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A calculation method of transmission line equivalent geometrical parameters based on power-frequency parameters
Electrical Power and Energy Systems 111 (2019) 152–159
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
A calculation method of transmission line equivalent geometrical parameters based on power-frequency parameters☆
T
⁎
Botong Li , Mingrui Lv Key Laboratory of Smart Grid (Tianjin University), Ministry of Education, School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Equivalent geometrical parameter Power- frequency parameter Carson formula Frequency-dependent parameter
The geometrical parameters of the transmission line are prerequisite for precisely calculating frequency-dependent parameters as well as studying the electromagnetic transient process. Considering that the equivalent geometrical parameters are difficult to obtain directly, this paper proposes a method to calculate geometrical parameters of overhead transmission line using impedance and capacitance matrices measured at power frequency. By applying the inverse derivation of the Carson formula, the expressions of the average height of the line and the conductor radius can be obtained if the self-impedance of the line is known, while the distance between two conductors and that between one conductor and the image of another can be found with the mutual impedance. On this basis, if both the impedance and capacitance of the transmission line at power frequency are given, iteration can be used for higher accuracy. Finally, MATLAB- and PSCAD/EMTDC-based simulations are carried out verifying the effectiveness and sensitivity of the method.
1. Introduction Transmission line parameters include resistance, inductance and capacitance. The impedance of the line is specifically composed of the external impedance, the earth-return impedance due to the soil effect, and the internal impedance due to the skin effect, which are all the functions of frequency [1–5]. It is essential to take the frequency-dependent characteristics of the impedance into consideration in the research of the electromagnetic transient process of the power system. One basic calculation method of frequency-dependent impedance is the Carson formula, requiring equivalent geometrical parameters concerning tower configuration and conductor characteristics [5]. However, practical difficulty lies in identifying the equivalent geometrical parameters due to the ever-changing terrain, environment and line transposition [6]. Usually, power-frequency line parameters are first measured and can be used as input data for calculating geometrical parameters. Therefore, for a better transient analysis of power system, it is important to study a method for solving the equivalent geometrical parameters with power-frequency line parameters. Progress has been made in the study of the relationship between geometrical and frequency-dependent parameters of the transmission line. A number of formulas have been employed to calculate frequencydependent line parameters in the frequency domain. The Carson
formula to compute the frequency-dependent line parameters was introduced in [5]. Deri-Semlyen formula as a simplified formula of Carson’s infinite integral term was described in [7,8]. A method for calculating earth impedance based on the Carson integral was presented in [9]. In [10], series impedance and inductance of transmission line were obtained via the approximation of Dubanton and Wait. In [11], power series and asymptotic series expansions for calculating self and mutual line impedances were derived. To approximate the frequency-dependent parameters in the time domain, some fitting methods have been proposed. In [12], the characteristic impedance and the propagation function were simulated by a series of resistance-capacitance parallel blocks, which were approximated by rational functions. An optimization fitting procedure used to obtain low-order approximation of rational functions for the frequencydependent parameters was shown in [13]. Ref. [14] approximated the frequency-dependent admittances with the vector fitting by optimal scaling using complex starting poles. In [15], the frequency-dependent line parameters were obtained by means of a convolution procedure based on the synthesized rational function. In [16], the frequency-dependent impedance of the Bergeron model was computed directly in the time domain by means of a single rational function and lumped elements. For the multi-conductor transmission lines, the fitting methods are
☆ ⁎
This research work was supported in part by the National Natural Science Foundation of China (NSFC) under Grant 51677125. Corresponding author. E-mail addresses: [email protected] (B. Li), [email protected] (M. Lv).
https://doi.org/10.1016/j.ijepes.2019.04.013 Received 2 December 2018; Received in revised form 28 February 2019; Accepted 9 April 2019 0142-0615/ © 2019 Elsevier Ltd. All rights reserved.
Electrical Power and Energy Systems 111 (2019) 152–159
B. Li and M. Lv
and ri is the equivalent radius of conductor i. The calculation formulas of hi, dij and dij,mir are described in Appendix A [3]. The radius changes with the number of conductors and the inter-conductor spacing in the case of the conductor bundle, whose specific calculation method is given in [23]. With these geometrical parameters, the Carson formula can be used to calculate frequency-dependent parameters precisely.
usually used in combination with modal decoupling to solve the frequency-dependent parameters in the modal domain. In [17], the impedance and admittance matrices of the transmission line were decoupled with the frequency-dependent transformation matrices evaluated by the Newton-Raphson method. A sequential quadratic programming method for calculating the transformation matrices was described in [18]. Some constant and real modal transformation matrices are also used to decouple the three-phase line into three independent propagation modes. In [19,20], the frequency-dependent parameters of the equivalent Bergeron circuit were solved using an accurate real modal decoupling and the vector fitting method. Ref. [21] built a new frequency-dependent multi-conductor transmission line model with a single real constant transformation matrix. In the literature above, equivalent geometrical parameters, such as the average above-ground height of the transmission lines and the interconductor distances, are a must for calculating the frequency-dependent parameters. However, it is not easy to get the equivalent geometrical parameters due to the complex transmission path and changeable conditions along the line. Therefore, it remains a key problem to determine the equivalent geometrical parameters. Ref. [22] solved the equivalent geometrical parameters by referring to zero- and positivesequence inductances and capacitances evaluated at the power frequency. However, the results are susceptible to errors in line parameter evaluation because of exponential operation. This paper proposes a Carson-formula-based method of calculating the equivalent geometrical parameters of overhead transmission line, using impedance and capacitance matrices measured at power frequency. Unlike [22] where a slight error in the line parameter evaluation can lead to a rather inaccurate result, this paper studies how to improve the calculation accuracy with the impedance and capacitance matrices both known at the power frequency.
