# PI Section Line

Implement transmission line or cable with lumped parameters

## Library

Simscape / Electrical / Specialized Power Systems / Power Grid Elements

## Description

The PI Section Line block implements an N-phase transmission line or cable 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. The following figure shows one PI section of a three-phase transmission line.

When the number of phases is greater than 1, the series resistance and inductance are implemented by a mutual inductance device defined by the R and L matrices.

When the number of phases is 1, the series resistance and inductance are modeled by a single resistance R and inductance L. The phase-to-phase capacitors, Cp, are not modeled.

The shunt capacitance Cg of each phase and the phase-to-phase capacitance Cp are defined as:

Cg = sum(C(:,p)) / 2;

Cp = -C(p,k)

where C is the capacitance matrix and p and k are the number of the phases to which Cp is connected.

Unlike the Distributed Parameters 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:

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

where

 Nbpi Number of PI sections v Propagation speed (km/s) = $\frac{1}{\sqrt{lc}}$; 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 Power Line Parameters app and the Power Cable Parameters app, which calculates resistance, inductance, and capacitance per unit of length based on the line or cable 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 Parameters 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, as an N-by-N matrix in ohms/km (Ω/km). Default is `0.01273`.

Inductance per unit length

The inductance per unit length of the line, as an N-by-N matrix in henries/km (H/km). The terms in the matrix diagonal 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, as an N-by-N matrix in farads/km (F/km). The terms in the matrix diagonal 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`.

Specify Conductance

Select to specify the shunt conductance per unit length. Default is unselected.

Conductance per unit length

Specify the shunt conductance per unit length as an N-by-N matrix in mho/km.

Shunt conductance is normally not considered for transmission line since leakage currents flowing across insulators and air is significantly smaller than nominal current. However, if you are using the Pi Section Line block to model a cable arrangement, the shunt conductance can be non negligible. Default is `0`.

This parameter is not accessible when Specify Conductance is not selected.

Measurements

When the Number of phases [N] parameter is greater than `1`, the only measurement available in the drop-down menu is `Phase-to-ground voltages`.

When the Number of phases [N] is `1`, you can select these options:

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:`

Phase-to-ground (block inputs and outputs)

```Us phase 1, 2, 3, ...N```

```Ur phase 1, 2, 3, ...N```

## Examples

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