Bus quantities and decoupled load flow steps

Specified and unspecified quantities

The slack bus supplies the losses and balances generation and load; a PV bus whose Q hits a limit is converted to a PQ bus.

Assumptions in decoupled load flow

1. $X \gg R$ for transmission lines, so $G_{ik} \ll B_{ik}$.
2. Angle differences across lines are small: $\sin(\delta_i - \delta_k) \approx 0$, $\cos(\delta_i - \delta_k) \approx 1$.
3. So P depends mainly on $\delta$ and Q mainly on $|V|$: $J_2 = \partial P/\partial|V| \approx 0$ and $J_3 = \partial Q/\partial\delta \approx 0$.

The N-R equations then split into:

$$\Delta P = J_1\,\Delta\delta,\qquad \Delta Q = J_4\,\Delta|V|$$

Steps

1. Form $Y_{bus}$; flat start ($\delta = 0$, $|V| = 1.0$ at PQ buses, $|V|^{sp}$ at PV buses).
2. Calculate $P_i$ for all non-slack buses and $Q_i$ for PQ buses.
3. Find mismatches $\Delta P$ and $\Delta Q$; stop if all are below tolerance.
4. Form $J_1$ and solve $\Delta\delta = J_1^{-1}\Delta P$; update $\delta$.
5. Form $J_4$ and solve $\Delta|V| = J_4^{-1}\Delta Q$; update $|V|$.
6. Check Q limits of PV buses; repeat from step 2.
7. After convergence, compute slack power, line flows and losses.