Decoupled equations and FDLF assumptions

Decoupled load-flow equations

In a transmission system, real power depends mainly on voltage angles and reactive power mainly on voltage magnitudes. Decoupled load-flow equations exploit this by dropping the weak couplings in the N-R equations.

Full N-R:

$$\begin{bmatrix} \Delta P \\ \Delta Q \end{bmatrix} = \begin{bmatrix} J_1 & J_2 \\ J_3 & J_4 \end{bmatrix} \begin{bmatrix} \Delta\delta \\ \Delta|V| \end{bmatrix}$$

Since $J_2 = \partial P/\partial|V|$ and $J_3 = \partial Q/\partial\delta$ are small, they are set to zero, giving two separate (decoupled) sets:

$$\Delta P = \left[\frac{\partial P}{\partial \delta}\right]\Delta\delta,\qquad \Delta Q = \left[\frac{\partial Q}{\partial |V|}\right]\Delta|V|$$

These are solved alternately: the P–δ equations give angle corrections, the Q–V equations give magnitude corrections. Each system is about a quarter the size of the full Jacobian.

Assumptions in the Fast Decoupled load flow method

1. Lines have high X/R ratio, so $G_{ik} \ll B_{ik}$ and $J_2$, $J_3$ are neglected.
2. Angle differences between connected buses are small: $\cos(\delta_i - \delta_k) \approx 1$, $\sin(\delta_i - \delta_k) \approx 0$.
3. $Q_i \ll B_{ii}|V_i|^2$, so $Q_i$ is neglected in the diagonal Jacobian terms.
4. $|V_i| \approx 1.0$ pu in the Jacobian terms.
5. In $B'$, elements that mainly affect Q (shunt reactances, line charging, off-nominal taps) and series resistances are omitted.

The result is

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

with constant matrices $B'$ and $B''$ that are formed and factorised only once.