Sensor Fault Diagnosis in Wind Turbines - Wind Turbine Control and Monitoring - page 263

Now, as k 0 is actually unknown, it is substituted by its maximum likelihood
estimate. The resulting decision function at time k can then be written as:

1 j k S j ð R 1 Þ

g ð k Þ¼ max ð 10 : 2 Þ

An alarm is triggered at the time instant t a defined as:

t a ¼ min f k : g ð k Þ [ h a g ð 10 : 3 Þ

and the estimated fault occurrence time can be determined from

k 0 ¼ arg

S j ð R 1 Þ max

1 j t a

ð 10 : 4 Þ

This test can be implemented recursively in order to handle the increasing
number of data as time elapses. The derivation of this recursive algorithm is
explained in [ 8 ], and we only describe the algorithm here.
CUSUM algorithm—recursive form

g ð k Þ¼ max ð 0 ; g ð k 1 Þþ s ð k ÞÞ g ð 0 Þ¼ 0 ð 10 : 5 Þ

d ð k Þ¼ d ð k 1 Þ 1 f g ð k 1 Þ [ 0 g þ 1 d ð 0 Þ¼ 0 ð 10 : 6 Þ

t a ¼ min f k : g ð k Þ [ h a g ð 10 : 7 Þ

k 0 ¼ d ð t a Þ

ð 10 : 8 Þ

where 1 {x} is the indicator function of event x. It is equal to 1 when x is true and to
zero otherwise.
As an example, let us consider the case where the residual is normally
distributed.

10.2.1.1 Example

In fault free mode, the probability law of the n r -dimensional residual vector is
assumed to be

L ð r ð i ÞÞ ¼ N ð l 0 ; R Þ

while upon occurrence of a fault, it is given by

L ð r ð i ÞÞ ¼ N ð l 1 ; R Þ