Linear State Estimation

Unlike traditional Weighted Least Squares (WLS) which requires multiple Newton-Raphson iterations, a linear estimator solves the system in a single step. By using complex phasor measurements from Phasor Measurement Units (PMUs) and expressing voltages in rectangular coordinates, the measurement function becomes z = Hx + e, where H is a constant matrix. This eliminates convergence issues entirely, making execution time predictable and bounded.

The linear formulation is fundamentally enabled by PMU data. PMUs provide time-synchronized, complex voltage and current phasors with precise GPS timestamps. Because these measurements are directly proportional to the complex bus voltages (the state vector), the Jacobian matrix H becomes constant, consisting only of network admittance parameters and transformer tap ratios, independent of the operating point.

Linear state estimation naturally extends to three-phase distribution networks without increasing algorithmic complexity. The state vector includes the real and imaginary components of all three phase voltages at each bus. The constant admittance matrix captures mutual coupling between phases, making it ideal for modern distribution grids with high single-phase solar photovoltaic penetration and unbalanced loads.

A linear system is observable if the Gain Matrix G = H^T R^{-1} H is non-singular. This drives optimal PMU placement problems to minimize infrastructure cost. Unlike iterative methods, observability analysis is a one-time topological check. If a network is unobservable, pseudo-measurements derived from historical AMI data can be injected to restore solvability without breaking the linear framework.

Linear estimators use the Normalized Residual Test for anomaly detection. The measurement residual r = z - Hx̂ is computed post-solution. Because the relationship is linear, the statistical properties of residuals are well-defined. A Chi-Square test on the sum of weighted squared residuals identifies gross errors, while individual normalized residuals pinpoint faulty sensors without iterative re-weighting.

The core computation involves solving a linear system via Cholesky decomposition or QR factorization of the constant gain matrix. Since H is fixed for a given topology, the gain matrix can be pre-factored and only re-computed on switching events. This provides sub-second execution times, critical for real-time Wide-Area Monitoring Systems and closed-loop control applications where latency is unacceptable.

The foundational sensor enabling linear state estimation. A PMU provides time-synchronized measurements of voltage and current phasors via GPS clocks, delivering the complex numbers required to construct a direct linear measurement model.

• Reports at 30-60 samples per second vs. 1 sample every 2-4 seconds for SCADA
• Eliminates the need for iterative Gauss-Newton methods
• Directly observes phase angles, a critical input for the linear formulation

The standard solution method for the overdetermined linear system. WLS minimizes the sum of weighted squared residuals between measured and estimated states, where weights are the inverse of measurement error variances.

• In the linear formulation, the gain matrix is constant, requiring only a single factorization
• Solution: x̂ = (HᵀR⁻¹H)⁻¹HᵀR⁻¹z
• Computationally deterministic with no convergence risk

Statistical validation applied post-solution to identify gross measurement errors. The Normalized Residual Test flags measurements where the residual exceeds a statistical threshold, typically 3σ.

• Uses the Chi-Square test on the objective function to detect the presence of bad data
• Largest normalized residual test identifies the specific offending measurement
• Essential for PMU data quality, as a single faulty phasor can skew the entire linear estimate

The extension of linear state estimation to model the full unbalanced, multi-phase nature of distribution networks. Unlike transmission systems, distribution grids require modeling of:

• Mutual coupling between phases
• Single-phase laterals and untransposed lines
• Wye and delta connected loads
• The linear formulation extends naturally to three-phase when PMU data is available for each phase