Linear State Estimation (LSE)

LSE mathematically determines if the grid is observable—whether the available PMU measurements are sufficient to solve for all unknown bus voltages. It exploits measurement redundancy (having more measurements than unknowns) to filter out noise and bad data.

• N-1 observability: The system remains solvable even if a single PMU or communication link fails.
• Critical measurement identification: Pinpoints which specific measurements are absolutely essential for a solution.
• Unobservable branch detection: Identifies isolated network sections lacking sufficient sensor coverage.

A core function of LSE is statistical gross error analysis. By comparing each measurement against the estimated state, the algorithm computes normalized residuals. Measurements with residuals exceeding a statistical threshold are flagged as suspect.

• Chi-squared test: Evaluates the overall fit of the measurement set to detect the presence of bad data.
• Largest normalized residual test: Iteratively identifies and removes the single worst measurement until the solution is clean.
• Leverage point analysis: Identifies measurements that have disproportionate influence on the estimate, making them harder to detect as bad.

LSE can detect errors in the assumed breaker-switch status of the network model. A topology error occurs when the digital model's connectivity does not match the physical grid, causing large, systematic estimation errors.

• Normalized Lagrange multiplier method: Tests the statistical validity of assumed zero-impedance branch flows to detect erroneous breaker statuses.
• Substation graph analysis: Models the internal bus-bar/breaker configuration to identify which specific switch status is incorrect.
• Suspected topology error flagging: Alerts operators to a mismatch between the SCADA-reported topology and the synchrophasor-derived physical reality.

Beyond state variables, LSE can be extended to estimate and correct network parameter errors, such as incorrect transmission line impedance values stored in the planning database.

• Augmented state vector: Treats suspicious branch impedances as additional unknown variables to be solved simultaneously with bus voltages.
• Residual sensitivity analysis: Calculates the sensitivity of measurement residuals to parameter errors to identify the most likely erroneous parameter.
• Off-nominal tap ratio detection: Identifies transformers operating at tap positions that differ from the telemetered or assumed value.

Standard Weighted Least Squares (WLS) is sensitive to outliers. Robust LSE methods use alternative objective functions that automatically suppress the influence of bad data without requiring iterative removal steps.

• Least Absolute Value (LAV): Minimizes the sum of absolute residuals, making it naturally resistant to outlier contamination.
• Huber M-estimator: Applies quadratic weighting to small residuals and linear weighting to large residuals, blending WLS efficiency with LAV robustness.
• Schweppe-type GM-estimator: Bounds the influence of both measurement residuals and leverage points simultaneously for maximum resilience.

Modern LSE extends beyond the traditional positive-sequence model to perform a full three-phase estimation, capturing the asymmetrical conditions common in distribution grids and during unbalanced faults.

• Mutual coupling modeling: Accurately represents the electromagnetic coupling between phases on the same tower.
• Single-phase laterals: Models the true unbalanced nature of distribution feeders with single-phase taps.
• Neutral voltage estimation: Calculates the neutral-to-ground voltage rise, a critical safety and power quality metric invisible to positive-sequence estimators.