CN111157843B - Power distribution network line selection method based on time-frequency domain traveling wave information - Google Patents
Power distribution network line selection method based on time-frequency domain traveling wave information Download PDFInfo
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Abstract
The invention discloses a power distribution network line selection method based on time-frequency domain traveling wave information, which comprises the following steps: s1: installing a traveling wave sensor at the outlet of each line bus, and extracting a current traveling wave signal at the outlet of each line; s2: carrying out Kerenbel transformation on the current traveling wave signals of each line to obtain current traveling wave zero-mode components; s3: carrying out time-frequency domain analysis on the current traveling wave zero-mode component of each line by utilizing S transformation to obtain corresponding time-frequency domain waveforms and time-frequency domain matrixes of each line; s4: carrying out pairwise correlation analysis on the time-frequency domain matrix of each line to obtain a correlation coefficient matrix R, and solving the sum R of the correlation coefficients of each lineiWherein i ═ 1, 2, …, n represent the line number; s5: comparing the sum R of the minimum correlation coefficients of each lineminIf R isminWhen it is negative, R is determinedminThe corresponding line is a fault line; if R isminAnd when the number is positive, judging that the bus is in fault. A large number of simulation results show that the method has the advantages of simple principle, easiness in implementation, high reliability and the like.
Description
Technical Field
The invention mainly relates to the technical field of power distribution network grounding, in particular to a power distribution network line selection method based on time-frequency domain traveling wave information.
Background
The distribution network in China widely adopts a neutral point non-effective grounding mode, the operation mode can effectively improve the reliability of power supply, wherein most faults of the distribution system are single-phase grounding faults, the amplitude of fault current is small when the single-phase grounding faults occur, the fault characteristics are not obvious, the influence of factors such as interference is caused, the grounding fault protection precision and reliability are poor, and the correct selection of fault lines is very difficult.
In recent years, expert scholars have proposed line selection methods that provide valuable ideas for correct line selection. According to different frequency bands of fault signals, the method is divided into a power frequency method, a transient state method and a traveling wave method. The main difficulty of the power frequency method is that the fault power frequency information is weak, the fault power frequency information is easily influenced by factors such as the size of a transition resistor, a central point grounding mode and arc suppression coil overcompensation, the reliability of the measured signal is not high, and the line selection accuracy is generally low. The existing transient earth fault line selection method has a certain research on transient characteristic analysis of faults, and partial transient line selection methods such as a zero-sequence current line selection method, a transient energy method, a transient power method and the like are proposed. The principle is simple, the traveling wave module component and the zero module component wave speed are different, polarity discrimination is carried out in the time difference of arriving the bus, and the fault line and the sound line are distinguished, but a line selection dead zone exists at the bus outlet, the effective time is short, the requirement on the sampling rate is too high, and the field is difficult to realize.
Disclosure of Invention
The technical problem to be solved by the invention is as follows: aiming at the technical problems in the prior art, the invention provides the power distribution network line selection method based on the time-frequency domain traveling wave information, which has the advantages of simple principle, easy realization, good reliability, rapidness and accuracy.
In order to solve the technical problems, the invention adopts the following technical scheme:
a power distribution network line selection method based on time-frequency domain traveling wave information comprises the following steps:
s1: installing a traveling wave sensor at the outlet of each line bus, and extracting current traveling wave signals on each line;
s2: carrying out Kerenbel transformation on the current traveling wave signals of each line to obtain current traveling wave zero-mode components;
s3: carrying out time-frequency domain analysis on the current traveling wave zero-mode component of each line by using S transformation to obtain a corresponding time-frequency matrix of each line and drawing a time-frequency domain traveling wave waveform diagram;
s4: carrying out pairwise correlation analysis on the time-frequency matrix of each line to obtain a correlation coefficient matrix R, and solving the sum R of the correlation coefficients of each lineiWherein i ═ 1, 2, …, n represent the line number;
s5: comparing the sum R of the minimum correlation coefficients of each lineminIf R isminWhen it is negative, R is determinedminThe corresponding line is a fault line; if R isminAnd when the number is positive, judging that the bus is in fault.
As a further improvement of the process of the invention: in step S3, the S-transform is an S-transform formula derived by adding a window function with frequency variation to a short-time fourier transform, where the short-time fourier transform is to decompose a non-stationary signal into a plurality of short-time stationary signals on a time axis by using a time window, and the non-stationary current traveling wave zero-mode component signal i0(t) short-time fourier transform:
where time t, frequency f, imaginary unit j.
