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Examoogle • User • info@examoogle.com • EE-2024-8821
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When WLSE method is applied for only active power injection measurement vector then the estimation of ‘x’ is
x e s t ( δ ) P + 1 = x e s t ( δ ) P + H 1 P T W P Z P - h p x e s t P
x e s t ( δ ) P + 1 = x e s t ( δ ) P + H 1 P T W P H 1 P H 1 P T W P Z P - h p x e s t P
x e s t ( δ ) P + 1 = x e s t ( δ ) P + H 2 P T W P H 2 P - 1 H 2 P T W P Z P - h p x e s t P
x e s t ( δ ) P + 1 = x e s t ( δ ) P + H 1 P T W P H 1 P - 1 H 1 P T W P Z P - h p x e s t P
xest(δ)P+1=xest(δ)P+H1PTWPH1P-1H1PTWPZP-hpxestP
Quick Summary: The Weighted Least Square Estimation (WLSE) method finds the state vector $x$ that minimizes the weighted sum of squared residuals. For a non-linear system, this requires an iterative Newton-Raphson approach where the update step involves the gain matrix $G = H^T W H$, leading to the standard iterative update rule.
The Weighted Least Square Estimation (WLSE) method finds the state vector x that minimizes the weighted sum of squared residuals. For a non-linear system, this requires an iterative Newton-Raphson approach where the update step involves the gain matrix G=HTWH, leading to the standard iterative update rule.
J(x)=[z−h(x)]TW[z−h(x)] — Objective function for WLSE
G=HTWH — Gain matrix of the estimator
xp+1=xp+G−1HTWΔz — Iterative state update equation
The measurement model is defined as z=h(x)+e. To solve for x, we minimize J(x)=[z−h(x)]TW[z−h(x)]. By taking the gradient and setting it to zero, we linearize the system using the Jacobian H (where H=∂x∂h). The state update is given by Δx=[HTWH]−1HTW[z−h(x)], which added to the current estimate yields the next estimate.
WLSE utilizes the Gaussian distribution assumption of measurement errors.
The gain matrix G must be non-singular for the solution to exist.
The Jacobian H1 is used specifically when considering only active power measurements.
The inverse term [HTWH]−1 characterizes the sensitivity of the estimation process.
Optimal estimation under Gaussian noise conditions.
Provides statistically unbiased results.
Computationally intensive due to matrix inversion at each iteration.
Requires a good initial guess to ensure convergence.
Power system state estimation for EMS (Energy Management Systems).
Real-time monitoring of bus voltage angles and magnitudes.
Option D is the standard formulation for the Newton-based state update.
Option B lacks the inverse matrix operator, which is mathematically incorrect for the Newton step.
Option C uses H2 which is typically associated with reactive power measurements.
D is correct — The formula represents the standard Newton-Raphson iteration for Weighted Least Square state estimation.
Always verify the Jacobian indices: H1 typically relates to active power (P) and H2 to reactive power (Q) when linearizing power flow equations.