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According to WLSE method for state estimation on nonlinear power system equation , the estimation of ‘x’ at ‘0th’ iteration is
Is not possible to derive
∆ X ( 0 ) e s t = H 0 T W H 0 - 1 Z - h X 0
∆ X ( 0 ) e s t = H 0 T W H 0 - 1 H 0 T Z - h X 0
∆ X ( 0 ) e s t = H 0 T W H 0 - 1 H 0 T W Z - h X 0
∆X(0)est=H0TWH0-1H0TWZ-hX0
Quick Summary: The Weighted Least Squares Estimation (WLSE) method aims to minimize the weighted sum of squared residuals between observed measurements and the system's nonlinear model. At the 0th iteration, the correction vector $\Delta X$ is computed using the gain matrix $G = H^T W H$ and the weighted residual vector.
The Weighted Least Squares Estimation (WLSE) method aims to minimize the weighted sum of squared residuals between observed measurements and the system's nonlinear model. At the 0th iteration, the correction vector ΔX is computed using the gain matrix G=HTWH and the weighted residual vector.
ΔX=(HTWH)−1HTWΔZ — Standard update step where ΔZ=Z−h(x)
W=R−1 — Weighting matrix defined as the inverse of the measurement error covariance matrix R
The WLSE approach linearizes the nonlinear system Z=h(x)+e using a Taylor series expansion around an initial state estimate x0. By taking the derivative of the objective function J(x)=[Z−h(x)]TW[Z−h(x)] and setting it to zero, we derive the normal equations. The update step ΔX=(HTWH)−1HTW[Z−h(x)] ensures the estimation iteratively converges to the true state.
The Jacobian matrix H contains the partial derivatives of the measurement functions with respect to the state variables.
The weighting matrix W is a diagonal matrix containing the reciprocal of the measurement variances, i.e., wi=1/σi2.
Iteration continues until the change in state ΔX falls below a specified threshold.
Statistically optimal if measurement errors are Gaussian.
Provides a systematic approach to handle redundant measurements.
Computationally expensive due to matrix inversion at each iteration.
Susceptible to divergence if the initial guess X0 is too far from the solution.
Real-time power system monitoring (SCADA).
Bad data detection and identification in power networks.
The term [Z−h(X0)] is the residual vector.
Option C is missing the weighting matrix W in the numerator term, making it dimensionally inconsistent with the Gauss-Newton derivation.
Option B is missing the transpose of the Jacobian matrix HT and the weighting matrix W inside the gain inverse term.
D is correct — The correct update formula is ΔX=(H0°TWH0)−1H0°TW[Z−h(X0)], which correctly applies the Gauss-Newton optimization step.
Always verify matrix dimensions in state estimation formulas: (HTWH)−1 must be a square matrix of size n×n, where n is the number of state variables.