CN105738770A - Cable hybrid line single-ended traveling wave fault location method based on fault traveling wave distribution characteristic along line - Google Patents

Abstract

The invention relates to a cable hybrid line single-ended traveling wave fault location method based on a fault traveling wave distribution characteristic along a line, and belongs to the technical field of the relay protection of an electric power system. When a line fault occurs, firstly, a high speed acquisition device obtains fault current traveling wave data at a measurement terminal, and intercepts the traveling wave data when fault initial traveling wave reaches a front (l1/vl +l2/vc) time window length and when the fault initial traveling wave reaches a rear 2(l1/vl +l2/vc) time window length. And then fault sections are judged by using a DWT-PCA-SVM judgment mechanism. Then saltation points of the distribution along the line of fault location functions fuI(x) and fuII(x) of the two successive time windows are calculated, and the fault location of the cable hybrid line is realized according to a distribution rule of the fault location function saltation along the line.

Description

A kind of cable joint line single-ended traveling wave based on fault traveling wave distribution character along the line is surveyed Away from method

Technical field

The present invention relates to a kind of cable joint line Single Terminal Traveling Wave Fault Location method based on fault traveling wave distribution character along the line, Belong to Relay Protection Technology in Power System field.

Background technology

The task of fault localization is exactly when the certain point of circuit breaks down, by measured current, the electricity at circuit two ends The parameters such as pressure and line impedance calculate fault distance.Generally, transmission line fault distance-finding method mainly has two classes, a class to be impedances Method, is directly to calculate fault impedance or the algorithm of its percentage ratio；Another kind of is traveling wave method, utilizes high frequency fault transient current, electricity The row ripple etc. of pressure carrys out the distance of indirect judgement trouble point.

Transmission line travelling wave fault localization experienced by traveling wave fault location and two stages of Modern Travelling Wave Fault Locating in early days. Recently as developing rapidly of hardware manufacturing level and computer technology, Modern Fault Location Techniques Based On Fault Generated Travelling Waves runs at a lot of aspects Predicament be obtained for breakthrough, but still suffer from some and not yet solve or be badly in need of problem to be improved, these problems mainly have: therefore How the identification precision of barrier row ripple improves, and the wavefront arrival measurement end moment catches the most accurately, different transmission lines of electricity And how velocity of wave corresponding to electric pressure choose, how utilize other to perfect fault message that circuit contains realizes wide area row ripple is surveyed Away from etc. aspect.Therefore, Modern Travelling Wave Fault Locating also will be in many technology and principal level in the road of future development Challenge.

Summary of the invention

The technical problem to be solved in the present invention is to propose a kind of cable blend based on fault traveling wave distribution character along the line Road Single Terminal Traveling Wave Fault Location method, in order to solve the problems referred to above.

The technical scheme is that a kind of cable joint line single-ended traveling wave based on fault traveling wave distribution character along the line Distance-finding method, when line failure, first, is obtained measuring end fault current row wave datum by high speed acquisition device, and cuts Take (l before fault initial row ripple arrives₁/v_l+l₂/v_c) time window length and fault initially arrive after 2 (l₁/v_l+l₂/v_c) time window length row Wave datum；Secondly, DWT-PCA-SVM is used to differentiate mechanism, it is achieved the differentiation of faulty section；Again, calculate two in succession time window Range function f_uI(x) and f_uIICatastrophe point of (x) distribution along the line, and the regularity of distribution suddenlyd change along the line according to range function realizes line The fault localization of cable joint line.

Concretely comprise the following steps:

The first step, when line failure, by high speed acquisition device obtain measuring end fault current row wave datum, and cut Take (l before fault initial row ripple arrives₁/v_l+l₂/v_c) time window length and fault initially arrive after 2 (l₁/v_l+l₂/v_c) time window length row Wave datum；

Second step, employing the method described below failure judgement section:

First, choosing row ripple after being taken absolute value by current traveling wave and arrive front 3 points of measuring end, row ripple arrives after measuring end 27 Individual, i.e. use 2 (l₁/v_l+l₂/v_c) time window length data, carry out wavelet decomposition (DWT), obtain a₀、d₁～d₈Wavelet reconstruction system Number；Secondly, a, d are chosen₂～d₈The fault current waveform reconstructing acquisition under yardstick carries out principal component analysis (PCA), obtains main constituent PC₁～PC₅The projection value q that main constituent is corresponding₁～q₅；Again, by q₁～q₅As the input quantity of support vector machine (SVM), if then SVM output 0, represents overhead transmission line fault；If SVM output 1, represent cable fault；And said method is called " DWT-PCA-SVM " Faulty section differentiates mechanism；

