Lab
IEEE 39-Bus Transient Stability
Transient stability on the same IEEE 39-bus system at three levels of model fidelity, from the classical model to full GENROU + AVR + PSS dynamics, plus small-signal modal analysis — apply a three-phase fault at any bus, choose which line trips to clear it, and watch every generator's rotor angle swing relative to the center of inertia. Includes a bisection search for the critical clearing time.
In plain terms
When a fault (like lightning striking a line) hits a power grid, generators briefly speed up or slow down relative to each other — like a group of runners jostled off their pace. If the fault is cleared quickly enough, they settle back into step; clear it too slowly, and one or more can permanently fall out of sync with the rest, which is how blackouts start. This tool simulates exactly that moment-by-moment for all ten generators in the IEEE 39-bus system, and can automatically find the 'critical clearing time' — the slowest a fault can be cleared before the grid loses synchronism for good.
How this is built — three fidelity levels sharing one network: Classical holds each generator as a constant-magnitude EMF behind its transient reactance x'd, with mechanical power Pm frozen at its pre-fault value — the same simplification the equal-area criterion is built on. + governor adds TGOV1N (droop → anti-windup valve lag → lead-lag, verified against ANDES's own implementation) so Pm responds to speed deviation instead of staying frozen; a droop governor alone doesn't damp the swing itself, it mainly stops Pm from staying frozen, and a small permanent offset after a disturbance is the expected signature of proportional-only droop control without secondary (AGC) correction. Full detailed replaces the constant EMF with GENROU, the standard 6th-order round-rotor machine model (two-axis flux dynamics behind sub-transient reactance x'', xd2 = xq2), adds IEEEX1 (a DC-type AVR: sensing → lead-lag → anti-windup-limited amplifier → saturating field-voltage integrator → stabilizing feedback washout) regulating terminal voltage, and IEEEST (a power- system stabilizer feeding a damping signal derived from electrical power back into the AVR reference) — all three transcribed and derived directly from ANDES's source, together with the same TGOV1N governor. This is what actually damps the electromechanical oscillation the classical model can only sustain forever. Loads are converted to constant- impedance shunts at the pre-fault operating point, generator internal nodes and the fault are added to the network, and every non-generator bus is eliminated by Kron reduction — separately for the pre-fault, faulted and post-fault topologies, and (for the detailed model) behind x'' instead of x'd — leaving a compact network whose generator- side phasors are reconstructed from the current flux state every step, integrated with fixed-step RK4. All machine, governor, exciter and PSS data is transcribed from the dynamic case bundled with ANDES (see references); the classical and +governor levels carry zero explicit mechanical damping, so a stable case there shows sustained undamped oscillation, not decay. "Critical clearing time" is found by bisection on clearing time alone, holding the fault location and tripped line fixed, re-running the selected fidelity level at each trial.
Explore: IEEE 39-Bus Power Flow
The same network's steady-state solution — scale load, take a line out for N-1, and see voltages and loadings on the diagram.
Further reading
- P. Kundur, Power System Stability and Control. New York: McGraw-Hill, 1994. — the classical model, swing equation and network reduction (Ch. 13) and small-signal stability, participation factors and PSS (Ch. 12)
- P. M. Anderson and A. A. Fouad, Power System Control and Stability, 2nd ed. Piscataway, NJ: IEEE Press / Wiley, 2003. — worked multi-machine classical-model transient stability examples on this same system
- H. Cui, F. Li and K. Tomsovic, “Hybrid symbolic-numeric framework for power system modeling and analysis” (the ANDES library), IEEE Transactions on Power Systems, vol. 36, no. 2, pp. 1373–1384, Mar. 2021. — the ANDES library, source of the generator, governor, exciter and PSS data used here
- T. Athay, R. Podmore and S. Virmani, “A practical method for the direct analysis of transient stability,” IEEE Transactions on Power Apparatus and Systems, vol. PAS-98, no. 2, pp. 573–584, Mar. 1979. — the original New England 39-bus dynamic data set
- IEEE Std 421.5-2016, IEEE Recommended Practice for Excitation System Models for Power System Stability Studies. New York: IEEE, 2016. — the standard the IEEEX1 exciter and IEEEST stabilizer models are named for