P3_2_Heraud

where x is the state variable, u is the input variable of the system, A, B are two matrices. In this paper, an extended method is presented to estimate gross errors. 2.2 Dynamic integral measurement test Considering the measurement adjustments variable k at time interval j: ˆ e jk = X jk − X jk (2)
(22)
0
SU . . . 0
Hale Waihona Puke −1. . .. . .
. .
.
−1 SU 0 0 . . . S X −1 0 0 . . .
δˆi = ∑ ρ k t k , δˆi ' = ∑ ρ 'k t k
14)
(13,
The reconciliation and gross error estimation problem becomes: Min{(Tβ β + E 'Tρ ' ρ '− u + )T RU −1 (Tβ β + E 'Tρ ' ρ '− u + )
(18,
19) All matrices are given by: B 0 . . . 0 0 B 0 . . 0 . . . Dm = . . . . . . B 0 0 (20)
T T E 'T RU −1E 'Tρ ' 0 U = ρ' −1 T T 0 Tρ E RX ETρ (28)
+ (Tαα + ETρ ρ − x + )T RX −1 (Tα α + ETρ ρ − x + )} Rmα = Dm β (15) where α 0 β0 . . α = . ,β = . . . α β s s 17) Measurements are: u0 + x0 + . . u + = . , x+ = . . . u + x + n n
S X −1 0 . RX −1 = . . 0
0 S X −1 . . . 0
. . .
. . .
. .
.
(23) where SU : variance matrix of u .
S X : variance matrix of x . I I . Tα = . . I 0 t1I . . . tn I . . . . . . . . . . . . 0 t1s I . = Tβ . . tn s I
(8,
I I . E = E' = . (10) and ˆ ˆ' δ δ 0 0 . . δˆ = . , δˆ ' = . . . ˆ ˆ δ n δ 'n 12)
The sample standard deviation is given by:
Sk =
1 n ∑ (e jk − ek )² n j =0
(3) where ek is the mean value of all adjustments, n − 1 is the number of measurements. If we suppose that there is no gross errors, the following variable: ek yk = Sk n + 1 (4) follows a t-student distribution: yk : t (n) . So, variable k will be suspected of containing a gross error if the following holds | ek | > t1− (η / 2) Sk n + 1 (5) where η is usually selected as 0.05; thus, the critical value t1− (η / 2) = 1.99 for n = 100 .
(16,
(24) Assume that the measurements are taken at equal intervals h. Then: 0 . . . 0 I I hI . . . h s I . . . . Tα = = Tβ . . . . . . . . s s I nhI . . . n h I (25) Let T T E 'T RU −1Tβ 0 PT = 2 ρ ' −1 T T 0 Tρ E RX Tα (26) qT = −2 ( (u + )T RU −1E 'Tρ ' ( x + )T RX −1ETρ ) (27)
2. Polynomial representation 2.1 Introduction An integral dynamic data reconciliation method is presented in a previous paper [1]. It concern dynamic linear systems with the following model:
s' s' k =0 k =0
.
I
0 − A I 0 − A 2I . . Rm = . . 0 . 0
. 0 . . .
0 0 . − A sI . − A . ..
(21)
(11,
SU −1 0 . RU −1 = . . 0
ρ ' y= ρ (29) then min{z T Q −1 z + wT z + y T PT z + qT y + yTUy} Mz = 0 (30) where ( R −1T )T u + β z = , w = −2 U −1 β T + α ( RX Tα ) x
x ˆ is the where X = is the measured variables; X u corresponding reconciled value.
x = ∑α k t k + δ
k =0
s
(6)
u = ∑ βk t k + δ '
k =0
s
(7)
for
where δ = Eδˆ , δ ' = E 'δˆ ' 9)
dx = Ax + Bu dt
(1)
2.3 Bias detection A method is presented to detect and estimate the bias corresponding to each variable [5]. Assume a set of gross errors is present. This is modeled as follows:
*
Summary Gross error detection is a companion technique to data reconciliation that has been developed to identify and eliminate gross errors. This paper deals with a method to identify and estimate gross errors for linear dynamic systems. An appliation is made in wind power, in particular the Doubly Fed Induction Generator (D.F.I.G) of a wind turbine. The method presented in a previous paper [1] is extended to allow gross error estimation.
1. Introduction Process measurements are inevitably corrupted by errors. These errors can lead to significant deterioration in plant performance. They can also drive the process into an unsafe operating regime. It is therefore important to reduce the effect of such errors. Several data processing techniques can be used to achieve this objective. An example of these techniques is data reconciliation and gross error detection which are applied together to improve accuracy of measured data [2]. In the domain of electric machines, several publications treated the diagnosis and fault detection [3, 4]. Most of these techniques permit to detect just parameter faults of the machine. In this paper, the method is used to detect sensors’s errors. The global aim is to combine both methods to make a good supervision of the system.