Fault location of cable hybrid transmission lines based on energy attenuation characteristics of traveling waves

The overhead-cable hybrid transmission lines are alternately connected by two types of lines, with a more complex structure and higher difficulty in fault location. This paper presents an ac-curate fault location method for overhead-cable hybrid lines based on traveling wave energy. Firstly, the basic concept of traveling wave energy is defined. Based on the attenuation charac-teristics of the traveling wave, the mapping relationship between traveling wave energy and fault location is analyzed. Secondly, considering the influence of S-transform error on the traveling wave energy propagation law, the traveling wave energy attenuation characteristics of common A-type and B-type hybrid lines are analyzed. Then, for the overhead-cable hybrid lines with different structures, the mapping relationship between the traveling wave energy at both ends of the line and the fault distance is quantitatively derived, and an accurate fault location method based on the initial traveling wave energy ratio at the same frequency at both ends of the line is proposed. Finally, a 110 kV hybrid transmission line fault simulation model is built in PSCAD/EMTDC, and the faults under different conditions are simulated in different line sections. The effectiveness and robustness of the proposed method are verified through the simulation.

(1) W u = 1 2 c�xu(x, t 0 , ω 0 ) 2 , W i = 1 2 l�xi(x, t 0 , ω 0 ) 2 www.nature.com/scientificreports/ The traveling wave voltage and current satisfy the relation: Bringing Eq. (2) into either Eq. (1) shows that the electric field energy stored on the line is essentially the same as the magnetic field energy. Thus, the electromagnetic energy W x in the line micro-element is: The electromagnetic energy W t per unit time at x 0 on the line is: For point x 0 on the line, the traveling wave energy passing through the point at time t 1 to time t 2 can be expressed as follows: Meanwhile, the power loss on the line will lead to the attenuation of traveling wave energy, and its value is: When the traveling wave with power P = i(x 0 ,t 0 ) 2 Z propagates on the line element dx, the power variation is ΔP = 2Zi(x 0 ,t 0 )di, due to the existence of R and G. Where, Z = R + jX, R is the line resistance, and X is the line reactance. Since the energy is decaying, the ΔP sign is negative, and in connection with Eq. (6) it is obtained that: By solving the differential equation about current and distance in Eq. (7), the following equation can be obtained: where γ is the propagation coefficient of traveling wave energy, which is related to the resistance, capacitance, inductance and conductance parameters of the line. Substitute Eq. (8) into Eq. (5) to obtain the propagation formula of traveling wave energy: Equation (9) is the attenuation law of traveling wave energy derived from the power loss of resistance. The equation shows that traveling wave energy also decays exponentially during transmission. Combining with the frequency-dependent characteristics of the line, the propagation of traveling wave energy has the following characteristics: The high frequency traveling wave component has a large attenuation coefficient, and the traveling wave energy decays quickly in the propagation process. Correspondingly, the low frequency traveling wave component has a small attenuation coefficient, and the traveling wave energy decays slowly. (2) u(x, t, ω 0 ) = l c i(x, t, ω 0 ) l c dt (6) P loss = i(x 0 , t 0 , ω 0 ) 2 R�x + u(x 0 , t 0 , ω 0 ) 2 G�x = i(x 0 , t 0 , ω 0 ) 2 (R + lG/c)�x (7) i(x 0 , t 0 , ω 0 ) 2 (R + lG/c)dx = −2i(x 0 , t 0 , ω 0 )Zdi www.nature.com/scientificreports/ Mapping between energy and fault location. According to the attenuation characteristics of travelling wave energy mentioned above, it can be seen qualitatively that the resistance and conductance of transmission lines will cause the reduction of travelling wave energy, and the longer the propagation distance, the smaller the value of travelling wave energy. Quantitatively, the attenuation of traveling wave signals conforms to the law of exponential attenuation, and the magnitude of traveling wave energy and the distance of traveling wave propagation can be described by an exponential function. Based on the above analysis, the mapping relationship between travelling wave energy and fault location at both ends of a single type of line is derived as follows: Mathematically, the following differential equation can be used to describe the reduction of traveling wave energy during propagation, where A refers to the value of traveling wave energy and λ to the decay constant of traveling wave energy.
One solution of the above equation is: where A(x) represents the value of traveling wave energy at the distance x from the starting point, and A 0 is the initial value of traveling wave energy.
