10km.Now suppose that within trunk 2 half line length, distance M holds 26km place (namely leaving B node 6km) that A phase earth fault occurs, and first utilizes PCA-SVM identification of defective to be positioned at cut set 2, then respectively at [k ₀, k ₀+ L ₁/ (2v)] and [k ₀+ L ₁/ (2v), k ₀+ L ₁/ v] time window in, utilize line mould current traveling wave and the line mode voltage row ripple of trunk 1 measuring end, material calculation along the line gets 0.1km, calculates the range function f of measuring end M according to Bei Jielong line transmission equation _ux () is along line length L ₁row ripple sudden change distribution as shown in Figures 2 and 3.As shown in Figure 2, [k ₀, k ₀+ L ₁/ (2v)] time window in, f _uIcatastrophe point A (the x)=6km of (x), and polarity is negative; As shown in Figure 3, [k ₀+ L ₁/ (2v), k ₀+ L ₁/ v] time window in, f _uIIcatastrophe point B (the x)=9km of (x), and polarity is just.Because A (x)+B (x)=6+9=15km, meet the line length constraint condition shown in formula (13), so fault is positioned within trunk 2 half line length.

Embodiment 2:

Adopt the multiple-limb radial power distribution network network topological structure as shown in Figure 1 with outgoing lines wiring, measuring end is positioned at head end M, perfects circuit L ₄=48km, L ₅=4km.Suppose that circuit MN is made up of trunk 1, trunk 2, trunk 3, spoke 1 and spoke 2, line length is respectively L ₁=20km, L ₂=15km, L ₃=15km, L ₆=10km, L ₇=10km.When now supposing that outside trunk 3 half line length, distance M holds 45km place (namely leaving C node 10km) that A phase earth fault occurs, PCA-SVM identification of defective is first utilized to be positioned at cut set 3, then respectively at [k ₀, k ₀+ L ₁/ (2v)] and [k ₀+ L ₁/ (2v), k ₀+ L ₁/ v] time window in, utilize line mould current traveling wave and the line mode voltage row ripple of trunk 1 measuring end, material calculation along the line gets 0.1km, calculates the range function f of measuring end M according to Bei Jielong line transmission equation _ux () is along line length L ₁row ripple sudden change distribution as shown in Figures 2 and 3.As shown in Figure 2, [k ₀, k ₀+ L ₁/ (2v)] time window in, f _uIcatastrophe point A (the x)=10km of (x), and polarity is negative; As shown in Figure 3, [k ₀+ L ₁/ (2v), k ₀+ L ₁/ v] time window in, f _uIIcatastrophe point B (the x)=5km of (x), and polarity is negative.Because A (x)+B (x)=10+5=15km, meet the line length constraint condition shown in formula (13), so fault is positioned at trunk 3.

Claims (2)

1. the radiation network Fault branch identification method based on voltage's distribiuting along the line and all standing of row ripple information, it is characterized in that: when multiple-limb radial distribution generation singlephase earth fault, adopt PCA-SVM to identify the fault branch place cut set of the radial distribution of multiple-limb; Then under fault cut set, line line ripple is asked for the fault traveling wave data that trunk 1 measuring end obtains, the line line ripple that application obtains also calculates voltage traveling wave along the line and current traveling wave distribution according to Bei Jielong circuit equation of transfer, voltage traveling wave along the line and current traveling wave along the line are carried out direction along the line row Wave Decomposition, obtain the direction row ripple of distribution along the line, recycle its direct wave be multiplied by backward-travelling wave and respectively at two L in succession ₁carry out integration to construct range function in/(2v) window length, finally according to range function along line length L ₁scope expert ripple sudden change distribution carrys out the fault branch of identification multiple-limb radiation network.

