In the last thirty years, separate electrical power transmission networks within and across countries have been linked together. As a result of deregulation, these linked networks now handle large power transfers across long distances as power is bought and sold in a deregulated market. While these linked networks provide many advantages, they also expose a larger number of users to a larger number of potential sources of network problems.
In order to protect users from network problems the monitoring of network stability is crucial. Instability can cause brown outs (low voltage conditions) or black outs (complete loss of power) to a portion or all of the network. Instability can result when loads (e.g. power required by a defined group of users) on the network increase unexpectedly, transmission lines are reduced in power carrying capacity, generators go offline, or transformers at generator, transmission, or distribution substations are reduced in power conversion capacity. Instability can take various forms such as voltage dips, frequency shifts, or phase changes which can last from milliseconds up to tens of seconds. Minimizing the propagation of instability across the network is crucial to providing continuous power to industrial, commercial, educational, residential and other users.
Network operators typically monitor the status of the network at various network locations. The amplitude, phase and frequency of the voltages and currents at these locations are measured in order to (a) identify problems before they occur so that a failing piece of equipment can be taken off-line, and (b) detect instability so that a small part of the network can be disconnected in order to reduce the load on the remaining generation or transmission capacity (known as “load shedding”). When the conditions giving rise to network problems are poorly understood, operators may shed more load than necessary causing a wider black out than needed, or they may be unable to isolate the problem such that the problem may spread across the network causing a widespread black out. Recent blackouts include the August 2003 cascading blackout that affected Ontario, Canada, the US Midwest, and the US Northeast. It has been estimated that the August 2003 blackout has an economic cost of between $6 and $10 billion.
One approach to protecting the network is to assess the stability of the network by determining a Thévenin equivalent of the network at particular substation buses in the network as seen from the bus. The Thévenin equivalent models the network as consisting of a Thévenin voltage and a Thévenin impedance that are connected in series to the bus and a load. Once determined, the Thévenin parameters can be utilized to determine the stability of the network at the particular substation bus and appropriate protective action can be taken.
The relationship between the Thévenin parameters is given by the following equation:Ēthevenin− Zthevenin×Īmeasured= Vmeasured  (1)
where,
Ēthevenin is the Thévenin voltage,
 Zthevenin is the Thévenin impedance,
Īmeasured is the measured current passing through the bus, and
 Vmeasured is the measured voltage at the bus.
The measured current, Īmeasured, and measured voltage, Vmeasured, are typically measured at the bus using current transformers and voltage transformers (not shown) that are usually installed at the substation housing the bus for use with overcurrent protection and other standard relays.
Equation 1 consists of two known values, Īmeasured and Vmeasured , and two unknown Thévenin parameters, Ēthevenin and Zthevenin.In order to solve for the two Thévenin parameters, previous solutions require two or more measurements of the measured current, Īmeasured, and measured voltage, Vmeasured, at different instants in time. These solutions assume that (a) the Thévenin parameters do not change between successive measurements of the measured current and measured voltage and (b) the measured current and measured voltage do change between successive measurements. The Thévenin parameters are then estimated and compared with the impedance of the load to assess the stability of the network at the bus. Because the time scale of voltage collapse in a network can be as short as a few seconds, the two measurements must be made very close to each other to protect the network from voltage collapse. However, when the measurements are made very close to each other, the differences in measured voltages and currents, particularly with respect to the phase angles, are usually not large enough to produce an accurate approximation of the Thévenin parameters. Further, if more time is allowed between measurements to improve the accuracy of the approximation, then the assumption that the Thévenin parameters do not change between successive measurements is less accurate and there is additional delay in taking protective action to address network instability.