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

# Decoupled equations and FDLF assumptions

Power System Analysis II · Chapter 2: Load Flow Analysis

- Asked in: 2072 Kartik, 6 marks
- 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-equations-fdlf-assumptions
- Whole chapter: https://www.bctengineer.com/md/semester_5/power_system_analysis_ii/chapter-2

## Question

What do you mean by decoupled load flow equations? List the assumptions to be made in Fast Decoupled load flow method.

## Short answer

Decoupled equations neglect J2 and J3, giving ΔP = J1Δδ and ΔQ = J4Δ|V|. Assumptions for Fast Decoupled: lines have high X/R ratio; angle differences small; Qi << Bii|Vi|^2; |Vi|≈1 pu; in B', omit shunt reactances, line charging, series resistance, off-nominal taps.

## Answer

#### 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.

## Check yourself

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

1. What does decoupling in load flow mean?
   A. P is linked to delta and Q to |V|, by setting J2 and J3 to zero
   B. P is linked to |V| and Q to delta
   C. Bus voltages are fixed and not calculated
   D. The slack bus is removed from the system
   Answer: A. The weak couplings dP/d|V| and dQ/d delta are dropped, giving separate P-delta and Q-V sets. (Key point: Meaning of decoupling)

2. How large is each decoupled system relative to the full Jacobian?
   A. Same size
   B. About a quarter
   C. Twice as large
   D. About one-tenth
   Answer: B. Each of J1 and J4 is roughly a quarter of the full Jacobian. (Key point: Size of decoupled systems)

3. Which set correctly lists FDLF assumptions?
   A. High R/X, large angles, Q much larger than Bii|V|^2
   B. High X/R, small angle differences, Q much smaller than Bii|V|^2, |V| about 1
   C. High X/R, large angles, |V| about 0
   D. Low X/R, small angles, |V| about 1
   Answer: B. These assumptions give constant B' and B'' in dP/|V| = -B' d delta and dQ/|V| = -B'' d|V|. (Key point: FDLF assumptions)

## Related in Power System Analysis II

- [Decoupled load flow assumptions and algorithm](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Adecoupled-assumptions-algorithm) — Power System Analysis II, chapter 2, asked 2×
- [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
- [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
- [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
- [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

## Related in other subjects

- [Unit hydrograph, assumptions, base flow separation](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Auh-assumptions-baseflow-methods) — Engineering Hydrology, chapter 5