# PI Section Line

Implement single-phase transmission line with lumped parameters

## Library

Simscape / Electrical / Specialized Power Systems / Fundamental Blocks / Elements

• ## Description

The PI Section Line block implements a single-phase transmission line with parameters lumped in PI sections.

For a transmission line, the resistance, inductance, and capacitance are uniformly distributed along the line. An approximate model of the distributed parameter line is obtained by cascading several identical PI sections, as shown in the following figure. Unlike the Distributed Parameter Line block, which has an infinite number of states, the PI section linear model has a finite number of states that permit you to compute a linear state-space model. The number of sections to be used depends on the frequency range to be represented.

An approximation of the maximum frequency range represented by the PI line model is given by the following equation:

`${f}_{\mathrm{max}}=\frac{N\cdot v}{8\cdot ltot}$`

where

 N Number of PI sections v Propagation speed (km/s) = $1=\sqrt{lc}$; l in H/km, c in F/km ltot Line length (km)

For example, for a 100 km aerial line having a propagation speed of 300,000 km/s, the maximum frequency range represented with a single PI section is approximately 375 Hz. For studying interactions between a power system and a control system, this simple model could be sufficient. However for switching surge studies involving high-frequency transients in the kHz range, much shorter PI sections should be used. In fact, you can obtain the most accurate results by using a distributed parameters line model.

### Note

The powergui block provides the RLC Line Parameters tool, which calculates resistance, inductance, and capacitance per unit of length based on the line geometry and the conductor characteristics.

### Hyperbolic Correction of RLC Elements

For short line sections (approximately lsec <50 km) the RLC elements for each line section are simply given by:

`$\begin{array}{l}R=r\cdot l\mathrm{sec}\\ L=l\cdot l\mathrm{sec}\\ C=c\cdot l\mathrm{sec}\end{array}$`

where

 r Resistance per unit length (Ω/km) l Inductance per unit length (H/km) c Capacitance per unit length (F/km) f Frequency (Hz) lsec Line section length = ltot / N (km)

However, for long line sections, the RLC elements given by the above equations must be corrected in order to get an exact line model at a specified frequency. The RLC elements are then computed using hyperbolic functions as explained below.

`$\omega =2\pi f$`

Per unit length series impedance at frequency f is

`$z=r+j\omega l$`

Per unit length shunt admittance at frequency f is

`$y=j\omega c$`

Characteristic impedance is

`${Z}_{c}=\sqrt{z/y}$`

Propagation constant is

`$\gamma =\sqrt{z\cdot y}$`
`$Z=R+j\omega L={Z}_{c}\cdot \mathrm{sinh}\left(\gamma \cdot l\mathrm{sec}\right)$`
`$R=\text{real}\left(Z\right)$`
`$L=\text{imag}\left(Z\right)/\omega$`
`$Y=\frac{2}{{Z}_{c}}\cdot \mathrm{tanh}\left(\gamma \cdot \frac{l\mathrm{sec}}{2}\right)$`
`$C=\text{imag}\left(Y\right)/\omega$`

Hyperbolic corrections result in RLC values slightly different from the non-corrected values. R and L are decreased while C is increased. These corrections become more important as line section length is increasing. For example, let us consider a 735 kV line with the following positive-sequence and zero-sequence parameters (these are the default parameters of the Three-Phase PI Section Line block and Distributed Parameter Line block):

Positive sequence

 r = 0.01273 Ω/km l = 0.9337×10−3H/km c = 12.74×10−9F/km

Zero sequence

 r = 0.3864 Ω/km l = 4.1264×10−3H/km c = 7.751×10−9F/km

For a 350 km line section, noncorrected RLC positive-sequence values are:

Hyperbolic correction at 60 Hz yields:

For these particular parameters and long line section (350 km), corrections for positive-sequence RLC elements are relatively important (respectively −6.8%, −3.4%, and + 1.8%). For zero-sequence parameters, you can verify that even higher RLC corrections must be applied (respectively −18%, −8.5%, and +4.9%).

The PI Section Line block always uses the hyperbolic correction, regardless of the line section length.

## Parameters

Frequency used for rlc specifications

Frequency f, in hertz (Hz), at which per unit length r, l, c parameters are specified. Hyperbolic correction is applied on RLC elements of each line section using this frequency. Default is `60`.

Resistance per unit length

The resistance per unit length of the line, in ohms/km (Ω/km). Default is `0.01273`.

Inductance per unit length

The inductance per unit length of the line, in henries/km (H/km). This parameter cannot be zero, because it would result in an invalid propagation speed computation. Default is `0.9337e-3`.

Capacitance per unit length

The capacitance per unit length of the line, in farads/km (F/km). This parameter cannot be zero, because it would result in an invalid propagation speed computation. Default is `12.74e-9`.

Length

The line length in km. Default is `100`.

Number of pi sections

The number of PI sections. The minimum value is 1. Default is `1`.

Measurements

Select `Input and output voltages` to measure the sending end (input port) and receiving end (output port) voltages of the line model.

Select `Input and output currents` to measure the sending end and receiving end currents of the line model.

Select `All pi-section voltages and currents` to measure voltages and currents at the start and end of each pi-section.

Select `All voltages and currents` to measure the sending end and receiving end voltages and currents of the line model.

Default is `None`.

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

Sending end voltage (block input)

`Us:`

Receiving end voltage (block output)

`Ur:`

Sending end current (input current)

`Is:`

Receiving end current (output current)

`Ir:`

## Examples

The `power_piline` example shows the line energization voltages and currents of a PI section line.