Fast Decoupled

The fast decoupled method is a fantastic power flow algorithm developed by Stott and Alsac [FD]. The method builds on a series of very clever simplifications and the decoupling of the Jacobian matrix of the canonical Newton-Raphson algorithm to yield the fast-decoupled method.

The method consists in the building of two admittance-based matrices B' and B'' which are independently factorized (using the LU method or any other) which then serve to find the increments of angle and voltage magnitude separately until convergence.

Finding B' and B''

To find B' we perform the following operations:

bs' = \frac{1}{X} bs'_{ff} = diag(bs') bs'_{ft} = diag(-bs') bs'_{tf} = diag(-bs') bs'_{tt} = diag(bs') B'_f = bs'_{ff} \times Cf + bs'_{ft} \times Ct B'_t = bs'_{tf} \times Cf + bs'_{tt} \times Ct B' = Cf^\top \times B'_f + Ct^\top \times B'_t

To find B'' we perform the following operations:

bs_{ff}^{''} = -Re \left\{\frac{bs' + B}{tap \cdot tap^*} \right\} bs_{ft}^{''} = -Re \left\{ \frac{bs'}{tap^*} \right\} bs_{tf}^{''} = -Re \left\{ \frac{bs'}{tap} \right\} bs_{tt}^{''} = - bs'' B''_f = bs''_{ff} \times Cf + bs''_{ft} \times Ct B''_t = bs''_{tf} \times Cf + bs''_{tt} \times Ct B'' = Cf^\top \times B''_f + Ct^\top \times B''_t

The fast-decoupled algorithm

  • Factorize B'

    J1 = factorization(B')

  • Factorize B''

    J2 = factorization(B'')

  • Compute the voltage module V_m = |V|

  • Compute the voltage angle V_a= atan \left ( \frac{V_i}{V_r} \right )

  • Compute the error

    S_{calc} = V \cdot \left( Ybus \times V - I_{bus} \right)^*

    \Delta S = \frac{S_{calc} - S_{bus}}{V_m}

    \Delta P = Re \left\{\Delta S[pqpv] \right\}

    \Delta Q = Im \left\{ \Delta S[pq] \right\}

  • Check the convergence

    converged = |\Delta P|_{\infty} < tol \quad \& \quad|\Delta Q|_{\infty} < tol

  • Iterate; While convergence is false and the number of iterations is less than the maximum:

    • Solve voltage angles (P-iteration)

      \Delta V_a = J1.solve( \Delta P)

    • Update voltage

      V_a[pqpv] = V_a[pqpv] - \Delta V_a V = V_m \cdot e^{j \cdot V_a}

    • Compute the error (follow the previous steps)

    • Check the convergence (follow the previous steps)

    • If the convergence is still false:

      • Solve voltage modules (Q-iteration)

        \Delta V_m = J2.solve( \Delta Q)

      • Update voltage

        V_m[pq] = V_m[pq] - \Delta V_m V = V_m \cdot e^{j \cdot V_a}

      • Compute the error (follow the previous steps)

      • Check the convergence (follow the previous steps)

    • Increase the iteration counter.

  • End

[FD]
  1. Stott and O. Alsac, 1974, Fast Decoupled Power Flow, IEEE Trasactions PAS-93 859-869.