--- title: "Fast decoupled method approximations and algorithm" subject: "Power System Analysis II" chapter: 2 key: "semester_5/power_system_analysis_ii:fast-decoupled-approximations" source: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/fast-decoupled-approximations site: BCT Engineering --- # Fast decoupled method approximations and algorithm Power System Analysis II · Chapter 2: Load Flow Analysis - Asked in: 2075 Chaitra, 2+2+4 marks - Video: [Fast Decoupled Load Flow - Load Flow Studies - Power System 3](https://www.youtube.com/watch?v=9BhsZClvkmc) — Ekeeda - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/fast-decoupled-approximations - Whole chapter: https://www.bctengineer.com/md/semester_5/power_system_analysis_ii/chapter-2 ## Question What are the approximations in fast decoupled method? How does this method improve the computational efficiency than the NR method in power flow problem? Write an algorithm for this method. ## Short answer Approximations: X>>R, small angle differences, neglect Qi term, |V|≈1 pu. This gives constant B' and B'' matrices, factorized once. Per-iteration fast, less memory; only forward/back substitution. Algorithm: form B', B''; flat start; compute ΔP/|V|, solve -B'Δδ=ΔP/|V|; compute ΔQ/|V|, solve -B''Δ|V|; update; repeat until convergence. ## Answer The **Fast Decoupled Load Flow (FDLF)** method (Stott and Alsac) simplifies the decoupled N-R method further so that the two Jacobian submatrices become **constant** matrices $B'$ and $B''$, formed once. #### Approximations 1. **Decoupling:** $X \gg R$, so $\partial P/\partial|V| \approx 0$ and $\partial Q/\partial\delta \approx 0$ ($J_2 = J_3 = 0$). 2. **Small angle differences:** $\cos(\delta_i - \delta_k) \approx 1$, $\sin(\delta_i - \delta_k) \approx 0$; with $G_{ik}\sin(\delta_i - \delta_k) \ll B_{ik}$. 3. $Q_i \ll B_{ii}|V_i|^2$, so the $Q_i$ term in the diagonal elements is neglected. 4. Voltages near 1.0 pu, so $|V_k| \approx 1$ in the Jacobian terms. With these, $J_1 \approx -|V|B'|V|$ and $J_4 \approx -|V|B''$, giving: $$ \frac{\Delta P}{|V|} = -B'\,\Delta\delta,\qquad \frac{\Delta Q}{|V|} = -B''\,\Delta|V| $$ - $B'$: imaginary part of $Y_{bus}$ for non-slack buses (order $N-1$). Usually shunts, line charging, off-nominal taps and series resistance are omitted from it. - $B''$: imaginary part of $Y_{bus}$ for PQ buses only (order $N-1-m$). #### Why it is more efficient than N-R 1. $B'$ and $B''$ are **constant**: formed and factorised (LU) **only once**; full N-R recomputes and refactorises the Jacobian each iteration. 2. The two matrices are **real, symmetric and sparse** and about half the size of the full Jacobian, so less memory and fewer operations. 3. Each iteration is only forward/back substitution, so it is several times faster per iteration. 4. More iterations are needed (linear-type convergence), but total time is much less; very suitable for on-line and contingency analysis (many repeated load flows). 5. Accuracy of the final answer is not lost, since the mismatches $\Delta P$, $\Delta Q$ are still computed exactly. #### Algorithm 1. Read data; form $Y_{bus}$, then form $B'$ and $B''$ and factorise them. 2. Flat start: $\delta = 0$; $|V| = 1.0$ at PQ buses and $|V|^{sp}$ at PV buses. 3. Compute $P_i$ at non-slack buses; $\Delta P_i = P_i^{sp} - P_i$; form $\Delta P/|V|$. 4. Solve $-B'\Delta\delta = \Delta P/|V|$; update $\delta \leftarrow \delta + \Delta\delta$. 5. Compute $Q_i$ at PQ buses with the new angles; $\Delta Q_i = Q_i^{sp} - Q_i$; form $\Delta Q/|V|$. (Check PV-bus Q limits.) 6. Solve $-B''\Delta|V| = \Delta Q/|V|$; update $|V| \leftarrow |V| + \Delta|V|$. 7. If $\max|\Delta P|$ and $\max|\Delta Q|$ are less than tolerance, go to step 8; else go to step 3. 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. Which is NOT an approximation of the fast decoupled method? A. cos(delta_i - delta_k) is about 1 B. Qi is much smaller than Bii|Vi|^2 C. Voltages are near 1.0 pu in the Jacobian terms D. R is much larger than X in the lines Answer: D. The method assumes X >> R (high X/R), not R >> X. (Key point: Approximations used) 2. What are the sizes of B' and B''? A. Both of order N B. B' of order N-1 (non-slack buses); B'' of order N-1-m (PQ buses) C. B' of order N-1-m; B'' of order N-1 D. Both of order 2N-2-m Answer: B. B' relates angles at all non-slack buses and B'' relates magnitudes at PQ buses only. (Key point: Matrices B' and B'') 3. Why is each fast decoupled iteration faster than an N-R iteration? A. B' and B'' are constant, factorised once, so only forward/back substitution is done B. It needs fewer buses C. It does not compute mismatches D. It uses complex voltages instead of polar form Answer: A. The constant, real, symmetric, sparse matrices are factorised once; the Jacobian is not rebuilt each time. (Key point: Efficiency over N-R) 4. How does the fast decoupled method compare in convergence and accuracy with N-R? A. Fewer iterations and lower accuracy B. More iterations but the same final accuracy, since mismatches are computed exactly C. Same iterations and lower accuracy D. Quadratic convergence and lower accuracy Answer: B. Convergence is slower (more iterations) but the accuracy is not lost because dP and dQ are exact. 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