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

Library

Elements Library

Description

The Distributed Parameter 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 and the phase velocity .

The model uses the fact that the quantity e+Zi, where e is line voltage and i is line current, entering one end of the line must arrive unchanged at the other end after a transport delay of , where d is the line length. By lumping R/4 at both ends of the line and R/2 in the middle and using the current injection method of the Power System Blockset, the following two port models are derived:

where     


          

For multi-phase 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 sections line model, the distributed line represents wave propagation phenomena and line end reflections with much less error. See the comparison between the two models in the example section.

Dialog Box

The first entry in the dialog box specifies the number of phases of the model. The block icon dynamically changes according to the number of phases that you specify. When you apply the parameters or close the dialog box, the number of inputs and outputs is updated. The icon also displays the individual conductors. If you specify more than three phases, only one conductor is displayed.

The second entry specifies the frequency used to compute the line parameters. This entry is needed for the computation of the modal impedance and admittance matrices.

The third, fourth, and fifth entries are the R(ohms/km) L(H/km) and C(F/km) matrices. For unsymmetrical lines, you must specify the complete R L C matrices of the line.

If you want to model your line as a symmetrical line (continuously transposed), you can also specify the sequence parameters. This is permitted for the following types of lines: two-phases, three-phases transposed and six-phases transposed (double-circuit three-phase line with zero sequence coupling only between the two circuits). Therefore the two- and three-phase lines require two dimension vector entries [R1 R0] [L1 L0] [C1 C0] while the six-phase line requires three-dimension vector entries [R1 R0 R0m] [L1 L0 L0m] [C1 C0 C0m], where the subscripts 1,0, and 0m hold respectively for positive-sequence, zero-sequence, and mutual-zero-sequence.

Finally, the last entry specifies the line length in Km.

Limitations

One limitation of this line model is its failure to represent accurately the frequency dependence of R L C parameters of real power lines. Indeed, because of skin effects in the conductors and ground, the R and L matrices exhibit strong frequency dependence, causing an attenuation of the high frequencies.

Example

Obtain the line energization voltages and current in the following circuit.

This circuit is available in the psbdistline.mdl file.

The sending end current and receiving end voltage obtained with the distributed parameter line are compared with a 10-pi-section line. On both graphs, the PI line model shows high frequency oscillatory modes superimposed on the 200Hz characteristic frequency of travelling waves. These oscillations due to PI sections are not found with the distributed parameter line with lumped losses model.

Note: Notice that the high current peak obtained with the PI line model at the breaker closing (due to the first section capacitor charging) does not exist with the distributed parameter line model.

References

[1] H. Dommel, "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

See Also

PI Section Line



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