--- title: "3-bus load flow with Q limits (cut off)" subject: "Power System Analysis II" chapter: 2 key: "semester_5/power_system_analysis_ii:load-flow-reference-load-generator-bus" source: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/load-flow-reference-load-generator-bus site: BCT Engineering --- # 3-bus load flow with Q limits (cut off) Power System Analysis II · Chapter 2: Load Flow Analysis - Asked in: 2074 Asoj - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/load-flow-reference-load-generator-bus - Whole chapter: https://www.bctengineer.com/md/semester_5/power_system_analysis_ii/chapter-2 ## Question Suppose you are given a 3-bus power system network with one reference bus, one load bus and one generator bus with reactive power limits and asked to perform the [rest of the question is cut off in the scan: ...unknown variables and total losses in the network]. ## Short answer Form Ybus and start with a flat voltage profile. For PV bus 2, compute Q2, check Q-limits; if violated fix Q and treat as PQ, else correct angle and reset |V2|. Update PQ bus 3 with the Gauss–Seidel formula. Iterate until convergence, then compute slack bus power and line flows to obtain total losses. ## Answer The scanned question is cut off. It is taken here as: *describe how to carry out the load flow of such a 3-bus system (Gauss-Seidel method) to find the unknown variables and the total losses.* Bus 1 is the reference (slack) bus, bus 2 the generator (PV) bus with Q-limits, and bus 3 the load (PQ) bus. #### Known and unknown quantities | Bus | Type | Known | Unknown | |---|---|---|---| | 1 | Reference (slack) | $\lvert V_1\rvert$, $\delta_1 = 0$ | $P_1$, $Q_1$ | | 2 | Generator (PV) | $P_2$, $\lvert V_2\rvert$, $Q_{min}$, $Q_{max}$ | $Q_2$, $\delta_2$ | | 3 | Load (PQ) | $P_3$, $Q_3$ | $\lvert V_3\rvert$, $\delta_3$ | #### Step-by-step procedure 1. **Form $Y_{bus}$** from the line data: $Y_{ii}$ = sum of admittances connected to bus $i$ (including half line charging), $Y_{ij} = -y_{ij}$. 2. **Net injections:** $P_i = P_{Gi} - P_{Li}$, $Q_i = Q_{Gi} - Q_{Li}$. 3. **Flat start:** $V_2^{(0)} = |V_2|_{spec}\angle 0^\circ$, $V_3^{(0)} = 1\angle 0^\circ$. 4. **PV bus 2:** find the reactive power $$ Q_2^{(k+1)} = -\text{Im}\left\{V_2^{(k)*}\sum_{j=1}^{3}Y_{2j}V_j\right\} $$ 5. **Check Q-limits.** If $Q_{min} \le Q_2 \le Q_{max}$, keep bus 2 as PV. If not, set $Q_2$ to the violated limit and treat bus 2 as a PQ bus (its voltage magnitude is then not reset). 6. **Update $V_2$:** $$ V_2^{(k+1)} = \frac{1}{Y_{22}}\left[\frac{P_2 - jQ_2}{V_2^{(k)*}} - Y_{21}V_1 - Y_{23}V_3^{(k)}\right] $$ If bus 2 is still PV, keep only the angle and reset the magnitude to $|V_2|_{spec}$. 7. **Update $V_3$** using the newest $V_2$: $$ V_3^{(k+1)} = \frac{1}{Y_{33}}\left[\frac{P_3 - jQ_3}{V_3^{(k)*}} - Y_{31}V_1 - Y_{32}V_2^{(k+1)}\right] $$ 8. **Convergence test:** stop when $|V_i^{(k+1)} - V_i^{(k)}| < \varepsilon$ (e.g. 0.0001) for all buses. Otherwise repeat from step 4. An acceleration factor of about 1.6 may be used on PQ buses. 9. **Slack bus power:** $$ P_1 - jQ_1 = V_1^{*}\left(Y_{11}V_1 + Y_{12}V_2 + Y_{13}V_3\right) $$ and $P_{G1} = P_1 + P_{L1}$, $Q_{G1} = Q_1 + Q_{L1}$. 10. **Line flows:** $S_{ij} = V_i\left[(V_i - V_j)y_{ij} + V_i\,y_{sh,ij}\right]^{*}$. 11. **Total losses:** $$ \begin{aligned} P_{loss} &= \sum_{i=1}^{3}P_i = P_1 + P_2 + P_3 \\ &= \sum_{\text{lines}}\text{Re}(S_{ij} + S_{ji}) \end{aligned} $$ and similarly $Q_{loss} = \sum Q_i$. ```text Read data -> form Ybus -> flat start | v +--> Q2 from present voltages | check Q-limits (PV or PQ?) | update V2 (reset |V2| if PV) | update V3 | converged? --no--+ | | +---------------------+ | yes v slack P1,Q1 -> line flows -> losses ``` ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which quantities are unknown at the generator (PV) bus 2? A. P2 and |V2| B. |V2| and delta2 C. P2 and Q2 D. Q2 and delta2 Answer: D. At a PV bus P and |V| are specified, so Q and the angle must be found. (Key point: Unknowns by bus type) 2. In Gauss-Seidel, what is done if calculated Q2 violates its limit? A. Q2 is set to the violated limit and bus 2 is treated as PQ B. The slack bus is changed C. The iteration is stopped D. Q2 is set to zero and |V2| is kept Answer: A. Q is fixed at the limit and the voltage magnitude is no longer reset, so the bus behaves as PQ. (Key point: Q-limit violation) 3. Which V2 is used when updating V3 in the same iteration? A. The initial V2(0) B. The newest V2(k+1) C. The average of V2(k) and V2(k+1) D. The specified magnitude with zero angle Answer: B. Gauss-Seidel uses the latest available values immediately, which speeds convergence. (Key point: Sequential update) 4. How is the total real power loss found after convergence? A. P1 alone B. The difference of the largest and smallest bus voltage C. The slack bus Q D. The sum of all bus injections P1 + P2 + P3 Answer: D. Total loss equals the algebraic sum of net bus injections, equal to the sum of line losses. 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