For the homogenized line shown in Fig. 2, the first and last ends of the line are noted as S and R. Assuming that the distance of the fault point from the S end of the line is x and the total length of the line is L, the energy of the traveling wave component at any frequency at the S and R ends of the line, which can be calculated by Eq. (12) : In the formula, W S (ω) and W R (ω) are respectively the energy of the traveling wave component with the frequency of ω at the S-end and R-end of the line, W F (ω) is the energy of the initial traveling wave component with the frequency of ω at the fault point. Since it is a uniform line, the initial traveling wave energy propagated to both ends of the line is consistent, and α(ω) is the energy attenuation coefficient of the traveling wave component with the frequency of ω.
By dividing the upper and lower equations of Eq. (12), the energy of the unknown initial travelling wave of the fault W F (ω) is eliminated, and the following equation can be obtained: By moving x in Eq. (13) to the left of the equation, the relationship between the fault location and travelling wave energy at both ends of the line can be obtained: The formula is the frequency ω, based on the traveling wave energy attenuation theory derived from the fault location calculation formula, from the formula can be seen, the line length L is a known quantity, traveling wave energy attenuation coefficient α(ω) can be calculated through the line parameters, as long as the line can be obtained at both ends of the traveling wave energy W S (ω), W R (ω), you can directly calculate the fault location.
Error analysis of line wave amplitude extraction based on S-transform. The analysis of the mapping relationship between energy and faults is carried out under ideal conditions. It is considered that the traveling wave component of a single frequency can be extracted from the initial traveling wave head. However, due to the limitation of the current time-frequency analysis algorithm, it is impossible to accurately extract the signal of a specific frequency in the multi-frequency aliasing signal. For the traveling wave signal with a continuous spectrum, a certain frequency component after decomposition still contains other frequency components. Therefore, under the same fault, when the time-frequency analysis algorithm is used to decompose the measured fault traveling wave signals at different locations and obtain the traveling wave energy of a certain frequency, the calculated results will deviate from the real value of the descending wave energy of this frequency, www.nature.com/scientificreports/ which leads to the following two problems: first, the attenuation coefficient and refraction coefficient of traveling wave energy cannot be calculated by using line parameters; Second, the mathematical relationship between the measured value of traveling wave energy and the propagation distance no longer strictly satisfies the form of the exponential function. In this case, the errors generated by the decomposition of traveling wave signals will definitely affect the positioning accuracy of the fault location algorithm based on Formula (14). Compared with uniform lines, the traveling wave energy changes on the non-fault section should also be considered for the cable mixed lines, and the correction of S-transform error should be paid more attention to during fault location. In theory, a mathematical formula can be used to describe the S-transform error and the relationship between the traveling wave propagation distance, eliminating the influence of S-transform on the positioning accuracy and fault traveling wave. However, one frequency component size is unknown, can't solve the S-transform error and its relationship with the propagation distance is fitting, but from another point of view, Fitting this relation is to describe the variation law of traveling wave energy propagating along the road more accurately. By fitting the relation between traveling wave energy attenuation coefficient α(ω) and traveling wave propagation distance x, the relation between S-transform error and traveling wave propagation distance can be indirectly reflected, and the accurate solution of traveling wave energy at the fault point can also be realized. Therefore, it is necessary to analyze and study the traveling wave energy variation rule under the effect of S-transform error according to the structural characteristics of cable mixed lines, so as to improve the accuracy of the fault location.
As the analysis in this paper is directed at the initial travelling wave of the fault, fault information needs to be extracted from the initial traveling wave head, so the selection of fault data needs to be discussed before analyzing the effect of the S-transform error on the travelling wave energy change pattern. Figure 3 is the initial traveling wave diagram of voltage measured 50 km away from the fault point when the overhead line is faulty. The arrival time of traveling wave is 0.033636 s, and the arrival time of the first reflected wave is 0.033803 s. Below, the S-transform algorithm is used to decompose the signal in the time-frequency domain, and the change characteristics of the S-transform results of each frequency signal component are compared when the fault signals at different starting and ending times are used. The length of the S-transform time window in this paper is chosen to be 2 ms, with a total of 2000 time points. The S-transform time window is continuously moved from right to left to reduce the signal length of the effective waveform of the fault line waveform, and the S-transform results for the fault voltage signal are shown in Table 1.
