--- title: "Equivalent swing equation, coherent machines" subject: "Power System Analysis II" chapter: 6 key: "semester_5/power_system_analysis_ii:equivalent-swing-equation-coherent" source: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/equivalent-swing-equation-coherent site: BCT Engineering --- # Equivalent swing equation, coherent machines Power System Analysis II · Chapter 6: Power System Stability - Asked in: 2082 Bhadra (new course), 4+3 marks - Video: [Lec 05 Power System Stability Swing equation of machines swinging coherently and non coherently](https://www.youtube.com/watch?v=HmlYSFSfQhM) — Chinmay Deshpande - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/equivalent-swing-equation-coherent - Whole chapter: https://www.bctengineer.com/md/semester_5/power_system_analysis_ii/chapter-6 ## Question Starting from the single machine swing equation, derive the expression for equivalent swing equation for multiple generators swinging together. Three generators in a power plant of 50 MVA, 100 MVA & 200 MVA have inertia constant of 6 MJ/MVA each. Compute the equivalent inertia constant at a base of 100 MVA if all units swing together. ## Short answer For coherent machines, express each H on the system base and add: Heq = sum Hi Si / Ssys. With H = 6 MJ/MVA for 50, 100 and 200 MVA units on a 100 MVA base, Heq = (6*50 + 6*100 + 6*200)/100 = 21 MJ/MVA. ## Answer When several machines in one plant swing together (**coherent machines**), they can be replaced by one equivalent machine. Its inertia constant is the MVA-weighted sum of the individual inertias, expressed on the system base. #### Derivation The swing equation of machine $i$ on its own rating $S_{i}$ is $$ \frac{2H_i}{\omega_s}\frac{d^2\delta_i}{dt^2} = P_{mi} - P_{ei}\quad \text{(pu on } S_i) $$ Convert it to a common system base $S_{sys}$ by multiplying both sides by $S_i/S_{sys}$: $$ \frac{2H_{i,sys}}{\omega_s}\frac{d^2\delta_i}{dt^2} = P_{mi} - P_{ei}\quad \text{(pu on } S_{sys}), \qquad H_{i,sys} = H_i\frac{S_i}{S_{sys}} $$ The machines are coherent, so they swing together: $\delta_1 = \delta_2 = \dots = \delta$. Adding the $n$ equations gives $$ \begin{aligned} \frac{2}{\omega_s}\left(\sum_{i=1}^{n} H_{i,sys}\right)\frac{d^2\delta}{dt^2} &= \sum P_{mi} - \sum P_{ei} \\ \frac{2H_{eq}}{\omega_s}\frac{d^2\delta}{dt^2} &= P_m - P_e \end{aligned} $$ where $$ H_{eq} = \sum_{i=1}^{n} \frac{H_i\,S_i}{S_{sys}}, \quad P_m = \sum P_{mi}, \quad P_e = \sum P_{ei} $$ This is the equivalent swing equation: a single machine with inertia $H_{eq}$ and the total power of the plant. #### Numerical Given $H = 6$ MJ/MVA for each unit, ratings 50, 100 and 200 MVA, and a base of 100 MVA: $$ \begin{aligned} H_{eq} &= \frac{6\times 50 + 6\times 100 + 6\times 200}{100} \\ &= \frac{300 + 600 + 1200}{100} = \frac{2100}{100} \\ &= 21\ \text{MJ/MVA} \end{aligned} $$ Check: the total stored energy is $2100$ MJ. On a 100 MVA base this gives $2100/100 = 21$ MJ/MVA. **Answer: $H_{eq} = 21$ MJ/MVA on a 100 MVA base.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What does it mean that machines swing coherently? A. They have the same MVA rating B. They have the same H C. They have equal rotor angles at all times, delta1 = delta2 = ... = delta D. They produce no electrical power Answer: C. Coherent machines swing together, so a single common delta can be used when adding their equations. (Key point: Coherent machines) 2. The equivalent inertia constant on the system base is: A. Sum of Hi without weighting B. Average of Hi C. Hi x Ssys / Si D. Sum of Hi x Si / Ssys Answer: D. Each Hi is converted to the system base by multiplying by Si/Ssys and then added. (Key point: Equivalent H formula) 3. Three units 50, 100, 200 MVA, each with H = 6, on a 100 MVA base give Heq of: A. 21 MJ/MVA B. 6 MJ/MVA C. 18 MJ/MVA D. 12 MJ/MVA Answer: A. Heq = 6(50+100+200)/100 ... computed as (300+600+1200)/100 = 21 MJ/MVA. 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