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Examoogle • User • info@examoogle.com • EE-2024-8821
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When WLSE method is applied for line flow only algorithm then the estimation of ‘x’ for power system is
xest=(ATWA)−1ATF(xest,Z)
xest=(ATWA)−1ATWF(xest,Z)
xest=(ATWA)−1ATW−1F(xest,Z)
None of above
xest=(ATWA)−1ATWF(xest,Z)
Quick Summary: The Weighted Least Squares Estimation (WLSE) method aims to minimize the weighted sum of squared residuals between observed measurements and the estimated state. For a system with line flow measurements defined by a non-linear function $F(x)$, the estimator updates the state vector $x_{est}$ using the gain matrix $(A^TWA)^{-1}$ to adjust based on the residual error weighted by the precision matrix $W$.
The Weighted Least Squares Estimation (WLSE) method aims to minimize the weighted sum of squared residuals between observed measurements and the estimated state. For a system with line flow measurements defined by a non-linear function F(x), the estimator updates the state vector xest using the gain matrix (ATWA)−1 to adjust based on the residual error weighted by the precision matrix W.
J(x)=[Z−F(x)]TW[Z−F(x)] — Objective function to be minimized
xk+1=xk+(ATWA)−1ATW[Z−F(xk)] — Iterative update law
The WLSE algorithm solves the problem by linearizing the measurement model around an initial state. By minimizing J(x)=[Z−F(x)]TW[Z−F(x)], we derive the normal equations. In the context of line flow algorithms, A represents the Jacobian of the measurement function, W is the diagonal matrix of measurement weights (inverse of covariance), and xest is iteratively updated via the Gauss-Newton approach to converge on the true system state.
WLSE provides the Best Linear Unbiased Estimator (BLUE) under Gaussian noise assumptions.
The weight matrix W is typically defined as R−1, where R is the measurement covariance matrix.
The Jacobian A reflects the sensitivity of line flows to voltage magnitude and phase angle changes.
Iterative convergence is required for non-linear power flow models.
High statistical accuracy for Gaussian measurement noise
Well-understood mathematical framework for convergence
High computational cost due to matrix inversion
Sensitivity to outliers and bad data measurements
Energy Management Systems (EMS)
Real-time power system security assessment
Option B is the correct representation of the iterative step in Gauss-Newton WLSE.
Option A is dimensionally inconsistent because it omits the weight matrix W in the residual multiplication.
Option C incorrectly uses the inverse weight matrix W−1 instead of the weights W.
B is correct — the expression represents the standard iterative update equation for the Weighted Least Squares Estimator in a power system.
Always ensure dimensions align in matrix operations: if A is m×n, then (ATWA)−1 must be n×n and ATW must be n×m to produce an n×1 state vector correction.