ElectricalPower Generation

The covariance of error of estimation is (L = H T W H - 1 H T W and R= covariance of error vector ‘r’)

$L R L^T$

$L R^T L$

$L R L$

None of above

Correct Answer

Concept & PrincipleElectricalPower Generation

Option A

$L R L^T$

Quick Summary: 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 $Cov(x) = L \cdot Cov(r) \cdot L^T$.

💡 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 $Cov(x) = L \cdot Cov(r) \cdot L^T$ .

🔢 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.

⚙️ 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 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 $e$ ).
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The transformation of covariance preserves the symmetric and positive semi-definite nature of the matrix.

✅ Advantages

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

❌ Disadvantages / Limitations

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

🛠️ Applications / Uses

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

📄 Additional Information

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The matrix $L$ is derived from the Gauss-Markov theorem for the Best Linear Unbiased Estimator (BLUE).
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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.

📊 Diagram / Illustration

✅

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

Core Concepts Used

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Weighted Least Squares (WLS) Linear Algebra in Estimation Covariance Propagation

💡 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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