3. Geometrical parameters solutions 3.1. Solution based on impedance matrix The impedance matrix Z in per unit length of the transmission line at power frequency is
RmAB + jXmAB RmAC + jXmCA ⎤ ⎡ RsA + jXsA RsB + jXsB RmBC + jXmBC ⎥ Z = ⎢ RmBA + jXmBA ⎢R + + jX R jX RsC + jXsC ⎥ mCA mCA mCB mCB ⎣ ⎦
(1)
where Rsi is the self-resistance and Xsi is the self-reactance of conductor i, while Rmij and Xmij are the mutual resistance and reactance respectively. According to the Carson formula, the self-resistance Rsi and the selfreactance Xsi of conductor i are obtained as
Rsi = Ri, ac + ΔRsi Xsi = ω
μ0 2h ln i + Xi, ac + ΔXsi 2π ri
(2)
(3)
where Ri,ac is the AC resistance of conductor i and Xi,ac the internal reactance; ΔRsi and ΔXsi are Carson’s correction terms for earth return effects; μ0 = 4π × 10 - 4 H/km is the uniform permeability for both the aerial space and the earth; andω is the angular frequency proportional to system frequency f (ω = 2πf ). The DC resistance of the line is marked as Ri,dc, yet the line resistance will increase with frequency due to the skin effect in the AC case. So the AC resistance Ri,ac is
2. Equivalent geometrical line parameters Carson formula is one basic method for calculating the frequencydependent impedances of the transmission line, but it requires geometrical parameters concerning the tower configuration and conductor radius as the input constants. The geometrical parameters defined in the plane rectangular coordinate system are shown in Fig. 1. The x axis is given as the same height of the ground. The positions of the three phases are PA = [xA, yA], PB = [xB, yB] and PC = [xC, yC], where xi is the abscissa of conductor i (i = A, B, C) and yi is the relative height of the conductor i to the ground. With the coordinates of all the conductors, it is easy to know the geometrical distance between any two conductors and the distance between the conductor and the ground. In Fig. 1, hi is the average above-ground height of conductor i; dij is the distance between conductors i and j (j = A, B, C and j ≠ i); dij,mir is the distance between conductor i and the image of conductor j;
Ri,ac = kR·Ri,dc
(4)
where kR is the ratio of the AC resistance to the DC resistance. Also as a result of the skin effect, the inductance decreases as frequency increases. The AC internal reactance Xi,ac can be negligible in practical application, but to improve the accuracy in the derivation of the radius, Xi,ac is taken into consideration and given as
Xi,ac = kL·Xi,dc
(5)
where kL is the ratio of AC internal reactance to the DC internal reactance. Calculation methods for both kR and kL are detailed in [24]. In [3], the DC internal reactance Xi,dc is calculated by the following formula
Xi,dc = ω
μ0 2πf μ0 1 1 ln 1 = = μ0 f . 2π 4 2π 4 e− 4
(6)
Carson’s correction terms ΔRsi and ΔXsi in (2) are the functions of both the angle ϕ and parameter as. According to Appendix B, when the average height of the conductor is less than 755 m, parameter as at the power frequency is less than 3. Therefore, according to the Carson formula, the first two parts of formula of Carson’s correction terms are
ΔR = k1 (
π − b1 as cos ϕ ) 8
1 ΔX = k1 ( (b0 − ln as ) + b1 as cos ϕ ) 2 Fig. 1. Equivalent geometrical parameters of transmission line.
where 153
(7) (8)
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B. Li and M. Lv
r=
k1 = 4ω × 10−4 , b0 = 0.6159315, and b1 = 2 /6.
π − b1 as ) 8
d=
π 8
−
R si − Ri, ac k1
b1
(10)
).
1 2
μ0 f
)
(13)
10−4
= 2μ0 f . where k1 = 4ω × Based on the Carson formula, the distance between conductors i and j and that between conductor i and the image of conductor j can be deduced from the mutual reactance Xmij according to dij,mir μ0 ln + ΔXmij 2π dij
Psi =
hi2 =
k2·exp
( 12 b0 + k2 b1 (hi + hj) ) ⎞
⎜ ⎝
μ0 f
⎟ ⎠
(15)
.
hiav = hi2
dij2 − (hi − hj )2
dij,mir =
sij2 + (hi + hj )2 .
riav =
h=
hA hB hC
(26)
2hiav exp(2πε0·Psi )
(27)
According to Section 3.1, equivalent geometrical parameters can be calculated using only the power-frequency impedance matrix. But in the simulation, the calculation is not accurate enough and the results are susceptible to the error of the impedance matrix. To achieve higher accuracy for hiav and riav, the impedance and capacitance matrices are both used in calculation in this section, providing more parameters for later iteration to reduce the effect of the errors in power-frequency parameters. The calculation steps of iteration are shown as follows. The Fig. 2 shows the flowchart of the calculation process.
(17) (18)
The heights hA, hB and hC as well as the radiuses rA, rB and rC for the three phases respectively can be obtained. After transposition, according to [3], the geometric average height and radius are respectively 3
(25)
Then the geometrical average of the radius riav is
(16)
Therefore, if the self-resistance is known, the average height of each conductor can be calculated by (11) and (12). Then with the mutual impedance already known, any inter-conductor distance can be obtained via (16). As such, the horizontal distance sij between two conductors and the distance dij,mir between conductor i and the image of conductor j are
sij =
ri·exp(2πε0·Psi ) . 2
One problem with calculating the average height of the line based on the impedance matrix only is that an inaccurate impedance value can lead to a large error in the final result. To minimize the error, hi2 should be introduced to get the average height of the line. The value of average height hiav is replaced by the value of hi2, which is
Derived from (14) and (15), dij is expressed as
1
(24)
F/km. where ε0 is the permittivity of free space, which is 8.85 × As shown in Section 3.1, the height hi1 and the radius ri of conductor i can be obtained if the diagonal elements of the impedance matrix (selfimpedance) are known. If ri is given, the height can also be calculated by (24), expressed as
where ΔXmij is Carson’s correction terms for earth return effects. For mutual impedance, ϕij = arccos((hi + hj )/ dij,mir ) and the formula for computing am is shown in Appendix B. With as replaced as am, ΔXmij can be calculated by (8) as
⎛ Xmij − k1
1 2h ln i 2πε0 ri 10−9
(14)
h i + hj ⎞ 1 ΔXmij = k1 ⎛⎜ (b0 − ln(k2 dij,mir )) + b1 k2 dij,mir ⎟ dij,mir ⎠ 2 ⎝
(23)
According to [3], it is assumed that the air is lossless and the earth is uniformly at zero potential. In addition, the radius should be no more than a tenth of the distance between two conductors. Under these conditions, the elements of P can be easily determined with the equivalent geometrical parameters. Then the diagonal element Psi of the potential coefficient matrix P can be calculated as
1 Xsi − Xi, ac − k1 ( b0 + as b1)
dij =
(22)
⎡ PsA PmAB PmAC ⎤ C −1 = P = ⎢ PmBA PsB PmBC ⎥ ⎢ PmCA PmCB PsC ⎥ ⎦ ⎣
where k2 = 4π 5 × 10−4 × f / ρ . Eqs. (3) and (10) combined, the radius ri for conductor i can be expressed as
Xmij = ω
dAB,mir dBC,mir dCA,mir
The potential coefficient matrix P is the inverse matrix of the capacitance matrix C in per unit length, which is (12)
k2·exp(
3
(21)
3.2. Solution based on both impedance and capacitance matrices
as 2k2
ri =
dAB dBC dCA
(11)
If as is given, the average height of the conductor i is
hi =
(20)
where dAB, dBC and dCA are the distances between two conductors, while dAB,mir, dBC,mir and dCA,mir are the distances between a conductor and the image of another conductor. In this calculation, only the first two parts of Carson’s correction terms are used to replace the infinite series, which can bring errors to the equivalent geometrical parameters. In order to minimize the error caused by only using the impedance matrix and improve precision, the power-frequency capacitance matrix is introduced to the calculation in Section 3.2.
