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2.5.9 Illustrative Example
ОглавлениеFor illustrative purposes, the state estimation procedures previously described are applied to a small 4‐bus system. Network topology and measurement configuration is depicted in Figure 2.8.
Figure 2.8 Example of alternative estimators: four‐bus system.
The network is composed of two generating buses, two load buses, and four lines. The set of measurements includes four voltage measurements, one active/reactive power injection measurement, and four active/reactive power flow measurements (labeled as Vi, Pi/Qi, and Pij/Qij). If we consider that the measurement vector comprises 14 elements, then the redundancy ratio is r = 14/(4 × 2 − 1) = 2. The network data (resistance, reactance, and total line charging susceptance) are provided in Table 2.8.
TABLE 2.8 Example of alternative estimators: line characteristics.
| Line | Resistance (p.u.) | Reactance (p.u.) | Susceptance (p.u.) |
|---|---|---|---|
| 1–2 | 0.010 08 | 0.0504 | 0.1025 |
| 1–3 | 0.007 44 | 0.0372 | 0.0775 |
| 1–4 | 0.007 44 | 0.0372 | 0.0775 |
| 1–4 | 0.012 72 | 0.0636 | 0.1275 |
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.
Figure 2.12 Example of alternative estimators: residuals of the QL solution.
In Figures 2.11–2.12, note that the largest absolute residual yi(x) is located in the non‐quadratic zone of the objective function for the QC and QL estimators. This large residual corresponds to measurement V2.
With regard to the LMS, LTS, and LMR procedures, Table 2.12 provides the optimal value of each binary variable bi corresponding to the ith measurement indicated in the first column of the same table.
TABLE 2.12 Example of alternative estimators: optimal values for the binary variables.
| Estimators | Estimators | ||||||
|---|---|---|---|---|---|---|---|
| LMS | LTS | LMR | LMS | LTS | LMR | ||
| V 1 | 1 | 1 | 0 | P 2, 4 | 1 | 1 | 0 |
| P 1 | 1 | 1 | 0 | Q 2, 1 | 1 | 1 | 0 |
| P 1, 2 | 1 | 1 | 0 | V 3 | 1 | 1 | 0 |
| P 1, 3 | 1 | 1 | 0 | P 3, 4 | 1 | 1 | 0 |
| Q 1 | 1 | 0 | 0 | V 4 | 1 | 1 | 0 |
| Q 1, 3 | 1 | 0 | 0 | Q 4, 2 | 0 | 0 | 0 |
| V 2 | 0 | 1 | 1 | Q 4, 3 | 0 | 1 | 0 |
In Table 2.12, note that the sum of the optimal binary variables for the LMS and LTS estimators is equal to 11, i.e. to the median. However, the LMR procedure only rejects one measurement. This residual corresponds to measurement V2.
Finally, Table 2.13 provides the estimated voltage magnitude and angle for the WLS, LAV, QC, QL, LMS, LTS, and LMR estimation procedures. For comparison purposes, the second column of this table shows the true state xtrue.
TABLE 2.13 Example of alternative estimators: true and estimated state vectors.
| Estimators | ||||||||
|---|---|---|---|---|---|---|---|---|
| x true | WLS | LAV | QC | QL | LMS | LTS | LMR | |
| V1 (p.u.) | 1.000 | 0.994 | 0.993 | 1.000 | 0.998 | 0.996 | 0.985 | 1.000 |
| V2 (p.u.) | 0.985 | 0.979 | 0.978 | 0.985 | 0.983 | 0.981 | 0.970 | 0.985 |
| V3 (p.u.) | 0.973 | 0.967 | 0.965 | 0.972 | 0.970 | 0.969 | 0.976 | 0.972 |
| V4 (p.u.) | 1.020 | 1.014 | 1.013 | 1.020 | 1.018 | 1.030 | 1.023 | 1.020 |
| θ1 (rad) | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| θ2 (rad) | −0.008 | −0.008 | −0.008 | −0.008 | −0.008 | −0.008 | −0.008 | −0.008 |
| θ3 (rad) | −0.025 | −0.025 | −0.025 | −0.025 | −0.025 | −0.025 | −0.028 | −0.025 |
| θ4 (rad) | 0.038 | 0.038 | 0.038 | 0.038 | 0.038 | 0.035 | 0.034 | 0.038 |
In Table 2.13, note that the best estimates are provided by the QC and LMR estimators, while the performance of the LTS procedure is poor. Note that this conclusion is withdrawn considering only one measurement scenario, a particular metering configuration, and a small 4‐bus system. In order to obtain statistically sound conclusions, the following section provides a detailed analysis of both the numerical and computational behaviors of the aforementioned algorithms, considering 100 measurement scenarios, different measurement configurations, and a large system.
