power_lineparam

Compute RLC parameters of overhead transmission line from its conductor characteristics and tower geometry

Syntax

DATA = power_lineparam('new')
DATA = power_lineparam(DATA)
DATA = power_lineparam('MYLINEDATA')
power_lineparam(DATA,'BLK')
power_lineparam('MYLINEDATA','BLK')


Description

DATA = power_lineparam('new') returns a structure variable with the default line geometry parameters. You can use DATA as a template variable to configure a new line geometry and to compute RLC line parameters.

DATA is a structure with the fields in this table.

Field	Description
comments	Specify user comments related to the line parameters.
units	The units of the line parameters. Set to metric to specify the conductor diameter, GMR, and bundle diameter in centimeters and the conductor position in meters. Set to English to specify the conductor diameter, GMR, and bundle diameter in inches and the conductor position in feet. The default is metric.
frequency	The frequency, in hertz, to evaluate the RLC line parameters. The default is 60.
groundResistivity	The ground resistivity, in ohm-meters. Set to 0 to specify a perfectly conducting ground.
Geometry	Structure specifying the line geometry and conductor types.
Conductors	Structure specifying the conductor characteristics for each type.
evaluatedFrom	Specify how the internal conductor inductance is evaluated. Set to 'T/D ratio' to compute the inductance from DATA.Conductors.ThickRatio field. When this option is selected, the DATA.Conductors.GMR and DATA.Conductors.Xa fields are not considered by the function, you leave the values as-is or to empty values. Set to 'GMR' to compute the inductance from DATA.Conductors.GMR field. When this option is selected, the DATA.Conductors.Xa field is not considered by the function, you leave the Xa value as-is or to empty value. Set to 'Xa' to compute the inductance from DATA.Conductors.Xa field. When this option is selected, the DATA.Conductors.GMR field is not considered by the function, you leave the GMR value as-is or to empty value.

Geometry is a structure with the fields in this table.

Field	Description
NPhaseBundle	The number of phase conductors (single conductors or bundles of subconductors). The default is 3.
NGroundBundle	The number of ground wires (single conductors or bundles of subconductors). The default is 2.
PhaseNumber	1-by-NPhaseBundle + NGroundBundle vector. This setting represents the phase number to which the conductors belong. Several conductors may have the same phase number. The phase number of ground is 0. The default is [1 2 3 0 0].
X	1-by-NPhaseBundle + NGroundBundle vector. This setting represents the horizontal position of the conductors, in meters or feet. The location of the zero reference position is arbitrary. For a symmetrical line, you typically choose X = 0 at the center of the line. The default is [-12 0 12 -8 8].
Ytower	1-by-NPhaseBundle + NGroundBundle vector. This setting represents the vertical position of the conductors (at the tower) with respect to ground, in meters or feet. The default is [20 20 20 33 33].
Ymin	1-by-NPhaseBundle + NGroundBundle vector. This setting represents the vertical position of the conductors with respect to ground at mid-span, in meters or feet. The default is [20 20 20 33 33].
ConductorType	1-by-NPhaseBundle + NGroundBundle vector. This setting represents the type of conductor. The numbers correspond to the elements of the vector defined in the Conductors structure. The default is [1 1 1 2 2].

Conductors is a structure with the fields in this table. The length of the Conductors structure corresponds to the number of conductor types in your line parameters:

Field	Description
Diameter	The conductor outside diameter, in cm. The default is [3.5500 1.2700].
ThickRatio	The T/D ratio of the hollow conductor. T is the thickness of the conducting material, and D is the outside diameter. This parameter can vary between 0 and 0.5. A T/D value of 0.5 indicates a solid conductor. For Aluminum Cable Steel Reinforced (ACSR) conductors, you can ignore the steel core and consider a hollow aluminum conductor (typical T/D ratios are between 0.3 and 0.4).The default is [0.3700 0.5000].
GMR	The Geometric Mean Radius.
Xa	The Reactance Xa, in ohms/km. The default is 0.
Res	The conductor DC resistance, in ohms/km. The default is [0.0430 3.1060].
Mur	The conductor relative permeability. The default is [1 1].
NConductors	The number of conductors per bundle. The default is [4 1].
BundleDiameter	The bundle diameter, in cm or inches. The default is [65 0].
AngleConductor1	The angle, in degrees, that determines the position of the first conductor in the bundle with respect to a horizontal line parallel to ground. This angle determines the bundle orientation. The default is [45 0].
skinEffect	Set to yes to include the impact of the frequency on conductor AC resistance and inductance (skin effect). If set to no, the resistance is kept constant at the value specified by the Res field, and the inductance is kept constant at the value computed in DC, using the Diameter field, and the ThickRatio field.