3rd step, structure range function:

First, voltage's distribiuting along the line is calculated according to formula (1) and (2)；

$\begin{array}{c}{u}_x(x,t)=\frac{1}{2}{\left(\frac{Z_c+rx/4}{Z_c}\right)}^2\[u_M(t+\frac{x}{v})-i_M(t+\frac{x}{v})(Z_c+\frac{rx}{4})\]\\ +\frac{1}{2}{\left(\frac{Z_c-rx/4}{Z_c}\right)}^2\[u_M(t-\frac{x}{v})+i_M(t-\frac{x}{v})(Z_c-rx)\]\\ -{\left(\frac{rx/4}{Z_c}\right)}^2{u}_M\left(t\right)-\frac{rx}{4}\left(\frac{Z_c+rx/4}{Z_c}\right)\left(\frac{Z_c-rx/4}{Z_c}\right)i_M\left(t\right)\end{array}---\left(1\right)$

$\begin{array}{c}{i}_x(x,t)=\frac{1}{2Z_c}\left(\frac{Z_c+rx/4}{Z_c}\right)\[u_M(t+x/v)-i_M(t+x/v)\·(Z_c+rx/4)\]\\ -\frac{1}{2Z_c}\left(\frac{Z_c-rx/4}{Z_c}\right)\[u_M(t-x/v)+i_M(t-x/v)\·(Z_c-rx/4)\]\\ =\frac{1}{2Z_c}\·\frac{rx}{2Z_c}\[u_M\left(t\right)-i_M\left(t\right)(tx/4)\]\end{array}---\left(2\right)$

In formula, u_M=i_k×Z_c, i_kThe popular ripple of line electricity is perfected for adjacent；

Secondly, calculated direction row ripple is distributed along circuit:

It is calculated voltage traveling wave and current traveling wave and formula (3) according to formula (1) and formula (2) and (4) calculate direct wave And backward-travelling wave.

Direct wave: u⁺ _x=(u_x+Z_ci_x)/2 (3)

Backward-travelling wave: u^- _x=(u_x-Z_ci_x)/2 (4)

Again, direct wave and the sudden change of backward-travelling wave are extracted: first use formula (5) and (6) calculus of differences to obtain With

$c_{dif\_u^{+}}\left(t\right)=\[u_x^{+}\left(t\right)-u_x^{+}(t-\mathrm{\Δ}t)\]/\mathrm{\Δ}t---\left(5\right)$

$c_{dif\_u^{-}}\left(t\right)=\[u_x^{-}\left(t\right)-u_x^{-}(t-\mathrm{\Δ}t)\]/\mathrm{\Δ}t---\left(6\right)$

For the difference result of direct wave,For the difference result of backward-travelling wave, Δ t is the sampling interval；

Calculate difference result c again_difEnergy S in a period of time_2u(x, t), it may be assumed that

$S_{2u^{+}}(x,t)=\underset{n=t-N\mathrm{\Δ}t+1}{\overset{t}{\Σ}}{\[c_{dif\_u^{+}}\left(t\right)\]}^3---\left(7\right)$

$S_{2u^{-}}(x,t)=\underset{n=t-N\mathrm{\Δ}t+1}{\overset{t}{\Σ}}{\[c_{dif\_u^{+}}\left(t\right)\]}^3---\left(8\right)$

In formula,For direct wave energy within a period of time,For backward-travelling wave within a period of time Energy；

Finally, range function is built:

Respectively at [t₀,t₀+ l/ (2v)] and [t₀+l/(2v),t₀+ l/v] in time window length, obtain range function f_uI(x) and f_uII(x)；

$f_{uI}\left(x\right)={\∫}_{t_0}^{t_0+l/\left(2v\right)}{S}_{2u^{+}}(x,t)\×S_{2u^{-}}(x,t)dt---\left(9a\right)$

$f_{uII}\left(x\right)={\∫}_{t_0+l/\left(2v\right)}^{t_0+l/v}{S}_{2u^{+}}(x,t)\×S_{2u^{-}}(x,t)dt---\left(9b\right)$

