Decoupled load flow assumptions and algorithm

The decoupled load-flow method is a simplification of Newton–Raphson in which the weak coupling between P and $|V|$ and between Q and $\delta$ is neglected, so the large N-R system splits into two smaller ones.

Assumptions

1. Transmission lines have high X/R ratio ($X \gg R$), so $G_{ik} \ll B_{ik}$.
2. Angle differences between adjacent buses are small, so $\sin(\delta_i - \delta_k) \approx 0$ (small) and $\cos(\delta_i - \delta_k) \approx 1$.
3. Hence real power depends mainly on angles and reactive power mainly on voltage magnitudes: $\partial P/\partial |V| \approx 0$ (submatrix $J_2 = 0$) and $\partial Q/\partial \delta \approx 0$ (submatrix $J_3 = 0$).

Derivation

Full N-R equations:

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

Setting $J_2 = J_3 = 0$:

$$\Delta P = J_1\,\Delta\delta = \left[\frac{\partial P}{\partial \delta}\right]\Delta\delta,\qquad \Delta Q = J_4\,\Delta|V| = \left[\frac{\partial Q}{\partial |V|}\right]\Delta|V|$$

where

$$\begin{aligned} \frac{\partial P_i}{\partial \delta_i} &= \sum_{k \ne i} |V_i||V_k||Y_{ik}|\sin(\theta_{ik} - \delta_i + \delta_k) \\ \frac{\partial P_i}{\partial \delta_k} &= -|V_i||V_k||Y_{ik}|\sin(\theta_{ik} - \delta_i + \delta_k) \\ \frac{\partial Q_i}{\partial |V_i|} &= -2|V_i||Y_{ii}|\sin\theta_{ii} - \sum_{k \ne i} |V_k||Y_{ik}|\sin(\theta_{ik} - \delta_i + \delta_k) \\ \frac{\partial Q_i}{\partial |V_k|} &= -|V_i||Y_{ik}|\sin(\theta_{ik} - \delta_i + \delta_k) \end{aligned}$$

$J_1$ is of order $(n-1)$ (all buses except slack) and $J_4$ is of order $(n-1-m)$ (PQ buses only, $m$ = number of PV buses).

Algorithm

1. Form $Y_{bus}$. Set flat start: $|V_i| = 1.0$ at PQ buses, specified $|V|$ at PV buses, $\delta_i = 0$.
2. Compute $P_i^{cal}$ (all non-slack buses) and $Q_i^{cal}$ (PQ buses).
3. Find mismatches $\Delta P_i = P_i^{sp} - P_i^{cal}$, $\Delta Q_i = Q_i^{sp} - Q_i^{cal}$.
4. If all $|\Delta P|, |\Delta Q| < \varepsilon$, go to step 8.
5. Form $J_1$ and solve $\Delta\delta = J_1^{-1}\Delta P$; update $\delta \leftarrow \delta + \Delta\delta$.
6. Form $J_4$ and solve $\Delta|V| = J_4^{-1}\Delta Q$; update $|V| \leftarrow |V| + \Delta|V|$. Check PV-bus Q limits.
7. Go to step 2.
8. Compute slack bus power, line flows and losses.