Advances in Electric Power and Energy. Группа авторов

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 y_i(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 y_i(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 b_i 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