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Clarke’s matrix has been applied as a phase-mode transformation matrix to three-phase transmission lines substituting the eigenvector matrices. Considering symmetrical untransposed three-phase lines, an actual symmetrical three-phase line on untransposed conditions is associated with Clarke’s matrix for error and frequency scan analyses in this paper. Error analyses are calculated for the eigenvalue diagonal elements obtained from Clarke’s matrix. The eigenvalue off-diagonal elements from the Clarke’s matrix application are compared to the correspondent exact eigenvalues. Based on the characteristic impedance and propagation function values, the frequency scan analyses show that there are great differences between the Clarke’s matrix results and the exact ones, considering frequency values from 10 kHz to 1 MHz. A correction procedure is applied obtaining two new transformation matrices. These matrices lead to good approximated results when compared to the exact ones. With the correction procedure applied to Clarke’s matrix, the relative values of the eigenvalue matrix off-diagonal element obtained from Clarke’s matrix are decreased while the frequency scan results are improved. The steps of correction procedure application are detailed, investigating the influence of each step on the obtained two new phase-mode transformation matrices.

Modal transformations are applied to transmission line analyses because, in mode domain, it is easier to represent the frequency influence on the line parameters. Using phase-mode transformation matrices, all electrical parameters and all line representative matrices are obtained in mode domain [1–4]. The line representtative matrices become diagonal and the frequency influence can independently be introduced for every mode because the mutual phase couplings are independently included at every mode. Applying frequency dependent line parameters also leads to frequency dependent phasemode transformation matrices. Because of this, to obtain voltages and currents in phase domain after signal mode propagation, it is necessary to use a convolution procedure [5–10].

An alternative is to change the exact transformation matrices into single real ones. With these single real transformation matrices, any values can be determined in phase or mode domain using only a matricial multiplication [3,11]. The single real transformation matrices can obtain exact modes and diagonal line representative matrices for ideally transposed lines [12–14]. For untransposed lines, the results are not exact. The errors related to the eigenvalues (l) can be considered negligible for some untransposed three‑phase line analyses when Clarke’s matrix is applied as the transformation matrix. The data obtained with Clarke’s matrix are called quasi-modes. Increasing the asymmetrical geometrical line characteristics, even though the errors of quasi-mode matrix diagonal elements are negligible, the relative values of the quasi-mode matrix off‑diagonal elements can be significant when compared to the correspondent eigenvalues (l).

Based on these hypotheses, Clarke’s matrix application is analyzed considering a symmetrical three-phase line and a frequency range from 10 Hz to 1 MHz. The quasi-mode errors related to the eigenvalues (l) are studied as well as the off‑diagonal elements of the l quasi-mode matrix. Improving the analyses, frequency scans are also made using the characteristic impedance (Z_{C}) and the propagation function (g) calculated from the exact mode values and the quasi-mode ones. Searching for the off-diagonal element relative value minimization, a perturbation approach corrector matrix is applied to Clarke’s matrix. The errors and frequency scan analyses are carried out again and the new results are compared to the previous error values. So, with a first-order approximation procedure, the l quasi-mode off-diagonal element relative values are highly decreased and the frequency spectrum of the processed signals is closer to that of the exact transformation. Neglecting the imaginary part of the new transformation matrix, frequency scan results similar to those from the first order matrix correction are obtained.

From the comparisons carried out using the 10 kHz frequency value, mode voltage and mode current vectors, it is suggested to extend the analyses shown in this paper considering one of the both modes that is not the homopolar reference of the system as the correction procedure application base. Another suggestion is to apply twice the correction procedure where the both modes related to the modal coupling for symmetrical threephase transmission lines are subsequently used.

