CN105403813A - Aerial line single-end travelling wave distance measuring method based on direction traveling wave decomposition and distance calibration - Google Patents

Abstract

The invention relates to an aerial line single-end travelling wave distance measuring method based on direction traveling wave decomposition and distance calibration, and belongs to the technical field of power system relay protection. First of all, a high-speed acquisition apparatus obtains fault current travelling wave data of a measurement end and intercepts a l/(2v) time window length before arrival of fault initial travelling waves and l/v time window length after the arrival of the fault initial travelling waves, i.e., travelling wave data of a 1.5l/v time window length all together; secondly, voltage travelling waves are constructed by use of adjacent sound line current travelling waves and wave impedance; then, according to the current travelling waves and the voltage travelling waves obtained from the above method, by use of a Bergeron formula, the voltage travelling waves and the current travelling waves along a line are calculated within the l/(2v) time window length; and finally, a distance measuring function is constructed, and a fault distance is obtained according to abrupt change points along the distance measuring function, wherein l is a total length of the line. According to the invention, automation of single-end distance measurement is easily realized by use of the high-pass filter effect of a Bergeron line model in terms of line length dimension.

Description

A kind of overhead transmission line Single Ended Fault Location based on direction row Wave Decomposition and distance calibration

Technical field

The present invention relates to a kind of overhead transmission line Single Terminal Traveling Wave Fault Location method based on direction row Wave Decomposition and distance calibration, belong to Relay Protection Technology in Power System field.

Background technology

The task of fault localization is exactly when certain of circuit a bit breaks down, and calculates fault distance by the measured current at circuit two ends, the parameter such as voltage and line impedance.Usually, transmission line fault distance-finding method mainly contains two classes, and a class is impedance method, is the algorithm directly calculating fault impedance or its number percent; Another kind of is traveling wave method, utilizes the row ripple etc. of high frequency fault transient current, voltage to carry out the distance of indirect judgement trouble spot.

Transmission line travelling wave fault localization experienced by early stage traveling wave fault location and two stages of Modern Travelling Wave Fault Locating.In recent years along with the develop rapidly of hardware manufacturing level and computer technology, the predicament that Modern Fault Location Techniques Based On Fault Generated Travelling Waves runs in a lot of is obtained for breakthrough, but still there are some problems not yet solving or be badly in need of to improve, these problems mainly contain: how the identification precision of fault traveling wave improves, how the wavefront arrival measuring junction moment catches accurately, how different transmission line of electricity and velocity of wave corresponding to electric pressure are chosen, and how failure message that circuit contains realizes the aspects such as wide area travelling wave ranging to utilize other to perfect.Therefore, Modern Travelling Wave Fault Locating also will in the face of the challenge in many new application aspects in the road of future development.A kind of overhead transmission line travelling wave ranging method based on direction row Wave Decomposition and distance calibration is now proposed.This location algorithm specific practice is in single-ended observation, the fault traveling wave data of measuring end are adopted to calculate voltage traveling wave along the line and current traveling wave, direction row ripple is obtained, recycling direct wave and backward-travelling wave structure range function according to voltage traveling wave, current traveling wave and wave impedance.

Summary of the invention

The technical problem to be solved in the present invention proposes a kind of overhead transmission line Single Terminal Traveling Wave Fault Location method based on direction row Wave Decomposition and distance calibration, in order to solve the problem.

Technical scheme of the present invention is: a kind of overhead transmission line Single Ended Fault Location based on direction row Wave Decomposition and distance calibration, first the capable wave datum of measuring end fault current is obtained by high speed acquisition device, and intercept fault initial row ripple arrive before l/ (2v) window length and fault initial row ripple arrive after l/v window length, i.e. the row wave datum of 1.5l/v window length altogether; Next utilizes the adjacent popular ripple of line electricity and the wave impedance of perfecting to construct voltage traveling wave; The current traveling wave again obtained according to said method and voltage traveling wave, utilize Bei Jielong formulae discovery to distribute at l/ (2v) window length voltage traveling wave and current traveling wave along the line; Finally build range function, and obtain fault distance according to the catastrophe point that range function is along the line.

