Modeling Branch Impedances

Due to the ability of the Phasor Analytics Platform to process three phase measurements, modeling of Ithe branch impedances is more thorough than practically every other piece of software besides perhaps that used in system protection studies. The series impedance (and shunt impedance) of every branch that assums a two-port pi-model will be represented as a 3x3 impedance matrix. When the line is perfectly balanced, the series impedance and shunt susceptance matrices takes the following general form.
Balanced Three Phase Impedance Matrix.png
Because transmission lines either are not transposed or are not perfectly transposed, the realistic series and shunt model for each branch contains 6 unique complex values representing the self and mutual impedance of each of the phases.
Unbalanced Three Phase Impedance Matrix.png
Therefore, the impedance model for every non-zero impedance network element will contain 6 unique resistance values, 6 unique reactance values, and 6 unique susceptance values. To demonstrate, consider the following example of an impedance entry from the network configuration XML file.

<Impedance R1="0.0016842" R2="0.001337" R3="0.0016673" R4="0.0013447" R5="0.001337" R6="0.0016842" X1="0.016655" X2="0.0079151" X3="0.016676" X4="0.0068129" X5="0.0079151" X6="0.016655" B1="0.570752" B2="-0.14096" B3="0.58482" B4="-0.0662" B5="-0.14096" B6="0.570752" />

Where the relationship between the XML Attributes and the matrix elements is given roughly by the following set of equations:

Three Phase Impedance Matrix with Numbering.png

With respect to the Impedance data structure and modeling, a three phase impedance model is used regardless of whether the implementation is positive sequence or three phase. For a positive sequence implementation, the desired branch impedance can be modeled in one of two ways:

  1. The full three phase branch impedance can be modeled and the application will calculate the positive sequence equivalent on-demand.
  2. If the three phase impedances are unavailable, the positive sequence equivalent impedance can be mapped to the balanced three phase impedance matrix in the following way:

Positive Sequence Impedance Matrix with Numbering.pngThe XML snippet from the configuration file would look like the following:

<Impedance R1="0.001678" R2="0" R3="0.001678" R4="0" R5="0" R6="0.001678" X1="0.016662" X2="0" X3="0.016662" X4="0" X5="0" X6="0.016662" B1="0.575441" B2="0" B3="0.575441" B4="0" B5="0" B6="0.575441" /> 


NOTE: The preference of using a three phase impedance or positive sequence equivalent mapped to a three phase impedance matrix is purely one of the modeling process; there is no computational benefit to modeling the positive sequence equivalent as the application always assumes a three phase matrix and computes the positive sequence equivalent on demand.

Last edited Apr 29, 2014 at 10:47 PM by kevinjones, version 21