DATA = power_lineparam(DATA) computes the RLC line parameters for the geometric line parameters in the DATA structure. The returned structure contains both geometric data and computed RLC line parameters.

The DATA structure is augmented with the RLC line parameters fields in this table:

Field	Description
R	The N-by-N resistance matrix of the line, in ohms/km.
L	The N-by-N inductance matrix of the line, in Henries/km.
C	The N-by-N capacitance matrix of the line, in Farads/km.
R10	For a three-phase line, this setting represents the positive-sequence and zero-sequence resistances, in ohms/km. For a six-phase line (two coupled three-phase circuits), this setting represents the positive-sequence and zero-sequence resistances of circuit 1, the mutual zero-sequence resistance between circuit 1 and circuit 2, and the positive-sequence and zero-sequence resistances of circuit 2.
L10	For a three-phase line, the positive-sequence and zero-sequence inductances, in Henries/km. For a six-phase line (two coupled three-phase circuits), this setting represents the positive-sequence and zero-sequence inductances of circuit 1, the mutual zero-sequence inductance between circuit 1 and circuit 2, and the positive-sequence and zero-sequence inductances of circuit 2.
C10	For a three-phase line, this setting represents the positive-sequence and zero-sequence capacitances, in Farads/km. For a six-phase line (two coupled three-phase circuits), this setting represents the positive-sequence and zero-sequence capacitances of circuit 1, the mutual zero-sequence capacitance between circuit 1 and circuit 2, and the positive-sequence and zero-sequence capacitances of circuit 2.

DATA = power_lineparam('MYLINEDATA') computes the RLC line parameters for the geometric line parameters in the specified file. 'MYLINEDATA' is the name of a MAT-file. The MAT-file must contain a structure variable of the same format as the DATA variable.

power_lineparam(DATA,'BLK'), and power_lineparam('MYLINEDATA','BLK') upload the RLC line parameters in the specified PI Section Line, PI Section Cable, Distributed Parameters Line, or Three-Phase PI Section Line block. 'BLK' is a block path name.

The power_lineparam function computes the resistance, inductance, and capacitance matrices of an arbitrary arrangement of conductors of an overhead transmission line. For a three-phase line, the symmetrical component RLC values are also computed.

The following figure shows a typical conductor arrangement for a three-phase double-circuit line. This line configuration illustrates the various line parameters you specify in the DATA structure.

Configuration of a Three-Phase Double-Circuit Line

For a set of N conductors, power_lineparam computes three N-by-N matrices: the series resistance and inductance matrices [R] and [L] and the shunt capacitance matrix [C]. These matrices are required by the Distributed Parameters Line block for modeling N-phase asymmetrical lines and by the single-phase PI Section Line block. power_lineparam also computes the symmetrical component RLC parameters required by the Three-Phase PI Section Line block. For two coupled conductors, i and k, the self and mutual terms of the R, L, and C matrices are computed using the concept of image conductors [1] .

The self and mutual resistance terms are:

$$\begin{array}{cc}{R}_{ii}=R_{\mathrm{int}}+\Delta R_{ii}& \Omega /\text{km}\end{array}$$

$$\begin{array}{cc}{R}_{ik}=\Delta R_{ik}& \Omega /\text{km}\end{array}$$

The self and mutual inductance terms are:

$$\begin{array}{cc}{L}_{ii}=L_{\mathrm{int}}+\frac{{\mu }_0}{2\pi }\cdot \log \frac{2h_i}{r_i}+\Delta L_{ii}& H/\text{km}\end{array}$$

$$\begin{array}{cc}{L}_{ik}=\frac{{\mu }_0}{2\pi }\cdot \log \frac{D_{ik}}{d_{ik}}+\Delta L_{ik}& H/\text{km}\end{array}$$

The self and mutual potential coefficients terms are:

$$\begin{array}{cc}{P}_{ii}=\frac{1}{2\pi {\epsilon }_0}\cdot \log \frac{2h_i}{r_i}& \text{km}/\text{F}\end{array}$$

$$\begin{array}{cc}{P}_{ik}=\frac{1}{2\pi {\epsilon }_0}\cdot \log \frac{D_{ik}}{d_{ik}}& \text{km}/\text{F}\end{array}$$

$$\begin{array}{cc}\left[C\right]={\left[P\right]}^{-1}& F/\text{km}\end{array}$$

where:

