Power System Stability - GATE EE Exam Preparation Notes

System Stability Concepts

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

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

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

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:

\[\int_{\delta_0}^{\delta_c} P_m d\delta = \int_{\delta_c}^{\delta_{\text{max}}} (P_{\text{max}}^{(pf)}\sin\delta - P_m) d\delta\]

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

Stability Improvement Methods

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 Focus Areas

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 Problems

Sample GATE Question 1

Example

A generator connected to infinite bus through reactance has steady-state stability limit at power angle:

  • \(0^\circ\)

  • \(45^\circ\)

  • \(90^\circ\)

  • \(180^\circ\)

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

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

Sample GATE Question 2

Example

The synchronizing power coefficient is maximum when the power angle is:

  • \(0^\circ\)

  • \(30^\circ\)

  • \(60^\circ\)

  • \(90^\circ\)

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

Example

A generator: \(P_{\text{max}} = 2.0\) pu, operating at \(\delta_0 = 30^\circ\). A fault reduces power transfer to zero. Find critical clearing angle if post-fault \(P_{\text{max}} = 1.5\) pu.

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