EQUAL AREA CRITERION – GATE LEVEL NOTES

Equal Area Criterion is a graphical method used to determine transient stability of a power system after a large disturbance like a fault.

It is mainly used for:

• Single machine infinite bus (SMIB) system

• Determining stability without solving swing equation

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1. BASIC PRINCIPLE

Stability depends on balance between:

• Accelerating energy

• Decelerating energy

Condition for stability:

$$\textbf{Accelerating Area} = \textbf{Decelerating Area}$$ $$A_1 = A_2$$

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2. POWER–ANGLE EQUATION (MOST IMPORTANT)

Electrical power:

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

Mechanical power:

$$P_m = \text{constant}$$

Accelerating power:

$$P_a = P_m - P_e$$

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3. OPERATING CONDITIONS DURING FAULT

Three stages:

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Stage 1: Before fault (Normal operation)

System stable at angle:

$$\delta_0$$

Condition:

$$P_m = P_e$$

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Stage 2: During fault

Electrical power reduces.

$$P_e \downarrow$$

So:

$$P_m > P_e$$

Rotor accelerates.

Accelerating area formed:

$$A_1$$

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Stage 3: After fault cleared

Electrical power increases.

$$P_e > P_m$$

Rotor decelerates.

Decelerating area formed:

$$A_2$$

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4. STABILITY CONDITION (MOST IMPORTANT)

Stable system:

$$\boxed{A_1 = A_2}$$

Unstable system:

$$A_1 > A_2$$

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5. MATHEMATICAL EXPRESSION

Accelerating area:

$$A_1 = \int_{\delta_0}^{\delta_c} (P_m - P_e) d\delta$$

Decelerating area:

$$A_2 = \int_{\delta_c}^{\delta_m} (P_e - P_m) d\delta$$

Where:

$\delta_0$ = initial angle
$\delta_c$ = clearing angle
$\delta_m$ = maximum angle

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6. GRAPHICAL EXPLANATION (VERY IMPORTANT)

Power-angle curve shows:

• Electrical power curve (sin δ)

• Mechanical power line (constant)

Areas formed between curves represent energy.

Accelerating area:

Between δ₀ and δc

Decelerating area:

Between δc and δm

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7. CLEARING ANGLE

Clearing angle:

$$\delta_c$$

Angle at which fault is cleared.

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8. CRITICAL CLEARING ANGLE

Maximum allowable clearing angle.

$$\delta_{cr}$$

If:

$$\delta_c < \delta_{cr}$$

System stable

If:

$$\delta_c > \delta_{cr}$$

System unstable

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9. CRITICAL CLEARING TIME

Maximum time allowed to clear fault.

If fault cleared before this time → stable

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10. SPECIAL CASE: COMPLETE LOSS OF POWER DURING FAULT

During fault:

$$P_e = 0$$

Then accelerating area:

$$A_1 = P_m (\delta_c - \delta_0)$$

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11. APPLICATIONS

Used for:

• Transient stability analysis

• Fault analysis

• Determining critical clearing angle

• SMIB system analysis

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12. IMPORTANT ASSUMPTIONS

Mechanical power constant

No losses

Voltage constant

System is SMIB

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13. GATE IMPORTANT RESULTS SUMMARY

Stability condition:

$$A_1 = A_2$$

If accelerating area > decelerating area → unstable

Clearing angle must be less than critical angle

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14. STEP-BY-STEP PROCEDURE

Step 1: Draw power-angle curve

Step 2: Draw mechanical power line

Step 3: Find accelerating area

Step 4: Find decelerating area

Step 5: Compare areas

If equal → stable

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15. GATE EXAM IMPORTANT POINTS

Equal area criterion used for:

Transient stability

Used only for:

Single machine infinite bus

Does not require solving swing equation

Graphical method

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16. ONE-PAGE REVISION SHEET

Power equation:

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

Stability condition:

$$A_1 = A_2$$

If:

$$\delta_c < \delta_{cr}$$

Stable

Used for transient stability