If judging that fault is positioned at overhead transmission line according to step 2, then formula (1) and (2) take v=v_l, l=l₁, i.e. formula (9a) being calculated with (9b) is that range function is along the sudden change distribution of overhead transmission line total length；

If judging that fault is positioned at cable according to step 2, then formula (1) and (2) take v=v_c, l=l₂, i.e. formula (9a) and (9b) being calculated is that range function is along the sudden change distribution of cable run total length；

4th step, determine fault distance:

By [t₀,t₀+ l/ (2v)] time window in be calculated range function f_uIX the catastrophe point of () is designated as disaggregation f of suddenling change_uI= [x_I1,x_I2,...], [t₀+l/(2v),t₀+ l/v] time window in be calculated range function f_uIIX the catastrophe point of () is designated as sudden change Disaggregation f_uII=[x_II1,x_II2,……]；

According to formula x^* _I+x^* _II=l, and combine polarity and the amplitude of catastrophe point, determine fault distance.

The model of structure DWT-PCA-SVM cable mixed line fault line segment identification and algorithm:

First, use the cable joint line analogue system shown in accompanying drawing 1, as a example by overhead transmission line origin or beginning is observed.Assume There is A phase metallic earthing fault in circuit, fault initial phase angle is set to+90 ° and-90 °.Abort situation is set from leaving origin or beginning 1km starts, and step-length is 1km, traversal cable and overhead transmission line total length.Obtain 68 fault current row wave datum, and to electric current row Ripple is chosen row ripple and is arrived front 3 points of measuring end after taking absolute value, row ripple arrives 27 points after measuring end, i.e. uses 2 (l₁/v_l+l₂/ v_c) time window length data.Secondly, current traveling wave data are carried out wavelet decomposition, and chooses a, d₂～d₈Wavelet coefficient under yardstick It is reconstructed；Again, the current traveling wave wavelet coefficient after reconstruct is carried out PCA (main constituent) cluster analysis, obtains PC₁～PC₅Main The projection value q that composition is corresponding₁～q₅.Finally, by q₁～q₅As the input quantity of SVM, it is trained, and specifies SVM output 0, table Show overhead transmission line fault；If SVM output 1, represent cable fault.Wherein, l₁For overhead transmission line total length, l₂For cable run total length, v_lFor overhead transmission line wave velocity, v_cFor cable run wave velocity.

The invention has the beneficial effects as follows:

(1) DWT-PCA-SVM is used to differentiate mechanism, it is achieved that the reliable identification of cable mixed line fault section；

(2) Bei Jielong circuit model is utilized to have the high pass filter effect in line length dimension so that distance-finding method is more Tool robustness and universality, it is easy to accomplish single end distance measurement practical.

Accompanying drawing explanation

Fig. 1 is the line assumption diagram in the embodiment of the present invention 1, embodiment 2 and embodiment 3, overhead transmission line total length 25km, electricity Cable total track length is 10km.

Detailed description of the invention

Below in conjunction with the accompanying drawings and detailed description of the invention, the invention will be further described.

A kind of cable joint line Single Terminal Traveling Wave Fault Location method based on fault traveling wave distribution character along the line, when circuit occurs During fault, first, by high speed acquisition device obtain measuring end fault current row wave datum, and intercept fault initial row ripple arrive before (l₁/v_l+l₂/v_c) time window length and fault initially arrive after 2 (l₁/v_l+l₂/v_c) time window length row wave datum；Secondly, DWT-is used PCA-SVM differentiates mechanism, it is achieved the differentiation of faulty section；Again, calculate two in succession time window range function f_uI(x) and f_uII(x) The catastrophe point of distribution, and the regularity of distribution suddenlyd change along the line according to range function along the line realizes the fault localization of cable joint line.

Concretely comprise the following steps:

The first step, when line failure, by high speed acquisition device obtain measuring end fault current row wave datum, and cut Take (l before fault initial row ripple arrives₁/v_l+l₂/v_c) time window length and fault initially arrive after 2 (l₁/v_l+l₂/v_c) time window length row Wave datum；

Second step, employing the method described below failure judgement section:

First, choosing row ripple after being taken absolute value by current traveling wave and arrive front 3 points of measuring end, row ripple arrives after measuring end 27 Individual, i.e. use 2 (l₁/v_l+l₂/v_c) time window length data, carry out wavelet decomposition (DWT), obtain a₀、d₁～d₈Wavelet reconstruction system Number；Secondly, a, d are chosen₂～d₈The fault current waveform reconstructing acquisition under yardstick carries out principal component analysis (PCA), obtains main constituent PC₁～PC₅The projection value q that main constituent is corresponding₁～q₅；Again, by q₁～q₅As the input quantity of support vector machine (SVM), if then SVM output 0, represents overhead transmission line fault；If SVM output 1, represent cable fault；And said method is called " DWT-PCA-SVM " Faulty section differentiates mechanism；

