The covariance of error of estimation is (L=HTWH-1HTW and R= covariance of error vector ‘r’)
A
$L R L^T$
B
$L R^T L$
C
$L R L$
D
None of above

Question

The covariance of error of estimation is (L=HTWH-1HTW and R= covariance of error vector ‘r’)

Examoogle · Accepted Answer

$L R L^T$

Explanation: In state estimation for power systems, the estimation error vector is linearly related to the measurement error vector through the gain matrix $L$. When a random vector $r$ undergoes a linear transformation $x = Lr$, the covariance matrix of the resulting vector is given by the sandwich formula $Cov(x) = L \cdot Cov(r) \cdot L^T$.

Working Principle: The state estimation process involves minimizing the weighted sum of squared residuals. Given the measurement model $z = Hx + e$, where $e$ is the measurement error with covariance $R$, the best linear unbiased estimator $x = Lz$ is derived using the weight matrix $W$. The propagation of the error covariance follows the property $Cov(Ax) = A \cdot Cov(x) \cdot A^T$. Thus, if $L$ is the gain matrix, the error covariance of the estimate $\hat{x}$ is calculated as $Cov(\hat{x}) = L \cdot R \cdot L^T$.

Key Formulas: $Cov(\hat{x}) = L R L^T$ — The formula for the covariance matrix of the estimated state vector.; $L = (H^T W H)^{-1} H^T W$ — The gain matrix used in weighted least squares state estimation.

Key Points: The estimator $\hat{x}$ is a linear function of the measurements $z$, represented by $\hat{x} = Lz$.; The covariance matrix $R$ represents the uncertainty in the measurement vector $r$ (or $e$).; The transformation of covariance preserves the symmetric and positive semi-definite nature of the matrix.

Advantages: Provides a systematic way to quantify uncertainty in state estimates.; Essential for assessing the quality and reliability of real-time power system monitoring.

Disadvantages: Requires accurate knowledge of measurement error covariance matrix $R$, which is often difficult to estimate precisely.; Computational complexity increases with the size of the power network.

Applications: Bad data detection in SCADA systems.; Optimal power flow and contingency analysis.

Additional Info: The matrix $L$ is derived from the Gauss-Markov theorem for the Best Linear Unbiased Estimator (BLUE).. Options B and C are mathematically incorrect as covariance matrices must satisfy the sandwich form $A R A^T$ to maintain dimensions and symmetric properties.

Answer: A is correct — The covariance of the estimation error is given by the matrix product $L R L^T$.

Exam Tip: Always remember that covariance transformations follow the sandwich rule: if $y = Ax$, then $Cov(y) = A Cov(x) A^T$. This is a fundamental result in statistics and signal processing.

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