Searching for more simplicity for phase-mode transformation applied to transposed three-phase lines, single real transformation can be used. One sample of these matrices is Clarke’s matrix [3,11]. The exact differential equations that relate the transversal voltages and the longitudinal currents are described below. In this case, the phase-mode transformations (T_{V} and T_{I}), the per unit length longitudinal impedance (Z) and the per unit length transversal admittance (Y) matrices are included [1, 5, 7, 10, 12–14].

u_{MD} and i_{MD} are the voltages and the currents in mode domain, respectively. For transposed three-phase lines, the T_{V} and T_{I} matrices are changed into Clarke’s matrix represented by T_{CL}. So, the u_{MD} and i_{MD} values are:

For exact eigenvectors (T_{V} and T_{I}), the phase-mode relations of Equation (2) are described by:

Using Equations (2) and (3) for transposed three-phase lines, the following relations are obtained:

The inverse Clarke’s matrix is equal to its transposed one and the initial differential equations are changed into:

The Z and Y matrices in mode domain are:

The Clarke’s matrix structure is [3, 11]:

Based on Equations (4–7), the eigenvalues of a transposed three-phase line are determined by:

In this case, the modes are called a, b and 0 (homopolar). The l_{CL} is described by:

The impedance characteristic (Z_{C}) is described by:

The propagation function is described by:

Considering a symmetrical untransposed three-phase line, l_{CL} is not diagonal. The results are called quasimodes. There is a modal coupling between a and 0 modes and the l_{CL} matrix becomes the following [

For the exact l matrix, the exact eigenvectors (T_{V} and T_{I}) are applied, obtaining the following:

The exact Z and Y matrices in mode domain are:

The exact modal Z_{C} matrix is:

The exact modal g matrix is:

Applying Clarke’s matrix to an actual symmetrical untransposed three-phase line, the quasi-mode results are compared to the exact values through the following equations:

The relative values of the l_{CL} off-diagonal elements are obtained with the following:

Regarding the frequency scan, the modal couplings among the quasi-modes are neglected. Every mode or quasi-mode is analyzed as in

The propagation wave in

The system of Equations (19) is applied for every mode considering three situations in the line receiving terminal (the B terminal): opened line, short-circuit and infinite line. The infinite line is calculated using an impedance with the Z_{C} value connected to the line receiving terminal. For the frequency scan, the line sending terminal is connected to a unitary step voltage source, considering the frequency domain. This voltage source is described by the next equation. The unitary step voltage

is chosen because it includes all frequency values. In the case of this paper, the frequency scan analyses are performed with a frequency range from 10 Hz to 1 MHz.

The interactions between any transposed three-phase line and Clarke’s matrix produce exact mode results and the modal representative matrices are diagonal. In case of untransposed three-phase lines, the T_{CL} results are not exact. These results are compared to the correspondent exact values using Equations (17) and (18) as well as the frequency scan.

The actual three-phase line analyzed in this paper has a vertical symmetry and shown in

The central phase conductor height is 27.67 m on the tower. The height of adjacent phase conductors is 24.07 m. Every phase is composed of four conductors distributed in a square shape with 0.4 m side length. Every conductor is an ACSR type one (ACSR‑26/7‑636 MCM). The phase conductor resistivity is 0.089899 W/m and the sag at the midspan is 13.43 m. The earth resistivity is considered constant (1000 W.m). The ground wires are EHS 3/8 with the resistivity of 4.188042 W/m. The height of these cables on the tower is 36.00 m. The sag of the ground wires at the midspan is 6.40 m.

From the Equations (8), (9), (12) and (13), the l_{NCL} quasi‑modes are compared to the eigenvalues using Equation (17). In this case, firstly, the eigenvectors are calculated applying the iterative Newton-Raphson’s method. The initial values for this method are frequency of 10 Hz and the Clarke’s matrix elements. For this first frequency

value, the iterative processing is started considering the eigenvectors equal to the Clarke’s elements. When the iterative processing converges to the exact values, it is restarted with the next frequency value and uses the exact values of the previous frequency value for the reinitialization of the new eigenvectors. After the determination of eigenvectors, the eigenvalues and the comparisons to the quasi-modes are performed for every frequency value [

The results of the comparisons between quasi-modes and the eigenvalues are shown in

