---
title: "Decoupled load flow assumptions and algorithm"
subject: "Power System Analysis II"
chapter: 2
key: "semester_5/power_system_analysis_ii:decoupled-assumptions-algorithm"
source: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/decoupled-assumptions-algorithm
site: BCT Engineering
---

# Decoupled load flow assumptions and algorithm

Power System Analysis II · Chapter 2: Load Flow Analysis

- Asked in: 2079 Bhadra, 2+4 marks; 2073 Chaitra, 6 marks (asked 2 times)
- Video: [Decoupled Load Flow Solution Studies - Load Flow Studies - Power System 3](https://www.youtube.com/watch?v=YLOpBRie_bI) — Ekeeda
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/decoupled-assumptions-algorithm
- Whole chapter: https://www.bctengineer.com/md/semester_5/power_system_analysis_ii/chapter-2

## Question

Describe about the assumptions to be made for decoupled load flow method, and hence derive the required equations and write down its algorithm.

## Short answer

Decoupled load flow neglects the weak coupling between P and |V| and Q and δ, assuming high X/R ratio and small angle differences. The Jacobian splits into J1 (∂P/∂δ) and J4 (∂Q/∂|V|). Algorithm: form Ybus, flat start, compute mismatches, solve Δδ from ΔP and Δ|V| from ΔQ separately, update variables, check PV Q limits, iterate to convergence, then compute slack power and line flows.

## Answer

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.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The decoupled method assumes weak coupling between:
   A. P and delta; Q and |V|
   B. P and |V|; Q and delta
   C. P and Q
   D. Delta and |V| only
   Answer: B. dP/d|V| and dQ/d delta are neglected. (Key point: Assumption)

2. Which Jacobian submatrices are set to zero?
   A. J1 and J4
   B. J2 and J3
   C. J1 and J2
   D. J3 and J4
   Answer: B. J2 = dP/d|V| and J3 = dQ/d delta are taken as zero. (Key point: Neglected submatrices)

3. J4 has order:
   A. n-1
   B. n-1-m (PQ buses only)
   C. n
   D. m
   Answer: B. J4 involves only PQ-bus voltage magnitudes. (Key point: Order of J4)

4. In the algorithm, delta is updated from:
   A. Delta Q and J4
   B. Delta P and J1
   C. Delta Q and J1
   D. Delta P and J4
   Answer: B. Delta delta = J1^-1 Delta P, and delta |V| = J4^-1 Delta Q. (Key point: Algorithm)

## Related in Power System Analysis II

- [Need for load flow; NR and decoupled equations](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Aload-flow-need-nr-decoupled) — Power System Analysis II, chapter 2
- [Decoupled equations and FDLF assumptions](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Adecoupled-equations-fdlf-assumptions) — Power System Analysis II, chapter 2
- [Bus quantities and decoupled load flow steps](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Abus-types-decoupled-steps) — Power System Analysis II, chapter 2
- [G-S vs N-R and decoupled assumptions](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Ags-vs-nr-decoupled-assumptions) — Power System Analysis II, chapter 2
- [Basis and advantages of decoupled load flow](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Adecoupled-basis-advantages) — Power System Analysis II, chapter 2
- [Fast decoupled method approximations and algorithm](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Afast-decoupled-approximations) — Power System Analysis II, chapter 2

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