A way to understand Weighted Least Square (WLS) based State Estimator

As the most commonly used state estimator in the power industry, the weighted least squares (WLS) based SE functions essentially as an average estimator. Consequently, the influence of bad data is distributed and often concealed among the neighboring buses around the location of the bad data. This characteristic of WLS-based SE makes detecting bad data very difficult, if not impossible.

The divergence issue of WLS based SE

1. Presence of Bad Data:

   • Erroneous measurements: The WLS state estimator can diverge when there are significant errors or outliers in the measurement data. Bad data can skew the residuals, leading to incorrect state estimates and causing the estimator to fail to converge.

2. Leverage Points:

   • Influence of leverage points: Leverage points, which are data points with a high influence on the estimation process, can cause divergence if they contain bad data. The WLS method minimizes the residuals of these points, potentially leading to biased or incorrect estimates.

3. Ill-Conditioned System Matrix:

   • Poorly conditioned system: An ill-conditioned system matrix, where the equations are poorly scaled or imbalanced, can cause numerical instability. This instability makes it difficult for the WLS algorithm to find a convergent solution.

4. Inconsistent Measurement Data:

   • Inconsistencies and inaccuracies: If the measurement data are inconsistent or inaccurate, the WLS estimator may not be able to reconcile the discrepancies, leading to divergence. This is especially problematic in systems with high variability or noise.

5. Poor Initialization:

   • Inadequate initial estimates: Starting with poor or inaccurate initial estimates for the state variables can hinder the convergence of the WLS estimator. Poor initialization increases the number of iterations needed and can lead to divergence.

6. Topology and Parameter Errors:

   • Errors in system topology and parameters: Errors in the system's topology (network configuration) or parameters (line impedances, transformer ratios, etc.) can cause the WLS estimator to diverge. Incorrect system models lead to incorrect interpretations of the measurements.

7. Heavy Load Conditions:

   • High system stress: During periods of heavy load or stress, the power system can exhibit nonlinear behaviors that are not well captured by the WLS method. These conditions can lead to significant estimation errors and divergence.

8. Lack of Redundant Measurements:

   • Insufficient redundancy: A lack of redundant measurements means there are fewer data points available to validate and correct inconsistencies. This can make the system more sensitive to errors and increase the likelihood of divergence.

9. Inadequate Weighting of Measurements:

   • Improper weights: If the weights assigned to different measurements do not accurately reflect their reliability, the WLS estimator may place undue emphasis on inaccurate data, leading to divergence.

10. Extreme Conditions and Emergencies:

    • Unusual operating conditions: The WLS estimator can struggle to converge during unusual or emergency conditions, such as during faults or rapid load changes. These conditions can introduce sudden and significant changes that the estimator is not equipped to handle.

By understanding and addressing these factors, improvements can be made to the state estimation process to reduce the likelihood of divergence and enhance the reliability of the WLS state estimator.

The difficulty of bad data detection

Bad data detection in power system state estimation is challenging due to several factors:

1. Measurements and System Model:

   • Low Measurement Redundancy and High Bad Data Percentage: Real-time data observations from various power utility companies show that many utilities have a low measurement redundancy ratio and a high percentage of bad data. This lack of redundancy means there are fewer measurements to cross-validate and identify errors, making it harder to detect and isolate bad data.

2. Algorithms:

   • Weighted Least Squares (WLS) Limitations: The WLS-based state estimator distributes the influence of bad data across neighboring buses. This dispersion makes bad data detection very difficult, especially given the nonlinearity of power grids, which poses significant challenges to residual-based bad data detection methods that rely on measurement residual linearization.
   • Leverage Points: Leverage points are a primary reason why bad data is hard to detect with WLS and Least Absolute Value (LAV) based state estimators. If bad data exist in a leverage point measurement, this data is often prioritized or selected first by current state estimators. Unfortunately, leverage points are very common in power system state estimation.

SE+: Solving the Leverage Point Problem

SE+ effectively addresses the issue of leverage points, which significantly enhances its capability to detect and reject bad data. By not depending solely on residual linearization and effectively managing leverage points, SE+ provides a more accurate and reliable state estimation, even in the presence of bad data.

By tackling these challenges, SE+ offers a robust solution that significantly improves the accuracy and reliability of power system state estimation, ensuring better performance under a variety of conditions.

Leverage Points dominate the solutions of today's state estimators

Leverage points present a significant challenge in power system state estimation due to their behavior and impact on the estimation process. These points act like critical measurements with near-zero residuals, making bad data on them nearly impossible to detect and leading to significant biases in the solution. The prevalence of leverage points in power systems exacerbates this issue.

Impact on Current State Estimators

Today's state estimators, including the commonly used Weighted Least Squares (WLS) and Least Absolute Value (LAV) estimators, are particularly affected by leverage points:

• Leverage Point Selection: Both WLS and LAV state estimators tend to satisfy or select leverage points first to achieve a minimum objective function value. For instance, LAV SE will prioritize leverage point measurements to form its base set, aiming to minimize the objective function.
• Consequences: When leverage points contain bad data, they can cause these state estimators to either diverge or produce significantly biased solutions. The inability to detect and manage bad data in leverage points undermines the reliability of the estimation process.

The SE+ Advantage

SE+ effectively addresses the problem of leverage points, ensuring more accurate and reliable state estimation:

• Enhanced Detection and Rejection: SE+ appropriately rejects bad data associated with leverage points, preventing these errors from skewing the results.
• Consistent Accuracy: By solving the leverage point problem, SE+ consistently reaches feasible and accurate solutions, even in the presence of bad data. This capability ensures that SE+ provides more reliable state estimation under various conditions.

In conclusion, the innovative approach of SE+ in handling leverage points marks a significant advancement over traditional WLS and LAV state estimators. By eliminating the adverse effects of leverage points, SE+ ensures more accurate, reliable, and robust state estimation, enhancing the overall performance and reliability of power system operations.

Characteristics of Accurate Non-Divergent State Estimator (SE+)

Unlike the Weighted Least Squares (WLS) based state estimator and all other existing ones, SE+ (or SEPlus) can obtain a feasible voltage solution once the system is observable. It possesses the following characteristics:

1. No Human Involvement:

   • Autonomous Operation: SE+ eliminates the need for human intervention in the estimation process. This means no manual adjustments are required, such as:
     • Adjusting measurement weights to ensure convergence.
     • Removing suspicious measurements.
     • Changing measurement values and/or system parameters.

2. Measurement-Dependent Voltage Estimates:

   • Accurate Representation: The voltage estimate provided by SE+ depends entirely on the given measurements and system data/parameters. This ensures that the state estimation process is objective and data-driven.

3. Improved Data Accuracy:

   • Enhanced Precision: SE+ significantly improves the data accuracy of power system operations, leading to more reliable and precise monitoring and control.

4. Robustness to Bad Data:

   • High Robustness: SE+ is notably robust, as it is not sensitive to bad data. This robustness is quantified by its breakdown point value of 0.5. The breakdown point is a statistical index used to evaluate the robustness of a state estimator, with values ranging from 0.0 to 0.5. A breakdown point of 0.5 indicates that SE+ can handle up to 50% bad data without compromising the accuracy and reliability of the state estimation.

In summary, SE+ stands out due to its autonomous operation, reliance on accurate measurements and system data, improved data accuracy, and exceptional robustness to bad data. These characteristics make SE+ a highly reliable and effective solution for power system state estimation.