Abstract
The partial differential equation (PDE) solution of the telegrapher is a promising fault location method among time-domain and model-based techniques. Recent research works have shown that the leap-frog process is superior to other explicit methods for the PDE solution. However, its implementation is challenged by determining the initial conditions in time and the boundary conditions in space. This letter proposes two implicit solution methods for determining the initial conditions and an analytical way to obtain the boundary conditions founded on the signal decomposition. The results show that the proposal gives fault location accuracy superior to the existing leap-frog scheme, particularly in the presence of harmonics.
FAULT location is an essential function in the operation of modern power systems for lessening the outage time following a fault event [
This letter proposes improvements to the LF method applied to the fault location. The first one is calculating initial values along the -axis (line length) using extracted phasors, which provides significant improvements, particularly when the system is contaminated with harmonics. The issue of harmonics has become critical due to the widespread use of power electronics. The second improvement is correctly estimating the initial voltage values for the first space-time stepping solution of LF. In particular, two new implicit-based schemes are introduced for calculating these initial values.
The equations of telegrapher are a pair of coupled PDEs that govern the voltage and current on the transmission line in the function of distance and time :
(1) |
The coefficients of the PDEs are in terms of distributed parameters: , , , and are the the series resistance, series inductance, shunt capacitance, and shunt conductance (per unit length), respectively. By differentiating the set of PDEs in (1) with respect to and and eliminating the terms involving , a second-order hyperbolic PDE can be derived as:
(2) |
The LF method operates on (2) and employs second-order central differences for approximating the space and time derivatives. Reference [
(3) |
Fig. 1 Required data, calculation progress, and calculation space for LF method.
where ; ; ; ; and the voltage subscripts and (without parenthesis) are integer values representing positions on the power grid with spacings and . As shown in (3), the LF calculation scheme requires the knowledge of values for and for all . Therefore, the calculation in (3) begins with and and computes all values along the -axis. After that, is increased by one, and the calculation is repeated for all possible values along the -axis, as shown in
The two-ended fault location technique solves PDEs twice using the LF method. The first solution starts at the sending end and the second at the receiving end. The fault location can be determined by comparing the voltage profiles computed from the sending and receiving ends; and the mismatch between these voltages is the least at the fault location. In the computation, the solution is found in the domain, making the goal function (whose components are shown in
(4) |
Fig. 2 Goal functions || in , , and components for BCG fault (phases B and C to ground) on 240 km line, fault at km.
Although the initial voltage value for and , i.e., , is given by the intelligent electronic device (IED) at the sending end of the line, the values along the -axis for are unavailable. Therefore, to compute , the LF implementation in [
Assuming that the system was in a steady state before the fault, one can consider the presence of a fundamental component voltage signal and its harmonics. This letter proposes an analytical procedure for calculating the pre-fault voltage and current profiles along the line length (-axis) starting from the phasor estimation of the fundamental component and harmonics described in [
(5) |
where , and is the fundamental system frequency; ; and is the number of harmonics.
The use of the wavelet transformation for double-ended fault location has been discussed in phasor extraction [
Providing initial values for the LF method requires a solution of PDEs of the telegrapher (1) for a single step in the discrete space-time representation. The process is solved implicitly for (scheme 1 in Section II-B-1)) or (scheme 2 in Section II-B-2)). The general algorithmic framework is as follows for sending-end calculations, and similar computations apply to the receiving end.
Step 1: initialization. Collect the sampled data and convert the sampled voltage data, current data, and the line parameters to the domain.
Step 2: implicit formulation. Formulate the diagonally dominant sparse matrix and the right-hand-side vector of the PDE discrete space-time representation using (6), (7) for scheme 1 and (8), (9) for scheme 2.
Step 3: iterative solution. Using an iterative solver, solve , where for scheme 1 and for scheme 2.
Step 4: LF method. Apply the LF method using the measured voltage initial values , the calculated initial values along the -axis for scheme 1 and for scheme 2, and the calculated initial voltage values for scheme 1 and for scheme 2.
To provide initial values required by the LF method, this scheme uses an implicit solution employing a centered derivative approximation with second-order accuracy along the -axis and a forward approximation of first-order along the -axis (represented by the green ellipses in
Fig. 3 Calculating initial values using unmodified space-time grid.
(6) |
(7) |
The solution to in Step 3 provides all values for (represented by the green dots in
Scheme 2 is similar to the first but differs from it by using a backward approximation of first-order along the -axis instead of a forward approximation [
(8) |
(9) |
The solution to provides all values for (represented by the green dots in
Fig. 4 Calculating initial values by extending space-time grid.
The two-terminal fault location method is implemented by comparing the components calculated using the samples from the sending- and receiving-end terminals. The calculation is done by solving the PDEs in the forward and backward directions, i.e., starting from the sending and receiving ends, using the LF method. The numerical methods were implemented in MATLAB using in-house developed solvers. Three initialization schemes were investigated, considering feasible combinations of two possibilities for the boundary condition () and two for the initial voltage values required to initiate the LF computation.
