📘 Power System Stability – Swing Equation & Power Angle
Stability means ability of system to return to steady state after disturbance. Most important for transient fault analysis.
🔹 1️⃣ What is Stability?
- Ability of generator to remain in synchronism.
- Occurs during faults, sudden load change, switching.
- Main focus: Rotor angle δ behavior.
🔹 2️⃣ Power Angle Equation (Derivation)
Consider simple system: Generator → Infinite Bus Ignoring resistance: Electrical power transmitted:P_e = (EV / X) sinδ
Where: E = Internal EMF V = Bus voltage X = Reactance δ = Power angle This shows: Power depends on sine of rotor angle. Maximum power: When δ = 90°P_max = EV / X
🔹 3️⃣ Mechanical vs Electrical Power
Mechanical power input: P_m Electrical output power: P_e If: P_m = P_e → Stable If: P_m > P_e → Rotor accelerates P_m < P_e → Rotor decelerates Difference power:P_a = P_m - P_e
This is accelerating power.🔹 4️⃣ Swing Equation Derivation
Newton's Law: Torque = J × angular acceleration Power form:M (d²Î´/dt²) = P_m - P_e
Where: M = Inertia constant δ = Rotor angle This is Swing Equation.🔹 5️⃣ Inertia Constant
In per unit:M = 2H / ω_s
Where: H = Inertia constant (MJ/MVA) ω_s = Synchronous speed🔹 6️⃣ Final Swing Equation
(2H/ω_s) d²Î´/dt² = P_m - P_e
This is second order differential equation.🔹 7️⃣ Important Stability Concepts
- Stable if rotor oscillations damped
- Unstable if δ increases continuously
- Critical clearing angle important
- Equal Area Criterion used
🎯 GATE Focus Points
- Power angle curve (sinδ)
- Maximum power at 90°
- Swing equation form
- Equal area concept
Stability = Rotor Angle Control After Disturbance
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