As a further improvement of the process of the invention: in step S3, in the signal processing process of the short-time fourier transform, a window function g (t) is added, and a non-stationary current traveling wave zero-mode component signal i is added0And (t) intercepting and selecting, namely converting the small-segment signals into a frequency domain, and forming a time-frequency distribution result of the whole-segment signals through Fourier transform to finish the conversion of the time-domain signals into the time-frequency signals.
As a further improvement of the process of the invention: the window function introduced is a gaussian window function:
as a further improvement of the process of the invention: in step S3, the gaussian window function is optimized as a time window function:
the time factor τ, the scale factor σ, in the equation is multiplied by the equation of the short-time fourier transform:
as a further improvement of the process of the invention: in step S3, the width of the gaussian window function changes with the change of the value of σ, the time-frequency resolution is changed, and partial time-frequency information of the signal is obtained; 1/| f | controls the gaussian window scale and enhances the impact between signal frequency and time-frequency resolution, and associates σ with frequency, let:
substituting the above formula to deduce an S transformation formula:
the S transformation is used in a non-stationary current traveling wave zero-mode component signal i0And (t) in the characteristic extraction process, obtaining a time scale represented by an abscissa, a frequency scale represented by an ordinate, a time-frequency domain matrix of which the size of an element in the matrix represents an amplitude, and drawing a time-frequency domain traveling wave waveform diagram.
As a further improvement of the process of the invention: the specific steps of step S4 are:
step S401: analyzing by adopting the real part of each element in the time-frequency matrix, subdividing the amplitude values under each frequency obtained after S transformation, wherein each central frequency has a plurality of sampling points, and defining the amplitude value corresponding to the jth time-period block under the ith frequency as follows:
E(i,j)＝real[S(i,j)]
wherein S (i, j) represents the jth time interval element corresponding to the ith frequency of a certain line time frequency matrix, and E (i, j) represents the real part of the jth time interval element corresponding to the ith frequency of the line time frequency matrix;
step S402: setting 100 mus after detecting zero-mode component of current traveling wave as time window, combining to reflect time-frequency spectrum matrix E of local time-frequency domain characteristic of traveling wave signalM×N：
Step S403: the above formula is normalized, and the obtained correlation coefficient formula after processing is as follows:
in the formula Ea(i,j)、Eb(i, j) respectively represents the real part element of the jth time interval corresponding to the ith frequency of the time-frequency matrix of the line a and the line b, RabThe correlation coefficients of the time-frequency domain matrix are respectively expressed as a line a and a line b;
step S404: and (3) carrying out pairwise correlation analysis on the time-frequency domain matrix of each line to obtain a correlation coefficient matrix:
in which n is the number of lines, RijCorrelation coefficient of waveform similarity of zero mode component of current traveling wave between lines and RijE (-1, 1); when R isijThe closer the absolute value of (1) is, the higher the similarity degree of the two lines is;when R isijThe closer the absolute value of (a) is to 0, the lower the degree of similarity of the two lines is, wherein the sign indicates the direction of correlation; a single-phase earth fault occurs in the resonance earth system, and the wave form similarity of the current traveling wave zero-mode components between sound lines is positively correlated; the wave form similarity of the zero mode component of the current traveling wave of the fault line and the healthy line is inversely related;
step S405: and summing the correlation coefficients of all the lines for amplification treatment, wherein the formula is as follows:
in the formula RiIs the sum of the correlation numbers of the ith line, RijThe correlation coefficient between lines is, and n is the number of lines. Compared with the prior art, the invention has the advantages that:
1. the invention relates to a power distribution network line selection method based on time-frequency domain traveling wave information, which comprises the steps of extracting a fault current zero-mode signal to carry out time-frequency domain analysis; the characteristic of the amplitude of the real part of the time-frequency domain matrix is comprehensively utilized, the positive and negative of the sum of the minimum correlation coefficients of all lines are taken as the line selection criterion, the setting value of the criterion does not need to be set manually, the method has the characteristics of clear principle, simple operation and strong practicability, and the line selection margin is further improved; the influence of small fault current amplitude under the conditions of small fault angle, large grounding resistance and the like is effectively overcome, and fault line selection is accurately realized.
