Building the System Matrices

Current Measurement-Bus Incidence Matrix

The current measurement-bus incidence matrix is a matrix that shows the location of the current flow measurements in the network relative to the observable busses. It is an m by b size matrix where m is the number of current measurements in the network and b is the number of observed busses. It is populated using a few simple rules.

These include:
  1. Each row of the matrix corresponds to a current phasor measurement in the network
  2. Each column of the matrix corresponds to an indirectly (via current phasor) or directly observed (via voltage phasor) bus in the system.
  3. The current phasor measurements and observed busses are enumerated.
  4. If measurement X (corresponding to row X ) leaves bus Y (corresponding to column Y ) then the matrix element ( X, Y ) will be a 1.
  5. If measurement X (corresponding to row X ) heads towards bus Z (corresponding to column Z ) then the matrix element ( X ,Z ) will be a -1.
  6. All remaining entries will be identically zero.

Consider the fictious 5 bus transmission network shown below. The presence of a
current measurement is represented by an arrow above a CT.

example5BusSystemShowingCurrentMeasurements.png

The current measurement bus index matrix for this system in a single phase
representation would look like the following:

currentMeasurementBusIncidenceMatrix.png


The Phasor Analytics matrix library can generate a current measurement-bus incidence matrix for a network by calling its designated constructor.

...
using SynchrophasorAnalytics.Matrices
using SynchrophasorAnalytics.Networks
...
Network network = Network.DeserializeFromXml("//pathToNetworkModel");
CurrentMeasurementBusIncidenceMatrix A = new CurrentMeasurementBusIncidenceMatrix(network);

Voltage Measurement-Bus Incidence Matrix

The voltage measurement-bus incidence matrix is very similar to the current measurement-bus incidence matrix. It shows the relationship between a voltage phasor measurement and its respective location in the network. Its purpose can most easily be understood by inspection of the state equation introduced in the Linear State Estimation with the Pi-Equivalent Wiki Documentation.

Voltage phasors measurements are actually a direct measurement of the system state and therefore only require a simple identity relationship between the measurement and the state. The voltage measurement-bus incidence matrix is an m by b matrix where m is the number of voltage phasor measurements in the network and b is the number of indirectly (via current phasor) or directly (via voltage phasor) observed busses.

This matrix is populated using the following rules.
  1. Each row of the matrix corresponds to a voltage measurement in the network
  2. Each column of the matrix corresponds to a bus in the system which has a voltage phasor measurement.
  3. The voltage phasors and the observed busses are enumerated.
  4. If measurement X (corresponding to row X ) is located at bus Y (corresponding to column Y ) then the matrix element ( X , Y ) will be a 1.
  5. All remaining entries will be identically 0.

Consider again the small fictitious system shown in the previous section. The location of the voltage measurements are shown by the circles on the busses. For a single phase representation, the voltage measurement-bus incidence matrix would look like the following:

voltageMeasurementBusIncidenceMatrix.png


The Phasor Analytics matrix library can generate a voltage measurement-bus incidence matrix for a network by calling its designated constructor.

...
using SynchrophasorAnalytics.Matrices
using SynchrophasorAnalytics.Networks
...
Network network = Network.DeserializeFromXml("//pathToNetworkModel");
VoltageMeasurementBusIncidenceMatrix II = new VoltageMeasurementBusIncidenceMatrix(network);

Series Admittance Matrix

The series admittance matrix is a diagonal matrix where the diagonal elements are the admittances of the lines being measured. It is an m by m matrix where m is the number of current phasor measurements in the network.

It is populated using a single rule:
  1. For measurement X , the ( X , X ) matrix element is the admittance of the branch being measured.

seriesAdmittanceMatrix.png

The Phasor Analytics matrix library can generate a series admittance matrix for a network by calling its designated constructor.

...
using SynchrophasorAnalytics.Matrices
using SynchrophasorAnalytics.Networks
...
Network network = Network.DeserializeFromXml("//pathToNetworkModel");
SeriesAdmittance Y = new SeriesAdmittanceMatrix(network);

Shunt Susceptance Matrix

The shunt admittance matrix is a matrix that relates the location of each current
measurement to the shunt admittance of the line that it is measuring. It is an m by b matrix
where m is the number of current measurements in the network and b is the number of
observed busses in the network.

It is populated using a single rule:
  1. For measurement X (corresponding to column X ) leaving bus Y (corresponding to row Y), the matrix element ( X , Y ) is the shunt admittance of the side of the line where measurement was taken.

The Phasor Analytics matrix library can generate a shunt susceptance matrix for a network by calling its designated constructor.

...
using SynchrophasorAnalytics.Matrices
using SynchrophasorAnalytics.Networks
...
Network network = Network.DeserializeFromXml("//pathToNetworkModel");
ShuntSusceptance Ys = new ShuntSusceptanceMatrix(network);

System Matrix

The system matrix is essentially the linear transformation from the measurement vector to the state vector. And is at its core, the same concept seen in the Linear State Estimation with the Two-Port Pi-Model Wiki Documentation.

systemMatrix1.png
The system matrix is a vertical concatenation of the Voltage Measurement-Bus Incidence Matrix (II), and the lower partition which is the sum of the Shunt Susceptance Matrix (Ys) and the product of the Current Measurement-Bus Incidence Matrix (A) and the Series Admittance Matrix (Y).
systemMatrix2.png

Shown below are the matrices for the above example system.

systemMatrix3.png

systemMatrix4.png

The Phasor Analytics matrix library can generate a system matrix for a network by calling its designated constructor.

...
using SynchrophasorAnalytics.Matrices
using SynchrophasorAnalytics.Networks
...
Network network = Network.DeserializeFromXml("//pathToNetworkModel");
SystemMatrix systemMatrix = new SystemMatrix(network);

Last edited Jul 24, 2013 at 5:41 PM by kevinjones, version 9