# Distributed Parameter Line

Implement N-phase distributed parameter transmission line model with lumped losses

## Library

Fundamental Blocks/Elements

## Description

The Distributed Parameter Line block implements an N-phase distributed parameter line model with lumped losses. The model is based on the Bergeron's traveling wave method used by the Electromagnetic Transient Program (EMTP) [1]. In this model, the lossless distributed LC line is characterized by two values (for a single-phase line): the surge impedance ${Z}_{c}=\sqrt{l/c}$ and the wave propagation speed $v=1/\sqrt{lc}$. l and c are the per-unit length inductance and capacitance.

The figure shows the two-port model of a single-phase line.

For a lossless line (r = 0), the quantity e + Zci, where e is the line voltage at one end and i is the line current entering the same end, must arrive unchanged at the other end after a transport delay τ.

$\tau ={\frac{d}{v}}_{}$

where d is the line length and v is the propagation speed.

The model equations for a lossless line are:

${e}_{r}\left(t\right)-{Z}_{c}\text{\hspace{0.17em}}{i}_{r}\left(t\right)={e}_{s}\left(t-\tau \right)+{Z}_{c}\text{\hspace{0.17em}}{i}_{s}\left(t-\tau \right)$

${e}_{s}\left(t\right)-{Z}_{c}\text{\hspace{0.17em}}{i}_{s}\left(t\right)={e}_{r}\left(t-\tau \right)+{Z}_{c}\text{\hspace{0.17em}}{i}_{r}\left(t-\tau \right)$

knowing that

${i}_{s}\left(t\right)=\frac{{e}_{s}\left(t\right)}{Z}-\text{\hspace{0.17em}}{I}_{sh}\left(t\right)$

${i}_{r}\left(t\right)=\frac{{e}_{r}\left(t\right)}{Z}-\text{\hspace{0.17em}}{I}_{rh}\left(t\right)$

In a lossless line, the two current sources Ish and Irh are computed as:

${I}_{s}{}_{h}\left(t\right)=\frac{2}{{Z}_{c}}{e}_{r}\left(t-\tau \right)-\text{\hspace{0.17em}}{I}_{rh}\left(t-\tau \right)$

${I}_{r}{}_{h}\left(t\right)=\frac{2}{{Z}_{c}}{e}_{s}\left(t-\tau \right)-\text{\hspace{0.17em}}{I}_{sh}\left(t-\tau \right)$

When losses are taken into account, new equations for Ish and Irh are obtained by lumping R/4 at both ends of the line and R/2 in the middle of the line:

R = total resistance = r × d

The current sources Ish and Irh are then computed as follows:

${I}_{s}{}_{h}\left(t\right)=\left(\frac{1+h}{2}\right)\left(\frac{1+h}{Z}{e}_{r}\left(t-\tau \right)-\text{\hspace{0.17em}}h\text{\hspace{0.17em}}{I}_{rh}\left(t-\tau \right)\right)+\left(\frac{1-h}{2}\right)\left(\frac{1+h}{Z}{e}_{s}\left(t-\tau \right)-\text{\hspace{0.17em}}h\text{\hspace{0.17em}}{I}_{sh}\left(t-\tau \right)\right)$

${I}_{r}{}_{h}\left(t\right)=\left(\frac{1+h}{2}\right)\left(\frac{1+h}{Z}{e}_{s}\left(t-\tau \right)-\text{\hspace{0.17em}}h\text{\hspace{0.17em}}{I}_{sh}\left(t-\tau \right)\right)+\left(\frac{1-h}{2}\right)\left(\frac{1+h}{Z}{e}_{r}\left(t-\tau \right)-\text{\hspace{0.17em}}h\text{\hspace{0.17em}}{I}_{rh}\left(t-\tau \right)\right)$

where

$\begin{array}{c}Z={Z}_{C}+\frac{r}{4}\\ h=\frac{{Z}_{C}-\frac{r}{4}}{{Z}_{C}+\frac{r}{4}}\\ {Z}_{C}=\sqrt{\frac{l}{c}}\\ \tau =d\sqrt{lc}\end{array}$

r, l, c are the per unit length parameters, and d is the line length. For a lossless line, r = 0, h = 1, and Z = Zc.

For multiphase line models, modal transformation is used to convert line quantities from phase values (line currents and voltages) into modal values independent of each other. The previous calculations are made in the modal domain before being converted back to phase values.

