Weighted least square is more accurate than least square because

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Examoogle · Accepted Answer

Weight on each error is define and use during estimation Explanation: Weighted Least Squares (WLS) is a method of state estimation that incorporates the relative uncertainty or confidence of individual measurements by assigning weights to them. Unlike Ordinary Least Squares (OLS) which assumes all measurements have equal precision, WLS prioritizes more reliable data points, leading to a more accurate estimate of the system state. Working Principle: In power system state estimation, the objective function to be minimized is $J(x) = [z - h(x)]^T W [z - h(x)]$, where $W$ is the weight matrix. Typically, $W = R^{-1}$, where $R$ is the measurement covariance matrix representing the variance of noise in each sensor. By setting $W_{ii} = \frac{1}{\sigma_i^2}$, the method effectively penalizes errors from less accurate sensors more heavily, ensuring the final estimate is robust against noisy data. Key Formulas: $J(x) = r^T W r$ — where $r$ is the residual vector and $W$ is the weight matrix; $W = R^{-1}$ — where $R$ is the diagonal covariance matrix of measurement errors Key Points: OLS is a special case of WLS where $W = I$ (the identity matrix).; WLS reduces the influence of measurements with high variance.; The weights are inversely proportional to the variance of the measurement error.; State estimation is a critical component of Energy Management Systems (EMS). Advantages: Accounts for different accuracies of measuring devices (PMUs, RTUs).; Improved robustness against measurement noise. Disadvantages: Requires prior knowledge of measurement error variances.; Computationally more complex than OLS due to matrix inversion. Applications: Power System State Estimation (PSSE).; Sensor fusion in navigation systems.; Control loop performance monitoring. Additional Info: In real-world power grids, sensors have varying levels of precision based on age, calibration, and noise environment.. Option B is incorrect because weights are generally derived from historical error statistics (covariance matrices) rather than being arbitrary 'found' values. Answer: A is correct — Weighted least square uses specific weights on each error during estimation to account for varying measurement precision. Exam Tip: Always remember that WLS is an optimization problem; the 'weights' act as a penalty function in the optimization space to minimize total system error.

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