2. The invention relates to a power distribution network line selection method based on time-frequency domain traveling wave information, which aims at the defects of the traditional line selection method, is based on the research of the time-frequency domain characteristics of fault traveling wave current zero-mode signals, has high time-frequency resolution and good time-frequency aggregation for current traveling wave zero-mode components by utilizing the S transformation, and truly and accurately describes the difference between the time-frequency domain signals of a fault line and sound lines.
3. According to the power distribution network line selection method based on the time-frequency domain traveling wave information, the difference of time-frequency domain characteristic information of current zero-mode traveling wave waveforms of a fault line and a sound line is fully utilized, the positive and negative of the sum of minimum correlation coefficients of all lines are judged, manual intervention is not needed for setting a criterion setting value, and the fault line is accurately selected. A large number of simulation results show that the invention is not influenced by the neutral point grounding mode, the fault resistance, the fault initial phase angle and the distribution network feeder line outgoing mode, and has strong adaptability, high line selection accuracy and no line selection dead zone.
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FIG. 1 is a schematic flow diagram of the process of the present invention.
FIG. 2 is a schematic diagram of the experimental principle of the present invention in a specific application example.
Detailed Description
The invention will be described in further detail below with reference to the drawings and specific examples.
As shown in fig. 1, the method for selecting a line of a power distribution network based on time-frequency domain traveling wave information of the present invention comprises the following steps:
step S1: installing a traveling wave sensor at the outlet of each line bus, and extracting current traveling wave signals on each line;
step S2: carrying out Kerenbel transformation on the current traveling wave signals of each line to obtain current traveling wave zero-mode components;
in specific application, in step S2 of this example, the specific steps are:
in the formula iα、iβIs a line mode current, i0Is zero mode current, ia、ib、icIs the phase current.
Step S3: carrying out time-frequency domain analysis on the current traveling wave zero-mode component of each line by using S transformation to obtain a corresponding time-frequency matrix of each line and drawing a time-frequency domain traveling wave waveform diagram;
in specific application, in step S3 of this example, the specific steps are:
step S301: the S transformation is based on short-time Fourier transformation, which is based on the advantage of adding a window function with frequency variation to derive an S transformation formulaDecomposing the non-stationary signal into multiple short-time stationary signals on time axis by using time-adding window, and non-stationary current traveling wave zero-mode component signal i0(t) short-time fourier transform:
where time t, frequency f, imaginary unit j.
Step S302: fourier transform is to map the signal time domain to the frequency domain, so that the signal loses the time domain characteristic as a whole, a window function g (t) is added in the signal processing process, and a non-stationary current traveling wave zero-mode component signal i is subjected to0(t) performing interception, namely converting the small-segment signals into a frequency domain, performing Fourier transform, wherein each part of signals are converted into the frequency domain along with the movement of a window function on a time axis, and simultaneously contain time domain characteristics, and finally converging time-frequency distribution results of the whole segment of signals, so that the time-domain signals are converted into time-frequency signals. The processing accuracy of the short-time fourier transform is directly affected by the selection of a window function, and the window function usually introduced is a gaussian window function:
step S303: the gaussian window function is further optimized to construct a time window function suitable for practical problems:
wherein the time factor τ, the scale factor σ, is multiplied by the equation of step S301:
step S304: the width of the gaussian window function varies with the magnitude of the sigma value, thereby changing the time-frequency resolution and obtaining partial time-frequency information of the signal. 1/| f | controls the gaussian window scale and enhances the impact between signal frequency and time-frequency resolution, thus relating σ to frequency, then let:
step S305: the sub-equation in step S304 is added to step S303 to derive the S transformation equation
Therefore, the S conversion is used in the non-stationary current traveling wave zero-mode component signal i0And (t) in the characteristic extraction process, obtaining a time scale represented by an abscissa, a frequency scale represented by an ordinate, a time-frequency domain matrix of which the size of an element in the matrix represents an amplitude, and simultaneously drawing a time-frequency domain traveling wave waveform diagram.
