📘 Multi-Machine Stability – Relative Swing Equation
In real power systems, multiple generators are interconnected. We study stability based on relative rotor motion.
🔹 1️⃣ Why Multi-Machine Analysis?
Single Machine Infinite Bus (SMIB) is simplified. Real systems: • Many generators • Different inertia • Different mechanical inputs • Interconnected through transmission network Stability depends on relative rotor angle differences.🔹 2️⃣ Swing Equation for Each Machine
For machine i:Mi d²Î´i/dt² = Pmi − Pei
Each generator has its own swing equation.🔹 3️⃣ Two Machine System
Let machines 1 and 2: M1 d²Î´1/dt² = Pm1 − Pe1 M2 d²Î´2/dt² = Pm2 − Pe2 We define relative angle:δ = δ1 − δ2
Subtract equations: Equivalent swing equation:M_eq d²Î´/dt² = Pm − Pe
Where: 1 / M_eq = 1/M1 + 1/M2 This reduces two-machine system into equivalent SMIB system.🔹 4️⃣ Center of Inertia (COI)
For multiple machines: COI angle: δ_coi = (Σ Mi δi) / (Σ Mi) We measure rotor angles relative to COI. This removes reference speed.🔹 5️⃣ Stability Interpretation
System stable if: • All relative rotor angles remain bounded • No machine slips pole • Oscillations are damped Unstable if: • Angle increases continuously • Loss of synchronism🔹 6️⃣ Important Observations
- Higher inertia improves stability
- Strong transmission network improves stability
- Fault location affects stability
- Generators may separate into groups (coherent machines)
🎯 GATE Focus
- Equivalent inertia calculation
- Relative angle concept
- COI formula
- Conceptual stability questions
Multi-Machine Stability = Relative Rotor Motion Stability
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