2., according to the radiation network Fault branch identification method based on voltage's distribiuting along the line and all standing of row ripple information according to claim 1, it is characterized in that concrete steps are:

(1), during multiple-limb radial distribution generation singlephase earth fault, application PCA-SVM differentiates the fault cut set of this radial distribution;

(2) under sampling rate 1MHz, voltage, the electric current that trunk 1 measuring end obtains is sampled, obtains phase current sampling value sequence i _m,a(k), i _m,b(k), i _m,c(k), phase voltage sampled value sequence u _m,a(k), u _m,b(k), u _m,c(k), wherein k represents sampled point, k=1,2, M represents measuring end;

(3) the discrete series i of wire finishing die electric current and line mode voltage is asked respectively according to formula (1) and formula (2) _m,s(k) and u _m,s(k):

i _M,s(k)＝i _M,a(k)-i _M,b(k)(1)

u _M,s(k)＝u _M,a(k)-u _M,b(k)(2)

(4) calculating of distribution along the line: utilize formula (3) and formula (4) to calculate the voltage's distribiuting along the line of measuring end place feeder line trunk 1 and distribution of current along the line respectively;

$\begin{array}{c}{u}_{M,x,s}\left(x,k\right)=\frac{1}{2}{\left(\frac{Z_{c,s}+r_{s}x/4}{Z_{c,s}}\right)}^2\[u_{M,s}\left(k+\frac{x}{v_s}\right)-i_{M,s}\left(k+\frac{x}{v_s}\right)\left(Z_{c,s}+\frac{r_{s}x}{4}\right)\]\\ +\frac{1}{2}{\left(\frac{Z_{c,s}-r_{s}x/4}{Z_{c,s}}\right)}^2\[u_{M,s}\left(k-\frac{x}{v_s}\right)+i_{M,s}\left(k-\frac{x}{v_s}\right)\left(Z_{c,s}-r_{s}x\right)\]\\ -{\left(\frac{r_{s}x/4}{Z_{c,s}}\right)}^2{u}_{M,s}\left(k\right)-\frac{r_{s}x}{4}\left(\frac{Z_{c,s}+r_{s}x/4}{Z_{c,s}}\right)\left(\frac{Z_{c,s}-r_{s}x/4}{Z_{c,s}}\right)i_{M,s}\left(k\right)\end{array}---\left(3\right)$

$\begin{array}{c}{i}_{M,x,s}\left(x,k\right)=\frac{1}{2Z_{c,s}}\left(\frac{Z_{c,s}+r_{s}x/4}{Z_{c,s}}\right)\[u_{M,s}\left(k+x/v_s\right)-i_{M,s}\left(k+x/v_s\right)\·\left(Z_{c,s}+r_{s}x/4\right)\]\\ -\frac{1}{2Z_{c,s}}\left(\frac{Z_{c,s}-r_{s}x/4}{Z_{c,s}}\right)\[u_{M,s}\left(k-x/v_s\right)+i_{M,s}\left(k-x/v_s\right)\·\left(Z_{c,s}-r_{s}x/4\right)\]\end{array}$

$-\frac{1}{2Z_{c,s}}\·\frac{r_{s}x}{2Z_{c,s}}\[u_{M,s}\left(k\right)-i_{M,s}\left(k\right)(r_{s}x/4)\]---\left(4\right)$

In formula, s is Aerial mode component; X is the distance of any point along the line to measuring end; V is the wave velocity of circuit; Z _c,sfor the characteristic impedance of circuit; r _sfor circuit resistance per unit length; u _m,sk line mode voltage row ripple that () is measuring end; i _m,sk line mould current traveling wave that () is measuring end; u _{m, x, s}(x, k) is for the k moment is apart from the voltage at measuring end x place; i _{m, x, s}(x, k) is for the k moment is apart from the electric current at measuring end x place;

(5) direct wave and the backward-travelling wave of distribution along the line is calculated: calculate the capable ripple of forward voltage of trunk 1 distribution along the line, the capable ripple of reverse voltage of distribution along the line respectively according to formula (5) and formula (6), that is:

u ⁺ _M,x,s＝(u _M,x,s+Z _c,si _M,x,s)/2(5)

u ^- _M,x,s＝(u _M,x,s-Z _c,si _M,x,s)/2(6)

(6) the direct wave gradient of distribution along the line and the calculating of backward-travelling wave gradient: the forward voltage gradient utilizing the difference structure distribution along the line of adjacent two sampled values of the capable ripple of forward voltage of distribution along the line, that is:

c ⁺ _M,dif—u(k)＝u ⁺ _k,x,s(k)-u ⁺ _k,x,s(k-1)(7)

Utilize the reverse voltage gradient of the difference structure distribution along the line of adjacent two sampled values of the capable ripple of reverse voltage of distribution along the line, that is:

c ^- _M,dif—u(k)＝u ^- _k,x,s(k)-u ^- _k,x,s(k-1)(8)

