Linear State Estimation with the Pi-Equivalent
To understand the fundamental difference between the measurements used in a traditional state estimator and the measurements used in a linear state estimator it is best to begin with a simple two-port pi-model equivalent of a transmission line. The state of
this simple system will be the voltage magnitude and angle at each end of the transmission line, represented here as a single complex variable. If there is a PMU at each end of the transmission line then it can be assumed that the measurement set for this
system will consists of the voltage phasors at each end of the line and the line flows leaving each end of the line.
Recall that because of the capacitance of transmission lines that the line current on each side of a single line will not be the same.
Consider the positive sequence pi-equivalent of a transmission line shown below.
The state equation for this two-port pi-model is given below where the measurement vector is the vertical concatenation of the set of voltage phasors and current phasors, the system matrix is separated into an upper section where the measurements are identically
related to the system state and a lower section where the measurements are related linearly through the line admittances.
To see how this concept expands to a network of measurements, see
Building the System Matrices
in the Wiki Documentation.