Power System Stability - Definition

• Definition: Ability of a power system to maintain synchronous operation after being subjected to a disturbance

• Rotor Angle Stability: Ability of synchronous machines to maintain synchronism

• Voltage Stability: Ability to maintain steady voltages at all buses

• Frequency Stability: Ability to maintain steady frequency following severe system imbalances

Classification of Stability

• Steady-State Stability:

  • Small, gradual disturbances

  • Slow changes in load/generation

  • Linear analysis applicable

• Transient Stability:

  • Large disturbances (faults, line switching)

  • First swing stability (0-1 second)

  • Non-linear analysis required

• Dynamic Stability:

  • Long-term oscillations (1-10 seconds)

  • Involves automatic controls

  • Multi-swing stability

Synchronous Machine Fundamentals

• Swing Equation: \(M\frac{d^2\delta}{dt^2} = P_m - P_e\)

• Where:

  • \(M\) = Inertia constant

  • \(\delta\) = Rotor angle (power angle)

  • \(P_m\) = Mechanical power input

  • \(P_e\) = Electrical power output

• Power-Angle Relationship: \(P_e = \frac{EV}{X}\sin\delta\)

• Synchronizing Power Coefficient: \(P_s = \frac{dP_e}{d\delta} = \frac{EV}{X}\cos\delta\)

Steady-State Stability Analysis

• Stability Criterion: \(\frac{dP_e}{d\delta} > 0\) at operating point

• Power-Angle Equation: \(P_e = P_{\text{max}}\sin\delta\)

• Where \(P_{\text{max}} = \frac{EV}{X}\) is the maximum power transfer

• Steady-State Stability Limit: \(\delta = 90^\circ\)

• Practical Operating Range: \(\delta = 30^\circ - 45^\circ\)

• Static Stability Margin: \(\frac{P_{\text{max}} - P_e}{P_{\text{max}}} \times 100\%\)

Factors Affecting Steady-State Stability

• System Reactance: Lower X increases stability limit

• Excitation System: Higher field current increases E

• Load Characteristics: Constant power loads reduce stability

• Transmission System: Series compensation improves stability

• Generator Characteristics: Higher inertia provides better stability

Transient Stability Concepts

• Definition: Ability to maintain synchronism following large disturbances

• Critical Clearing Time: Maximum time a fault can persist without losing synchronism

• Critical Clearing Angle: Rotor angle at which fault must be cleared

• First Swing Stability: Stability during first oscillation after disturbance

• Analysis Methods:

  • Equal area criterion (single machine)

  • Time-domain simulation (multi-machine)

Equal Area Criterion - Theory

• Principle: Based on energy balance during transient

• Assumption: Classical model (E’ = constant, X = constant)

• Stability Condition: Kinetic energy gained = Kinetic energy lost

• Mathematical Form: \(A_{\text{acc}} = A_{\text{dec}}\)

• Where areas represent energy per unit of inertia

Equal Area Criterion - Application

During Fault Period (\(\delta_0\) to \(\delta_c\)):

• Accelerating area: \(A_{\text{acc}} = \int_{\delta_0}^{\delta_c} (P_m - P_{\text{fault}}) d\delta\)

• Generator accelerates as \(P_m > P_{\text{fault}}\)

Post-Fault Period (\(\delta_c\) to \(\delta_{\text{max}}\)):

• Decelerating area: \(A_{\text{dec}} = \int_{\delta_c}^{\delta_{\text{max}}} (P_{\text{post}} - P_m) d\delta\)

• Generator decelerates as \(P_{\text{post}} > P_m\)

Critical Condition: \(A_{\text{acc}} = A_{\text{dec}}\)

Equal Area Criterion - Three Power Curves

• Pre-fault curve: \(P_{\text{pre}} = P_{\text{max}}\sin\delta\)

• During fault curve: \(P_{\text{fault}} = P_{\text{max}}^{(f)}\sin\delta\)

• Post-fault curve: \(P_{\text{post}} = P_{\text{max}}^{(pf)}\sin\delta\)

• Where \(P_{\text{max}}^{(f)} < P_{\text{max}}^{(pf)} < P_{\text{max}}\)

Initial Conditions:

• \(P_m = P_{\text{pre}}\sin\delta_0\) (initial operating point)

• \(\delta_0 = \sin^{-1}(P_m/P_{\text{max}})\)

Critical Clearing Angle Calculation

For a fault at generator terminal (3-phase fault):

• During fault: \(P_{\text{fault}} = 0\)

• Post-fault: \(P_{\text{post}} = P_{\text{max}}^{(pf)}\sin\delta\)

Equal Area Equation:

Solving: \(P_m(\delta_c - \delta_0) = P_{\text{max}}^{(pf)}[\cos\delta_c - \cos\delta_{\text{max}}] - P_m(\delta_{\text{max}} - \delta_c)\)

Where \(\delta_{\text{max}} = \pi - \sin^{-1}(P_m/P_{\text{max}}^{(pf)})\)

Methods to Improve Stability

• Fast Fault Clearing: Reduce critical clearing time

• Automatic Reclosing: Restore transmission lines quickly

• Excitation System Control: Fast-acting AVR systems

• Power System Stabilizers (PSS): Damp oscillations

• FACTS Devices: SVC, STATCOM, TCSC, UPFC

• Series Compensation: Reduce effective reactance

• Generator Inertia: Higher H constant improves stability

• Load Shedding: Reduce system stress during emergencies

GATE Important Topics

• Numerical Problems:

  • Equal area criterion calculations

  • Critical clearing angle/time

  • Stability margin calculations

  • Power-angle curve analysis

• Conceptual Questions:

  • Swing equation interpretation

  • Stability improvement methods

  • Classification of stability

  • Synchronizing power coefficient

Sample GATE Question 1

Answer: C) \(90^\circ\)

Explanation: At \(\delta = 90^\circ\), \(\frac{dP_e}{d\delta} = 0\), which is the stability limit.

Sample GATE Question 2

Answer: A) \(0^\circ\)

Explanation: \(P_s = \frac{EV}{X}\cos\delta\) is maximum when \(\cos\delta = 1\), i.e., \(\delta = 0^\circ\).

Sample Calculation

Solution:

1. \(P_m = 2.0 \sin(30^\circ) = 1.0\) pu

2. \(\delta_{\text{max}} = 180^\circ - \sin^{-1}(1.0/1.5) = 138.2^\circ\)

3. Using equal area: \(1.0(\delta_c - 30^\circ) = 1.5[\cos\delta_c - \cos(138.2^\circ)] - 1.0(138.2^\circ - \delta_c)\)

4. Solving: \(\delta_c \approx 83.2^\circ\)

Key Formulas Summary

• Swing Equation: \(M\frac{d^2\delta}{dt^2} = P_m - P_e\)

• Power-Angle: \(P_e = \frac{EV}{X}\sin\delta\)

• Synchronizing Power: \(P_s = \frac{EV}{X}\cos\delta\)

• Stability Criterion: \(\frac{dP_e}{d\delta} > 0\)

• Equal Area: \(A_{\text{acc}} = A_{\text{dec}}\)

• Maximum Swing: \(\delta_{\text{max}} = \pi - \sin^{-1}(P_m/P_{\text{max}}^{(pf)})\)