• µ₀ is the permeability of free space = 4π.10⁻⁴ H/km

• ɛ₀ is the permittivity of free space = 8.8542.10⁻⁹ F/km

• r_i is the radius of conductor i in meters

• d_ik is the distance between conductors i and k, in meters

• D_ik is the distance between conductor i and image of k, in meters

• h_i is the average height of conductor i above ground, in meters

• Rint, Lint are the internal resistance and inductance of conductor

• ΔR_ii, ΔR_ik are the Carson R correction terms due to ground resistivity

• ΔL_ii, ΔL_ik are the Carson L correction terms due to ground resistivity

The conductor self-inductance is computed from the magnetic flux circulating inside and outside the conductor, and produced by the current flowing in the conductor itself. The part of flux circulating inside the conducting material contributes to the internal inductance Lint, which is dependent on the conductor geometry. Assuming a hollow or solid conductor, the internal inductance is computed from the T/D ratio, where D is the conductor diameter and T is the thickness of the conducting material. See the figure Configuration of a Three-Phase Double-Circuit Line. The conductor self-inductance is computed by means of modified Bessel functions from the conductor diameter, T/D ratio, resistivity, and relative permeability of conducting material and specified frequency [1].

The conductor self-inductance can also be computed from parameters that are usually found in tables provided by conductor manufacturers: the Geometric Mean Radius (GMR) or the "Reactance at one-foot spacing."

The GMR is the radius of the equivalent hollow conductor with zero thickness, producing no internal flux, giving the same self-inductance. The conductor self-inductance is then derived from the GMR using the following equation:

$$\begin{array}{cc}{L}_{ii}=\frac{{\mu }_0}{2\pi }\cdot \log \frac{2h_i}{GMR}+\Delta L_{ii}& H/\text{km}\end{array}$$

For a solid conductor (T/D=0.5), the GMR is

$$GMR=r\cdot e^{-{\mu }_r/4}$$

r = radius of conductor

μr = relative permeability of conducting material

The GMR obtained from this equation assumes a uniform current density in the conductor. This assumption is strictly valid in DC. In AC, the GMR is slightly higher. For example, for a 3 cm diameter solid aluminum conductor (Rdc = 0.040 Ω/km), the GMR increases from 1.1682 cm in DC to 1.1784 cm at 60 Hz. Manufacturers usually give the GMR at the system nominal frequency (50 Hz or 60 Hz).

The reactance X_a at 1-foot spacing (or 1-meter spacing, if metric units are used) is the positive-sequence reactance of a three-phase line having one foot (or one meter) spacing between the three phases and infinite conductor heights. The reactance at 1-foot spacing (or 1-meter spacing) at frequency f is related to the GMR by the following equation:

$$\begin{array}{cc}{X}_a=\omega \cdot \frac{{\mu }_0}{2\pi }\cdot \log \frac{1}{GMR}& \Omega /\text{km}\end{array}$$

GMR = Geometric Mean Radius in feet or meters

ω = 2π.f in rad/s

f = frequency in hertz

The conductor resistance matrix at a particular frequency depends on the DC resistance of the conductor corrected for skin effect and ground resistivity. Both the resistance matrix and the inductance matrix are dependent on the ground resistivity and frequency. Correction terms for the R and L terms as developed by J.R. Carson in 1926 [2] are implemented in power_lineparam.

Examples

The first example uses a simple line consisting of two conductors spaced by 1 meter at an average height of 8 meters above a perfect ground (ground resistivity ρ_g = 0). The two conductors are solid aluminum conductors (resistivity ρ_c = 28.3 10⁻⁹ Ω.m at 20^º C) that have a 15-mm diameter.

The DC resistance per km of each conductor is:

$$\begin{array}{cc}r=\frac{{\rho }_c\cdot l}{A}=\frac{28.3\times {10}^{-9}\times 1000}{\pi \cdot {\left(\frac{15\times {10}^{-3}}{2}\right)}^2}=0.1601& \Omega /\text{km}\end{array}$$

As ground is supposed to be perfect, the off-diagonal terms of the R matrix are zero and the diagonal terms represent the conductor resistances:

$$R=\left[\begin{array}{cc}0.1601& 0\\ 0& 0.1601\end{array}\right]\Omega /\text{km}$$

For the solid conductors, the GMR is:

$$\begin{array}{cc}GMR=r\cdot e^{-{\mu }_r/4}=\frac{1.5}{2}{e}^{-1/4}=0.5841& \text{cm}\end{array}$$

The self- and mutual-inductances are computed as follows. The ΔL correction terms are ignored because ground resistivity is zero.