Since (2) and (9) hold simultaneously, as for conductor i can be expressed as
(
3
d mir =
(9)
1 ΔXsi = k1 ⎛ (b0 − ln as ) + b1 as ⎞. ⎝2 ⎠
as =
rA rB rC
After transposition, the geometric average distance d and the imaginary average of distance dmir can be expressed as
As Carson’s series converge rather fast at a low frequency [3], (7) and (8) are expected to give sufficiently accurate result. For self-impedance, the angle ϕ is zero and the formula for computing the parameter as is shown in Appendix B, so the Eqs. (7) and (8) are transformed as
ΔRsi = k1 (
3
Step 1: Get the line impedance matrix Z and the potential coefficient matrix P at power frequency. Step 2: Let n be a counter of rounds of iteration and n = 1 for the
(19) 154
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Fig. 3. Flowchart of calculating average distance between conductors i and j, and then the average distance between conductor i and the image of conductor j.
inter-conductor distance dijav between conductor i and the image of conductor j as
dijav,mir =
(29)
4. Numerical results
first iteration. Set the initial value of average height hiav[0] and radius riav[0] of the conductor as 0. Step 3: Use the self-impedance to calculate as[n] according to (11), which is the value of as in the nth round of iteration. And then obtain the average height using impedance matrix and (12), and record the value as hi1. Step 4: Calculate radius ri with mutual impedance and (13). Obtain the average height using potential coefficient matrix and (25), and record the value as hi2. Step 5: Get the height hiav[n]. When the known capacitance matrix is more accurate than the impedance matrix, hiav[n] equals to hi2. Then calculate the average conductor radius riav[n] through (27). Step 6: Check whether the value of hiav in the nth round of iteration is less than 0.01% different from that in the n-1th round and whether the same has been observed for riav simultaneously. If both are true, let hiav = hiav[n] and riav = riav[n], and then end the iteration. Otherwise, let n = n + 1 and as [n] = 2k2 hiav [n − 1], then return to Step 4 and repeat the iteration.
A frequency-dependent model of a 500-km three-phase overhead transmission line is built on the PSCAD/EMTDC. The tower configuration is that PA = [−10 m, 30 m], PB = [0 m, 35 m], PC = [10 m, 30 m] and the sag is 10 m. The DC resistance of the conductor is Rdc = 0.03206 Ω/km and the DC inductance is Lint,dc = 0.05 mH/km. The conductor radius is 0.0203454 m, and the power frequency is 50 Hz. It is assumed that there are no ground wires and the circuit is not ideally transposed. According to [27], the impedance matrix can be obtained through the direct numerical integration or the Deri-Semlyen formula in PSCAD/EMTDC. Both the impedance matrices based on the direct numerical integration and the Deri-Semlyen formula in the output file TLine.out after the simulation are shown in Appendix C. Because the direct numerical integration is more precise than the Deri-Semlyen formula [27], the impedance matrix based on the direct numerical integration is regarded as accurate and thus used as a benchmark to measure the effectiveness of the proposed method. The calculated equivalent geometrical parameters based on the solution discussed in Section 3.1 using the accurate impedances are compared with the parameters set in the model, as shown in Table 1. It can be seen from Table 1 that with only the accurate impedance matrix used, the error is at least 0.184%, shown by the geometric average of inter-conductor distances. And the error can also be as much as 2.077%, seen in the geometric average height of the conductors. The
The off-diagonal element Pmij of the potential coefficient matrix P is expressed as
dij,mir 1 ln 2πε0 dij
.
With the average heights hiav and hjav for conductors i and j respectively, the inter-conductor distance dijav between the two conductors is calculated by (16). And then the horizontal distance sijav between the two conductors can be computed by (17). Based on (29), the distance dijav,mir between conductor i and the image of conductor j can be obtained with the off- diagonal elements of P and the interconductor distance. Fig. 3 shows the calculation process. In practical application, the accurate impedances measured by PMU technology or the method in [25–26] can be used to compute the equivalent geometrical parameters of the line.
Fig. 2. Flowchart of calculating average height and radius of conductor i.
Pmij = Pmji =
dijav exp(2πε0·Pmij )
(28)
according to which, the distance dijav,mir can be calculated using the 155
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Table 1 Comparison between set parameters and the parameters calculated by accurate impedance matrix.
Table 4 Comparison between set parameters and the parameters calculated by approximate impedance and capacitance matrices.
Average of geometric parameters
Set parameters (m)
Equivalent geometric parameters (m)
Error (%)
Average of geometric parameters
Set parameters (m)
Equivalent geometric parameters (m)
Error (%)
h r d dmir
24.893 0.0203 13.572 52.000
25.410 0.0204 13.597 52.979
2.077 0.493 0.184 1.883
h r d dmir
24.893 0.0203 13.572 52.000
23.313 0.0191 12.674 48.560
6.347 5.911 6.617 6.615
operation in (13) and (16) of the method discussed in Section 3.1. The equivalent geometrical line parameters derived from the approximate impedance matrix and capacitance matrix with the formula introduced in Section 3.2 are compared with the set parameters, as shown in Table 4. From Table 4, when both the approximate impedance and capacitance matrices are used for derivation, the maximum error is in the inter-conductor distance and is only 6.617%. The accuracy is improved. The self-impedance and the mutual impedance at different frequencies calculated with equivalent geometrical parameters in Table 1 and Table 2 using Carson formula are compared with the impedances computed with set parameters using Carson formula. Since conductors A and C are set symmetric with respect to the y-axis, the self-impedance of conductor A, ZsA is equal to that of conductor C, ZsC. Similarly, the mutual impedance ZmAB equals ZmBC. The resistances and reactances change with the frequency of ZsA, ZsB, ZmAB and ZmCA in the semilogarithmic coordinate system, as shown in Figs. 4–7. From Fig. 4 to 7, it can be seen that the errors of the resistances and reactances derived from impedance matrix at power frequency are larger than those derived from impedance and capacitance matrices at power frequency. The errors of resistances and reactances at different frequencies based on the two solutions are shown in Figs. 8 and 9. From Fig. 8, it can be seen that the errors in the frequency-dependent impedances solved only by the impedance matrix are small. The largest errors at 1000 kHz are 2.79%, 3.14%, 0.34%, and 1.53% for selfresistance, mutual resistance, self-reactance and mutual reactance respectively. Meanwhile, it can be seen from Fig. 9 that the accuracy of the frequency-dependent impedances solved by both the impedance and capacitance matrices has improved a lot. The largest errors at
Table 2 Comparison between set parameters and the parameters calculated by accurate impedance and capacitance matrices. Average of geometric parameters
Set parameters (m)
Equivalent geometric parameters (m)
Error (%)
h r d dmir
24.893 0.0203 13.572 52.000
24.996 0.0204 13.587 52.059
0.414 0.493 0.111 0.113
errors can be accounted for by the Carson’s correction terms, since only the first two terms are used to simplify the derivation and calculation. With the formula introduced in Section 3.2, the equivalent geometrical parameters of the transmission line are also derived from the accurate impedance and capacitance matrices. The results are shown in comparison with the set parameters in Table 2. Seen from Table 2, when both the impedance and capacitance matrices are used for derivation, the errors of the equivalent geometrical parameters are all less than 0.5%. The maximum error is in the geometric mean of radiuses, but only 0.493%. The accuracy of the geometric average height h and the imaginary average distance dmir has significantly improved compared with the results of the impedancematrix-based-only calculation. To validate that the sensitivity of the methods introduced in Sections 3.1 and 3.2 to the error in the given impedances, the way adopted in this paper is to use approximate impedances to calculate the equivalent geometrical parameters using the method discussed in Sections 3.1 and 3.2 and compare the results with those in Table 1 and Table 2, correspondingly. It is noted that the approximate impedances are generated by the Deri- Semlyen formula. The errors in the approximate impedances compared with the accurate impedances are shown in Appendix C. When only the approximate impedance matrix is used to calculate the equivalent geometrical parameters, the calculated geometric parameters are compared with the parameters set in the model, as shown in Table 3. In Table 3, when only the approximate impedance matrix is used in calculation, the error in the geometric average height of the conductors reaches 15.759%. It can be seen that minor errors in self- and mutual impedances at power frequency will be enlarged through exponential
Table 3 Comparison between set parameters and the parameters calculated by approximate impedance matrix. Average of geometric parameters
Set parameters (m)
Equivalent geometric parameters (m)
Error (%)
h r d dmir
24.893 0.0203 13.572 52.000
20.970 0.0190 12.618 44.172
15.759 6.404 7.029 15.054
Fig. 4. Self-impedance of conductor A. (a) Resistance. (b) Reactance. 156
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Fig. 7. Mutual impedance between conductor C and conductor A. (a) Resistance. (b) Reactance.
