# Distributed Parameters Line

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

**Libraries:**

Simscape /
Electrical /
Specialized Power Systems /
Power Grid Elements

## Description

The Distributed Parameters 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* +
*Z _{c}i*, 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:

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

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

knowing that

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

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

In a lossless line, the two current sources
*I _{sh}* and

*I*are computed as:

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

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

When losses are taken into account, new equations for
*I _{sh}* and

*I*are obtained by lumping

_{rh}*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 *I _{sh}* and

*I*are then computed as follows:

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

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

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* =
*Z _{c}*.

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.

## Examples

## Assumptions and 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.

## Ports

### Conserving

## Parameters

## 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.

## Extended Capabilities

## Version History

**Introduced before R2006a**