(7) direct wave sudden change and the backward-travelling wave sudden change of distribution along the line is calculated: the capable ripple sudden change of forward voltage of extracting trunk 1 distribution along the line according to formula (9), that is:

${S^{+}}_{M,2u}(x,k)=\underset{n=k-R+1}{\overset{k}{\Σ}}{\[{c^{+}}_{M,dif\_u}\left(k\right)\]}^3---\left(9\right)$

The capable ripple sudden change of reverse voltage of trunk 1 distribution is along the line extracted, that is: according to formula (10)

${S^{-}}_{M,2u}(x,k)=\underset{n=k-R+1}{\overset{k}{\Σ}}{\[{c^{-}}_{M,dif\_u}\left(k\right)\]}^3---\left(10\right)$

In formula, R is taken as 3;

(8) structure of range function: adopt formula (11) and formula (12), window [k when direct wave step (6) obtained sudden change is multiplied with backward-travelling wave sudden change and observes respectively at row ripple ₀, k ₀+ L ₁/ (2v)] and [k ₀+ L ₁/ (2v), k ₀+ L ₁/ v] in carry out integration, obtain range function f _uI(x) and f _uIIthe row ripple sudden change along the line of (x):

$f_{uI}\left(x\right)={\∫}_{k_0}^{k_0+L_1/2v}{S}_{M,2u}^{+}(x,k)\×S_{M,2u}^{-}(x,k)dk,x\∈\[0,L_1/2\]---\left(11\right)$

$f_{uII}\left(x\right)={\∫}_{k_0+L_1/2v}^{k_0+L_1/v}{S}_{M,2u}^{+}(x,k)\×S_{M,2u}^{-}(x,k)dk,x\∈\[L_1/2,L_1\]---\left(12\right)$

In formula, k ₀represent the fault initial row ripple due in that measuring end M detects; L ₁for the line length of trunk 1;

(9) structure of localization of fault criterion:

[k is calculated according to step (8) ₀, k ₀+ L ₁/ (2v)] and [k ₀+ L ₁/ (2v), k ₀+ L ₁/ v] two in succession time window in, range function f _uI(x) and f _uIIx the distribution catastrophe point along the line of (), its respective distances is designated as [x respectively _i1, x _i2... ] and [x _iI1, x _iI2... ];

If in step (1), SVM exports is 1, then failure judgement occurs on trunk 1;

If SVM exports in step (1) is 2, then failure judgement occurs in fault cut set 2 (that is: trunk 2 or branch 1), if now [x _i1, x _i2... ] in mutation distance x ^* _i[x _iI1, x _iI2... ] in mutation distance x ^* _iImeet:

x ^* _I+x ^* _II＝L ₂(13)

And work as x ^* _icatastrophe point polarity be negative, x ^* _iIcatastrophe point polarity be timing, failure judgement is positioned within trunk 2 half line length; Work as x ^* _icatastrophe point polarity be just, x ^* _iIcatastrophe point polarity for time negative, failure judgement is positioned at outside trunk 2 half line length, in formula (13), L ₂for the line length of trunk 2;

If [x _i1, x _i2... ] in mutation distance x ^* _i[x _iI1, x _iI2... ] in mutation distance x ^* _iImeet:

x ^* _I+x ^* _II＝L ₆(14)

And x ^* _iand x ^* _iIcatastrophe point polarity be negative polarity, then failure judgement occurs in branch 1, in formula (14), L ₆for the line length of branch 1;

If SVM exports in step (1) is 3, then failure judgement occurs in fault cut set 3 (that is: trunk 3 or branch 2), if now [x _i1, x _i2... ] in mutation distance x ^* _i[x _iI1, x _iI2... ] in mutation distance x ^* _iImeet constraint condition shown in formula (13), and x ^* _iand x ^* _iIcatastrophe point polarity be negative polarity, then failure judgement occurs in trunk 3;

If [x _i1, x _i2... ] in mutation distance x ^* _i[x _iI1, x _iI2... ] in mutation distance x ^* _iImeet constraint condition shown in formula (14), and x ^* _iand x ^* _iIcatastrophe point polarity be negative polarity, then failure judgement occurs in branch 1.