Fig. 5. Self-impedance of conductor B. (a) Resistance. (b) Reactance.
Fig. 8. Errors of resistances and reactances solved by impedance matrix.
Fig. 6. Mutual impedance between conductor A and conductor B. (a) Resistance. (b) Reactance.
1000 kHz are 0.38%, 0.35%, 0.023% and 0.30% for self-resistance, mutual resistance, self-reactance and mutual reactance respectively.
Fig. 9. Errors of resistances and reactances solved by impedance and capacitance matrices.
5. Conclusion given, the equivalent geometrical parameters can be calculated. However, the precision of the results will be reduced, if the measured impedance matrix is not accurate enough. (2) When the power-frequency impedance and capacitance matrices are both given, the precision of the calculated equivalent geometrical parameters has been improved. With the two matrices, the errors of the results caused by only using the impedance matrix can be minimized. (3) For both the impedance and capacitance matrices, all elements are independent from one another, which means the equivalent geometrical parameters of the three phases are calculated separately. A
The equivalent geometrical parameters of the line are the basic inputs for calculating the frequency-dependent parameters in the research of the power system transient process. Based on the Carson formula, the paper proposed a method to obtain the equivalent geometrical parameters of the line using the power-frequency parameters. Combined with the measurement of accurate power-frequency parameters, the proposed method can be used to calculate the frequency-dependent parameters at the range of 0 Hz-1000 kHz with Carson formula. The highlights of the method are: (1) When only the power-frequency impedance matrix of the line is 157
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recalculated according to the frequency.
possible large measurement error in the self-impedance, mutual impedance or capacitance of one conductor has no bearing on others, making the calculation more error-resistant to some extent. (4) The impedance matrix at other frequencies (less than 104 Hz) can also be used to obtain the equivalent geometrical parameters by the method proposed, only the Ri,ac and Xi,ac of the equations need to be
It is noted that the method is only proposed for the AC overhead transmission line and thus cannot be applicable to cable and HVDC projects. Further study is needed for calculating the equivalent geometrical parameters for cable and HVDC projects.
Appendix A The geometrical parameters of transmission line in Fig. 1 can be obtained as follows:
hi = |yi | − SAG + dij =
1 2 SAG = |yi | − SAG 3 3
(A.1)
(x i − x j )2 + (hi − hj )2
dij, mir =
(A.2)
(x i − x j )2 + (hi + hj )2
(A.3)
where SAG is the sag for all three conductors. Appendix B The parameter as and am are calculated by
as = 4π 5 × 10−4 × 2hi ×
f /ρ
am = 4π 5 × 10−4 × dij,mir ×
(A.4)
f / ρ = k2 dij,mir
(A.5)
where ρ is the resistivity of the earth with a default of 100 Ωm. According to (A.4), it is known that the value of as is proportional to hi at a given f. When hi is 755 m and the frequency is the power-frequency (50 Hz), as is equal to 3.0. Appendix C The impedance matrix Zaccura based on the direct numerical integration is given as
⎡ 0.0810 + j0.692 0.0466 + j0.281 0.0468 + j0.244 ⎤ Zaccura = ⎢ 0.0466 + j0.281 0.0805 + j0.693 0.0466 + j0.281⎥ ⎢ ⎥ ⎣ 0.0468 + j0.244 0.0466 + j0.281 0.0810 + j0.692 ⎦
(A.6)
and the impedance matrix Zapprox based on the Deri-Semlyen formula is that
⎡ 0.0815 + j0.696 0.0472 + j0.285 0.0473 + j0.248⎤ Zapprox = ⎢ 0.0472 + j0.285 0.0811 + j0.697 0.0472 + j0.285⎥ ⎢ ⎥ ⎣ 0.0473 + j0.248 0.0472 + j0.285 0.0815 + j0.696 ⎦
(A.7)
Compared with Zaccura, the maximum errors of Zapprox at 50 Hz are 0.7143%, 1.1608%, 0.6102% and 1.7254% for self-resistance, mutual resistance, self-reactance and mutual reactance respectively.
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References [1] Hedman DE. Propagation on overhead transmission lines II: earth conduction effects and practical results. IEEE Trans Power App Syst 1965;PAS-84(3):205–11. [2] Budner Alan. Introduction of frequency-dependent line parameters into an electromagnetic transients program. IEEE Trans Power App Syst 1970;89(1):88–95. [3] Dommel HW. EMTP theory book. Portland, OR, USA: BPA; 1986. p. 41–99. [4] Ney MM. Striction and skin effects on the internal impedance value of flat conductors. IEEE Trans Electromagn Compat 1991;33(4):321–7. [5] Carson JR. Wave propagation in overhead wires with ground return. Bell Syst Tech J 1926;5(4):539–54. [6] Zhirui Liang, et al. Current status and development trend of ac transmission line parameter measurement. Automat Electric Power Syst 2017;41(11):181–91. [7] Deri A, et al. The complex ground return plane a simplified model for homogeneous and multi-layer earth return. IEEE Trans Power App Syst 1981;PAS100(8):3686–93. [8] Semlyen A, Deri A. Time domain modeling of frequency dependent three-phase transmission line impedance. IEEE Trans Power App Syst 1985;PAS104(6):1549–55. [9] Ramos-Leaños Octavio, et al. Accurate and approximate evaluation of power-line earth impedances through the Carson integral. IEEE Trans Electromagn Compat 2017;59(5):1465–73. [10] Brandão Faria JA. Dubanton’s and Wait’s approximations of the earth return impedance used for evaluating transmission-line wave propagation parameters. IEEE Trans Power Del 2015;30(6):2600–1.