Concrete steps are as follows:

The first step, read line wave datum: the capable wave datum of measuring end fault current obtained by high speed acquisition device, and intercept fault initial row ripple arrive before l/ (2v) window length and fault initial row ripple arrive after l/v window length, i.e. the row wave datum of 1.5l/v window length altogether;

Second step, utilize the adjacent popular ripple of line electricity and the wave impedance of perfecting to construct voltage traveling wave, that is:

u _M＝i _k×Z _c(1)

In formula, u _mfor measuring end voltage, i _kperfect circuit measuring end electric current for the longest, Zc is surge impedance of a line.

3rd step, calculated direction row ripple distribute along circuit: the current traveling wave obtained according to step (1) and step (2) and voltage traveling wave, utilize Bei Jielong formulae discovery at [t ₀, t ₀+ l/ (2v)] window length voltage traveling wave and current traveling wave distribute along the line.Wherein t ₀for namely fault initial row ripple arrives the moment of measuring end:

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

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

In formula, subscript s represents modulus, s=1,2.U _m,sfor measuring end line mode voltage, i _m,sfor measuring end line mould electric current, x is the distance of the amount of leaving side, r _sthe line mould resistance of unit length, Z _c,sfor line mould wave impedance, v _sline mould wave velocity.

4th step, calculating direct wave and backward-travelling wave:

The capable ripple of forward voltage is:

u ⁺ _x,s＝(u _x,s+Z _c,si _x,s)/2(4)

The capable ripple of reverse voltage is

u ^- _x,s＝(u _x,s-Z _c,si _x,s)/2(5)

In formula, u ⁺ _x,sfor the direct wave at distance measuring end x place, u ^- _x,sfor the backward-travelling wave that distance measuring end is x place, u _x,sfor the voltage traveling wave at distance measuring end x place, i _x,sfor the current traveling wave at distance measuring end x place.

5th step, adopt the sudden change of extracting direct wave and anti-row ripple such as formula (4) and (5): first, adopt calculus of differences to obtain with

$c_{dif\_u^{+}}\left(t\right)=\[u_{x,s}^{+}\left(t\right)-u_{x,s}^{+}(t-\mathrm{\Δ}t)\]/\mathrm{\Δ}t---\left(6\right)$

$c_{dif\_u^{-}}\left(t\right)=\[u_{x,s}^{-}\left(t\right)-u_{x,s}^{-}(t-\mathrm{\Δ}t)\]/\mathrm{\Δ}t---\left(7\right)$

for the difference result of direct wave, for the difference result of backward-travelling wave.Δ t is sampling interval.

Secondly, difference result c is calculated _difat the energy S of a period of time _2u(x, t), that is:

$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)$

$S_{2u^{-}}(x,t)=\underset{n=t-N\mathrm{\Δ}t+1}{\overset{t}{\Σ}}{\[c_{dif\_u^{+}}\left(t\right)\]}^3---\left(9\right)$

In formula, for the energy of direct wave within a period of time, for the energy of backward-travelling wave within a period of time.

6th step, structure range function: obtain according to formula (8) and formula (9) with and at [t ₀, t ₀+ l/ (2v)] in window length, obtain range function according to formula (10), that is:

$f_u\left(x\right)={\∫}_{t_0}^{t_0+l/v}{S}_{2u^{+}}(x,t)\×S_{2u^{-}}(x,t)dt---\left(10\right)$

In formula, t ₁, t ₂for integration upper and lower limit.

7th step, acquisition fault distance: if at [t ₀, t ₀+ l/ (2v)] time window long in, range function only has a catastrophe point, can according to the polarity of catastrophe point judge fault be positioned at half line length within or outside, then obtain fault distance.

Even range function only has a negative catastrophe point in circuit length range, then fault is positioned within half line length, and trouble spot is left M and held x;

If range function only has a positive catastrophe point in circuit length range, then fault is positioned at outside half line length, and M end x '=l-x is left in trouble spot.