Table1 shows that the waveform data in the first row of the time window contains the initial travelling wave and reflected travelling wave, while the second row of time window only contains the initial travelling wave data. Comparing the S-transform results of the three frequencies, the results are completely the same. Since both ends of the line are completely reflected, the reflection coefficients are equal and cancel each other, so the reflected wave has no effect on the extraction of the initial traveling wave component. Comparison table of the same frequency of S-transform results under different time window, when the time to move out of the window to the left, the length of the fault effectively reduced, the transformation results gradually becomes poor, comparing the signals of different frequency, the greater the impact on the low frequency of signal, sometimes even unable to extract  www.nature.com/scientificreports/ low frequency signal amplitude. As shown in Table 1, when the extraction frequency is 20 kHz, the fourth to sixth rows cannot display data, which means that with the left shift of the time window, the S-transform cannot extract amplitude at low frequency. Considering the attenuation characteristics of traveling wave, the components of traveling wave with higher frequencies decay quickly and are not easy to detect, while the frequencies with too low are easy to be aliased together and not easy to distinguish. Three frequencies of 20 kHz, 50 kHz and 80 kHz are selected from small to large, and the extraction effects of S transform under the three frequencies are compared and analyzed, and the results are shown in Table 1. The traveling wave component with 80 kHz frequency can obtain better decomposition effect. It can be seen that the faulty high-frequency components are concentrated in the front of the traveling wave head, that is, the part where the wave head rises quickly. The flatter part of the wave head has less influence on the extraction results of high frequency components. Therefore, when solving the traveling wave energy, it is necessary to ensure that the complete wave head of the initial traveling wave of the fault is contained in the S-transform time window.

Precise positioning of hybrid cable circuits
Classification of traveling wave energy variation laws. From the structure of the type A and type B hybrid lines and the location of the fault occurrence, the propagation path of the initial traveling wave of the fault can be divided into five categories as shown in Table 2.
Comparing the similarities and differences of the five types of propagation paths, the variation in traveling wave energy from the fault point to both ends of the line can be classified as a combination of the following six types of laws: As long as the above-mentioned variation laws are clarified, the fault point data can be deduced from the data at both ends of the line to realize fault location.
Determination of the attenuation coefficient. The B-type hybrid line simulation model is built in PSCAD to analyze and fit the variation law of traveling wave energy under different scenarios. The S-transform time window selected in this section is consistent with the previous paper. At the same time, considering the attenuation characteristics of traveling waves, the traveling wave components of higher frequencies decay quickly and are not easy to detect, while those of too low frequencies are easily mixed and not easy to distinguish. Therefore, the traveling wave components of 80 kHz frequency are used to obtain a better decomposition effect.
Overhead line fault, the variation of traveling wave energy in this section. Set a traveling wave measuring point every 10 km in the overhead line section of the simulation model. There are 10 points in total, the sampling frequency is 1 MHz, and the traveling wave measuring points are named M1 ~ M10 respectively. As shown in Fig. 4, an A-phase grounding fault is set on the left side of the measuring point M1, and the fault resistance is 10Ω, and   www.nature.com/scientificreports/ the voltage and current waveforms at each measuring point are recorded. As shown in Fig. 5a,b from left to right show the initial traveling wave waveforms of voltage and current at M1-M10 respectively. In this paper, the S-transform is used to process the voltage and current traveling waves at the measuring points M 1 -M 10 . We calculate the modulus of each element in the complex matrix. Since the frequency of 80 kHz not only ensures the smoothness of the curve, but also reduces the influence of stray waves, so extract the frequency component of 80 kHz, and the results are shown in Fig. 6. The amplitude in Fig. 6a represents the voltage amplitude of 10 measurement points after undergoing S transformation, and the amplitude in (b) represents the current amplitude of 10 measurement points after undergoing S transformation. The nonlinear arc fault is considered in this paper. In the figure, from left to right are the S-transform results of traveling wave waveforms at M 1 -M 10 . Next, the traveling wave energy at each measurement point is calculated according to Eq. (15), where W Mi represents the traveling wave energy at measurement point M i , and S U (d) and S I (d) are the values of d-column of the voltage and current traveling wave S-transformation mode matrix, respectively. And D i is the column number corresponding to the maximum value of the faulty traveling wave S-transformation waveform at M i . Therefore S U (D i ) and S I (D i ) are used to characterize the amplitude of 80 kHz frequency component, and n characterizes the duration of the 80 kHz frequency component. For n, in the actual simulation, we set n = 5, n = 10 and n = 20 respectively, and observe the influence of different n values on the processing results of S transformation. It is found that when n is 5, the fitting results of S transformation are not smooth enough. With the increase of n, the curve is gradually smoothed, and when n is 10, the fitting degree of the curve has met the requirements. If n is too large, the required computing power and time will also be greatly increased, and the cost is too high. Therefore, after comprehensive consideration, n = 10 is finally selected.
The results of the calculations are shown in Table 3:    Table 4.
Using the cubic function in MATLAB to fit the relationship between the energy attenuation coefficient and the propagation distance of the traveling wave, the results are as follows: where x 1 is the distance from the head end to the fault point.