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Wiley; 1922. p. 1431–3. [25] Yang X, et al. Method for accurately measuring the power-frequency parameters of EHV/UHV transmission lines. IET Gener Transm Distrib 2017;11(7):1726–34. [26] Das Swagata, et al. Estimating zero-sequence impedance of three- terminal transmission line and Thevenin impedance using relay measurement data. Prot Contr Mod Power Syst 2018;3(3):373–82. [27] EMTDC transient analysis for PSCAD power system simulation. Manitoba HVDC Research Center, 2011. [Online]. Available: https://hvdc.ca/uploads/ck/files/ reference_material/EMTDC_User_Guide_v4_3_1.pdf [accessed on: September 19, 2017].
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Contents lists available at ScienceDirect
Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
A calculation method of transmission line equivalent geometrical parameters based on power-frequency parameters☆
T
⁎
Botong Li , Mingrui Lv Key Laboratory of Smart Grid (Tianjin University), Ministry of Education, School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Equivalent geometrical parameter Power- frequency parameter Carson formula Frequency-dependent parameter
The geometrical parameters of the transmission line are prerequisite for precisely calculating frequency-dependent parameters as well as studying the electromagnetic transient process. Considering that the equivalent geometrical parameters are difficult to obtain directly, this paper proposes a method to calculate geometrical parameters of overhead transmission line using impedance and capacitance matrices measured at power frequency. By applying the inverse derivation of the Carson formula, the expressions of the average height of the line and the conductor radius can be obtained if the self-impedance of the line is known, while the distance between two conductors and that between one conductor and the image of another can be found with the mutual impedance. On this basis, if both the impedance and capacitance of the transmission line at power frequency are given, iteration can be used for higher accuracy. Finally, MATLAB- and PSCAD/EMTDC-based simulations are carried out verifying the effectiveness and sensitivity of the method.
1. Introduction Transmission line parameters include resistance, inductance and capacitance. The impedance of the line is specifically composed of the external impedance, the earth-return impedance due to the soil effect, and the internal impedance due to the skin effect, which are all the functions of frequency [1–5]. It is essential to take the frequency-dependent characteristics of the impedance into consideration in the research of the electromagnetic transient process of the power system. One basic calculation method of frequency-dependent impedance is the Carson formula, requiring equivalent geometrical parameters concerning tower configuration and conductor characteristics [5]. However, practical difficulty lies in identifying the equivalent geometrical parameters due to the ever-changing terrain, environment and line transposition [6]. Usually, power-frequency line parameters are first measured and can be used as input data for calculating geometrical parameters. Therefore, for a better transient analysis of power system, it is important to study a method for solving the equivalent geometrical parameters with power-frequency line parameters. Progress has been made in the study of the relationship between geometrical and frequency-dependent parameters of the transmission line. A number of formulas have been employed to calculate frequencydependent line parameters in the frequency domain. The Carson
formula to compute the frequency-dependent line parameters was introduced in [5]. Deri-Semlyen formula as a simplified formula of Carson’s infinite integral term was described in [7,8]. A method for calculating earth impedance based on the Carson integral was presented in [9]. In [10], series impedance and inductance of transmission line were obtained via the approximation of Dubanton and Wait. In [11], power series and asymptotic series expansions for calculating self and mutual line impedances were derived. To approximate the frequency-dependent parameters in the time domain, some fitting methods have been proposed. In [12], the characteristic impedance and the propagation function were simulated by a series of resistance-capacitance parallel blocks, which were approximated by rational functions. An optimization fitting procedure used to obtain low-order approximation of rational functions for the frequencydependent parameters was shown in [13]. Ref. [14] approximated the frequency-dependent admittances with the vector fitting by optimal scaling using complex starting poles. In [15], the frequency-dependent line parameters were obtained by means of a convolution procedure based on the synthesized rational function. In [16], the frequency-dependent impedance of the Bergeron model was computed directly in the time domain by means of a single rational function and lumped elements. For the multi-conductor transmission lines, the fitting methods are
☆ ⁎
This research work was supported in part by the National Natural Science Foundation of China (NSFC) under Grant 51677125. Corresponding author. E-mail addresses: [email protected] (B. Li), [email protected] (M. Lv).
https://doi.org/10.1016/j.ijepes.2019.04.013 Received 2 December 2018; Received in revised form 28 February 2019; Accepted 9 April 2019 0142-0615/ © 2019 Elsevier Ltd. All rights reserved.
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and ri is the equivalent radius of conductor i. The calculation formulas of hi, dij and dij,mir are described in Appendix A [3]. The radius changes with the number of conductors and the inter-conductor spacing in the case of the conductor bundle, whose specific calculation method is given in [23]. With these geometrical parameters, the Carson formula can be used to calculate frequency-dependent parameters precisely.
usually used in combination with modal decoupling to solve the frequency-dependent parameters in the modal domain. In [17], the impedance and admittance matrices of the transmission line were decoupled with the frequency-dependent transformation matrices evaluated by the Newton-Raphson method. A sequential quadratic programming method for calculating the transformation matrices was described in [18]. Some constant and real modal transformation matrices are also used to decouple the three-phase line into three independent propagation modes. In [19,20], the frequency-dependent parameters of the equivalent Bergeron circuit were solved using an accurate real modal decoupling and the vector fitting method. Ref. [21] built a new frequency-dependent multi-conductor transmission line model with a single real constant transformation matrix. In the literature above, equivalent geometrical parameters, such as the average above-ground height of the transmission lines and the interconductor distances, are a must for calculating the frequency-dependent parameters. However, it is not easy to get the equivalent geometrical parameters due to the complex transmission path and changeable conditions along the line. Therefore, it remains a key problem to determine the equivalent geometrical parameters. Ref. [22] solved the equivalent geometrical parameters by referring to zero- and positivesequence inductances and capacitances evaluated at the power frequency. However, the results are susceptible to errors in line parameter evaluation because of exponential operation. This paper proposes a Carson-formula-based method of calculating the equivalent geometrical parameters of overhead transmission line, using impedance and capacitance matrices measured at power frequency. Unlike [22] where a slight error in the line parameter evaluation can lead to a rather inaccurate result, this paper studies how to improve the calculation accuracy with the impedance and capacitance matrices both known at the power frequency.