It is shown in Equation (12) that there is a modal coupling between the α and 0 quasi-modes. Because the line representative matrices are symmetrical independently of whether the line is symmetrical, or not, the off-diagonal element in the intersection between the matrix first line and the matrix third row is equal to the off-diagonal in the third line and the first row intersection. In

In _{Nα0} off-diagonal element is compared to the correspondent eigenvalues from Equation (13). When it is compared to the λ_{α} eigenvalue, the λ_{Nα0} relative curve gets higher values than the curve obtained from the λ_{0} eigenvalue. The highest peak of the curve related to the λ_{α} eigenvalue is close to 18 % and it is associated to the initial frequency values of the consideredfrequency range. The curve related to the λ_{0} eigenvalue has softer variations than the other curve. Considering both curves presented in _{Nα0} relative values decrease when the frequency increases. For low frequency values, the λ_{Nα0} modal coupling is more significant then for high frequency values. Based on these results, signal phase-mode transformation comparisons are carried out for analyzed frequency range using Clarke’s matrix and the eigenvectors (T_{V} and T_{I}). From phase vectors and with the Clarke’s matrix application, it is obtained mode vectors where only one mode has non-null value. It is shown in the following Equation:

The non-null values are compared to the results obtained from the application of the eigenvectors:

The results of comparisons between the non-null voltage values of Equation (21) and its correspondent values of Equation (22) are shown in _{V} eigenvector matrix. These are voltage vector transformations detailed in Equations (21) and (22). In case of

For modal transformations of Z and Y matrices as well as for obtaining of eigenvalues, a transformation matrix and an inverse transformation one are used. Applying the modal transformations to the voltage and current vectors, only one transformation matrix, or its inverse matrix, is necessary. In this case, the transformation could increase the errors related to the quasi-modes. On the other hand, based on mentioned results, the use of quasi mode Z and Y matrices for determining other variables in mode domain could increase the errors observed in _{CL} and Y_{CL} matrices and compared to the correspondent values obtained from Z_{MD} and Y_{MD} matrices. The comparisons are made through Equation (17).

_{C} modulus. The Z_{C} angle is null because this variable is real. Figures 10 and 11 are

associated to the γ modulus and the γ angle errors, respectively.

Considering the Z_{C} modulus and the γ angle errors, there are curves with inverse signals (the α and 0 quasimodes). This characteristic can also be observed in

The range error for the Z_{C} modulus (from -2 % to 2 %) is about 10 times higher than the γ modulus error range (from -0.05 % to 0.25 %) and it is about 100 times higher than the γ angle error range (from ‑0.015 % to 0.03 %). So, the Z_{C} modulus is more sensitive to the errors introduced by Clarke’s matrix. The relative values of the off-diagonal element (the Nα0 modal coupling) for Z_{C} modulus are shown in _{C-Nα0} off-diagonal element influence is significant on the correspondent Z_{C} quasi-modes.

In _{Nα0} angle relative values are in a similar range to the γ_{Nα0} modulus relative values. In this case, the values do not tend to be crescent ones.

Frequency scan analyses are based on _{A} voltage is described by Equation (20). Three situations of the line receiving terminal (the B terminal) are checked: open line (I_{B}=0), short-circuit (E_{B}=0) and infinite line. The infinite line is obtained with an impedance equal to the Z_{C} value connected in the B terminal of

from quasi-modes are equal or very close to the exact values obtained from eigenvectors. The main differences are related to the angle values and the a mode. Showing the results related to the modulus values,

Considering the short-circuit analyses in

When the infinite line is analyzed (

Based on frequency scan analyses, Clarke’s matrix could be applied to transient simulations considering symmetrical untransposed three-phase lines and phenomena with a frequency spectrum concentrated below 10 kHz. For general phenomena, there are classical solutions for

this problem based on eigenvector applications. An alternative is to apply a perturbation approach corrector matrix, improving the results of the Clarke’s matrix application above 10 kHz. This alternative is described in the next item and based on the homopolar mode.

