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).
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  in what is called a continuation power ﬂow29 . 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
are restructured as
where and is a vector of power transfer given by
The eﬀects of the variation of loading or generation can be investigated using the continuation method by composing the vector appropriately.