#### 5.1 Parameterization

The values of along the solution curve can parameterized in a number of ways . Parameterization is a mathematical way of identifying each solution so that the next solution or previous solution can be quantiﬁed. Matpower includes three parameterization scheme options to quantify this relationship, detailed below, where is the continuation step size parameter.

• Natural parameterization simply uses directly as the parameter, so the new is simply the previous value plus the step size. (5.5)

• Arc length parameterization results in the following relationship, where the step size is equal to the 2-norm of the distance from one solution to the next. (5.6)

• Pseudo arc length parameterization  is Matpower’s default parameterization scheme, where the next point on the solution curve is constrained to lie in the hyperplane running through the predicted solution orthogonal to the tangent line from the previous corrected solution . This relationship can be quantiﬁed by the function (5.7)

where is the normalized tangent vector at and is the continuation step size parameter.