χ2 test and on normalized residuals, respectively (see [18]).
According to the traditional bad measurement detection procedure, if the objective function value at the estimated state
Therefore, if a bad measurement is corrupting the measurement set but its magnitude is not sufficiently large to satisfy
In the study below, two bad measurements are present in each measurement scenario. The magnitude of the corresponding error is sufficiently small to satisfy the condition
One hundred measurement scenarios have been considered, and Tables 2.16 and 2.17 provide the results concerning estimation accuracy and computational performance, respectively. Note that the format of these tables is similar to that of Tables 2.14 and 2.15.
TABLE 2.16 Case study: estimation accuracy results with bad measurements.
| Method |
|
|
|
|
|---|---|---|---|---|
| WLS | 0.0019 | 0.0014 | 0.0017 | 0.0014 |
| LAV | 0.0019 | 0.0016 | 0.0019 | 0.0016 |
| QC | 0.0016 | 0.0012 | 0.0017 | 0.0014 |
| QL | 0.0016 | 0.0012 | 0.0017 | 0.0014 |
| LMS | 0.0103 | 0.0106 | 0.0053 | 0.0047 |
| LTS | 0.0050 | 0.0048 | 0.0025 | 0.0022 |
| LMR | 0.0018 | 0.0013 | 0.0017 | 0.0014 |
TABLE 2.17 Case study: computational performance results with bad measurements.
| Method | Minimum (s) | Mean (s) | Maximum (s) | Std. dev. (s) |
|---|---|---|---|---|
| WLS | 0.94 | 1.70 | 2.28 | 0.18 |
| LAV | 0.59 | 0.94 | 1.29 | 0.12 |
| QC | 0.22 | 0.31 | 0.45 | 0.05 |
| QL | 1.00 | 1.74 | 2.71 | 0.27 |
| LMS | 3.80 | 8.21 | 12.64 | 1.42 |
| LTS | 1.28 | 2.36 | 3.96 | 0.36 |
| LMR | 0.94 | 2.73 | 34.84 | 4.99 |
The following observations can be made about Tables 2.16 and 2.17:
1 As expected, the WLS approach does not provide the most accurate results. The estimates computed using the QC and QL techniques are more precise than that obtained with the conventional WLS method.
2 The QC and LAV approaches are the most efficient ones from the computational perspective. The computational burden of the LMS technique is higher than that of any of the other procedures.
2.5.12 Conclusions
Considering recent advances in computational techniques, this work addresses the electric state estimation problem from a mathematical programming perspective.
In this chapter, the most common state estimators are formulated as optimization problems and implemented, proving to be computationally efficient and numerically accurate.
From the computational point of view, QC and LAV techniques perform faster than the conventional WLS estimator, saving up to 75% CPU time (compared with the WLS method directly solved as an optimization problem). On the other hand, mathematical programming formulation of some estimators (such as LMS and LTS approaches) involves non‐convexities and a significant number of binary variables, resulting in higher computational burdens.
With regard to estimation accuracy, numerical simulations denote that the LMR and QL techniques provide
