and (2.25):
The normalized residual (N(0, 12)) of measurement i is
The largest residual identifies a bad measurement with a 1 − α confidence level (e.g. 0.99) if
Example 2.4 Bad Measurement Identification Example
The considered covariance matrix R is
Matrices P and Ω are computed using expressions (2.19) and (2.27), respectively. The residual (from (2.21)) and the normalized residual of measurements (from (2.28)) are also calculated and provided in Table 2.6.
The threshold for detection at a 0.99 confidence level (α = 0.01) is Φ−1(1 − 0.01/2) = 2.5758. The normalized residual with highest absolute value in Table 2.6 (column 3) corresponds to v1, and, because its value is larger than 2.5758, it is identified as a bad measurement, and, therefore, it is removed from z .
After removing this bad measurement, observability is checked again. In this case, the system remains observable, and the state estimation is carried out again. After estimating the state, the objective function
TABLE 2.6 Residuals and normalized residuals.
| Measurement | r i |
 |
|---|---|---|
| P 1 | −0.0154 | 2.2757 |
| P 3 | −0.0183 | 1.5836 |
| P 14 | 0.0078 | −0.5386 |
| P 32 | −0.0127 | 0.8996 |
| P 34 | 0.0275 | −1.9219 |
| v 1 | 0.0258 | −3.6530 |
| v 2 | −0.0245 | 3.5700 |
| Q 1 | −0.0032 | 2.6631 |
| Q 3 | −0.0018 | 1.0033 |
| Q 34 | 0.0041 | −1.0586 |
2.4 MATHEMATICAL PROGRAMMING SOLUTION
Problem can be directly solved by using mathematical programming techniques through a nonlinear solver. This approach is considered because current available mathematical programming solvers treat sparsity efficiently and are robust. Moreover, they are computationally efficient and provide highly accurate results. In addition, these solvers allow incorporating easily inequality constraints representing physical limits [9].
For instance, problem can be solved by using solver CONOPT [10] or MINOS [11] under the General Algebraic Modeling System (GAMS) [12], which is a high‐level modeling system for mathematical programming. It consists of a language compiler and a set of integrated high‐performance solvers. GAMS is tailored for complex, large‐scale modeling applications and allows building large maintainable models that can be adapted quickly to new situations. Modeling systems similar to GAMS are AMPL [13] and AIMMS [14].
Example 2.5 Mathematical Programming Problem
In order to solve the example in Figure 2.1 through mathematical programming techniques, the solver CONOPT under GAMS is used [10]. The solution is provided in Tables 2.2–2.4. The objective function evaluated at the estimated state
2.5 ALTERNATIVE STATE ESTIMATORS
The most common state estimation method within electric energy systems is WLS technique. This well‐known procedure was first developed by Schweppe et al. [4–6] in the 1970s and was formulated as an optimization problem.
State of the art of current available nonlinear optimization solvers, recent advances in computational speed, and emerging multi‐core Hyper‐Threading Technology (HTT) processors allow the estimation problem to be solved directly [15]. Two main advantages can be led by proceeding in this manner: (i) alternative estimators (such as least absolute value, quadratic‐constant, or least median of squares, among others) can be easily used, and (ii) decomposition techniques can be applied, resulting in decentralized estimators.
Bearing in mind that recent progress in nonlinear and