3rd step, structure range function:

First, voltage's distribiuting along the line is calculated according to formula (1) and (2)；

$\begin{array}{c}{u}_x(x,t)=\frac{1}{2}{\left(\frac{Z_c+rx/4}{Z_c}\right)}^2\[u_M(t+\frac{x}{v})-i_M(t+\frac{x}{v})(Z_c+\frac{rx}{4})\]\\ +\frac{1}{2}{\left(\frac{Z_c-rx/4}{Z_c}\right)}^2\[u_M(t-\frac{x}{v})+i_M(t-\frac{x}{v})(Z_c-rx)\]\\ -{\left(\frac{rx/4}{Z_c}\right)}^2{u}_M\left(t\right)-\frac{rx}{4}\left(\frac{Z_c+rx/4}{Z_c}\right)\left(\frac{Z_c-rx/4}{Z_c}\right)i_M\left(t\right)\end{array}---\left(1\right)$

$\begin{array}{c}{i}_x(x,t)=\frac{1}{2Z_c}\left(\frac{Z_c+rx/4}{Z_c}\right)\[u_M(t+x/v)-i_M(t+x/v)\·(Z_c+rx/4)\]\\ -\frac{1}{2Z_c}\left(\frac{Z_c-rx/4}{Z_c}\right)\[u_M(t-x/v)+i_M(t-x/v)\·(Z_c-rx/4)\]\\ =\frac{1}{2Z_c}\·\frac{rx}{2Z_c}\[u_M\left(t\right)-i_M\left(t\right)(tx/4)\]\end{array}---\left(2\right)$

In formula, u_M=i_k×Z_c, i_kThe popular ripple of line electricity is perfected for adjacent；

Secondly, calculated direction row ripple is distributed along circuit:

It is calculated voltage traveling wave and current traveling wave and formula (3) according to formula (1) and formula (2) and (4) calculate direct wave And backward-travelling wave.

Direct wave: u⁺ _x=(u_x+ Z_ci_x)/2 (3)

Backward-travelling wave: u^- _x=(u_x-Z_ci_x)/2 (4)

Again, direct wave and the sudden change of backward-travelling wave are extracted: first use formula (5) and (6) calculus of differences to obtain With

$c_{dif\_u^{+}}\left(t\right)=\[u_x^{+}\left(t\right)-u_x^{+}(t-\mathrm{\Δ}t)\]/\mathrm{\Δ}t---\left(5\right)$

$c_{dif\_u^{-}}\left(t\right)=\[u_x^{-}\left(t\right)-u_x^{-}(t-\mathrm{\Δ}t)\]/\mathrm{\Δ}t---\left(6\right)$

For the difference result of direct wave,For the difference result of backward-travelling wave, Δ t is the sampling interval；

Calculate difference result c again_difEnergy S in a period of time_2u(x, t), it may be assumed that

$S_{2u^{+}}(x,t)=\underset{n=t-N\mathrm{\Δ}t+1}{\overset{t}{\Σ}}{\[c_{dif\_u^{+}}\left(t\right)\]}^3---\left(7\right)$

$S_{2u^{-}}(x,t)=\underset{n=t-N\mathrm{\Δ}t+1}{\overset{t}{\Σ}}{\[c_{dif\_u^{+}}\left(t\right)\]}^3---\left(8\right)$

In formula,For direct wave energy within a period of time,For backward-travelling wave within a period of time Energy；

Finally, range function is built:

Respectively at [t₀,t₀+ l/ (2v)] and [t₀+l/(2v),t₀+ l/v] in time window length, obtain range function f_uI(x) and f_uII(x)；

$f_{uI}\left(x\right)={\∫}_{t_0}^{t_0+l/\left(2v\right)}{S}_{2u^{+}}(x,t)\×S_{2u^{-}}(x,t)dt---\left(9a\right)$

$f_{uII}\left(x\right)={\∫}_{t_0+l/\left(2v\right)}^{t_0+l/v}{S}_{2u^{+}}(x,t)\×S_{2u^{-}}(x,t)dt---\left(9b\right)$