$$\begin{array}{c}{L}_{11}=L_{22}=\frac{{\mu }_0}{2\pi }\cdot \log \frac{2h_i}{GMR}=2\times {10}^{-4}\log \frac{2\times 8}{0.5841\times {10}^{-2}}=1.583\times {10}^{-3}H/\text{km}\\ L_{12}=L_{21}=\frac{{\mu }_0}{2\pi }\cdot \log \frac{D_{12}}{d_{12}}=2\times {10}^{-4}\log \frac{\sqrt{{16}^2+1}}{1}=0.5549\times {10}^{-3}H/\text{km}\\ L=\left[\begin{array}{cc}{L}_{11}& L_{12}\\ L_{21}& L_{22}\end{array}\right]=\left[\begin{array}{cc}1.583\times {10}^{-3}& 0.5549\times {10}^{-3}\\ 0.5549\times {10}^{-3}& 1.583\times {10}^{-3}\end{array}\right]H/\text{km}\end{array}$$

The self- and mutual-capacitances are computed as follows:

$$\begin{array}{c}{P}_{11}=P_{22}=\frac{1}{2\pi {\epsilon }_o}\log \frac{2h}{r_i}=1.7975\times {10}^7\log \frac{2\times 8}{0.75\times {10}^{-2}}=1.3779\times {10}^8\text{km / F}\\ P_{12}=P_{21}=\frac{1}{2\pi {\epsilon }_o}\log \frac{D_{12}}{d_{12}}=1.7975\times {10}^7\log \frac{\sqrt{{16}^2+1}}{1}=4.9872\times {10}^7\text{km / F}\\ C={\left[\begin{array}{cc}{P}_{11}& P_{12}\\ P_{21}& P_{22}\end{array}\right]}^{-1}=\left[\begin{array}{cc}8.352\times {10}^{-9}& -3.023\times {10}^{-9}\\ -3.023\times {10}^{-9}& 8.352\times {10}^{-9}\end{array}\right]F/\text{km}\end{array}$$

You first create a template DATA structure to define the line geometry and the type of conductors of this line. At the MATLAB^® prompt, type:

DATA = power_lineparam('new');

Then set the following fields with specific values:

 DATA.frequency = 50;
 DATA.groundResistivity = 0;
 DATA.Geometry.NPhaseBundle = 2;
 DATA.Geometry.NGroundBundle = 0;
 DATA.Geometry.PhaseNumber = [1,2];
 DATA.Geometry.X = [0,1];
 DATA.Geometry.Ytower = [8,8];
 DATA.Geometry.Ymin = [8,8];
 DATA.Geometry.ConductorType = [1,1];
 % define conductor Type 1:
 DATA.Conductors.Diameter = [1.5];
 DATA.Conductors.ThickRatio = [0.5];
 DATA.Conductors.GMR = [0.5841];
 DATA.Conductors.Res = [0.1601];
 DATA.Conductors.Mur = [1];
 DATA.Conductors.Nconductors = [1];
 DATA.Conductors.BundleDiameter =[0];
 DATA.Conductors.AngleConductor1 = [0];
 DATA.Conductors.skinEffect = 'no';

Then you compute the RLC line parameters with the command:

 DATA = power_lineparam(DATA);
 disp(DATA.R),disp(DATA.L),disp(DATA.C)

The PI model for a 1-km line is obtained from the R, L, and C matrices. The PI RLC values are deduced from the self- and mutual-terms of the R, L, and C matrices. Subscripts s and m designate the self- and mutual-terms in the R, L, and C matrices.

R_p1 = R_p2 = R_s = 0.1601 Ω/km

L_p1 = L_p2 = L_s = 1.583 mH/km

C_p1 = C_p2 = C_s + C_m = 8.352 – 3.023 = 5.329 nF/km (2.664 nF at each end of PI section)

C_p1p2 = – C_m = 3.023 nF/km (1.511 nF at each end of PI section)

You can also vary the ground resistivity and the frequency to observe their impact on the resistance and inductance of the conductor and of the ground return.

The following code varies the ground resistivity from zero to 10000 Ω.m:

Rground = [0,10,100,10000]
for k=1:4
   DATA.groundResistivity = Rground(k);
   DATA = power_lineparam(DATA);
   ConductorRS_RM(k) = DATA.R(1,1)-DATA.R(1,2);
   GroundRm(k) = DATA.R(1,2);
   ConductorLs_Lm(k) = DATA.L(1,1)-DATA.L(1,2);
   GroundLm(k) = DATA.L(1,2);
end

You should get the values listed in the following table. The expressions Rs-Rm and Ls-Lm represent respectively the resistance and the inductance of the conductor, whereas Rm and Lm are the resistance and the inductance of the ground return.

