Rotor dynamics and the swing equation
Inertia, power imbalance and electromechanical generator oscillations.
Before you begin: lessons 16–17, differential equations and RK4.
After this lesson you can: form a two-state rotor model and estimate acceleration and small-oscillation frequency.
Work through the explanation, examples, practical experiment and independent exercises.
Two states
The states are electrical angle δ in radians and per-unit speed deviation Δω. Inertia constant H is kinetic energy at rated speed divided by Sb, measured in seconds. The frequency of the rotor field is fr = f₀(1 + Δω); the infinite-bus grid frequency remains f₀.
dδ/dt = 2πf₀Δω
2H dΔω/dt = Pm − Pe − DΔω
Reading the balance
This classical model assumes speed close to synchronous, using a power balance in place of the exact torque balance. D acts on per-unit speed deviation, not rad/s. If Pm > Pe, the rotor accelerates and its angle grows. On the stable branch, the larger angle raises electrical power, producing a restoring effect. Inertia and synchronizing torque create oscillations.
Flux-linkage transients, saturation, automatic excitation and turbine control are omitted. The model illustrates electromechanical motion, not instantaneous short-circuit currents.
Small oscillations
ωn = √(2πf₀Ks/(2H)); ζ = D/(4Hωn)
For E = 1.2, V = 1, X = 0.6 and P₀ = 0.8, Ks = 1.8330. With H = 4 s and f₀ = 50 Hz, ωn ≈ 8.484 rad/s and the undamped period is about 0.741 s. Increasing H slows oscillations without changing the initial equilibrium.
Independent exercises
1. Immediately after a +0.1 pu Pm step, find dΔω/dt at H = 4 and Δω = 0.
Show solution
0.1/(2·4) = 0.0125 s⁻¹. At 50 Hz the initial rotor-field frequency slope is 0.625 Hz/s.
2. What happens to ωn if H is quadrupled?
Show solution
It halves when Ks stays fixed. With unchanged D, the damping ratio also changes.
Practical work
Apply a +0.1 pu power step without a voltage dip. Compare H = 2, 4 and 8 s, then D = 0 and 10. Record period and amplitude and distinguish the effects of inertia and damping.
Further reading: OpenIPSL: classical GENCLS model. Notation may differ from this course.
Key takeaways
Mechanical power surplus accelerates the rotor and increases angle. Synchronizing power restores the rotor and damping dissipates oscillation energy.
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