advanced Power Systems

Transient Stability — Swing Equation

Single-machine-infinite-bus (SMIB) transient stability with a 3-phase line fault. Slide the inertia, mechanical power, and clearing time and watch the rotor angle trajectory either return to a new equilibrium or run away. The equal-area criterion visualised on the P-δ curve gives the intuition.

Rotor angle δ vs time

Power-angle curve (P_e vs δ) — equal-area visualisation

✓ Stableclearing time: 150 ms · δ_crit (post): 89.4°

H (inertia, s)4.00Pm (mech. power, pu)0.70D (damping, pu)0.00X_post (post-fault X, pu)0.60

Fault inception (s)0.10Fault clearing (s)0.25

Fault is a 3-phase short at the line midpoint. Xduring → ∞ (P_e ≈ 0 during the fault). Post-fault X reflects the surviving topology after the fault is cleared.

Run the default case: H = 4 s, Pm = 0.7 pu, fault from 0.10 s to 0.25 s. The rotor angle swings out during the fault, then oscillates and settles around the new (post-fault) equilibrium because the post-fault topology is weaker. The system is STABLE.

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Equilibrium shifts because the post-fault X is larger (a line was tripped). Pm is unchanged so δ must rise to maintain Pm = Pe.
Push the clearing time to 0.40 s and re-run. The rotor angle now keeps growing past 180° — first-swing instability. The fault was on for too long; the machine accelerated past the unstable equilibrium and the deceleration area can't catch up. This is what 'critical clearing time' bounds.

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Equal-area criterion: the kinetic energy gained during the fault must be returned during the post-fault deceleration. If the available deceleration area (between Pm and Pe(δ)) is too small, the machine slips a pole.
Bring clearing back to 0.20 s but drop H to 1.5 s (a small machine, or a converter-interfaced source acting like one). With less inertia, the angle accelerates faster during the fault. The system that was stable at H = 4 may now go unstable.

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Inertia is becoming scarce on systems with high renewables penetration — synthetic inertia from wind/PV converters and grid-forming converters are the modern responses.
Reset, then increase D (damping) from 0 to 3. The post-fault oscillations damp out far faster. Damping comes physically from amortisseur windings and PSS (power system stabilizers). Without it, even a stable case rings on for many seconds.

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PSS = Power System Stabilizer, an additional control loop that injects a damping signal into the AVR. Standard kit on transmission-connected synchronous machines.
Push Pm up to 0.95 pu (heavy loading). The pre-fault δ₀ is now near 70°, leaving very little deceleration area on the post-fault P-δ curve. Even short faults can de-stabilise a heavily-loaded machine — this is the operating-point dependence of stability.

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This is also why operators de-load critical machines when storms or system disturbances are forecast — they're trading economic dispatch for stability margin.