3. Geometrical parameters solutions 3.1. Solution based on impedance matrix The impedance matrix Z in per unit length of the transmission line at power frequency is
RmAB + jXmAB RmAC + jXmCA ⎤ ⎡ RsA + jXsA RsB + jXsB RmBC + jXmBC ⎥ Z = ⎢ RmBA + jXmBA ⎢R + + jX R jX RsC + jXsC ⎥ mCA mCA mCB mCB ⎣ ⎦
(1)
where Rsi is the self-resistance and Xsi is the self-reactance of conductor i, while Rmij and Xmij are the mutual resistance and reactance respectively. According to the Carson formula, the self-resistance Rsi and the selfreactance Xsi of conductor i are obtained as
Rsi = Ri, ac + ΔRsi Xsi = ω
μ0 2h ln i + Xi, ac + ΔXsi 2π ri
(2)
(3)
where Ri,ac is the AC resistance of conductor i and Xi,ac the internal reactance; ΔRsi and ΔXsi are Carson’s correction terms for earth return effects; μ0 = 4π × 10 - 4 H/km is the uniform permeability for both the aerial space and the earth; andω is the angular frequency proportional to system frequency f (ω = 2πf ). The DC resistance of the line is marked as Ri,dc, yet the line resistance will increase with frequency due to the skin effect in the AC case. So the AC resistance Ri,ac is
2. Equivalent geometrical line parameters Carson formula is one basic method for calculating the frequencydependent impedances of the transmission line, but it requires geometrical parameters concerning the tower configuration and conductor radius as the input constants. The geometrical parameters defined in the plane rectangular coordinate system are shown in Fig. 1. The x axis is given as the same height of the ground. The positions of the three phases are PA = [xA, yA], PB = [xB, yB] and PC = [xC, yC], where xi is the abscissa of conductor i (i = A, B, C) and yi is the relative height of the conductor i to the ground. With the coordinates of all the conductors, it is easy to know the geometrical distance between any two conductors and the distance between the conductor and the ground. In Fig. 1, hi is the average above-ground height of conductor i; dij is the distance between conductors i and j (j = A, B, C and j ≠ i); dij,mir is the distance between conductor i and the image of conductor j;
Ri,ac = kR·Ri,dc
(4)
where kR is the ratio of the AC resistance to the DC resistance. Also as a result of the skin effect, the inductance decreases as frequency increases. The AC internal reactance Xi,ac can be negligible in practical application, but to improve the accuracy in the derivation of the radius, Xi,ac is taken into consideration and given as
Xi,ac = kL·Xi,dc
(5)
where kL is the ratio of AC internal reactance to the DC internal reactance. Calculation methods for both kR and kL are detailed in [24]. In [3], the DC internal reactance Xi,dc is calculated by the following formula
Xi,dc = ω
μ0 2πf μ0 1 1 ln 1 = = μ0 f . 2π 4 2π 4 e− 4
(6)
Carson’s correction terms ΔRsi and ΔXsi in (2) are the functions of both the angle ϕ and parameter as. According to Appendix B, when the average height of the conductor is less than 755 m, parameter as at the power frequency is less than 3. Therefore, according to the Carson formula, the first two parts of formula of Carson’s correction terms are
ΔR = k1 (
π − b1 as cos ϕ ) 8
1 ΔX = k1 ( (b0 − ln as ) + b1 as cos ϕ ) 2 Fig. 1. Equivalent geometrical parameters of transmission line.
where 153
(7) (8)
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B. Li and M. Lv
r=
k1 = 4ω × 10−4 , b0 = 0.6159315, and b1 = 2 /6.
π − b1 as ) 8
d=
π 8
−
R si − Ri, ac k1
b1
(10)
).
1 2
μ0 f
)
(13)
10−4
= 2μ0 f . where k1 = 4ω × Based on the Carson formula, the distance between conductors i and j and that between conductor i and the image of conductor j can be deduced from the mutual reactance Xmij according to dij,mir μ0 ln + ΔXmij 2π dij
Psi =
hi2 =
k2·exp
( 12 b0 + k2 b1 (hi + hj) ) ⎞
⎜ ⎝
μ0 f
⎟ ⎠
(15)
.
hiav = hi2
dij2 − (hi − hj )2
dij,mir =
sij2 + (hi + hj )2 .
riav =
h=
hA hB hC
(26)
2hiav exp(2πε0·Psi )
(27)
According to Section 3.1, equivalent geometrical parameters can be calculated using only the power-frequency impedance matrix. But in the simulation, the calculation is not accurate enough and the results are susceptible to the error of the impedance matrix. To achieve higher accuracy for hiav and riav, the impedance and capacitance matrices are both used in calculation in this section, providing more parameters for later iteration to reduce the effect of the errors in power-frequency parameters. The calculation steps of iteration are shown as follows. The Fig. 2 shows the flowchart of the calculation process.
(17) (18)
The heights hA, hB and hC as well as the radiuses rA, rB and rC for the three phases respectively can be obtained. After transposition, according to [3], the geometric average height and radius are respectively 3
(25)
Then the geometrical average of the radius riav is
(16)
Therefore, if the self-resistance is known, the average height of each conductor can be calculated by (11) and (12). Then with the mutual impedance already known, any inter-conductor distance can be obtained via (16). As such, the horizontal distance sij between two conductors and the distance dij,mir between conductor i and the image of conductor j are
sij =
ri·exp(2πε0·Psi ) . 2
One problem with calculating the average height of the line based on the impedance matrix only is that an inaccurate impedance value can lead to a large error in the final result. To minimize the error, hi2 should be introduced to get the average height of the line. The value of average height hiav is replaced by the value of hi2, which is
Derived from (14) and (15), dij is expressed as
1
(24)
F/km. where ε0 is the permittivity of free space, which is 8.85 × As shown in Section 3.1, the height hi1 and the radius ri of conductor i can be obtained if the diagonal elements of the impedance matrix (selfimpedance) are known. If ri is given, the height can also be calculated by (24), expressed as
where ΔXmij is Carson’s correction terms for earth return effects. For mutual impedance, ϕij = arccos((hi + hj )/ dij,mir ) and the formula for computing am is shown in Appendix B. With as replaced as am, ΔXmij can be calculated by (8) as
⎛ Xmij − k1
1 2h ln i 2πε0 ri 10−9
(14)
h i + hj ⎞ 1 ΔXmij = k1 ⎛⎜ (b0 − ln(k2 dij,mir )) + b1 k2 dij,mir ⎟ dij,mir ⎠ 2 ⎝
(23)
According to [3], it is assumed that the air is lossless and the earth is uniformly at zero potential. In addition, the radius should be no more than a tenth of the distance between two conductors. Under these conditions, the elements of P can be easily determined with the equivalent geometrical parameters. Then the diagonal element Psi of the potential coefficient matrix P can be calculated as
1 Xsi − Xi, ac − k1 ( b0 + as b1)
dij =
(22)
⎡ PsA PmAB PmAC ⎤ C −1 = P = ⎢ PmBA PsB PmBC ⎥ ⎢ PmCA PmCB PsC ⎥ ⎦ ⎣
where k2 = 4π 5 × 10−4 × f / ρ . Eqs. (3) and (10) combined, the radius ri for conductor i can be expressed as
Xmij = ω
dAB,mir dBC,mir dCA,mir
The potential coefficient matrix P is the inverse matrix of the capacitance matrix C in per unit length, which is (12)
k2·exp(
3
(21)
3.2. Solution based on both impedance and capacitance matrices
as 2k2
ri =
dAB dBC dCA
(11)
If as is given, the average height of the conductor i is
hi =
(20)
where dAB, dBC and dCA are the distances between two conductors, while dAB,mir, dBC,mir and dCA,mir are the distances between a conductor and the image of another conductor. In this calculation, only the first two parts of Carson’s correction terms are used to replace the infinite series, which can bring errors to the equivalent geometrical parameters. In order to minimize the error caused by only using the impedance matrix and improve precision, the power-frequency capacitance matrix is introduced to the calculation in Section 3.2.
