As per weighted least square estimation, the estimated value of X (Xest) is represented by (Where Z

Question

As per weighted least square estimation, the estimated value of X (Xest) is represented by (Where Z is measurement vector ,H is the linear relation between Z and X, W is Weighted error matrix)

Examoogle · Accepted Answer

$X_{est} = (H^T W H)^{-1} H^T W Z$

Explanation: Weighted Least Squares (WLS) is an optimization method used in power system state estimation to minimize the sum of the weighted squares of the residual differences between measured values and estimated values. The objective function is $J(X) = [Z - H(X)]^T W [Z - H(X)]$, where $W$ is the inverse of the measurement covariance matrix, representing the weight assigned to each measurement.

Working Principle: To find the optimal estimate $X_{est}$, we take the derivative of the objective function $J(X)$ with respect to $X$ and equate it to zero. Solving the resulting normal equations $\frac{dJ}{dX} = 0$ yields the solution $X_{est} = (H^T W H)^{-1} H^T W Z$. The weighting matrix $W$ ensures that measurements with lower variance (higher accuracy) contribute more significantly to the final estimate.

Key Formulas: $J(X) = (Z - HX)^T W (Z - HX)$ — The weighted objective function to be minimized; $X_{est} = (H^T W H)^{-1} H^T W Z$ — The optimal state estimation result

Key Points: WLS reduces the impact of noise by assigning weights proportional to the reciprocal of measurement variance.; The matrix $W$ is typically a diagonal matrix $W = diag(1/\sigma_i^2)$ for independent measurements.; State estimation is essential in Energy Management Systems (EMS) for real-time monitoring.

Advantages: Filters measurement noise effectively.; Provides optimal estimation in the presence of Gaussian noise.

Disadvantages: Computationally intensive due to matrix inversion of $(H^T W H)$.; Sensitive to bad data points if not accompanied by a detection mechanism.

Applications: Power System State Estimation (PSSE).; Sensor fusion in navigation and tracking systems.

Additional Info: The term $(H^T W H)^{-1}$ is often referred to as the Gain matrix.. Option B is incorrect because it includes an extra transpose $(W^T)$ which does not appear in the derivation. Option D is mathematically inconsistent as it adds an extra $H$ without correct inversion logic.

Answer: C is correct — The weighted least square estimator is derived by setting the gradient of the weighted residual sum of squares to zero, resulting in $X_{est} = (H^T W H)^{-1} H^T W Z$.

Exam Tip: In competitive exams, remember that for WLS, if all measurements have equal confidence, W becomes the Identity matrix, reducing the formula to the standard Ordinary Least Squares (OLS) estimator: $(H^T H)^{-1} H^T Z$.

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