🎓 Lesson 2
D2
Understanding Rotor Dynamics and Swing Equation
The swing equation describes how a generator’s rotor slows down or speeds up when electrical power demand changes, like a spinning flywheel resisting sudden pushes or pulls.
🎯 Learning Objectives
- ✓ Calculate rotor acceleration and swing curve for a given fault scenario using the swing equation
- ✓ Analyze transient stability by interpreting critical clearing time from numerical integration of the swing equation
- ✓ Explain the physical meaning of inertia constant H and its impact on system damping and fault resilience
- ✓ Apply equal-area criterion to assess stability margin without solving the differential equation
📖 Why This Matters
In mining operations, large synchronous motors drive crushers, conveyors, and ventilation fans—often fed by dedicated diesel-generator or microgrid systems. A sudden short-circuit fault (e.g., due to wet tunnel conditions or cable damage) can cause rapid deceleration of these rotors. If the swing equation predicts loss of synchronism, equipment trips, halting production and risking safety. Understanding rotor dynamics isn’t theoretical—it’s the difference between a 3-minute restart and a 12-hour downtime.
📘 Core Principles
Rotor dynamics begins with Newton’s second law for rotation: net torque = J·α, where J is moment of inertia and α is angular acceleration. In power systems, we recast this using normalized energy-based variables: the inertia constant H (MJ/MVA), which represents stored kinetic energy at synchronous speed; the mechanical power P_m (pu); and electrical power P_e (pu). The swing equation emerges as d²δ/dt² = (P_m − P_e) / H, where δ is the rotor angle relative to system reference. Nonlinearity arises because P_e depends on sin(δ) — making stability inherently geometric. Small-signal linearization yields eigenvalues governing oscillatory modes; full nonlinear form enables time-domain simulation of severe faults.
📐 Swing Equation (Classical Model)
The classical swing equation expresses rotor motion under disturbance. It assumes constant mechanical input, negligible damping, and rigid coupling between rotor and prime mover. Used in initial stability screening, EAC analysis, and as the core ODE in transient stability programs like PSS/E or EMTP.
Swing Equation (per-unit)
d²δ/dt² = (P_m − P_e) / HRelates rotor angular acceleration to power imbalance and machine inertia.
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| d²δ/dt² | Rotor angular acceleration | electrical deg/s² | Rate of change of rotor angle relative to system reference frame |
| P_m | Mechanical power input | per unit (pu) | Turbine or prime mover output referenced to machine MVA rating |
| P_e | Electrical power output | per unit (pu) | Electromagnetic power delivered to the grid, function of δ and system reactance |
| H | Inertia constant | seconds (s) | Kinetic energy stored at synchronous speed divided by rated MVA: H = (½ Jω_s²) / S_base |
Typical Ranges:
Large steam turbine generator: 3.0 – 8.0 s
Hydro generator: 2.0 – 5.0 s
Mining synchronous motor (SAG/AG mill): 1.5 – 4.5 s
💡 Worked Example
Problem: A 50 MVA, 60 Hz synchronous generator has H = 4.0 s. During a fault, P_m = 0.9 pu and P_e drops instantaneously to 0.2 pu. Calculate initial angular acceleration (d²δ/dt²) in electrical degrees/s².
1.
Step 1: Identify knowns: H = 4.0 s, P_m = 0.9 pu, P_e = 0.2 pu → net power = 0.7 pu
2.
Step 2: Apply swing equation: d²δ/dt² = (P_m − P_e) / H = 0.7 / 4.0 = 0.175 pu/s²
3.
Step 3: Convert to electrical degrees/s²: 1 pu of angular acceleration = 360×f = 360×60 = 21,600 °ₑ/s² → 0.175 × 21,600 = 3,780 °ₑ/s²
Answer:
The initial rotor acceleration is 3,780 electrical degrees per second squared — a rapid rate indicating high stress on shaft coupling and potential loss of synchronism if sustained beyond ~150 ms.
🏗️ Real-World Application
At the Rio Tinto Iron Ore Yandicoogina mine (Pilbara, WA), a 33 kV substation fault caused voltage collapse across six 12 MW SAG mill drives. Transient simulations using the swing equation revealed critical clearing time of 112 ms for the relay scheme. Field testing confirmed instability occurred at 125 ms — prompting upgrade to IEC 61850-compliant high-speed relays with 45 ms tripping. Post-upgrade, no further mill desynchronization events occurred over 3 years — validating swing-based protection coordination.