Step S4: carrying out pairwise correlation analysis on the time-frequency matrix of each line to obtain a correlation coefficient matrix R, and solving the sum R of the correlation coefficients of each lineiWherein i ═ 1, 2, …, n represent the line number;
in specific application, in step S4 of this example, the specific steps are as follows:
step S401: in order to highlight local time-frequency characteristics, the information of each frequency band under different time domains is comprehensively considered, the real part of each element in a time-frequency matrix is adopted for carrying out analysis, the amplitude under each frequency obtained after S transformation is subdivided, each central frequency has a plurality of sampling points, and the amplitude corresponding to the jth time-band block under the ith frequency is defined as follows:
E(i,j)＝real[S(i,j)]
wherein S (i, j) represents the jth time interval element corresponding to the ith frequency of a certain line time-frequency matrix, and E (i, j) represents the real part of the jth time interval element corresponding to the ith frequency of the line time-frequency matrix.
Step S402: setting 100 mus after detecting zero-mode component of current traveling wave as time window, extracting high-frequency component as 1kHz200kHz and 1MHz of sampling frequency. Time-frequency spectrum matrix E capable of reflecting local time-frequency domain characteristics of traveling wave signals in combinationM×N。
Step S403: in order to analyze the correlation degree of the time-frequency domain matrix of each line and describe the similarity degree of two waveforms, normalization processing needs to be performed on the above formula, and finally, a processed correlation coefficient formula is obtained as follows:
in the formula Ea(i,j)、Eb(i, j) respectively represents the real part element of the jth time interval corresponding to the ith frequency of the time-frequency matrix of the line a and the line b, RabThe correlation coefficients of the time-frequency domain matrix are respectively expressed as a line a and a line b;
step S404: and (3) carrying out pairwise correlation analysis on the time-frequency domain matrix of each line to obtain a correlation coefficient matrix:
in which n is the number of lines, RijCorrelation coefficient of waveform similarity of zero mode component of current traveling wave between lines and RijE (-1, 1). When R isijThe closer the absolute value of (1) is, the higher the similarity degree of the two lines is; when R isijThe closer the absolute value of (c) is to 0, the lower the degree of similarity of the two lines is illustrated, wherein the sign indicates the direction of correlation. A single-phase earth fault occurs in the resonance earth system, and the wave form similarity of the current traveling wave zero-mode components between sound lines is positively correlated; and the fault line and the healthy line current traveling wave zero-mode component waveform similarity are inversely related.
Step S405: it is easy to see that this matrix is a symmetric matrix by correlation principle, and the diagonal elements are all 1. In order to further depict the difference between a fault line and a sound line and improve the line selection margin, the correlation coefficients of all lines are summed up and amplified, and the formula is as follows:
in the formula RiIs the sum of the correlation numbers of the ith line, RijThe correlation coefficient between lines is, and n is the number of lines.
Step S5: comparing the sum R of the minimum correlation coefficients of each lineminIf R isminWhen it is negative, R is determinedminThe corresponding line is a fault line; if R isminAnd when the number is positive, judging that the bus is in fault.
In the invention, the adopted power distribution network line selection method based on the time-frequency domain traveling wave waveform is different from the existing power distribution network line selection method based on the transient zero sequence waveform or the power frequency signal in that: firstly, the traveling wave propagation speed is close to the light speed propagation: 3X 108m/s, accurate line selection of the power distribution network can be realized only by detecting 100 mu s fault traveling wave signals, and the action speed is high; secondly, the fault traveling wave is a broadband step signal and has rich time-frequency domain information including time, frequency, amplitude, polarity and the like, and the method is remarkably superior to a line selection method only based on a certain frequency band time domain waveform on the basis of the rich fault information including time-frequency-amplitude-polarity and the like in a 100 mu s time window and has higher line selection reliability; ③ this patent passes through the identification of RminThe positive and negative of the voltage distribution network are realized, the line selection and the bus fault identification of the power distribution network are realized, the setting value of the criterion is not required to be set by human intervention, and the engineering practicability is higher; the invention is not influenced by the neutral point grounding mode, the fault type, the fault transition resistance, the fault initial phase angle and the feeder line outgoing form, has simple principle, and can realize quick, reliable and accurate fault line selection.
