Continuation methods or branch tracing methods are used to trace a curve given an initial point on the curve. These are also called predictor-corrector methods since they involve the prediction of the next solution point and correcting the prediction to get the next point on the curve.

Consider a system of nonlinear equations . By adding a continuation parameter and one more equation to the system, can be traced by varying . The resulting system has dimensions. The additional equation is a parameterized equation which identiﬁes the location of the current solution with respect to the previous or next solution.

The continuation process can be diagrammatically shown by (5.1).

| (5.1) |

where, represents the current solution at step , is the predicted solution for the next step, and is the next solution on the curve.

Continuation methods are employed in power systems to determine
steady state stability limits [22] in what is called a continuation power
ﬂow^{29} .
The limit is determined from a nose curve where the nose represents the maximum power
transfer that the system can handle given a power transfer schedule. To determine the
steady state loading limit, the basic power ﬂow equations

| (5.2) |

are restructured as

| (5.3) |

where and is a vector of power transfer given by

| (5.4) |

The eﬀects of the variation of loading or generation can be investigated using the continuation method by composing the vector appropriately.

5.1 Parameterization

5.2 Predictor

5.3 Corrector

5.4 Step Length Control

5.5 Event Detection and Location

5.6runcpf

5.6.1 CPF Callback Functions

5.6.2 CPF Example

5.2 Predictor

5.3 Corrector

5.4 Step Length Control

5.5 Event Detection and Location

5.6

5.6.1 CPF Callback Functions

5.6.2 CPF Example