If range function has multiple catastrophe point in distribution along the line, Simulation after test determination abort situation catastrophe point need be adopted.

If range function has multiple catastrophe point in distribution along the line, need to adopt " Simulation after test " to determine abort situation catastrophe point.Concrete grammar is: the catastrophe point distributed along the line by range function is designated as x=[x ₁, x ₂... ].Suppose x ₁for abort situation catastrophe point, and time window [t ₀, t ₀+ t ₁] interior calculating range function.Wherein, t ₁span is [x ₁/ v, (x ₁+ l _k)/v _s].If range function only has a catastrophe point in the long scope in all fronts, and the x ' that this catastrophe point is corresponding ₁=x ₁, then suppose to set up, if the polarity of this catastrophe point is negative, then fault is positioned within half line length, and fault distance M holds x=x ₁if the polarity of this catastrophe point is just, then fault is positioned at outside half line length, and fault distance N holds l-x=x ₁; If range function does not have catastrophe point or more than one catastrophe point in the long scope in all fronts, then hypothesis is false, and this catastrophe point is not the catastrophe point of abort situation, then continue hypothesis x ₂for abort situation catastrophe point, and time window [t ₀, t ₀+ t ₂] interior calculating range function, and t ₂span is [x ₂/ v, (x ₂+ l _k)/v)], if range function only has a catastrophe point in the long scope in all fronts, and the x ' that this catastrophe point is corresponding ₂=x ₂, then suppose to set up, if the polarity of this catastrophe point is negative, then fault is positioned within half line length, and fault distance M holds x=x ₂if the polarity of this catastrophe point is just, then fault is positioned at outside half line length, and fault distance N holds l-x=x ₂.If range function does not have catastrophe point or more than one catastrophe point in the long scope in all fronts, then hypothesis is false, and this catastrophe point is not the catastrophe point of abort situation, then continue hypothesis x ₃for abort situation catastrophe point, like this until hypothesis is set up, obtain fault distance.Wherein l _kfor the shortest length perfecting circuit.

The invention has the beneficial effects as follows: utilize Bei Jielong circuit model to have along the Hi-pass filter effect in line length dimension, make distance-finding method have more robustness and universality, be easy to the robotization realizing single end distance measurement.

Accompanying drawing explanation

Fig. 1 is the line assumption diagram in the embodiment of the present invention 1, and total track length is 71.9km;

Fig. 2 is the distribution of range function within the scope of all fronts in the embodiment of the present invention 1;

Fig. 3 is the line construction in the embodiment of the present invention 2, and total track length is 93.11km;

Fig. 4 is the distribution of the embodiment of the present invention 2 range function within the scope of all fronts;

Fig. 5 is the line construction in the embodiment of the present invention 3, and total track length is 150km;

Fig. 6 is the distribution of the embodiment of the present invention 3 range function within the scope of all fronts.

Embodiment

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

A kind of overhead transmission line Single Ended Fault Location based on direction row Wave Decomposition and distance calibration, first the capable wave datum of measuring end fault current is obtained by high speed acquisition device, and intercept fault initial row ripple arrive before l/ (2v) window length and fault initial row ripple arrive after l/v window length, i.e. the row wave datum of 1.5l/v window length altogether; Next utilizes the adjacent popular ripple of line electricity and the wave impedance of perfecting to construct voltage traveling wave; The current traveling wave again obtained according to said method and voltage traveling wave, utilize Bei Jielong formulae discovery to distribute at l/ (2v) window length voltage traveling wave and current traveling wave along the line; Finally build range function, and obtain fault distance according to the catastrophe point that range function is along the line.

