📘 Two Machine Stability – Numerical (Equivalent Swing Equation)
We reduce a two-machine system into equivalent single machine system using relative rotor angle method.
🔹 Problem Statement
Two generators connected through transmission line. Generator 1: H1 = 6 MJ/MVA Generator 2: H2 = 4 MJ/MVA Mechanical powers: Pm1 = 1 pu Pm2 = 0.8 pu Electrical power transfer between machines: Pe = Pmax sin(δ1 − δ2) Find: 1️⃣ Equivalent inertia constant 2️⃣ Equivalent swing equation🔹 Step 1 – Calculate M for Each Machine
Formula: M = 2H / ωs Assume: f = 50 Hz ωs = 2Ï€f = 314 rad/sec Machine 1: M1 = 2×6 / 314 = 12 / 314 = 0.0382 Machine 2: M2 = 2×4 / 314 = 8 / 314 = 0.0255🔹 Step 2 – Equivalent Inertia
Formula: 1 / M_eq = 1/M1 + 1/M2 = 1/0.0382 + 1/0.0255 = 26.18 + 39.21 = 65.39 So: M_eq = 1 / 65.39M_eq ≈ 0.0153
🔹 Step 3 – Relative Angle
Define: δ = δ1 − δ2 Electrical power transfer: Pe = Pmax sin δ🔹 Step 4 – Equivalent Swing Equation
Equivalent mechanical power difference: Pm = Pm1 − Pm2 = 1 − 0.8 = 0.2 pu So swing equation:M_eq d²Î´/dt² = Pm − Pe
Substitute: 0.0153 d²Î´/dt² = 0.2 − Pmax sin δ🔹 Interpretation
If: Pm > Pe → acceleration Pm < Pe → deceleration Stability depends on balance.🎯 Key Learning
- Two-machine system reduced to SMIB form
- Use relative rotor angle
- Equivalent inertia smaller than individual
- Stability depends on power difference
Two Machine Stability = Relative Swing Stability
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