0.0636
Table 2.9 shows the true state of the network, obtained from a converged power flow solution [15]. Bearing in mind that the measurement standard deviations are 0.01 p.u., the actual measurement values are provided in Table 2.10.
TABLE 2.9 Example of alternative estimators: operating point.
| Bus no. | Voltage magnitude (p.u.) | Voltage angle (rad) |
|---|---|---|
| 1 | 1.000 | 0.000 |
| 2 | 0.985 | −0.008 |
| 3 | 0.973 | −0.025 |
| 4 | 1.020 | 0.038 |
TABLE 2.10 Example of alternative estimators: measurements.
| Measurement | Value (p.u.) | Measurement | Value (p.u.) |
|---|---|---|---|
| V 1 | 0.993 | P 2, 4 | −1.350 |
| P 1 | 0.963 | Q 2, 1 | −0.293 |
| P 1, 2 | 0.204 | V 3 | 0.974 |
| P 1, 3 | 0.750 | P 3, 4 | −1.076 |
| Q 1 | 0.762 | V 4 | 1.025 |
| Q 1, 3 | 0.556 | Q 4, 2 | 0.673 |
| V 2 | 0.963 | Q 4, 3 | 0.506 |
Since the proposed 4‐bus system does not comprise any zero‐injection buses, no equality constraints are considered. The tolerance T used for the QC, QL, and LMR estimators is set to 1, and parameter M, which is used for the LMS, LTS, and LMR procedures, is set to 100. From (2.40), the median ν is computed using the following equation:
(2.46)
where function int(x) denotes the integer part of x.
Table 2.11 provides a brief description of the computational characteristics of each optimization problem, detailing the number and type of optimization variables, and the number of additional constraints. Nonlinear problems (WLS and LAV estimators) are solved using MINOS 5.5 [11] under GAMS 23.5 [12, 30], whereas mixed integer nonlinear problems (QC, QL, LMS, LTS, and LMR procedures) are solved using SBB [31] under GAMS.
TABLE 2.11 Example of alternative estimators: characterization.
| Continuous variables | Binary variables | Additional constraints | |
|---|---|---|---|
| WLS | 7 | — | 0 |
| LAV | 21 | — | 28 |
| QC | 7 | 14 | 0 |
| QL | 21 | 14 | 28 |
| LMS | 8 | 14 | 29 |
| LTS | 21 | 14 | 29 |
| LMR | 7 | 14 | 28 |
Figures 2.9–2.12 depict the weighted measurement residuals yi(x) for the WLS, LAV, QC, and QL estimators (blue data points). The objective function shapes corresponding to each estimator are represented with dotted lines.
Figure 2.9 Example of alternative estimators: residuals of the WLS solution.
Figure 2.10 Example of alternative estimators: residuals of the LAV solution.
Figure 2.11 Example of alternative estimators: residuals of the QC solution.
