Advances in Electric Power and Energy. Группа авторов

✓ ✗ n + 1 m 2m + 1 MINLP LTS ✓ ✗ n + m m 2m + 1 MINLP LMR ✓ ✓ n m 2m MINLP

      The acronyms NLP and MINLP represent nonlinear problem and mixed integer nonlinear problem, respectively. The symbol ✓ indicates “needed,” while the symbol ✗ indicates “not needed.”

Figure 2.7 provides a graphical comparison between methods in terms of number of continuous/binary variables and quantity of additional constraints.

Figure 2.7 Problem size comparison for different estimators.

      In Figure 2.7, observe that the LMS and LTS estimators require the highest number of both continuous/binary variables and additional constraints. The computational burden of these algorithms is thus expected to be heavier than that of the others.

      2.5.9 Illustrative Example

For 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 V_i, P_i/Q_i, and P_ij/Q_ij). 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
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