Cable line fault, the variation of traveling wave energy in this section. Referring to Fig. 7, set 10 traveling wave measuring points as M 1 -M 10 , and the interval of which is 2 km. Then setting the A-phase core sheath fault on the left side of the measuring point M1, and the fault resistance is 10Ω. The cable used in this section is the most common high voltage cable: cross-linked polyethylene cable, which is laid horizontally and directly buried. The length of the cable is more than 1 km, so the metal sheath of the cable is often cross-connected. According to formula (15) and formula (16), the traveling wave energy and attenuation coefficient are calculated respectively, and the cubic function is used for fitting. The calculated data and fitting results are shown in Tables 5 and 6: The fitting formula of the traveling wave energy attenuation coefficient and the propagation distance of the cable line and the schematic diagram of the fitting curve are as follows, where x 1 is the distance from the fault point: The remaining three types of propagation paths can be adaptive to various situations using the same method, and finally get: α 1 ~ α 5 . ① α 1 (x 1 ) is the attenuation coefficient of traveling wave energy under the scenario of     From the influence of S-transform error, it is not only necessary to consider the change of the traveling wave energy attenuation coefficient on the faulty section, but also to accurately describe the change of the attenuation coefficient on the non-faulty section. Based on the above two reasons, it is necessary to modify the homogeneous line fault location method deduced above to make it suitable for precise location of line-cable hybrid line faults. Fig. 8, the length of overhead line SP is L SP , and the length of cable line PR is L PR . The attenuation coefficients of the traveling wave energy at the fault distance x 1 for the traveling wave component with frequency ω on the overhead line and the cable line as the fault section are α 1ω (x 1 ) and α 2ω (x 1 ) respectively. The attenuation coefficients of the traveling wave energy at the fault distance x 1 for the traveling wave component with frequency ω on the overhead line and the cable line as the non fault section are α 3ω (x 1 ,x 2 ) and α 4ω (x 1 ,x 3 ) respectively. Since the lengths of overhead line and cable line × 2 and × 3 are determined values, α 3ω (x 1 ,x 2 ) and α 4ω (x 1 ,x 3 ) can be reduced to α 3ω (x 1 ) and α 4ω (x 1 ).

Precise positioning method. As shown in
In addition, the special fault location of the cable connection point has been judged in the fault section location, so it is not analyzed below.
For overhead line fault F1, the initial traveling wave energy with frequency ω propagated from the fault point to both ends of the line is W F (ω). When the traveling wave propagates to the S end of the line and the cable connection point P, the traveling wave energy at these two places can be calculated by the following formula: In the formula, x represents the distance from the S end of the line. When the traveling wave propagates from the point P to the R end of the line, the traveling wave energy at the R end can be calculated by the following formula: www.nature.com/scientificreports/ In the formula, γ2(ω) represents the traveling wave energy refraction coefficient of the traveling wave component with frequency ω when it passes from the overhead line side to the cable side. The meaning of this coefficient is shown in formula (8), where W Pf (ω) and W Pb (ω) respectively represent the traveling wave energy of the traveling wave component with frequency ω before and after passing through the cable connection P-point.
Similarly, by dividing the two equations of Eq. (19), the energy W F (ω) of the unknown traveling wave when initial fault is eliminated. And then combining Eqs. (20) and (21), a mathematical relation only about W S (ω) and W R (ω) with fault distance x 1 can be obtained as follows: From the above equation, it can be seen that W S (ω), W R (ω) and L SP in the expression of W P (ω) are known quantities or can be obtained by actual measurement. And the variation law of γ 2 (ω), α 1ω (x 1 ), α 3ω (x 1 ) has been obtained by fitting. In identifying the fault is located in the overhead line section, the fault location can be calculated based on the traveling wave energy relationship between the S and R ends of the line shown in Eq. (22) to achieve the precise location of the overhead line fault on the A-type cable hybrid line.
Similarly, the fault location formula of the cable line can be obtained as: Similarly, the expression of the mapping relationship between traveling wave energy and fault location can be derived for B-type hybrid line faults with different fault locations. As shown in Fig. 9, the length of the overhead lines SP 1 and P 2 R are L SP1 and L P2R respectively, and the length of the cable P 1 P 2 is L P1P2 . When a fault occurs on the overhead line SP 1 /P 2 R, the traveling wave energy attenuation coefficient of the traveling wave component with the frequency ω on the SP 1 /P 2 R section at the fault distance x 1 is α 1ω (x 1 ). And the traveling wave energy attenuation coefficient of the traveling wave component with the frequency ω on the P 2 /P 1 point at the fault distance x 1 is α 3ω (x 1 ). Also, the traveling wave energy attenuation coefficient of the traveling wave component with the frequency ω on the R-end/S-end at the fault distance x 1 is α 5ω (x 1 ). When a fault occurs on the cable line P 1 P 2 , the traveling wave energy attenuation coefficient of the traveling wave component with the frequency ω on the P 1 P 2 section at the fault distance x 1 is α 2ω (x 1 ). And the traveling wave energy attenuation coefficient of the traveling wave component with the frequency ω on the R-end/S-end at the fault distance x 1 is α 4ω (x 1 ).
