POWER SYSTEM STABILITY – GATE LEVEL NOTES

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1. DEFINITION OF POWER SYSTEM STABILITY

Power system stability is the ability of the system to return to normal operating condition after a disturbance.

Disturbances include:

• Faults
• Sudden load change
• Line tripping
• Generator outage

Stable system → returns to equilibrium
Unstable system → loses synchronism

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2. SYNCHRONISM CONCEPT (VERY IMPORTANT)

Generator stability depends on maintaining synchronism.

Condition:

$$\delta = \text{constant or bounded}$$

Where:

$$\delta = \text{power angle (rotor angle)}$$

If δ increases continuously → system unstable

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3. POWER-ANGLE EQUATION (MOST IMPORTANT FOR GATE)

Power transferred:

$$P = \frac{EV}{X} \sin \delta$$

Where:

E = Generator voltage
V = Bus voltage
X = Reactance
δ = Power angle

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Maximum Power Transfer

Occurs when:

$$\delta = 90^\circ$$ $$P_{max} = \frac{EV}{X}$$

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Stability condition:

$$\delta < 90^\circ$$

If δ > 90° → unstable

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4. SWING EQUATION (MOST IMPORTANT)

Describes rotor motion:

$$\frac{2H}{\omega_s} \frac{d^2 \delta}{dt^2} = P_m - P_e$$

Where:

H = inertia constant
Pm = mechanical power
Pe = electrical power

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If:

$$P_m = P_e$$

System stable

If:

$$P_m \ne P_e$$

Rotor accelerates or decelerates

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5. TYPES OF POWER SYSTEM STABILITY

VERY IMPORTANT FOR GATE

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1. Steady-State Stability

Ability to maintain synchronism for small gradual disturbances.

Example:

Slow load increase

Maximum power:

$$P_{max} = \frac{EV}{X}$$

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2. Transient Stability (MOST IMPORTANT)

Ability to maintain synchronism after large disturbance.

Examples:

• Fault
• Line tripping
• Generator outage

Occurs in first few seconds.

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3. Dynamic Stability

Stability under small oscillations over longer time.

Time range:

Several seconds

Includes effect of control systems.

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SUMMARY TABLE

Stability Type	Disturbance	Time
Steady state	Small disturbance	Long term
Transient	Large disturbance	Short term
Dynamic	Small oscillations	Medium term

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6. EQUAL AREA CRITERION (VERY IMPORTANT FOR GATE)

Used to determine transient stability.

Condition:

Accelerating area = Decelerating area

$$A_1 = A_2$$

If:

$$A_1 > A_2$$

System unstable

If:

$$A_1 \le A_2$$

System stable

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Explanation:

Accelerating power:

$$P_a = P_m - P_e$$

If Pa > 0 → rotor accelerates
If Pa < 0 → rotor decelerates

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7. CRITICAL CLEARING ANGLE

Maximum allowable angle before system becomes unstable.

Denoted by:

$$\delta_{cr}$$

If fault cleared before this → stable

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8. CRITICAL CLEARING TIME

Maximum time allowed to clear fault.

If fault cleared before critical clearing time → stable

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9. FACTORS AFFECTING STABILITY

VERY IMPORTANT THEORY

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Improves stability:

Increase inertia (H)

Reduce reactance (X)

Increase voltage (V)

Fast fault clearing

Use AVR

Use FACTS devices

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Reduces stability:

High reactance

Low voltage

Slow fault clearing

Heavy loading

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10. STABILITY LIMIT

Maximum power system can transmit without losing stability.

$$P_{max} = \frac{EV}{X}$$

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11. TYPES OF ROTOR ANGLE STABILITY

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1. Small Signal Stability

Small disturbances

Linear analysis used

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2. Transient Stability

Large disturbances

Nonlinear analysis

Equal area criterion used

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12. PRACTICAL METHODS TO IMPROVE STABILITY

VERY IMPORTANT FOR GATE THEORY

Fast circuit breakers

Automatic voltage regulator (AVR)

Reduce transmission reactance

Use series compensation

Use FACTS devices

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13. GATE IMPORTANT FORMULAS SUMMARY

Power-angle equation:

$$P = \frac{EV}{X} \sin \delta$$

Maximum power:

$$P_{max} = \frac{EV}{X}$$

Swing equation:

$$\frac{2H}{\omega_s} \frac{d^2 \delta}{dt^2} = P_m - P_e$$

Equal area criterion:

$$A_1 = A_2$$

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14. STABILITY TYPES ORDER (IMPORTANT)

Fastest instability:

Transient stability

Then:

Dynamic stability

Then:

Steady state stability

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15. ONE-PAGE REVISION SHEET

Stability → ability to maintain synchronism

Power equation:

$$P = \frac{EV}{X} \sin \delta$$

Maximum power at:

$$\delta = 90^\circ$$

Equal area criterion:

$$A_1 = A_2$$

Swing equation:

$$\frac{2H}{\omega_s} \frac{d^2 \delta}{dt^2} = P_m - P_e$$