If judging that fault is positioned at overhead transmission line according to step 2, then formula (1) and (2) take v=v_l, l=l₁, i.e. formula (9a) being calculated with (9b) is that range function is along the sudden change distribution of overhead transmission line total length；

If judging that fault is positioned at cable according to step 2, then formula (1) and (2) take v=v_c, l=l₂, i.e. formula (9a) and (9b) being calculated is that range function is along the sudden change distribution of cable run total length；

4th step, determine fault distance:

By [t₀,t₀+ l/ (2v)] time window in be calculated range function f_uIX the catastrophe point of () is designated as disaggregation f of suddenling change_uI= [x_I1,x_I2,...], [t₀+l/(2v),t₀+ l/v] time window in be calculated range function f_uIIX the catastrophe point of () is designated as sudden change Disaggregation f_uII=[x_II1,x_II2,……]；

According to formula x^* _I+x^* _II=l, and combine polarity and the amplitude of catastrophe point, determine fault distance.

The model of structure DWT-PCA-SVM cable mixed line fault line segment identification and algorithm:

First, use the cable joint line analogue system shown in accompanying drawing 1, as a example by overhead transmission line origin or beginning is observed.Assume There is A phase metallic earthing fault in circuit, fault initial phase angle is set to+90 ° and-90 °.Abort situation is set from leaving origin or beginning 1km starts, and step-length is 1km, traversal cable and overhead transmission line total length.Obtain 68 fault current row wave datum, and to electric current row Ripple is chosen row ripple and is arrived front 3 points of measuring end after taking absolute value, row ripple arrives 27 points after measuring end, i.e. uses 2 (l₁/v_l+l₂/ v_c) time window length data.Secondly, current traveling wave data are carried out wavelet decomposition, and chooses a, d₂～d₈Wavelet coefficient under yardstick It is reconstructed；Again, the current traveling wave wavelet coefficient after reconstruct is carried out PCA (main constituent) cluster analysis, obtains PC₁～PC₅Main The projection value q that composition is corresponding₁～q₅.Finally, by q₁～q₅As the input quantity of SVM, it is trained, and specifies SVM output 0, table Show overhead transmission line fault；If SVM output 1, represent cable fault.Wherein, l₁For overhead transmission line total length, l₂For cable run total length, v_lFor overhead transmission line wave velocity, v_cFor cable run wave velocity.

Embodiment 1: as a example by the transmission line of electricity shown in Fig. 1, there is earth fault in air line distance M end 2.5km.

According to step 2 in description, choose row ripple after being taken absolute value by current traveling wave and arrive front 3 points of measuring end, row ripple Arrive 27 points after measuring end, i.e. use 2 (l₁/v_l+l₂/v_c) time window length data, carry out wavelet decomposition, obtain a₀、d₁～d₈ Wavelet reconstruction coefficient.Secondly, a, d are chosen₂～d₈The fault current waveform reconstructing acquisition under yardstick carries out PCA cluster analysis, obtains PC₁～PC₅The projection value q that main constituent is corresponding₁～q₅=[3.05 0.0054 0.35 0.024-0.016] is as the input of SVM Attribute, SVM output 0, represent overhead transmission line fault.According to step 3, four obtain f_uI=[-2.41-4.96], f_uII=[+ 22.35], it is known that, x_I1+x_II1=2.41+22.35=24.76 ≈ l₁, therefore obtain fault and be positioned at overhead transmission line, and leave M end 2.41km。

Embodiment 2: as a example by the transmission line of electricity shown in Fig. 1, there is earth fault in air line distance M end 10.5km.

According to step 2 in description, choose row ripple after being taken absolute value by current traveling wave and arrive front 3 points of measuring end, row ripple Arrive 27 points after measuring end, i.e. use 2 (l₁/v_l+l₂/v_c) time window length data, carry out wavelet decomposition, obtain a₀、d₁～d₈ Wavelet reconstruction coefficient.Secondly, a, d are chosen₂～d₈The fault current waveform reconstructing acquisition under yardstick carries out PCA cluster analysis, obtains PC₁～PC₅The projection value q that main constituent is corresponding₁～q₅=[1.81 0.12 0.02 0.013 0.014] belong to as the input of SVM Property, SVM output 0, represent overhead transmission line fault.According to step 3, four obtain f_uI=[+14.30], f_uII=[-10.33], can Know, x_I1+x_II1=14.30+10.33=24.63 ≈ l₁, therefore obtain fault and be positioned at overhead transmission line, and leave M end 10.70km.