Since (2) and (9) hold simultaneously, as for conductor i can be expressed as
(
3
d mir =
(9)
1 ΔXsi = k1 ⎛ (b0 − ln as ) + b1 as ⎞. ⎝2 ⎠
as =
rA rB rC
After transposition, the geometric average distance d and the imaginary average of distance dmir can be expressed as
As Carson’s series converge rather fast at a low frequency [3], (7) and (8) are expected to give sufficiently accurate result. For self-impedance, the angle ϕ is zero and the formula for computing the parameter as is shown in Appendix B, so the Eqs. (7) and (8) are transformed as
ΔRsi = k1 (
3
Step 1: Get the line impedance matrix Z and the potential coefficient matrix P at power frequency. Step 2: Let n be a counter of rounds of iteration and n = 1 for the
(19) 154
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Fig. 3. Flowchart of calculating average distance between conductors i and j, and then the average distance between conductor i and the image of conductor j.
inter-conductor distance dijav between conductor i and the image of conductor j as
dijav,mir =
(29)
4. Numerical results
first iteration. Set the initial value of average height hiav[0] and radius riav[0] of the conductor as 0. Step 3: Use the self-impedance to calculate as[n] according to (11), which is the value of as in the nth round of iteration. And then obtain the average height using impedance matrix and (12), and record the value as hi1. Step 4: Calculate radius ri with mutual impedance and (13). Obtain the average height using potential coefficient matrix and (25), and record the value as hi2. Step 5: Get the height hiav[n]. When the known capacitance matrix is more accurate than the impedance matrix, hiav[n] equals to hi2. Then calculate the average conductor radius riav[n] through (27). Step 6: Check whether the value of hiav in the nth round of iteration is less than 0.01% different from that in the n-1th round and whether the same has been observed for riav simultaneously. If both are true, let hiav = hiav[n] and riav = riav[n], and then end the iteration. Otherwise, let n = n + 1 and as [n] = 2k2 hiav [n − 1], then return to Step 4 and repeat the iteration.
A frequency-dependent model of a 500-km three-phase overhead transmission line is built on the PSCAD/EMTDC. The tower configuration is that PA = [−10 m, 30 m], PB = [0 m, 35 m], PC = [10 m, 30 m] and the sag is 10 m. The DC resistance of the conductor is Rdc = 0.03206 Ω/km and the DC inductance is Lint,dc = 0.05 mH/km. The conductor radius is 0.0203454 m, and the power frequency is 50 Hz. It is assumed that there are no ground wires and the circuit is not ideally transposed. According to [27], the impedance matrix can be obtained through the direct numerical integration or the Deri-Semlyen formula in PSCAD/EMTDC. Both the impedance matrices based on the direct numerical integration and the Deri-Semlyen formula in the output file TLine.out after the simulation are shown in Appendix C. Because the direct numerical integration is more precise than the Deri-Semlyen formula [27], the impedance matrix based on the direct numerical integration is regarded as accurate and thus used as a benchmark to measure the effectiveness of the proposed method. The calculated equivalent geometrical parameters based on the solution discussed in Section 3.1 using the accurate impedances are compared with the parameters set in the model, as shown in Table 1. It can be seen from Table 1 that with only the accurate impedance matrix used, the error is at least 0.184%, shown by the geometric average of inter-conductor distances. And the error can also be as much as 2.077%, seen in the geometric average height of the conductors. The
The off-diagonal element Pmij of the potential coefficient matrix P is expressed as
dij,mir 1 ln 2πε0 dij
.
With the average heights hiav and hjav for conductors i and j respectively, the inter-conductor distance dijav between the two conductors is calculated by (16). And then the horizontal distance sijav between the two conductors can be computed by (17). Based on (29), the distance dijav,mir between conductor i and the image of conductor j can be obtained with the off- diagonal elements of P and the interconductor distance. Fig. 3 shows the calculation process. In practical application, the accurate impedances measured by PMU technology or the method in [25–26] can be used to compute the equivalent geometrical parameters of the line.
Fig. 2. Flowchart of calculating average height and radius of conductor i.
Pmij = Pmji =
dijav exp(2πε0·Pmij )
(28)
according to which, the distance dijav,mir can be calculated using the 155
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Table 1 Comparison between set parameters and the parameters calculated by accurate impedance matrix.
Table 4 Comparison between set parameters and the parameters calculated by approximate impedance and capacitance matrices.
