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The Gain Matrix ‘G’ of power system becomes more ill-condition if the condition number is
Decrease in number
Moderate in number
Increase in number
1
Increase in number
Quick Summary: The condition number of a matrix measures how sensitive the output of a system is to small changes in the input data. A higher condition number indicates that the matrix is ill-conditioned, meaning that small errors in measurements or numerical rounding can lead to large inaccuracies in the state estimate.
The condition number of a matrix measures how sensitive the output of a system is to small changes in the input data. A higher condition number indicates that the matrix is ill-conditioned, meaning that small errors in measurements or numerical rounding can lead to large inaccuracies in the state estimate.
k(G)=∣∣G∣∣⋅∣∣G−1∣∣
k(G)=σminσmax — Ratio of singular values for the Gain Matrix
In power system state estimation, the gain matrix G=HTWH (where H is the measurement Jacobian and W is the weight matrix) must be inverted. If the condition number k(G)=λminλmax is large, the matrix G is near-singular, making the system of linear equations GΔx=b extremely sensitive to noise and numerical instability, which degrades convergence in the Gauss-Newton method.
An ideal condition number is 1, indicating a perfectly conditioned matrix.
Ill-conditioning often occurs in power systems with high R/X ratio lines or redundant measurements.
Matrix inversion techniques like Cholesky decomposition are sensitive to high condition numbers.
State estimation accuracy decreases as the condition number increases.
High stability when condition number is low
Precise numerical solutions
Computational errors due to floating point precision
Slow convergence in iterative solvers
Power System State Estimation (PSSE)
Load Flow Analysis
Control System Stability Studies
A condition number of 1 represents an orthogonal matrix which is perfectly well-conditioned.
Options A and B describe conditions that improve stability rather than causing ill-conditioning.
C is correct — The Gain Matrix becomes more ill-conditioned as the condition number increases, leading to potential numerical instability and poor precision.
Always recall that the condition number is inversely proportional to the 'distance' of the matrix from the set of singular matrices.