Embodiment 3: as a example by the transmission line of electricity shown in Fig. 1, there is earth fault in cable run distance M end 28.5km.

According to step 2 in description, choose row ripple after being taken absolute value by current traveling wave and arrive front 3 points of measuring end, row ripple Arrive 27 points after measuring end, i.e. use 2 (l₁/v_l+l₂/v_c) time window length data, carry out wavelet decomposition, obtain a₀、d₁～d₈ Wavelet reconstruction coefficient.Secondly, a, d are chosen₂～d₈The fault current waveform reconstructing acquisition under yardstick carries out PCA cluster analysis, obtains PC₁～PC₅The projection value q that main constituent is corresponding₁～q₅Defeated as SVM of=[0.96-0.35-0.25 0.006-0.0049] Enter attribute, SVM output 0, represent overhead transmission line fault.According to step 3, four obtain f_uI=[+3.63], f_uII=[+6.22], can Know, x_I1+x_II1=3.63+6.22=9.85 ≈ l₂, therefore obtain fault and be positioned at cable run, and leave M end 28.63km.

In the above-described example, "+" representing that the polarity suddenlyd change just is, "-" represents that the polarity of sudden change is negative.

Above in association with accompanying drawing, the detailed description of the invention of the present invention is explained in detail, but the present invention is not limited to above-mentioned Embodiment, in the ken that those of ordinary skill in the art are possessed, it is also possible to before without departing from present inventive concept Put that various changes can be made.

Claims (2)

1. a cable joint line Single Terminal Traveling Wave Fault Location method based on fault traveling wave distribution character along the line, it is characterised in that: When line failure, first, high speed acquisition device obtain measuring end fault current row wave datum, and it is initial to intercept fault (l before the arrival of row ripple₁/v_l+l₂/v_c) time window length and fault initially arrive after 2 (l₁/v_l+l₂/v_c) time window length row wave datum；Its Secondary, use DWT-PCA-SVM to differentiate mechanism, it is achieved the differentiation of faulty section；Again, calculate two in succession time window range function f_uI (x) and f_uIICatastrophe point of (x) distribution along the line, and the regularity of distribution suddenlyd change along the line according to range function realizes cable joint line Fault localization.

Cable joint line Single Terminal Traveling Wave Fault Location side based on fault traveling wave distribution character along the line the most according to claim 1 Method, it is characterised in that concretely comprise the following steps:

The first step, when line failure, by high speed acquisition device obtain measuring end fault current row wave datum, and intercept therefore (l before barrier initial row ripple arrival₁/v_l+l₂/v_c) time window length and fault initially arrive after 2 (l₁/v_l+l₂/v_c) time window length row wave number According to；

Second step, employing the method described below failure judgement section:

First, choosing row ripple after being taken absolute value by current traveling wave and arrive front 3 points of measuring end, row ripple arrives after measuring end 27 Point, i.e. uses 2 (l₁/v_l+l₂/v_c) time window length data, carry out wavelet decomposition, obtain a₀、d₁～d₈Wavelet reconstruction coefficient；Its Secondary, choose a, d₂～d₈The fault current waveform reconstructing acquisition under yardstick carries out principal component analysis, obtains main constituent PC₁～PC₅Main The projection value q that composition is corresponding₁～q₅；Again, by q₁～q₅As the input quantity of support vector machine, if then SVM output 0, represent frame Empty line fault；If SVM output 1, represent cable fault；And said method is called that " DWT-PCA-SVM " faulty section differentiates mechanism；

3rd step, structure range function:

First, voltage's distribiuting along the line is calculated according to formula (1) and (2)；

$\begin{array}{c}{u}_x(x,t)=\frac{1}{2}{\left(\frac{Z_c+rx/4}{Z_c}\right)}^2\[u_M\left(t+\frac{x}{v}\right)-i_M\left(t+\frac{x}{v}\right)\left(Z_c+\frac{rx}{4}\right)\]\\ +\frac{1}{2}{\left(\frac{Z_c-rx/4}{Z_c}\right)}^2\[u_M\left(t-\frac{x}{v}\right)+i_M\left(t-\frac{x}{v}\right)(Z_c-rx)\]\\ -{\left(\frac{rx/4}{Z_c}\right)}^2{u}_M\left(t\right)-\frac{rx}{4}\left(\frac{Z_c+rx/4}{Z_c}\right)\left(\frac{Z_c-rx/4}{Z_c}\right)i_M\left(t\right)\end{array}---\left(1\right)$

