Robust State Estimator (Smart Electric Grid, LLC)
Robust State Estimator(Smart Electric Grid, LLC) 

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

As the most commonly used state estimator in power industry, weighted least square (WLS) based SE is essentially an average estimator, and therefore, the influence of a bad data is shared (or hidden) by the neighbor buses around the location of bad data. This feature of WLS based SE makes the bad data detection very hard, if not impossible. 

The convergence issue of WLS based SE

Considering WLS is essentially an average estimator, if the average value  goes beyond the stop criterion due to the huge inconsistence caused by the bad data, the state estimator will definitely diverge. In addition, WLS SE will force the residuals of the commonly existing leverage points as small as possible to obtain a minimum objective function. That bad data are present in those leverage points will cause WLS SE to diverge or at least a biased voltage estimate.

The difficulty of bad data detection

The difficulty of bad data detection lies in the following aspects:

1) Measurements and System Model

    Our observation on real-time data from limited power utility companies

    shows that power utility companies have low measurement redundancy

    ratio and high percentage of bad data.

2) Algorithms

  • Weighted Least Square based state estimator forces the influence of bad data shared by its neighbor buses, which makes the bad data detection very hard, especially the nonlinearity of power grids give a big challenge to residual-based bad data detection which fully depends on measurement residual linearization.
  • Leverage Point is one of the main reasons that bad data is hard to be detected by Weighted Least Square based SE and Least Absolute Value based SE. If a bad data (including three types) exists in a leverage point measurement, this bad data will be satisfied or select First by today's state estimators. Unfortunately, leverage points very commonly exist in power system state estomation. SE+ has effectively solved the problem of Leverage Point.

Leverage Points dominate the solutions of today's state estimators

Leverage points behave like critical measurements due to their close to zero residuals, therefore, bad data on them are impossible to detect and the solutions will have big bias. It is a fact that leverage points are very common in power systems.

Today's state estimators including the commonly used WLS and LAV state estimators, when leverage points exist, will satisfy or select those leverage points First in order to obtain a minimum objective function value. Using LAV SE as an example, it will select leverage point measurements first to form its base set in order to obtain a minimum objective function value. Therefore, leverage points with bad data will cause today's state estimators to diverge or at least a biased solution.

SE+ effectively solves the problem of leverage points and therefore it is always able to reach a feasible and more accurate solution.

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

Unlike WLS based state estimator and all other existing ones, SE+ or SEPlus can obtain a feasible voltage solution once the system is observable, and it has the following charcteristics:

1) No human involvement. Examples of human involvement include:

    adjusting the measurement weights to make it convergent; removing

    suspecious measurements; changing the measurement values and/or

    system parameters, etc.

2) The voltage estimate completely depends on the given measurements

    and system data/parameters.

3) SE+ can largely improve the data accuracy of power system


4) SE+ is robust because it is Not sensitive to bad data. Its breakdown

    point value is 0.5. Breakdown point is a statistic index to evaluate the

    robustness of a state estimator, which takes value from 0.0 to 0.5.





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