Concrete steps are as follows:

The first step, read line wave datum: the capable wave datum of measuring end fault current obtained by high speed acquisition device, and intercept fault initial row ripple arrive before l/ (2v) window length and fault initial row ripple arrive after l/v window length, i.e. the row wave datum of 1.5l/v window length altogether;

Second step, utilize the adjacent popular ripple of line electricity and the wave impedance of perfecting to construct voltage traveling wave, that is:

u _M＝i _k×Z _c(1)

In formula, u _mfor measuring end voltage, i _kperfect circuit measuring end electric current for the longest, Zc is surge impedance of a line.

3rd step, calculated direction row ripple distribute along circuit: the current traveling wave obtained according to step (1) and step (2) and voltage traveling wave, utilize Bei Jielong formulae discovery at [t ₀, t ₀+ l/ (2v)] window length voltage traveling wave and current traveling wave distribute along the line.Wherein t ₀for namely fault initial row ripple arrives the moment of measuring end:

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

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

In formula, subscript s represents modulus, s=1,2.U _m,sfor measuring end line mode voltage, i _m,sfor measuring end line mould electric current, x is the distance of the amount of leaving side, r _sthe line mould resistance of unit length, Z _c,sfor line mould wave impedance, v _sline mould wave velocity.

4th step, calculating direct wave and backward-travelling wave:

The capable ripple of forward voltage is:

u ⁺ _x,s＝(u _x,s+Z _c,si _x,s)/2(4)

The capable ripple of reverse voltage is

u ^- _x,s＝(u _x,s-Z _c,si _x,s)/2(5)

In formula, u ⁺ _x,sfor the direct wave at distance measuring end x place, u ^- _x,sfor the backward-travelling wave that distance measuring end is x place, u _x,sfor the voltage traveling wave at distance measuring end x place, i _x,sfor the current traveling wave at distance measuring end x place.

5th step, adopt the sudden change of extracting direct wave and anti-row ripple such as formula (4) and (5): first, adopt calculus of differences to obtain with

$c_{dif\_u^{+}}\left(t\right)=\[u_{x,s}^{+}\left(t\right)-u_{x,s}^{+}(t-\mathrm{\Δ}t)\]/\mathrm{\Δ}t---\left(6\right)$

$c_{dif\_u^{-}}\left(t\right)=\[u_{x,s}^{-}\left(t\right)-u_{x,s}^{-}(t-\mathrm{\Δ}t)\]/\mathrm{\Δ}t---\left(7\right)$

for the difference result of direct wave, for the difference result of backward-travelling wave.Δ t is sampling interval.

Secondly, difference result c is calculated _difat the energy S of a period of time _2u(x, t), that is:

$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)$

$S_{2u^{-}}(x,t)=\underset{n=t-N\mathrm{\Δ}t+1}{\overset{t}{\Σ}}{\[c_{dif\_u^{+}}\left(t\right)\]}^3---\left(9\right)$

In formula, for the energy of direct wave within a period of time, for the energy of backward-travelling wave within a period of time.

6th step, structure range function: obtain according to formula (8) and formula (9) with and at [t ₀, t ₀+ l/ (2v)] in window length, obtain range function according to formula (10), that is:

$f_u\left(x\right)={\∫}_{t_0}^{t_0+l/v}{S}_{2u^{+}}(x,t)\×S_{2u^{-}}(x,t)dt---\left(10\right)$

In formula, t ₁, t ₂for integration upper and lower limit.

7th step, acquisition fault distance: if at [t ₀, t ₀+ l/ (2v)] time window long in, range function only has a catastrophe point, can according to the polarity of catastrophe point judge fault be positioned at half line length within or outside, then obtain fault distance.

Even range function only has a negative catastrophe point in circuit length range, then fault is positioned within half line length, and trouble spot is left M and held x;

If range function only has a positive catastrophe point in circuit length range, then fault is positioned at outside half line length, and M end x '=l-x is left in trouble spot.

If range function has multiple catastrophe point in distribution along the line, Simulation after test determination abort situation catastrophe point need be adopted.

Embodiment 1: for the transmission line of electricity shown in Fig. 1, there is earth fault in circuit distance Zhanyi County 49km.

