point in the system during steady state operation. The power available is depict Ion o n the plane and the boundaries of the region represent the critical operating limits of the system. The best known example of a capability chart is the operating chart of a synchronous machine. The power available from the generator is restricted by limiting values of the rotor current, stator current, turbine power(if a generator), and synchronous stability limits. Capability charts have been produced for transmission lines and Hvdc converters Where the capability chart is extended to cover more than one power system component, the two-dimensional capability chart associated with a single busbar can be regarded as being a single slice of an overall 2n dimensional capability chart for the n busbars that make up a general power system. If the system is small, a contour plotting approach can be used to gradually trace out the locus on the complex power plane. a load flow algorithm used to iteratively solve the operating equations at each point on the contour, without having to resort to an explicit closed form solution The good contour behavior near the operating region has allowed a faster method to be adopted. A seed load flow solution, corresponding to the nominal system state, is obtained to begin drawing the chart. A region rowing process is then used to locate the region in which all constrained variables are less than 10% beyond their limits. This process is similar to a technique used in computer vision systems to recognize shapes of objects. The region grows by investigating the six nearest lattice vertices to any unconstrained vertex. Linear interpolation along the edges between vertices is then used to estimate the points of intersection between the contour and the lattice. This method has a second advantage in that it can detect holes and islands in the chart. However, it should be noted that these regions are purely speculative and have not been found in practice. Protection The need to analyze protection schemes has resulted in the development of protection coordination programs. Protection schemes can be divided into two major groupings: unit and non-unit schemes The first group contains schemes that protect a specific area of the system, i.e., a transformer, transmission line, generator, or busbar. The most obvious example of unit protection schemes is based on Kirchhoffs current law-the sum of the currents entering an area of the system must be zero. Any deviation from this must indicate an abnormal current path. In these schemes, the effects of any disturbance or operating condition outside the rea of interest are totally ignored and the protection must be designed to be stable above the maximum possible fault current that could flow through the protected area. Schemes can be made to extend across all sides of a transformer to account for the different currents at different voltage levels. Any analysis of these schemes ar thus of more concern to the protection equipment manufacturers. The non-unit schemes, while also intended to protect specific areas, have no fixed boundaries. As well as protecting their own designated areas, the protective zones can overlap into other areas. While this can be very beneficial for backup purposes, there can be a tendency for too great an area to be isolated if a fault is detected by different non-unit schemes. The most simple of these schemes measures current and incorporates an inverse time characteristic into the protection operation to allow protection nearer to the fault to operate first. while this is relatively straightforward for radial schemes, in networks, where the current paths can be quite different lepending on operating and maintenance strategies, protection can be difficult to set and optimum settings are probably impossible to achieve. It is in these areas where protection software has become useful to manu- facturers consultants, and utilities The very nature of protection schemes has changed from electromechanical devices, through electronic equivalents of the old devices, to highly sophisticated system analyzers. They are computers in their own right and thus can be developed almost entirely by computer analysis techniques. Other Uses for Load Flow analysis It has already been demonstrated that load flow analysis is necessary in determining the economic operation of the power system and it can also be used in the production of capability charts. Many other types of analyses require load flow to be embedded in the program. c 2000 by CRC Press LLC© 2000 by CRC Press LLC point in the system during steady state operation. The power available is depicted as a region on the plane and the boundaries of the region represent the critical operating limits of the system. The best known example of a capability chart is the operating chart of a synchronous machine. The power available from the generator is restricted by limiting values of the rotor current, stator current, turbine power (if a generator), and synchronous stability limits. Capability charts have been produced for transmission lines and HVdc converters. Where the capability chart is extended to cover more than one power system component, the two-dimensional capability chart associated with a single busbar can be regarded as being a single slice of an overall 2n dimensional capability chart for the n busbars that make up a general power system. If the system is small, a contour plotting approach can be used to gradually trace out the locus on the complex power plane. A load flow algorithm is used to iteratively solve the operating equations at each point on the contour, without having to resort to an explicit closed form solution. The good contour behavior near the operating region has allowed a faster method to be adopted. A seed load flow solution, corresponding to the nominal system state, is obtained to begin drawing the chart. A region growing process is then used to locate the region in which all constrained variables are less than 10% beyond their limits. This process is similar to a technique used in computer vision systems to recognize shapes of objects. The region grows by investigating the six nearest lattice vertices to any unconstrained vertex. Linear interpolation along the edges between vertices is then used to estimate the points of intersection between the contour and the lattice. This method has a second advantage in that it can detect holes and islands in the chart. However, it should be noted that these regions are purely speculative and have not been found in practice. Protection The need to analyze protection schemes has resulted in the development of protection coordination programs. Protection schemes can be divided into two major groupings: unit and non-unit schemes. The first group contains schemes that protect a specific area of the system, i.e., a transformer, transmission line, generator, or busbar. The most obvious example of unit protection schemes is based on Kirchhoff 's current law—the sum of the currents entering an area of the system must be zero. Any deviation from this must indicate an abnormal current path. In these schemes, the effects of any disturbance or operating condition outside the area of interest are totally ignored and the protection must be designed to be stable above the maximum possible fault current that could flow through the protected area. Schemes can be made to extend across all sides of a transformer to account for the different currents at different voltage levels. Any analysis of these schemes are thus of more concern to the protection equipment manufacturers. The non-unit schemes, while also intended to protect specific areas, have no fixed boundaries. As well as protecting their own designated areas, the protective zones can overlap into other areas. While this can be very beneficial for backup purposes, there can be a tendency for too great an area to be isolated if a fault is detected by different non-unit schemes. The most simple of these schemes measures current and incorporates an inverse time characteristic into the protection operation to allow protection nearer to the fault to operate first. While this is relatively straightforward for radial schemes, in networks, where the current paths can be quite different depending on operating and maintenance strategies, protection can be difficult to set and optimum settings are probably impossible to achieve. It is in these areas where protection software has become useful to manu￾facturers, consultants, and utilities. The very nature of protection schemes has changed from electromechanical devices, through electronic equivalents of the old devices, to highly sophisticated system analyzers. They are computers in their own right and thus can be developed almost entirely by computer analysis techniques. Other Uses for Load Flow Analysis It has already been demonstrated that load flow analysis is necessary in determining the economic operation of the power system and it can also be used in the production of capability charts. Many other types of analyses require load flow to be embedded in the program