$\begin{array}{c}{i}_x(x,t)=\frac{1}{2Z_c}\left(\frac{Z_c+rx/4}{Z_c}\right)\[u_M\left(t+x/v\right)-i_M\left(t+x/v\right)\·\left(Z_c+rx/4\right)\]\\ -\frac{1}{2Z_c}\left(\frac{Z_c-rx/4}{Z_c}\right)\[u_M\left(t-x/v\right)+i_M\left(t-x/v\right)\·(Z_c-rx/4)\]\\ -\frac{1}{2Z_c}\·\frac{rx}{2Z_c}\[u_M\left(t\right)-i_M\left(t\right)(rx/4)\]\end{array}---\left(2\right)$

In formula, u_M=i_k×Z_c, i_kThe popular ripple of line electricity is perfected for adjacent；

Secondly, calculated direction row ripple is distributed along circuit:

It is calculated voltage traveling wave and current traveling wave and formula (3) according to formula (1) and formula (2) and (4) calculate direct wave with anti- Xiang Hangbo.

Direct wave: u⁺ _x=(u_x+Z_ci_x)/2 (3)

Backward-travelling wave: u^- _x=(u_x-Z_ci_x)/2 (4)

Again, direct wave and the sudden change of backward-travelling wave are extracted: first use formula (5) and (6) calculus of differences to obtainWith

$c_{dif\_u^{+}}\left(t\right)=\[u_x^{+}\left(t\right)-u_x^{+}(t-\mathrm{\Δ}t)\]/\mathrm{\Δ}t---\left(5\right)$

$c_{dif\_u^{-}}\left(t\right)=\[u_x^{-}\left(t\right)-u_x^{-}(t-\mathrm{\Δ}t)\]/\mathrm{\Δ}t---\left(6\right)$

For the difference result of direct wave,For the difference result of backward-travelling wave, Δ t is the sampling interval；

Calculate difference result c again_difEnergy S in a period of time_2u(x, t), it may be assumed that

$S_{2u^{+}}(x,t)=\underset{n=t-N\mathrm{\Δ}t+1}{\overset{t}{\Σ}}{\[c_{dif\_u^{+}}\left(t\right)\]}^3---\left(7\right)$

$S_{2u^{-}}(x,t)=\underset{n=t-N\mathrm{\Δ}t+1}{\overset{t}{\Σ}}{\[c_{dif\_u^{+}}\left(t\right)\]}^3---\left(8\right)$

In formula,For direct wave energy within a period of time,For backward-travelling wave energy within a period of time Amount；

Finally, range function is built:

Respectively at [t₀,t₀+ l/ (2v)] and [t₀+l/(2v),t₀+ l/v] in time window length, obtain range function f_uI(x) and f_uII (x)；

$f_{uI}\left(x\right)={\∫}_{t_0}^{t_0+l/\left(2v\right)}{S}_{2u^{+}}(x,t)\×S_{2u^{-}}(x,t)dt---\left(9a\right)$

$f_{uII}\left(x\right)={\∫}_{t_0+l/\left(2v\right)}^{t_0+l/v}{S}_{2u^{+}}(x,t)\×S_{2u^{-}}(x,t)dt---\left(9b\right)$

If judging that fault is positioned at overhead transmission line according to step 2, then formula (1) and (2) take v=v_l, l=l₁, i.e. formula (9a) and (9b) being calculated is that range function is along the sudden change distribution of overhead transmission line total length；

If judging that fault is positioned at cable according to step 2, then formula (1) and (2) take v=v_c, l=l₂, i.e. formula (9a) and (9b) Be calculated is that range function is along the sudden change distribution of cable run total length；

4th step, determine fault distance:

By [t₀,t₀+ l/ (2v)] time window in be calculated range function f_uIX the catastrophe point of () is designated as disaggregation f of suddenling change_uI=[x_I1, x_I2...], [t₀+l/(2v),t₀+ l/v] time window in be calculated range function f_uIIX the catastrophe point of () is designated as disaggregation of suddenling change f_uII=[x_II1,x_II2,……]；

According to formula x^* _I+x^* _II=l, and combine polarity and the amplitude of catastrophe point, determine fault distance.