Got the row wave datum of 1.5l/v window length by measuring end according to step one in instructions; The adjacent popular ripple of line electricity and the wave impedance of perfecting is utilized to construct voltage traveling wave and obtain u according to step 2 _m=i _k× Z _c; To distribute u along circuit according to the capable ripple of step 3 calculating voltage and current traveling wave _x,s(x, t) and i _x,s(x, t); Direct wave and backward-travelling wave u is calculated according to step 4 ⁺ _x,sand u ^- _x,s; The sudden change of extracting direct wave and anti-row ripple is calculated according to step 5 with and energy with range function f is built according to step 6 _u(x).The result that the sudden change of range function distributes along the line as shown in Figure 2.As shown in Figure 2, at [t ₀, t ₀+ l/ (2v)] time window long in, range function only has a catastrophe point, and the polarity of this catastrophe point is just, known fault distance x=l-23.7=48.2km.

Embodiment 2: for the transmission line of electricity shown in Fig. 3, there is earth fault in circuit distance Soviet Union village 43.1km.

Got the row wave datum of 1.5l/v window length by measuring end according to step one in instructions; The adjacent popular ripple of line electricity and the wave impedance of perfecting is utilized to construct voltage traveling wave and obtain u according to step 2 _m=i _k× Z _c; To distribute u along circuit according to the capable ripple of step 3 calculating voltage and current traveling wave _x,s(x, t) and i _x,s(x, t); Direct wave and backward-travelling wave u is calculated according to step 4 ⁺ _x,sand u ^- _x,s; The sudden change of extracting direct wave and anti-row ripple is calculated according to step 5 with and energy with range function f is built according to step 6 _u(x).The result that the sudden change of range function distributes along the line as shown in Figure 4.As shown in Figure 4, at [t ₀, t ₀+ l/ (2v)] time window long in, range function only has a catastrophe point, and the polarity of this catastrophe point is negative, known fault distance x _f=42.9km.

Embodiment 3: for the transmission line of electricity shown in Fig. 5, circuit holds 70km that earth fault occurs apart from M.

Got the row wave datum of 1.5l/v window length by measuring end according to step one in instructions; The adjacent popular ripple of line electricity and the wave impedance of perfecting is utilized to construct voltage traveling wave and obtain u according to step 2 _m=i _k× Z _c; To distribute u along circuit according to the capable ripple of step 3 calculating voltage and current traveling wave _x,s(x, t) and i _x,s(x, t); Direct wave and backward-travelling wave u is calculated according to step 4 ⁺ _x,sand u ^- _x,s; The sudden change of extracting direct wave and anti-row ripple is calculated according to step 5 with and energy with range function f is built according to step 6 _u(x).The result that the sudden change of range function distributes along the line as shown in Figure 6.As shown in Figure 6, at [t ₀, t ₀+ l/ (2v)] time window long, range function only has a catastrophe point, and the polarity of this catastrophe point is negative, known fault distance x _f=69.8km.

Below by reference to the accompanying drawings the specific embodiment 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 possess, various change can also be made under the prerequisite not departing from present inventive concept.

Claims (2)

1. the overhead transmission line Single Ended Fault Location based on direction row Wave Decomposition and distance calibration, it is characterized in that: first obtain the capable wave datum of measuring end fault current by high speed acquisition device, and intercept fault initial row ripple arrive before l/ (2v) window length and fault initial row ripple arrive after l/v window length, i.e. the row wave datum of 1.5l/v window length altogether; Next utilizes the adjacent popular ripple of line electricity and the wave impedance of perfecting to construct voltage traveling wave; The current traveling wave again obtained according to said method and voltage traveling wave, utilize Bei Jielong formulae discovery to distribute at l/ (2v) window length voltage traveling wave and current traveling wave along the line; Finally build range function, and obtain fault distance according to the catastrophe point that range function is along the line, wherein l is total track length.

