PI Section Line
Implement transmission line with lumped parameters
Library
Simscape / Electrical / Specialized Power Systems / Power Grid Elements
Description
The PI Section Line block implements an Nphase 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. The following figure shows one PI section of a threephase 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 phasetophase capacitors, Cp, are not modeled.
The shunt capacitance Cg of each phase and the phasetophase 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 statespace 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
N_{bpi}  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 highfrequency 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, 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 noncorrected 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 positivesequence and zerosequence parameters (these are the default parameters of the ThreePhase PI Section Line block and Distributed Parameters Line block):
Positive sequence 
 
Zero sequence 

For a 350 km line section, noncorrected RLC positivesequence values are:
$$\begin{array}{l}R=0.01273\times 350=4.455\text{}\Omega \\ L=0.9337\times {10}^{3}\times 350=0.3268\text{H}\\ C=12.74\times {10}^{9}\times 350=4.459\times {10}^{6}\text{F}\end{array}$$
Hyperbolic correction at 60 Hz yields:
$$\begin{array}{l}R=4.153\text{}\Omega \\ L=0.3156\text{H}\\ C=4.538\times {10}^{6}\text{F}\end{array}$$
For these particular parameters and long line section (350 km), corrections for positivesequence RLC elements are relatively important (respectively −6.8%, −3.4%, and + 1.8%). For zerosequence 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 NbyN matrix in ohms/km (Ω/km). Default is
0.01273
. Inductance per unit length
The inductance per unit length of the line, as an NbyN 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
0.9337e3
. Capacitance per unit length
The capacitance per unit length of the line, as an NbyN 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
12.74e9
. 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
When the Number of phases [N] parameter is greater than
1
, the only measurement available in the dropdown menu isPhasetoground 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 pisection voltages and currents
to measure voltages and currents at the start and end of each pisection.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:
Phasetoground (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 singlephase PI section line.