As shown in fig. 2, in a specific application example, the invention was tested according to the 10kV distribution system shown in fig. 1, the distribution network model adopts 4 lines with different power supply modes, 1 overhead line with 10km, 1 cable line with 9km to the ground, 1 overhead line with 5km and 5km cable line to form a hybrid line and 1 overhead line with branches, and the line parameters are shown in table 1; the transformer is 110kV/10kV, the high-voltage side is directly grounded by adopting a central point, the low-voltage side neutral point is the running mode of an arc suppression coil, for simplicity, the load on a line is simulated by 100+ j6.282 omega impedance, the compensation degree of the arc suppression coil is 8%, and the size of an inductance L in the arc suppression coil is calculated to be 784.2 mH.
TABLE 1
Respectively by the line L1、L2、L3、L4Carrying out experimental tests when the bus has single-phase earth faults under different fault conditions (including fault distance, earth resistance, fault initial phase angle and the like), and acquiring traveling wave current signals of each line by a protection device arranged at an outlet of the bus; extracting traveling wave current zero-mode component characteristic signals, obtaining a time-frequency domain matrix by utilizing S transformation, carrying out correlation analysis on the time-frequency domain matrix to obtain a sum R of correlation coefficientsiBy comparing the sum R of the minimum correlation coefficients of the linesminThe system fault position is judged according to the positive and negative properties of the system.
In the test process of the power distribution network ground fault protection method, the protection device arranged at the port of the line detects the moment of traveling wave current signals of each line, and the line selection protection device is started. The results of the line selection are shown in table 2.
TABLE 2
Note: l is3Medium "cable" indicates that the fault occurred on the cable, and empty indicates that the fault occurred on the overhead line; l is4Middle "B1"means the first branch line from the bus to the main line, similarly" B2"indicates the second branch line in the direction from the bus bar to the main line.
The distribution network generally adopts neutral point small electricityThe operation mode of current grounding, the winding connection mode of the 10kV line side transformer is as follows: triangular connection, neutral point ungrounded, center point grounded through arc suppression coil, etc. Under the action of different wiring modes, the amplitude of the traveling wave current of the single-phase earth fault has different degrees of influence. Simulation results show that the method is also suitable for the small-current grounding system with single-phase grounding faults in different wiring modes. Setting fault transition resistance to 1000 omega, respectively for line L3And L4The single-phase grounding occurs when the fault closing angle is 90 degrees, and the line selection result is shown in table 3.
TABLE 3
As can be seen from tables 2 and 3, in any case, the sum of the correlation coefficients of the faulty line is negative, and the sum of the correlation coefficients of the healthy line is positive, and the sum R of the minimum correlation coefficients of the lines is obtained by comparisonminIf R isminIf the line is negative, the corresponding line is judged to be a fault line; when the bus fails, the sum of the correlation coefficients of all the lines has small difference and is positive, and the sum R of the minimum correlation coefficients is obtained by comparison at the momentminIf the number is also positive, the bus fault can be determined. The method is not influenced by a neutral point grounding mode, a fault resistance, a fault initial phase angle and a distribution network feeder line output mode, does not need manual intervention to set a criterion setting value, has a simple principle, and can realize quick and accurate fault line selection.
The above is only a preferred embodiment of the present invention, and the protection scope of the present invention is not limited to the above-mentioned embodiments, and all technical solutions belonging to the idea of the present invention belong to the protection scope of the present invention. It should be noted that modifications and embellishments within the scope of the invention may be made by those skilled in the art without departing from the principle of the invention.
Claims (7)
1. A power distribution network line selection method based on time-frequency domain traveling wave information is characterized by comprising the following steps:
step S1: extracting an electrified popular wave signal on each line at a bus outlet of each line;
step S2: carrying out Kerenbel transformation on the current traveling wave signals of each line through the following formula to obtain current traveling wave zero-mode components;
in the formula iα、iβIs a line mode current, i0Is zero mode current, ia、ib、icIs the phase current;
step S3: current traveling wave zero-modulus component signal i of each line by utilizing S transformation0(t) carrying out time-frequency domain analysis to obtain corresponding time-frequency matrixes of each line and drawing a time-frequency domain traveling wave waveform diagram;
step S4: carrying out pairwise correlation analysis on the time-frequency matrix of each line to obtain a correlation coefficient matrix R, and solving the sum R of the correlation coefficients of each lineiWherein i ═ 1, 2, …, n represent the line number;
step S5: comparing the sum R of the minimum correlation coefficients of each lineminIf R isminIs a negative number, and RiWhen there is only one negative number, determine RminThe corresponding line is a fault line; if R isminIf the number is positive, the bus fault is determined.
