System Stability Concepts
Power System Stability - Definition
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Definition: Ability of a power system to maintain synchronous operation after being subjected to a disturbance
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Rotor Angle Stability: Ability of synchronous machines to maintain synchronism
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Voltage Stability: Ability to maintain steady voltages at all buses
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Frequency Stability: Ability to maintain steady frequency following severe system imbalances
Classification of Stability
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Steady-State Stability:
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Small, gradual disturbances
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Slow changes in load/generation
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Linear analysis applicable
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Transient Stability:
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Large disturbances (faults, line switching)
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First swing stability (0-1 second)
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Non-linear analysis required
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Dynamic Stability:
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Long-term oscillations (1-10 seconds)
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Involves automatic controls
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Multi-swing stability
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Synchronous Machine Fundamentals
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Swing Equation: \(M\frac{d^2\delta}{dt^2} = P_m - P_e\)
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Where:
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\(M\) = Inertia constant
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\(\delta\) = Rotor angle (power angle)
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\(P_m\) = Mechanical power input
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\(P_e\) = Electrical power output
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Power-Angle Relationship: \(P_e = \frac{EV}{X}\sin\delta\)
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Synchronizing Power Coefficient: \(P_s = \frac{dP_e}{d\delta} = \frac{EV}{X}\cos\delta\)
Steady-State Stability
Steady-State Stability Analysis
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Stability Criterion: \(\frac{dP_e}{d\delta} > 0\) at operating point
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Power-Angle Equation: \(P_e = P_{\text{max}}\sin\delta\)
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Where \(P_{\text{max}} = \frac{EV}{X}\) is the maximum power transfer
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Steady-State Stability Limit: \(\delta = 90^\circ\)
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Practical Operating Range: \(\delta = 30^\circ - 45^\circ\)
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Static Stability Margin: \(\frac{P_{\text{max}} - P_e}{P_{\text{max}}} \times 100\%\)
Factors Affecting Steady-State Stability
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System Reactance: Lower X increases stability limit
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Excitation System: Higher field current increases E
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Load Characteristics: Constant power loads reduce stability
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Transmission System: Series compensation improves stability
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Generator Characteristics: Higher inertia provides better stability
Transient Stability
Transient Stability Concepts
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Definition: Ability to maintain synchronism following large disturbances
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Critical Clearing Time: Maximum time a fault can persist without losing synchronism
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Critical Clearing Angle: Rotor angle at which fault must be cleared
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First Swing Stability: Stability during first oscillation after disturbance
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Analysis Methods:
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Equal area criterion (single machine)
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Time-domain simulation (multi-machine)
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Equal Area Criterion
Equal Area Criterion - Theory
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Principle: Based on energy balance during transient
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Assumption: Classical model (E’ = constant, X = constant)
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Stability Condition: Kinetic energy gained = Kinetic energy lost
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Mathematical Form: \(A_{\text{acc}} = A_{\text{dec}}\)
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Where areas represent energy per unit of inertia
Equal Area Criterion - Application
During Fault Period (\(\delta_0\) to \(\delta_c\)):
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Accelerating area: \(A_{\text{acc}} = \int_{\delta_0}^{\delta_c} (P_m - P_{\text{fault}}) d\delta\)
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Generator accelerates as \(P_m > P_{\text{fault}}\)
Post-Fault Period (\(\delta_c\) to \(\delta_{\text{max}}\)):
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Decelerating area: \(A_{\text{dec}} = \int_{\delta_c}^{\delta_{\text{max}}} (P_{\text{post}} - P_m) d\delta\)
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Generator decelerates as \(P_{\text{post}} > P_m\)
Critical Condition: \(A_{\text{acc}} = A_{\text{dec}}\)
Equal Area Criterion - Three Power Curves
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Pre-fault curve: \(P_{\text{pre}} = P_{\text{max}}\sin\delta\)
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During fault curve: \(P_{\text{fault}} = P_{\text{max}}^{(f)}\sin\delta\)
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Post-fault curve: \(P_{\text{post}} = P_{\text{max}}^{(pf)}\sin\delta\)
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Where \(P_{\text{max}}^{(f)} < P_{\text{max}}^{(pf)} < P_{\text{max}}\)
Initial Conditions:
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\(P_m = P_{\text{pre}}\sin\delta_0\) (initial operating point)
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\(\delta_0 = \sin^{-1}(P_m/P_{\text{max}})\)
Critical Clearing Angle Calculation
For a fault at generator terminal (3-phase fault):
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During fault: \(P_{\text{fault}} = 0\)
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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 \(\delta_{\text{max}} = \pi - \sin^{-1}(P_m/P_{\text{max}}^{(pf)})\)
Stability Improvement Methods
Methods to Improve Stability
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Fast Fault Clearing: Reduce critical clearing time
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Automatic Reclosing: Restore transmission lines quickly
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Excitation System Control: Fast-acting AVR systems
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Power System Stabilizers (PSS): Damp oscillations
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FACTS Devices: SVC, STATCOM, TCSC, UPFC
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Series Compensation: Reduce effective reactance
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Generator Inertia: Higher H constant improves stability
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Load Shedding: Reduce system stress during emergencies
GATE Focus Areas
GATE Important Topics
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Numerical Problems:
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Equal area criterion calculations
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Critical clearing angle/time
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Stability margin calculations
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Power-angle curve analysis
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Conceptual Questions:
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Swing equation interpretation
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Stability improvement methods
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Classification of stability
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Synchronizing power coefficient
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Sample Problems
Sample GATE Question 1
Example
A generator connected to infinite bus through reactance has steady-state stability limit at power angle:
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\(0^\circ\)
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\(45^\circ\)
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\(90^\circ\)
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\(180^\circ\)
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:
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\(0^\circ\)
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\(30^\circ\)
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\(60^\circ\)
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\(90^\circ\)
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:
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\(P_m = 2.0 \sin(30^\circ) = 1.0\) pu
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\(\delta_{\text{max}} = 180^\circ - \sin^{-1}(1.0/1.5) = 138.2^\circ\)
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Using equal area: \(1.0(\delta_c - 30^\circ) = 1.5[\cos\delta_c - \cos(138.2^\circ)] - 1.0(138.2^\circ - \delta_c)\)
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Solving: \(\delta_c \approx 83.2^\circ\)
Key Formulas Summary
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Swing Equation: \(M\frac{d^2\delta}{dt^2} = P_m - P_e\)
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Power-Angle: \(P_e = \frac{EV}{X}\sin\delta\)
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Synchronizing Power: \(P_s = \frac{EV}{X}\cos\delta\)
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Stability Criterion: \(\frac{dP_e}{d\delta} > 0\)
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Equal Area: \(A_{\text{acc}} = A_{\text{dec}}\)
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Maximum Swing: \(\delta_{\text{max}} = \pi - \sin^{-1}(P_m/P_{\text{max}}^{(pf)})\)