Parameterization

5.1 Parameterization

The values of $(x,λ)$ along the solution curve can parameterized in a number of ways [23, 24]. 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 $σ$ is the continuation step size parameter.

Natural parameterization simply uses $λ$ directly as the parameter, so the new $λ$ is simply the previous value plus the step size.
$pj(x, λ) = λ − λj − σj = 0$ (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.
$∑ pj(x,λ ) = (x − xj)2 + (λ − λj)2 − (σj)2 = 0 i i i$ (5.6)
Pseudo arc length parameterization [26] is Matpower’s default parameterization scheme, where the next point $(x,λ)$ on the solution curve is constrained to lie in the hyperplane running through the predicted solution $j+1 ˆj+1 (ˆx ,λ )$ orthogonal to the tangent line from the previous corrected solution $(xj,λj)$ . This relationship can be quantiﬁed by the function
$( [ ] [ ] ) j x xj T j j p (x,λ) = λ − λj z¯ − σ = 0,$ (5.7)

where $j ¯z$ is the normalized tangent vector at $j j (x ,λ )$ and $j σ$ is the continuation step size parameter.

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