In comparison to the PI section line model, the distributed line represents wave propagation phenomena and line end reflections with much better accuracy. See the comparison between the two models in the Example section.

## Dialog Box and Parameters

Number of phases N

Specifies the number of phases, N, of the model. The block icon dynamically changes according to the number of phases that you specify. When you apply the parameters or close the dialog box, the number of inputs and outputs is updated.

Frequency used for rlc specifications

Specifies the frequency used to compute the per unit length resistance r, inductance l, and capacitance c matrices of the line model.

Resistance per unit length

The resistance r per unit length, as an N-by-N matrix in ohms/km (Ω/km).

For a symmetrical line, you can either specify the N-by-N matrix or the sequence parameters. For a two-phase or three-phase continuously transposed line, you can enter the positive and zero-sequence resistances [r1 r0]. For a symmetrical six-phase line you can enter the sequence parameters plus the zero-sequence mutual resistance [r1 r0 r0m].

For asymmetrical lines, you must specify the complete N-by-N resistance matrix.

Inductance per unit length

The inductance l per unit length, as an N-by-N matrix in henries/km (H/km).

For a symmetrical line, you can either specify the N-by-N matrix or the sequence parameters. For a two-phase or three-phase continuously transposed line, you can enter the positive and zero-sequence inductances [l1 l0]. For a symmetrical six-phase line, you can enter the sequence parameters plus the zero-sequence mutual inductance [l1 l0 l0m].

For asymmetrical lines, you must specify the complete N-by-N inductance matrix.

Capacitance per unit length

The capacitance c per unit length, as an N-by-N matrix in farads/km (F/km).

For a symmetrical line, you can either specify the N-by-N matrix or the sequence parameters. For a two-phase or three-phase continuously transposed line, you can enter the positive and zero-sequence capacitances [c1 c0]. For a symmetrical six-phase line you can enter the sequence parameters plus the zero-sequence mutual capacitance [c1 c0 c0m].

For asymmetrical lines, you must specify the complete N-by-N capacitance matrix.

 Note   The powergui block provides a graphical tool for the calculation of the resistance, inductance, and capacitance per unit length based on the line geometry and the conductor characteristics.
Line length

The line length, in km.

Measurements

Select `Phase-to-ground voltages` to measure the sending end and receiving end voltages for each phase of the line model.

Place a Multimeter block in your model to display the selected measurements during the simulation.

In the Available Measurements list box of the Multimeter block, the measurement is identified by a label followed by the block name:

Measurement

Label

Phase-to-ground voltages, sending end

`Us_ph1_gnd:, Us_ph2_gnd:, Us_ph3_gnd:`, etc.

Phase-to-ground voltages, receiving end

`Ur_ph1_gnd:, Ur_ph2_gnd:, Ur_ph3_gnd:`, etc.

## Limitations

This model does not represent accurately the frequency dependence of RLC parameters of real power lines. Indeed, because of the skin effects in the conductors and ground, the R and L matrices exhibit strong frequency dependence, causing an attenuation of the high frequencies.

## Example

The `power_monophaseline``power_monophaseline` example illustrates a 200 km line connected on a 1 kV, 60 Hz infinite source. The line is de-energized and then reenergized after 2 cycles. The simulation is performed simultaneously with the Distributed Parameter Line block and with the PI Section Line block.

The receiving end voltage obtained with the Distributed Parameter Line block is compared with the one obtained with the PI Section Line block (two sections).

Open the Powergui. Click the Impedance vs Frequency Measurement button. A new window appears, listing the two Impedance Measurement blocks connected to your circuit. Set the parameters of Impedance vs Frequency Measurement to compute impedance in the [0,2000] Hz frequency range, select the two measurements in the list, then click the Update button.

The distributed parameter line shows a succession of poles and zeros equally spaced, every 486 Hz. The first pole occurs at 243 Hz, corresponding to frequency f = 1/(4 * T), where

```T = traveling time = $l\sqrt{lc}$ = 1.028 ms ```

The PI section line only shows two poles because it consists of two PI sections. Impedance comparison shows that a two-section PI line gives a good approximation of the distributed line for the 0 to 350 Hz frequency range.

## References

[1] Dommel, H., "Digital Computer Solution of Electromagnetic Transients in Single and Multiple Networks," IEEE® Transactions on Power Apparatus and Systems, Vol. PAS-88, No. 4, April, 1969.