The procedure shown in this section is based on a first‑order perturbation theory approach [

The N_{22} matrix is defined as:

The normalization matrix is applied to the l_{NCL} matrix:

The described procedure is applied to the T_{V} and the T_{I} matrices. For the T_{I} matrix, the procedure is similar to the T_{V} one with a change in the position of the Z and Y matrices in Equation (38):

The structure of the A matrix is determined from:

The last Equation leads to:

The l_{CL-}_{a} element is equal to the l_{CL-}_{b}. Because of this, the a_{12}, a_{21}, q_{12} and q_{21} elements are null. Dividing the l_{NCL} matrix into blocks, the portioned structure can be described by:

The l_{P22} is:

In this case, despite the symmetry of the line representative matrices, small numeric differences are considered between symmetrical elements of the l_{P} matrix. The small numeric values of the l_{P}_{a}_{b} and l_{P}_{b}_{a} are also considered. Using Equations (42) and (43), the a_{a} and a_{b} elements are determined by:

The N_{22} matrix elements are determined by:

In this case, only the Q matrix elements of the third line and the third row can not be null. These elements correspond to the 0 mode and are calculated by:

The perturbation approach corrector matrix is described by:

The corrected transformation matrix is described by:

Checking the changes into the Clarke’s matrix results carried out applying only the N matrix, it is used the flowchart shown in

The changes obtained from the N matrix application are mainly related to the mode coupling relative values. The peak value is decreased from 18 % to 2 %. It is about a 10 time reduction. The off-diagonal relative obtained after applying the N matrix are shown in

Completing the analysis of the influence of N matrixthe results of Figures 22–25 show that the application of this matrix balances equally the phase-mode transformation results obtained from Equation (26) where the T_{CL} matrix is changed into the A_{V} and A_{I} matrices.

Another analysis about the correction procedure ap-

plication to Clarke’s matrix is about the Q matrix application. In this case,

In Figures 27 and 28, the shown values decrease when compared to the values presented in Figures 20 and 21. The peak value shown in

For the l errors, comparing Figures 3, 21 and 28, the reduction is about 150 times considering the negative peak values shown in these cases.

Applying the Q matrix, the off-relative and the l errors become negligible. Analyzing the results of both matrices applications, N and Q, it can be concluded that the N matrix mainly acts on the off‑diagonal relative values, decreasing them. On the other hand, the Q matrix acts on the λ error and the off-diagonal relative value decreasing.

Analyzing the results shown in Figures 27 and 28, these values can be considered negligible because they are in a very low range of relative values. The peak values reach 0.12 % and -0.002 % in mentioned figures. The shown curves also present some oscillations which, probably, are introduced by the used numeric method.

Based, on the results of _{C}, g and other electrical variables.

Changing eigenvector matrices into Clarke’s matrix for untransposed symmetrical three-phase lines leads to small errors related to the exact modulus eigenvalues.

The off-diagonal element of the matrix obtained from the Clarke’s matrix application, the quasi-mode eigenvalue matrix, has high relative values when compared to the correspondent exact eigenvalues. Based on these element results, the frequency scan analyses are carried out, showing that there are great differences between the quasimode current angles and the exact ones for frequency values above 10 kHz. In this casse, three situations of the line receiving terminal are checked: open line, short-circuit and infinite line.

A correction procedure is applied and new phase‑mode transformation matrices are determined: one matrix for voltages and another one for currents.

It is detailed the steps of the correction procedure application, describing the influence of each step on the decreasing of the off-diagonal quasi-mode relative value elements and the quasi-mode matrix ones. One of these steps is the normalization matrix application that carries out balanced voltage and current vectors, obtaining a 150 time reduction of the off-diagonal quasi-mode eigenvalue matrix relative values when compared to the Clarke’s matrix application. After this step, the final one reduces about 150 times the l errors when compared to the Clarke’s matrix results. Using the new phase-mode transformation matrices, obtained from the applied correction procedure, the off-diagonal element relative values and the l errors are highly decreased and could be considered negligible.