presentation for electrical engieering.pptx

PPt for electrical engineering

1. BUDGE BUDGE INSTITUTE OF TECHNOLOGY NAME : PRANIT KUMAR SAHOO DEPARTMENT : ELECTRICAL ENGINEERING YEAR : 4th SEMESTER: 8TH UNIVERSITY ROLL NO: 27601622004 SUBJECT : POWER SYSTEM DYNAMICS & CONTROL SUBJECT CODE : PE – EE801B FACULTY NAME : Dr. A.B. Chattopadhyay

2. INTRODUCTION TO VOLTAGE STABILITY Voltage stability refers to the ability of a power system to maintain steady acceptable voltages at all buses under normal operating conditions and after being subjected to a disturbance. DEFINATION: - IMPORTANCE IN POWER SYSTEM: - Continuous Power Supply: Ensures reliable delivery of electricity to consumers without interruptions. Preventing Voltage Collapse: Critical for avoiding widespread blackouts and equipment damage. System Reliability: Maintains the integrity and performance of the electrical grid.

3. JACOBIAN MATRIX BASIC The Jacobian matrix is a mathematical tool used in multivariable calculus. It’s a grid of numbers showing how small changes in inputs affect the outputs of multiple functions at once. WHAT IS JACOBIAN MATRIX ? MATHEMATICAL REPRESENTATION: - The power flow problem can be expressed as: F(X)=0 Where F is a vector of mismatch equations, and XX is the state vector containing voltage magnitudes and angles. The Jacobian J is defined as:- J=  

4. The Jacobian matrix is derived from power flow equations and plays a crucial role in solving them using the Newton-Raphson method . POWER FLOW EQUATIONS AND JACOBIAN MATRIX: - POWER FLOW EQUATION: - Active Power (P): -   JACOBIAN MTRAIX IN POWER FLOW: -   J 11 ​ relates active power to voltage angles. J12 ​ relates active power to voltage magnitudes. J21 ​ relates reactive power to voltage angles. J22 ​ relates reactive power to voltage magnitudes. Reactive power (Q): -  

5. The Jacobian matrix is essential in power flow analysis and helps determine voltage stability. Its properties indicate whether a power system is stable or approaching voltage collapse . Newton-Raphson Method The Jacobian matrix is used in the Newton-Raphson method to iteratively solve power flow equations. It relates changes in voltage magnitude and phase angle to changes in active and reactive power . Determinant of the Jacobian Matrix (det (J)) If det( J ) is close to zero , the system is near voltage instability. A singular Jacobian matrix (det( J )=0) indicates a voltage collapse condition. Sensitivity Analysis The J 22 ​ submatrix of the Jacobian determines how voltage magnitudes respond to reactive power injections. If J -1 22 ​ (inverse of J 22 ​) has large values, the system is weak and prone to voltage collapse. ROLE OF JACOBIAN MATRIX IN VOLTAGE STABILITY

6. The Jacobian matrix's eigenvalues provide insight into a power system’s voltage stability. If an eigenvalue approaches zero or becomes negative , the system is at risk of voltage collapse . What are Eigenvalues? Eigenvalues (λ) represent system stability characteristics. They satisfy the equation: Jv = λ v Eigenvalues and Stability: If all eigenvalues are positive , the system is stable . If any eigenvalue approaches zero , the system is at the voltage stability limit . If an eigenvalue is negative , the system is unstable . Voltage Collapse Condition: As power demand increases, the smallest eigenvalue of the Jacobian gradually decreases . When the smallest eigenvalue reaches zero, voltage collapse occurs. EIGENVALUES AND VOLTAGE STABILITY

7. PV Curve Analysis (P-V Curve) Plots active power (P) vs. voltage (V) at a particular bus. As system loading increases, voltage decreases. The "knee point" of the curve indicates the voltage collapse limit . QV Curve Analysis (Q-V Curve ) Plots reactive power (Q) vs. voltage (V) to analyze reactive power support. A steeper QV curve means the system needs more reactive power support to maintain stability. The point where dV / dQ becomes very large indicates voltage instability. Continuation Power Flow (CPF) Method Tracks voltage stability limits by increasing system load beyond normal power flow solutions. Detects the maximum loading point of the system before voltage collapse occurs. Singularity of the Jacobian Matrix When the Jacobian matrix becomes singular (det(J)=0), voltage collapse is imminent. The smallest eigenvalue of JJJ approaching zero signals instability. M ETHOD FOR VOLTAGE STABILITY ANALYSIS USING JACOBIAN

8. APPLICATION & CASE STUDY: - APPLICATION :- Power Flow Studies: Helps in solving load flow equations using Newton-Raphson method . Voltage Stability Assessment: Determines stability margins and collapse points using eigenvalues. Grid Operation & Planning: Ensures the secure operation of power networks by identifying weak areas. Fault Analysis & Blackout Prevention: Helps utilities predict and mitigate voltage instability issues. CASE STUDY :- 2003 North American Blackout One of the largest blackouts in history, affecting 50 million people . Cause: Voltage instability due to insufficient reactive power support .

9. CONCLUSION The Jacobian matrix plays a crucial role in voltage stability analysis. Key indicators such as determinant, singularity, and eigenvalues help predict voltage collapse. Methods like PV curves, QV curves, and Continuation Power Flow (CPF) assist in assessing stability. Practical applications in power flow analysis, grid security, and blackout prevention.

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