Average of geometric parameters
Set parameters (m)
Equivalent geometric parameters (m)
Error (%)
Average of geometric parameters
Set parameters (m)
Equivalent geometric parameters (m)
Error (%)
h r d dmir
24.893 0.0203 13.572 52.000
25.410 0.0204 13.597 52.979
2.077 0.493 0.184 1.883
h r d dmir
24.893 0.0203 13.572 52.000
23.313 0.0191 12.674 48.560
6.347 5.911 6.617 6.615
operation in (13) and (16) of the method discussed in Section 3.1. The equivalent geometrical line parameters derived from the approximate impedance matrix and capacitance matrix with the formula introduced in Section 3.2 are compared with the set parameters, as shown in Table 4. From Table 4, when both the approximate impedance and capacitance matrices are used for derivation, the maximum error is in the inter-conductor distance and is only 6.617%. The accuracy is improved. The self-impedance and the mutual impedance at different frequencies calculated with equivalent geometrical parameters in Table 1 and Table 2 using Carson formula are compared with the impedances computed with set parameters using Carson formula. Since conductors A and C are set symmetric with respect to the y-axis, the self-impedance of conductor A, ZsA is equal to that of conductor C, ZsC. Similarly, the mutual impedance ZmAB equals ZmBC. The resistances and reactances change with the frequency of ZsA, ZsB, ZmAB and ZmCA in the semilogarithmic coordinate system, as shown in Figs. 4–7. From Fig. 4 to 7, it can be seen that the errors of the resistances and reactances derived from impedance matrix at power frequency are larger than those derived from impedance and capacitance matrices at power frequency. The errors of resistances and reactances at different frequencies based on the two solutions are shown in Figs. 8 and 9. From Fig. 8, it can be seen that the errors in the frequency-dependent impedances solved only by the impedance matrix are small. The largest errors at 1000 kHz are 2.79%, 3.14%, 0.34%, and 1.53% for selfresistance, mutual resistance, self-reactance and mutual reactance respectively. Meanwhile, it can be seen from Fig. 9 that the accuracy of the frequency-dependent impedances solved by both the impedance and capacitance matrices has improved a lot. The largest errors at
Table 2 Comparison between set parameters and the parameters calculated by accurate impedance and capacitance matrices. Average of geometric parameters
Set parameters (m)
Equivalent geometric parameters (m)
Error (%)
h r d dmir
24.893 0.0203 13.572 52.000
24.996 0.0204 13.587 52.059
0.414 0.493 0.111 0.113
errors can be accounted for by the Carson’s correction terms, since only the first two terms are used to simplify the derivation and calculation. With the formula introduced in Section 3.2, the equivalent geometrical parameters of the transmission line are also derived from the accurate impedance and capacitance matrices. The results are shown in comparison with the set parameters in Table 2. Seen from Table 2, when both the impedance and capacitance matrices are used for derivation, the errors of the equivalent geometrical parameters are all less than 0.5%. The maximum error is in the geometric mean of radiuses, but only 0.493%. The accuracy of the geometric average height h and the imaginary average distance dmir has significantly improved compared with the results of the impedancematrix-based-only calculation. To validate that the sensitivity of the methods introduced in Sections 3.1 and 3.2 to the error in the given impedances, the way adopted in this paper is to use approximate impedances to calculate the equivalent geometrical parameters using the method discussed in Sections 3.1 and 3.2 and compare the results with those in Table 1 and Table 2, correspondingly. It is noted that the approximate impedances are generated by the Deri- Semlyen formula. The errors in the approximate impedances compared with the accurate impedances are shown in Appendix C. When only the approximate impedance matrix is used to calculate the equivalent geometrical parameters, the calculated geometric parameters are compared with the parameters set in the model, as shown in Table 3. In Table 3, when only the approximate impedance matrix is used in calculation, the error in the geometric average height of the conductors reaches 15.759%. It can be seen that minor errors in self- and mutual impedances at power frequency will be enlarged through exponential
Table 3 Comparison between set parameters and the parameters calculated by approximate impedance matrix. Average of geometric parameters
Set parameters (m)
Equivalent geometric parameters (m)
Error (%)
h r d dmir
24.893 0.0203 13.572 52.000
20.970 0.0190 12.618 44.172
15.759 6.404 7.029 15.054
Fig. 4. Self-impedance of conductor A. (a) Resistance. (b) Reactance. 156
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Fig. 7. Mutual impedance between conductor C and conductor A. (a) Resistance. (b) Reactance.
Fig. 5. Self-impedance of conductor B. (a) Resistance. (b) Reactance.
Fig. 8. Errors of resistances and reactances solved by impedance matrix.
Fig. 6. Mutual impedance between conductor A and conductor B. (a) Resistance. (b) Reactance.
1000 kHz are 0.38%, 0.35%, 0.023% and 0.30% for self-resistance, mutual resistance, self-reactance and mutual reactance respectively.
Fig. 9. Errors of resistances and reactances solved by impedance and capacitance matrices.
5. Conclusion given, the equivalent geometrical parameters can be calculated. However, the precision of the results will be reduced, if the measured impedance matrix is not accurate enough. (2) When the power-frequency impedance and capacitance matrices are both given, the precision of the calculated equivalent geometrical parameters has been improved. With the two matrices, the errors of the results caused by only using the impedance matrix can be minimized. (3) For both the impedance and capacitance matrices, all elements are independent from one another, which means the equivalent geometrical parameters of the three phases are calculated separately. A
The equivalent geometrical parameters of the line are the basic inputs for calculating the frequency-dependent parameters in the research of the power system transient process. Based on the Carson formula, the paper proposed a method to obtain the equivalent geometrical parameters of the line using the power-frequency parameters. Combined with the measurement of accurate power-frequency parameters, the proposed method can be used to calculate the frequency-dependent parameters at the range of 0 Hz-1000 kHz with Carson formula. The highlights of the method are: (1) When only the power-frequency impedance matrix of the line is 157
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recalculated according to the frequency.
possible large measurement error in the self-impedance, mutual impedance or capacitance of one conductor has no bearing on others, making the calculation more error-resistant to some extent. (4) The impedance matrix at other frequencies (less than 104 Hz) can also be used to obtain the equivalent geometrical parameters by the method proposed, only the Ri,ac and Xi,ac of the equations need to be
It is noted that the method is only proposed for the AC overhead transmission line and thus cannot be applicable to cable and HVDC projects. Further study is needed for calculating the equivalent geometrical parameters for cable and HVDC projects.
Appendix A The geometrical parameters of transmission line in Fig. 1 can be obtained as follows:
hi = |yi | − SAG + dij =
1 2 SAG = |yi | − SAG 3 3
(A.1)
(x i − x j )2 + (hi − hj )2
dij, mir =
(A.2)
(x i − x j )2 + (hi + hj )2
(A.3)
where SAG is the sag for all three conductors. Appendix B The parameter as and am are calculated by
as = 4π 5 × 10−4 × 2hi ×
f /ρ
am = 4π 5 × 10−4 × dij,mir ×
(A.4)
f / ρ = k2 dij,mir
(A.5)
where ρ is the resistivity of the earth with a default of 100 Ωm. According to (A.4), it is known that the value of as is proportional to hi at a given f. When hi is 755 m and the frequency is the power-frequency (50 Hz), as is equal to 3.0. Appendix C The impedance matrix Zaccura based on the direct numerical integration is given as
⎡ 0.0810 + j0.692 0.0466 + j0.281 0.0468 + j0.244 ⎤ Zaccura = ⎢ 0.0466 + j0.281 0.0805 + j0.693 0.0466 + j0.281⎥ ⎢ ⎥ ⎣ 0.0468 + j0.244 0.0466 + j0.281 0.0810 + j0.692 ⎦
(A.6)
and the impedance matrix Zapprox based on the Deri-Semlyen formula is that
⎡ 0.0815 + j0.696 0.0472 + j0.285 0.0473 + j0.248⎤ Zapprox = ⎢ 0.0472 + j0.285 0.0811 + j0.697 0.0472 + j0.285⎥ ⎢ ⎥ ⎣ 0.0473 + j0.248 0.0472 + j0.285 0.0815 + j0.696 ⎦
(A.7)
Compared with Zaccura, the maximum errors of Zapprox at 50 Hz are 0.7143%, 1.1608%, 0.6102% and 1.7254% for self-resistance, mutual resistance, self-reactance and mutual reactance respectively.
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