For overhead line fault F 1 , when the initial traveling wave with the frequency ω propagates to the S-end and R-end of the line, the traveling wave energy is as follows: Dividing the two formulas in formula (24) and the result is as follows: In the formula, x represents the distance between the fault point F 1 and the S-end of the line, and γ 12 (ω) represents the traveling wave energy refraction coefficient when the traveling wave component with frequency ω passes from the cable side to the overhead line side under the overhead line fault.
For cable line fault F 2 , when the initial traveling wave with the frequency ω propagates to the S-end and R-end of the line, the traveling wave energy is as follows: In the formula, x represents the distance between the fault point F 2 and P 1 point. For overhead line fault F 3 , when the initial traveling wave with the frequency ω propagates to the S-end and R-end of the line, the traveling wave energy is as follows: Dividing the two formulas in formula (28) and the result is as follows: In the formula, x represents the distance between the fault point F3 and R-end of the line.

Simulation verification
Precise positioning simulation. The line in the simulation adopts the B-type connection mode. The total length of the transmission line is 112 km, of which the length of the two overhead lines is 60 km and 40 km respectively, and the length of the cable line is 12 km. The segment location method described in the literature is used to locate the segment firstly, and then the precise fault location is solved through the loop iteration of the attenuation coefficient and the virtual fault point.
Firstly, the overhead line fault is used as a case study to demonstrate the fault location process. We set an A-phase ground fault at 22 km from the S-end of the line, and then measure initial fault traveling wave at both ends of the line. The waveform is shown in Fig. 10.
Using the S-transform to extract the 80 kHz traveling wave component, the S-transform results are shown in Fig. 11. The amplitude in Fig. 11 represents the amplitude results of the voltage and current at the initial and end of the line after S transformation. And calculating the traveling wave energy according to the method described in "Determination of the attenuation coefficient" section, the traveling wave energy at the S-end and R-end of the line is 472.227 and 0.216063 respectively. The iteration of the ranging algorithm is implemented in MATLAB by combining the variation law of the traveling wave energy attenuation coefficient of the overhead line and (26)   The results in Table 7 show that the fault initial phase angle and the fault transition resistance variation have little effect on the positioning results. This is because the initial phase angle and the transition resistance only change the amplitude of the initial wave head of the fault and do not change the rising process of the wave head. That is, the S-transformation of the fault traveling waves under different conditions will only proportionally reduce the traveling wave energy of the voltage and current traveling waves. However, if the fault resistance is high enough, and the fault transient characteristics are weak, or the traveling wave signal has basically decayed and disappeared after long distance transmission, the initial traveling wave cannot be accurately detected, and the method cannot accurately locate the fault location.
From the positioning results in the Table 8, it can be conclude that in the uncorrected method has relatively large errors in the positioning results, especially in the case of long line lengths, and the positioning results cannot meet the actual engineering needs. The method in this paper considers the influence of S-transform on the extraction of the single frequency component of the traveling wave, and has a good localization effect on any point fault on the line.
The analysis and results from Fig. 3 and Table 1 show that the distortion of the waveform has a small effect on the S-transform results. Table 9 shows the location results of the constant resistance fault and arc fault under different fault locations. From the data in the table, it can be seen that the arc fault location error has increased compared with constant resistance faults. However, it can still meet the requirements of practical engineering applications.

Conclusion
Based on the defects of the traditional localization method about traveling wave arrival time, this paper searches for the characteristic quantity that can be applied to fault localization from the perspective of traveling wave energy attenuation characteristics. And analyzing in detail the mapping relationship between the characteristic quantity and the fault location to achieve precise localization. In principle, this method is completed by using the difference of traveling wave energy at both ends of the line. It only needs to simply measure and calculate the fault traveling wave energy. In application, the method doesn't need to ensure strict time synchronization between the measuring points, nor does it need to use the information of the fault reflection wave head, which reduces the introduction of errors and can achieve precise and reliable fault location.

Data availability
All data generated or analysed during this study are included in this article and its supplementary information file.

Funding
This work was supported by the project ZNKJ-20-32.