Continuation power flow¶

The continuation power flow is a technique that traces a trajectory from a base situation given to a combination of power $S_0$ and voltage $V_0$ , to another situation determined by another combination of power $S'$ . When the final power situation is undefined, then the algorithm continues until the Jacobian is singular, tracing the voltage collapse curve.

The method uses a predictor-corrector technique to trace this trajectory.

Predictor¶

$\begin{bmatrix} \theta \\ V \\ \lambda \\ \end{bmatrix}^{predicted} = \begin{bmatrix} \theta \\ V \\ \lambda \\ \end{bmatrix}^{i} + \sigma \cdot \begin{bmatrix} J11 & J12 & P_{base} \\ J21 & J22 & Q_{base} \\ 0 & 0 & 1 \\ \end{bmatrix}^{-1} \times \begin{bmatrix} \hat{0} \\ \hat{0} \\ 1\\ \end{bmatrix}$

Corrector¶

$\begin{bmatrix} d\theta \\ dV \\ d\lambda \\ \end{bmatrix} = \begin{bmatrix} d\theta_0\\ dV_0 \\ d\lambda_0 \\ \end{bmatrix} + \sigma \cdot \begin{bmatrix} J11 & J12 & P_{base} \\ J21 & J22 & Q_{base} \\ 0 & -1 & 0 \\ \end{bmatrix}^{-1} \times \begin{bmatrix} \hat{0} \\ \hat{0} \\ 1\\ \end{bmatrix}$