2. the overhead transmission line Single Ended Fault Location based on direction row Wave Decomposition and distance calibration according to claim 1, is characterized in that concrete steps are:

The first step, read line wave datum: the capable wave datum of measuring end fault current obtained by high speed acquisition device, and intercept fault initial row ripple arrive before l/ (2v) window length and fault initial row ripple arrive after l/v window length, i.e. the row wave datum of 1.5l/v window length altogether;

Second step, utilize the adjacent popular ripple of line electricity and the wave impedance of perfecting to construct voltage traveling wave, that is:

u _M＝i _k×Z _c(1)

In formula, u _mfor measuring end voltage, i _kperfect circuit measuring end electric current for the longest, Zc is surge impedance of a line;

3rd step, calculated direction row ripple distribute along circuit: the current traveling wave obtained according to step (1) and step (2) and voltage traveling wave, utilize Bei Jielong formulae discovery at [t ₀, t ₀+ l/ (2v)] window length voltage traveling wave and current traveling wave distribute along the line, wherein t ₀for namely fault initial row ripple arrives the moment of measuring end:

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

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

In formula, subscript s represents modulus, s=1,2, u _m,sfor measuring end line mode voltage, i _m,sfor measuring end line mould electric current, x is the distance of the amount of leaving side, r _sthe line mould resistance of unit length, Z _c,sfor line mould wave impedance, v _sline mould wave velocity;

4th step, calculating direct wave and backward-travelling wave:

The capable ripple of forward voltage is:

u ⁺ _x,s＝(u _x,s+Z _c,si _x,s)/2(4)

The capable ripple of reverse voltage is

u ^- _x,s＝(u _x,s-Z _c,si _x,s)/2(5)

In formula, u ⁺ _x,sfor the direct wave at distance measuring end x place, u ^- _x,sfor the backward-travelling wave that distance measuring end is x place, u _x,sfor the voltage traveling wave at distance measuring end x place, i _x,sfor the current traveling wave at distance measuring end x place;

5th step, adopt the sudden change of extracting direct wave and anti-row ripple such as formula (4) and (5): first, adopt calculus of differences to obtain with

$c_{dif\_u^{+}}\left(t\right)=\[u_{x,s}^{+}\left(t\right)-u_{x,s}^{+}(t-\mathrm{\Δ}t)\]/\mathrm{\Δ}t---\left(6\right)$

$c_{dif\_u^{-}}\left(t\right)=\[u_{x,s}^{-}\left(t\right)-u_{x,s}^{-}(t-\mathrm{\Δ}t)\]/\mathrm{\Δ}t---\left(7\right)$

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

Secondly, difference result c is calculated _difat the energy S of a period of time _2u(x, t), that is:

$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)$

$S_{2u^{-}}(x,t)=\underset{n=t-N\mathrm{\Δ}t+1}{\overset{t}{\Σ}}{\[c_{dif\_u^{+}}\left(t\right)\]}^3---\left(9\right)$

In formula, for the energy of direct wave within a period of time, for the energy of backward-travelling wave within a period of time;

6th step, structure range function: obtain according to formula (8) and formula (9) with and at [t ₀, t ₀+ l/ (2v)] in window length, obtain range function according to formula (10), that is:

$f_u\left(x\right)={\∫}_{t_0}^{t_0+l/v}{S}_{2u^{+}}\left(x,t\right)\×S_{2u^{-}}(x,t)dt---\left(10\right)$

In formula, t ₁, t ₂for integration upper and lower limit;

7th step, acquisition fault distance: if at [t ₀, t ₀+ l/ (2v)] time window long in, range function only has a catastrophe point, can according to the polarity of catastrophe point judge fault be positioned at half line length within or outside, then obtain fault distance;

Even range function only has a negative catastrophe point in circuit length range, then fault is positioned within half line length, and trouble spot is left M and held x;

If range function only has a positive catastrophe point in circuit length range, then fault is positioned at outside half line length, and M end x '=l-x is left in trouble spot;

If range function has multiple catastrophe point in distribution along the line, Simulation after test determination abort situation catastrophe point need be adopted.