Impact of Ground Resistivity (Frequency = 50 Hz; Skin Effect Not Included)

Ground Resistivity (Ω.m)	Conductor Rs-Rm (Ω/km)	Ground Rm (Ω/km)	Conductor Ls-Lm (mH/km)	Ground Lm (mH/km)
0	0.1601	0	1.028	0.5549
10	0.1601	0.04666	1.029	1.147
100	0.1601	0.04845	1.029	1.370
10 000	0.1601	0.04925	1.029	1.828

When the ground resistivity varies in a normal range (between 10 Ω.m for humid soil and 10 000 Ω.m for dry, rocky ground), the ground resistance remains almost constant at 0.05 Ω/km, whereas its inductance increases from 1.15 mH/km to 1.83 mH/km.

Using the following code, repeat the computation with different frequencies ranging from 0.05 Hz to 50 kHz and with the conductor skin effect modeled for a ground resistivity of 100 Ohms-m.

DATA.groundResitivity = 100;
DATA.Conductors.skinEffect = 'yes';
Frequencies = [0.05,50,500,5000,50000]
for k=1:5
   DATA.frequency = Frequencies(k);
   DATA = power_lineparam(DATA);
   ConductorRS_RM(k) = DATA.R(1,1)-DATA.R(1,2);
   GroundRm(k) = DATA.R(1,2);
   ConductorLs_Lm(k) = DATA.L(1,1)-DATA.L(1,2);
   GroundLm(k) = DATA.L(1,2);
end

Impact of Frequency (Ground Resistivity = 100 Ω.m; with Conductor Skin Effect)

Frequency (Hz)	Conductor Rs-Rm (Ω/km)	Ground Rm (Ω/km)	Conductor Ls-Lm (mH/km)	Ground Lm (mH/km)
0.05	0.1601	4.93e-5	1.029	2.058
50	0.1606	0.04844	1.029	1.370
500	0.2012	0.4666	1.022	1.147
5000	0.5442	4.198	0.9944	0.9351
50 000	1.641	32.14	0.9836	0.7559

This table shows that frequency has a very large impact on ground resistance, but a much lower influence on ground inductance. Because of skin effect in the ground, when frequency increases, the ground current flows closer to the surface, reducing the equivalent section of the ground conductor and thereby increasing its resistance. As ground current travels at a lower depth at high frequencies, the loop inductance of the conductor plus ground return (or the self-inductance, Ls) decreases.

Because of conductor skin effect, frequency has a noticeable impact on conductor resistance from a few hundreds of hertz, but a negligible impact on conductor inductance. At nominal system frequency (50 Hz or 60 Hz), the increase of conductor resistance with respect to DC resistance (0.1601 Ω/km) is only 0.3%.

The second example corresponds to a 500 kV, three-phase, double-circuit line.

Power is transmitted over six phase conductors forming the two three-phase circuits. The line is protected against lightning by two ground wires. The phase conductors use bundles of three subconductors. Subconductors are located at the top of an equilateral triangle of 50-cm sides, corresponding to a 57.735 cm bundle diameter. This line configuration corresponds to the one shown in the figure Configuration of a Three-Phase Double-Circuit Line.

Phase numbering has been set to obtain the line parameters of the three-phase line equivalent to the two circuits connected in parallel. In the Compute section, click RLC Line Parameters to display the R, L, and C matrices and sequence parameters.

The positive-sequence and zero-sequence parameters of the transposed line are displayed in the Computed Parameters section in the R10 and L10 vectors:

R1 = 0.009009 Ω/km R0=0.2556 Ω/km

L1 = 0.4408 mH/km L0= 2.601 mH/km

C1 =25.87 nF/km C0=11.62 nF/km

You can also get the parameters of the two individual circuits and have access to the six phase conductors. Change the phase numbers of conductors p4, p5, and p6 (circuit 2) to 6, 5, and 4, respectively. The positive-sequence, zero-sequence, and mutual zero-sequence parameters of the transposed line are:

R1 = 0.01840 Ω/km R0 =0.2649 Ω/km R0m = 0.2462 Ω/km

L1 = 0.9296 mH/km L0 = 3.202 mH/km L0m = 2.0 mH/km

C1 =12.57 nF/km C0 =7.856 nF/km C0m = −2.044 nF/km

As the line is symmetrical, the positive- and zero-sequence parameters for circuit 2 are identical to the parameters of circuit 1.

References

[1] Dommel, H., et al., Electromagnetic Transients Program Reference Manual (EMTP Theory Book), 1986.

[2] Carson, J. R., "Wave Propagation in Overhead Wires with Ground Return," Bell Systems Technical Journal, Vol. 5, pp 539-554, 1926.