Advances in Electric Power and Energy. Группа авторов

Скачать книгу

850.5529 562.7272 4.3639 4.2674 4.2674

      2.3.1 Bad Measurement Detection

      If measurements are Gaussian distributed and independent, and if the weighting factors w_i correspond with the inverse of the measurement variances (

), the distribution of J(x ) is a χ² with m + r − n degrees of freedom [8], where r is twice the number of transit buses (exact measurements). Thus, we can write

(2.13)

      where 1 − α is the confidence level.

      Therefore, for a given α (e.g. 0.01), the value χ²(1 − α, m + r − n) can be computed and the χ² test applied, i.e. if

there is no bad measurement at the 1 − α confidence level; otherwise there is. Further details can be found in [7].

      Example 2.3 Bad Measurement Detection Example

      The voltage measurement at bus 1 (Figure 2.1) is considered to be 1.05 instead of 1.0933, i.e. a bad measurement is introduced. Solving the estimation problem, the optimal objective function value is

.

      The test threshold at 0.99 confidence level (α = 0.01) with 10 + 2 − 7 = 5 degrees of freedom is χ²(0.99,5) = 15.0863. Therefore, since χ²(0.99,5) < 16.6008, we conclude that bad data plague the measurement set with a 0.99 confidence level.

      For the initial case and since

, no bad data affect the measurement set with at 0.99 confidence level.

      2.3.2 Identification of Erroneous Measurements

      If the bad measurement test detects bad measurements, these measurements should be identified. This is accomplished below.

      Consider the nonlinear measurement model

(2.14)

      where x^true is the unknown true state vector and e is the measurement error vector. Note that

      (2.15)

      which is a typical assumption in state estimation.

      Consider the differential measurement equation:

      (2.16)

      If measurements are Gaussian distributed and independent, the least squares estimator

can be obtained by minimizing the weighted sum of square deviations of the differential errors:

      (2.17)

      This minimization problem leads to the system of equations:

      (2.18)

known as the system of normal equations [8]. Assuming that the gain matrix

(2.19)

      has an inverse, the least squares estimates

can be written explicitly as

      (2.20)

      from which we conclude that

is a linear function of dz.

      The residual and the differential residual are defined as

(2.21)

      (2.22)

      Thus

(2.23)

      where P = HG⁻¹H^TW is known as hat or projection matrix.

Integrating (2.23), the linear transformation from z to r at the optimum is

(2.24)

      where k is a constant vector and S is known as the residual sensitivity matrix.

If measurements z are Gaussian distributed and independent with zero mean and covariance matrix R (R = W⁻¹) and

is an unbiased estimator of h(x^true), i.e. , the residual vector r provided by the linear transformation (2.24) is Gaussian distributed with parameters given by

(2.25)

      (2.26)

and

Скачать книгу