Fast decoupled method approximations and algorithm

The Fast Decoupled Load Flow (FDLF) method (Stott and Alsac) simplifies the decoupled N-R method further so that the two Jacobian submatrices become constant matrices $B'$ and $B''$, formed once.

Approximations

1. Decoupling: $X \gg R$, so $\partial P/\partial|V| \approx 0$ and $\partial Q/\partial\delta \approx 0$ ($J_2 = J_3 = 0$).
2. Small angle differences: $\cos(\delta_i - \delta_k) \approx 1$, $\sin(\delta_i - \delta_k) \approx 0$; with $G_{ik}\sin(\delta_i - \delta_k) \ll B_{ik}$.
3. $Q_i \ll B_{ii}|V_i|^2$, so the $Q_i$ term in the diagonal elements is neglected.
4. Voltages near 1.0 pu, so $|V_k| \approx 1$ in the Jacobian terms.

With these, $J_1 \approx -|V|B'|V|$ and $J_4 \approx -|V|B''$, giving:

$$\frac{\Delta P}{|V|} = -B'\,\Delta\delta,\qquad \frac{\Delta Q}{|V|} = -B''\,\Delta|V|$$

• $B'$: imaginary part of $Y_{bus}$ for non-slack buses (order $N-1$). Usually shunts, line charging, off-nominal taps and series resistance are omitted from it.
• $B''$: imaginary part of $Y_{bus}$ for PQ buses only (order $N-1-m$).

Why it is more efficient than N-R

1. $B'$ and $B''$ are constant: formed and factorised (LU) only once; full N-R recomputes and refactorises the Jacobian each iteration.
2. The two matrices are real, symmetric and sparse and about half the size of the full Jacobian, so less memory and fewer operations.
3. Each iteration is only forward/back substitution, so it is several times faster per iteration.
4. More iterations are needed (linear-type convergence), but total time is much less; very suitable for on-line and contingency analysis (many repeated load flows).
5. Accuracy of the final answer is not lost, since the mismatches $\Delta P$, $\Delta Q$ are still computed exactly.

Algorithm

1. Read data; form $Y_{bus}$, then form $B'$ and $B''$ and factorise them.
2. Flat start: $\delta = 0$; $|V| = 1.0$ at PQ buses and $|V|^{sp}$ at PV buses.
3. Compute $P_i$ at non-slack buses; $\Delta P_i = P_i^{sp} - P_i$; form $\Delta P/|V|$.
4. Solve $-B'\Delta\delta = \Delta P/|V|$; update $\delta \leftarrow \delta + \Delta\delta$.
5. Compute $Q_i$ at PQ buses with the new angles; $\Delta Q_i = Q_i^{sp} - Q_i$; form $\Delta Q/|V|$. (Check PV-bus Q limits.)
6. Solve $-B''\Delta|V| = \Delta Q/|V|$; update $|V| \leftarrow |V| + \Delta|V|$.
7. If $\max|\Delta P|$ and $\max|\Delta Q|$ are less than tolerance, go to step 8; else go to step 3.
8. Compute slack bus power, line flows and losses.