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

`LDATA = power_lineparam('new')`

LDATA = power_lineparam(LDATA)

LDATA = power_lineparam('MYLINEDATA')

power_lineparam(LDATA,'BLK')

power_lineparam('MYLINEDATA','BLK')

`LDATA = power_lineparam('new')`

returns a structure variable with the
default line geometry parameters. You can use `LDATA`

as a template
variable to configure a new line geometry and to compute RLC line parameters.

`LDATA`

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. |

`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. |

`LDATA = power_lineparam(LDATA)`

computes
the RLC line parameters for the geometric line parameters in the `LDATA`

structure.
The returned structure contains both geometric data and computed RLC
line parameters.

The `LDATA`

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. |

`LDATA = 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 `LDATA`

variable.

`power_lineparam(LDATA,'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
`LDATA`

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:

$$\begin{array}{cc}{L}_{ii}={L}_{\mathrm{int}}+\frac{{\mu}_{0}}{2\pi}\cdot \mathrm{log}\frac{2{h}_{i}}{{r}_{i}}+\Delta {L}_{ii}& H/\text{km}\end{array}$$

$$\begin{array}{cc}{L}_{ik}=\frac{{\mu}_{0}}{2\pi}\cdot \mathrm{log}\frac{{D}_{ik}}{{d}_{ik}}+\Delta {L}_{ik}& H/\text{km}\end{array}$$

The self and mutual potential coefficients terms are:

$$\begin{array}{cc}{P}_{ii}=\frac{1}{2\pi {\epsilon}_{0}}\cdot \mathrm{log}\frac{2{h}_{i}}{{r}_{i}}& \text{km}/\text{F}\end{array}$$

$$\begin{array}{cc}{P}_{ik}=\frac{1}{2\pi {\epsilon}_{0}}\cdot \mathrm{log}\frac{{D}_{ik}}{{d}_{ik}}& \text{km}/\text{F}\end{array}$$

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

where:

µ

_{0}is the permeability of free space = 4π.10^{−4}H/kmɛ

_{0}is the permittivity of free space = 8.8542.10^{−9}F/kmr

_{i}is the radius of conductor i in metersd

_{ik}is the distance between conductors i and k, in metersD

_{ik}is the distance between conductor i and image of k, in metersh

_{i}is the average height of conductor i above ground, in metersRint, Lint are the internal resistance and inductance of conductor

ΔR

_{ii}, ΔR_{ik}are the Carson R correction terms due to ground resistivityΔL

_{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:

$$\begin{array}{cc}{L}_{ii}=\frac{{\mu}_{0}}{2\pi}\cdot \mathrm{log}\frac{2{h}_{i}}{GMR}+\Delta {L}_{ii}& H/\text{km}\end{array}$$

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

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

r = radius of conductor |

μ = relative
permeability of conducting material_{r} |

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:

$$\begin{array}{cc}{X}_{a}=\omega \cdot \frac{{\mu}_{0}}{2\pi}\cdot \mathrm{log}\frac{1}{GMR}& \Omega /\text{km}\end{array}$$

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`

.

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.

$$\begin{array}{c}{L}_{11}={L}_{22}=\frac{{\mu}_{0}}{2\pi}\cdot \mathrm{log}\frac{2{h}_{i}}{GMR}=2\times {10}^{-4}\mathrm{log}\frac{2\times 8}{0.5841\times {10}^{-2}}=1.583\times {10}^{-3}H/\text{km}\\ {L}_{12}={L}_{21}=\frac{{\mu}_{0}}{2\pi}\cdot \mathrm{log}\frac{{D}_{12}}{{d}_{12}}=2\times {10}^{-4}\mathrm{log}\frac{\sqrt{{16}^{2}+1}}{1}=0.5549\times {10}^{-3}H/\text{km}\\ L=\left[\begin{array}{cc}{L}_{11}& {L}_{12}\\ {L}_{21}& {L}_{22}\end{array}\right]=\left[\begin{array}{cc}1.583\times {10}^{-3}& 0.5549\times {10}^{-3}\\ 0.5549\times {10}^{-3}& 1.583\times {10}^{-3}\end{array}\right]H/\text{km}\end{array}$$

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

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

You first create a template `LDATA`

structure to define the line
geometry and the type of conductors of this line. At the MATLAB^{®} prompt, type:

LDATA = power_lineparam('new');

Then set the following fields with specific values:

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

Then you compute the RLC line parameters with the command:

LDATA = power_lineparam(LDATA); disp(LDATA.R),disp(LDATA.L),disp(LDATA.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 LDATA.groundResitivity = Rground(k); LDATA = power_lineparam(LDATA); ConductorRS_RM(k) = LDATA.R(1,1)-LDATA.R(1,2); GroundRm(k) = LDATA.R(1,2); ConductorLs_Lm(k) = LDATA.L(1,1)-LDATA.L(1,2); GroundLm(k) = LDATA.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 | Conductor | Ground | Conductor | Ground |
---|---|---|---|---|

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.

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

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

Frequency (Hz) | Conductor | Ground | Conductor | Ground |
---|---|---|---|---|

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.

[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.