Sensor Fault Diagnosis in Wind Turbines - Wind Turbine Control and Monitoring - page 275

y a ð t Þþ y b ð t Þþ y c ð t Þ¼ 0 ð 10 : 29 Þ

which means that one of the signals y j ð t Þ , for j 2f a ; b ; c g , can be computed from
the other two.
Taking into account this property, and exploiting model described in
Eqs. 10.26 - 10.27 , any balanced three-phase sinusoidal system can be generated by
a state-space model of the following form:

x ð t Þ¼ A ð x e ð t ÞÞ x ð t Þ ð 10 : 30 Þ

y ð t Þ¼ Cx ð t Þ ð 10 : 31 Þ

with state vector x ð t Þ¼½ x 1 ð t Þ; x 2 ð t Þ T , output vector y ð t Þ¼½ y a ð t Þ; y b ð t Þ; y c ð t Þ T ,
and initial state x ð 0 Þ¼½ M sin ð / a Þ; M cos ð / a Þ T . Matrices A ð x e ð t ÞÞ and C are
defined by:

2 3

1 0

;

3

p
2

0 x e ð t Þ

4 5

2

A ð x e ð t ÞÞ ¼ C ¼ ð 10 : 32 Þ

x e ð t Þ 0

p
2

2

Model described by Eqs. 10.30 - 10.32 can be extended to a balanced three-
phase system with multiple harmonics, for which the monitoring methodology
described below extends in a straightforward way [ 16 ]. The above model is used
for the design of residual generators in the following section.

10.5.2 Residual Generation

The presence of sensor faults in a balanced three-phase system can be modeled as
follows. First, model given by Eqs. 10.30 - 10.32 is discretized with sampling
period T s . Adding the effect of electromagnetic disturbances and measurement
noise to the resulting discrete-time model yields:

x ð k þ 1 Þ¼ U ð x e ð k ÞÞ x ð k Þþ w ð k Þ ð 10 : 33 Þ

y m ð k Þ¼ Cx ð k Þþ v ð k Þþ f ð k Þ

ð 10 : 34 Þ

with U ð x e ð k ÞÞ ¼ exp ð A ð x e ð k ÞÞ T s Þ , where x e ð k Þ is assumed to be constant over
the sampling period T s . Vectors w ð k Þ and v ð k Þ are uncorrelated zero-mean
Gaussian white noise sequences with covariance matrices R w and R v , respectively.
f ð k Þ¼½ f a ð k Þ; f b ð k Þ; f c ð k Þ T is a vector containing the faults, with f i ð k Þ the fault in
the i-th sensor, for i 2f a ; b ; c g . The three-phases will alternatively be indexed
with i 2f 1 ; 2 ; 3 g below.