Weighted Least Squares (WLS)

WLS minimizes the sum of weighted squared residuals: min J(x) = [z - h(x)]^T W [z - h(x)], where z is the measurement vector, h(x) is the nonlinear measurement function, and W is the weight matrix. The weight matrix is the inverse of the measurement error covariance matrix: W = R^{-1}. This formulation ensures that high-precision measurements (low variance) exert greater influence on the final estimate than noisy, low-confidence data points.

Because power flow equations are nonlinear, WLS solves the estimation problem iteratively. The normal equation at each iteration is: G Δx = H^T W [z - h(x)], where H is the Jacobian matrix of measurement functions, and G = H^T W H is the gain matrix. The state vector is updated via x^{k+1} = x^k + Δx until convergence. The gain matrix's sparsity structure mirrors the network topology, enabling efficient sparse triangular factorization.

When measurement errors follow a zero-mean Gaussian distribution, the WLS estimator is the Maximum Likelihood Estimator (MLE). It produces the Best Linear Unbiased Estimate (BLUE), achieving the Cramér-Rao lower bound. This means no other unbiased estimator can achieve lower variance. The covariance of the estimated state is given by the inverse gain matrix: cov(x̂) = G^{-1}, providing direct uncertainty quantification for every bus voltage angle and magnitude.

The quadratic penalty on residuals makes WLS highly sensitive to gross errors. A single corrupted measurement can skew the entire state estimate—a phenomenon called smearing. This is the estimator's primary operational weakness. Mitigation requires post-estimation bad data detection using the Chi-Square test on the objective function value or the Normalized Residual Test on individual measurement residuals to identify and remove outliers before re-estimation.

WLS requires the network to be algebraically observable. The gain matrix G must be nonsingular (full rank). If the measurement set is insufficient, G becomes singular and the normal equations cannot be solved. Observability analysis identifies observable islands and unobservable branches. In distribution systems with sparse real-time sensors, pseudo-measurements—synthetic data from historical load profiles or forecasts—are injected to restore numerical observability.

Weights are typically chosen as the reciprocal of measurement error variance: w_i = 1/σ_i². Standard assumptions include: 0.2% accuracy for PMU voltage measurements, 0.5-2% for RTU power flow measurements, and 20-50% for pseudo-measurements. The large variance assigned to pseudo-measurements reflects their low confidence, ensuring they only lightly constrain the solution. Proper weight tuning is critical—overly optimistic weights on bad data amplify estimation errors.

The overarching algorithmic process that WLS solves. DSSE infers the complete voltage and current state of an unbalanced distribution network from a limited set of real-time sensor measurements and pseudo-measurements. Unlike transmission systems, DSSE must handle high R/X ratios, three-phase imbalance, and radial or weakly meshed topologies.

The core computational object in the WLS normal equation: G = Hᵀ W H. It represents the information content of the measurement set, where:

• H: Jacobian matrix of measurement functions
• W: Inverse covariance matrix of measurement errors The condition number of G dictates numerical stability. Ill-conditioned gain matrices cause convergence failure and require regularization or observability restoration.

Post-estimation statistical tests that identify gross measurement errors before they corrupt operational decisions. The Normalized Residual Test flags measurements whose residuals exceed a statistical threshold. The Chi-Square Test evaluates the overall fit. WLS assumes Gaussian noise, making it vulnerable to outliers—robust alternatives like the Huber M-Estimator or Least Absolute Value (LAV) are used when data quality is poor.

Determines whether a unique WLS solution exists given the available measurements and network topology. An unobservable system has a singular gain matrix. The analysis identifies:

• Observable islands: Solvable sub-networks
• Unobservable branches: Require pseudo-measurements
• Critical measurements: Single points of failure for observability Without full observability, WLS cannot converge to a unique state vector.

Defines the weighting philosophy of WLS. The diagonal elements are the error variances (σ²) of each measurement. High-variance measurements receive low weight; precision sensors receive high weight. The matrix is typically assumed diagonal, ignoring error correlations. Proper variance assignment is critical—overconfident weights on bad pseudo-measurements can bias the entire state estimate.

A dynamic extension of static WLS that incorporates time-series forecasts of load and generation as prior information. This bridges the gap between snapshot estimation and real-time tracking. The forecast provides a predicted state with associated uncertainty, which is fused with new measurements using a Kalman Filter framework, improving accuracy during rapid ramping events from solar and wind.