5.2 Predictor

The predictor is used to produce an estimate for the next solution. The better the prediction, the faster is the convergence to the solution point. Matpower uses a tangent predictor for estimating the curve to the next solution. The tangent vector  j   [         ] T
z  =   dx  d λ  j   at the current solution   j  j
(x ,λ )  is found by solving the linear system

[            ]     [    ]
   fx    fλ     j     0
  pj−1  pj−1  z  =    1  .
   x     λ

The matrix on the left-hand side is simply the standard power flow Jacobian with an additional column and row added. The extra column fλ  is simply the negative of the power transfer vector b  and the extra row, required to make the system non-singular and define the magnitude of  j
z  , is the derivative of the the parameterization function at the previous solution point pj− 1   .

The resulting tangent vector is then normalized

  j   -zj---
z¯ =  ||zj||2

and used to compute the predicted approximation (ˆxj+1,ˆλj+1)  to the next solution (xj+1,λj+1) using

[ ˆxj+1 ]    [  j ]
    j+1   =    xj   + σj¯zj,
  ˆλ           λ

where  j
σ  is the continuation step size.