1) Int+ refers to linear interpolation for the initial voltage profile and estimation of the voltages using scheme 1. Int+ is closest to the initialization method in [
2) Extr+ refers to phasor extraction followed by the analytical voltage solution (described in Section II-A) for the initial voltage profile and estimation of the voltages using scheme 1. Extr- differs from Extr+ by estimating the voltages using scheme 2.
The tests were carried out on a transmission line (as shown in
Fig. 5 Test setup of fault location.
(km) | (°) | Error (%) | IF | |||
---|---|---|---|---|---|---|
Int+ | Int- | Extr+ | Extr- | |||
20 | 0 | 0.716 | 0.688 | 0.627 | 1.141 | |
45 | 1.147 | 1.064 | 1.008 | 1.138 | ||
90 | 1.322 | 1.193 | 1.163 | 1.137 | ||
65 | 0 | 1.163 | 1.086 | 1.034 | 1.124 | |
45 | 1.205 | 1.107 | 1.079 | 1.117 | ||
90 | 1.970 | 1.876 | 1.789 | 1.101 | ||
120 | 0 | 0.683 | 0.610 | 0.607 | 1.126 | |
45 | 0.831 | 0.764 | 0.731 | 1.137 | ||
90 | 0.998 | 0.951 | 0.899 | 1.110 |
Note: indicates that Int- cannot be applied.
(km) | (°) | Average error (%) | IF | |||
---|---|---|---|---|---|---|
Int+ | Int- | Extr+ | Extr- | |||
20 | 0 | 15.075 | 3.765 | 3.033 | 4.971 | |
45 | 17.089 | 4.325 | 4.182 | 4.086 | ||
90 | 20.711 | 5.802 | 4.959 | 4.177 | ||
65 | 0 | 13.801 | 3.774 | 2.965 | 4.655 | |
45 | 15.943 | 3.765 | 3.127 | 5.098 | ||
90 | 17.805 | 4.097 | 3.436 | 5.182 | ||
120 | 0 | 11.066 | 3.876 | 2.139 | 5.173 | |
45 | 13.721 | 3.096 | 1.996 | 6.875 | ||
90 | 15.097 | 2.973 | 1.890 | 7.987 |
Note: indicates that Int- cannot be applied.
(km) | (°) | Estimated (km) | Error (km) | Error (%) | IF | |||
---|---|---|---|---|---|---|---|---|
MC | LF | MC | LF | MC | LF | |||
20 | 0 | 24.812 | 22.147 | 4.812 | 2.147 | 2.005 | 0.895 | 2.241 |
45 | 25.713 | 22.618 | 5.713 | 2.618 | 2.380 | 1.091 | 2.182 | |
90 | 26.440 | 21.976 | 6.440 | 1.976 | 2.683 | 0.823 | 3.259 | |
135 | 25.913 | 22.633 | 5.913 | 2.633 | 2.464 | 1.097 | 2.246 | |
180 | 26.728 | 17.914 | 6.728 | 2.086 | 2.803 | 0.869 | 3.225 | |
65 | 0 | 59.276 | 62.215 | 5.724 | 2.785 | 2.385 | 1.160 | 2.055 |
45 | 58.301 | 62.416 | 6.699 | 2.584 | 2.791 | 1.077 | 2.592 | |
90 | 61.855 | 63.971 | 3.145 | 1.029 | 1.310 | 0.429 | 3.056 | |
135 | 71.029 | 68.102 | 6.029 | 3.102 | 2.512 | 1.293 | 1.944 | |
180 | 70.312 | 66.993 | 5.312 | 1.993 | 2.213 | 0.830 | 2.665 | |
120 | 0 | 125.013 | 122.711 | 5.013 | 2.711 | 2.089 | 1.130 | 1.849 |
45 | 124.997 | 122.323 | 4.997 | 2.323 | 2.082 | 0.968 | 2.151 | |
90 | 126.713 | 123.192 | 6.713 | 3.192 | 2.797 | 1.330 | 2.103 | |
135 | 114.992 | 122.175 | 5.008 | 2.175 | 2.087 | 0.906 | 2.303 | |
180 | 124.947 | 122.611 | 4.947 | 2.611 | 2.061 | 1.088 | 1.895 |
Fault type | Involved component | Average computational time (ms) | ||
---|---|---|---|---|
0 | ||||
AG | 448.3 | |||
BC | 262.1 | |||
BCG | 603.7 | |||
ABC | 452.8 |
Note: and indicate that the component is involved and not involved, respectively.
Among time-domain and model-based fault location techniques, the PDE solution of the telegrapher is the most effective. This letter builds upon recent research [
Acknowledgements
The work of Izudin Džafić was supported by the Federal Ministry of Education and Science, Bosnia, through funding.
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