Implement single-phase transmission line with lumped parameters
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:
|N||Number of PI sections|
|v||Propagation speed (km/s) = ; 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 a graphical tool for the calculation of the resistance, inductance, and capacitance per unit length based on the line geometry and the conductor characteristics.
Let us assume the following line parameters:
|r||Resistance per unit length (Ω/km)|
|l||Inductance per unit length (H/km)|
|c||Capacitance per unit length (F/km)|
|lsec||Line section length = ltot / N (km)|
For short line sections (approximately lsec <50 km) the RLC elements for each line section are simply given by:
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.
Per unit length series impedance at frequency f is
Per unit length shunt admittance at frequency f is
Characteristic impedance is
Propagation constant is
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):
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.
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.
The resistance per unit length of the line, in ohms/km (Ω/km).
The inductance per unit length of the line, in henries/km (H/km). This parameter can not be zero, because it would result in an invalid propagation speed computation.
The capacitance per unit length of the line, in farads/km (F/km). This parameter can not be zero, because it would result in an invalid propagation speed computation.
The line length in km.
The number of PI sections. The minimum value is 1.
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.
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.
Sending end voltage (block input)
Receiving end voltage (block output)
Sending end current (input current)
Receiving end current (output current)
The results obtained with the line modeled by one PI section of 100 km and 10 PI sections of 10 km are shown.