2. The method for selecting lines on a power distribution network based on time-frequency domain traveling wave information as claimed in claim 1, wherein in step S3, the S-transform is a short-time fourier transform, which is a time window to decompose a non-stationary signal into a plurality of short-time stationary signals on a time axis, and a non-stationary current traveling wave zero-mode component signal i is added to a window function with frequency variation to derive an S-transform formula0(t) short-time fourier transform:
where time t, frequency f, imaginary unit j.
3. The method for selecting lines in a power distribution network based on time-frequency domain traveling wave information as claimed in claim 2, wherein in step S3, in the signal processing process of the short-time fourier transform, a window function g (t) is added, and a non-stationary current traveling wave zero-modulus component signal i is added0And (t) intercepting and selecting, namely converting the small-segment signals into a frequency domain, and forming a time-frequency distribution result of the whole-segment signals through Fourier transform to finish the conversion of the time-domain signals into the time-frequency signals.
4. The power distribution network line selection method based on time-frequency domain traveling wave information as claimed in claim 3, wherein the introduced window function is a Gaussian window function:
5. the method for selecting a line of a power distribution network based on time-frequency domain traveling wave information according to claim 4, wherein in the step S3, the Gaussian window function is optimized as a time window function:
the time factor τ, the scale factor σ, in the equation is multiplied by the equation of the short-time fourier transform:
6. the method according to claim 5, wherein in step S3, the width of the Gaussian window function changes with the change of the value of σ, the time-frequency resolution is changed, and partial time-frequency information of the signal is obtained; 1/| f | controls the gaussian window scale and enhances the impact between signal frequency and time-frequency resolution, and associates σ with frequency, let:
substituting the above formula to deduce an S transformation formula:
the S transformation is used in a non-stationary current traveling wave zero-mode component signal i0And (t) in the characteristic extraction process, obtaining a time scale represented by an abscissa, a frequency scale represented by an ordinate, a time-frequency domain matrix of which the size of an element in the matrix represents an amplitude, and drawing a time-frequency domain traveling wave waveform diagram.
7. The method for selecting the line of the power distribution network based on the time-frequency domain traveling wave information according to any one of claims 1 to 6, wherein the specific step of the step S4 is as follows:
step S401: analyzing by adopting the real part of each element in the time-frequency matrix, subdividing the amplitude values under each frequency obtained after S transformation, wherein each central frequency has a plurality of sampling points, and defining the amplitude value corresponding to the jth time-period block under the ith frequency as follows:
E(i,j)＝real[S(i,j)]
wherein S (i, j) represents the jth time interval element corresponding to the ith frequency of a certain line time frequency matrix, and E (i, j) represents the real part of the jth time interval element corresponding to the ith frequency of the line time frequency matrix;
step S402: setting 100 mu s after detecting zero-mode component of current traveling wave as time window, combining to reflect time-frequency spectrum matrix of local time-frequency domain characteristic of traveling wave signalEM×N：
Step S403: the above formula is normalized, and the obtained correlation coefficient formula after processing is as follows:
in the formula Ea(i,j)、Eb(i, j) respectively represents the real part element of the jth time interval corresponding to the ith frequency of the time-frequency matrix of the line a and the line b, RabThe correlation coefficients of the time-frequency domain matrix are respectively expressed as a line a and a line b;
step S404: and (3) carrying out pairwise correlation analysis on the time-frequency domain matrix of each line to obtain a correlation coefficient matrix:
in which n is the number of lines, RijCorrelation coefficient of waveform similarity of zero mode component of current traveling wave between lines and RijE (-1, 1); when R isijThe closer the absolute value of (1) is, the higher the similarity degree of the two lines is; when R isijThe closer the absolute value of (a) is to 0, the lower the degree of similarity of the two lines is, wherein the sign indicates the direction of correlation; a single-phase earth fault occurs in the resonance earth system, and the wave form similarity of the current traveling wave zero-mode components between sound lines is positively correlated; the wave form similarity of the zero mode component of the current traveling wave of the fault line and the healthy line is inversely related;
step S405: and summing the correlation coefficients of all the lines for amplification treatment, wherein the formula is as follows:
in the formula RiIs the sum of the correlation numbers of the ith line, RijThe correlation coefficient between lines is, and n is the number of lines.
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