turboshaft engine condition monitoring by bayesian ...

TURBOSHAFT ENGINE CONDITION MONITORING BY BAYESIAN IDENTIFICATION L. Biagioni\ and R. Cinotti•• Centrospazio-CPR, I-56014 Pisa, Italy

L. d'Agostino· Dept. ofAerospace Engineering, University ofPisa, I-56100 Pisa, Italy The application of a Maximum Likelihood Estimate technique has been Implemented in an Engine Condition Monitoring framework for a small turboshaft for power generation purposes. The turboshaft has been modeled in a fully non-linear way, by using actual turbomachine performance maps obtained from the manufacturer: the accuracy of the simulation proved to be very good with respect to real operating data. The model was used both to generate sample synthetic dataset (by adding Gaussian noise to the selected outputs-measurements) and as the core computational engine in the identification process. The results obtain show the very good robustness of the proposed identification process, and its capability of dealing with noisy or even malfunctioning transducers. This capability is provided by the possibility of determining a mathematically sound statistical framework, which is not only capable of identifying the most likely fault configuration but also to indicate the confidence level with which the identification is performed.

List of Symbols f! A B D 'l m

state vector of the system (health factors) throttle matrix fault matrix standard deviation of the transducer noise efficiency air mass flow rate propellant mass flow rate

m1 M number of transducers N rotation speed, number of samples N number of unknown elements in the state vector p identification probability (confidence) 11

E..

performance parameter vector

P

pressure pressure loss pressure ratio vector output of the system

ff

I1 q £

T ~

standard deviation of the identified state vector temperature vector input of the system

Professor. Graduate Student. Undergraduate Student.

Subscripts

nominal operating condition total conditions

Introduction Since the early 1970's the maintenance process of turbomachines (especially aeronautical ones, or more in general complex engineering systems such as structures, vehicles, etc.), has undergone a deep change in attitude, with the introduction of Engine Condition Monitoring (ECM) methods for the prediction of upcoming failures of the system. In previous years, preventive maintenance on the machine was scheduled to occur at fixed time intervals, which were set by the manufacturer based on historical or nominal (design-derived) MTBF data. On the other hand, the introduction of ECM methods was aiming at tailoring the most efficient (usually meaning less frequent) intervention schedule based on the monitoring of specific parameters on each of the installed machines: the hypothesis underlying ECM is that the damage building up in one of the components is causing measurable changes in at least one of the monitored

Copyright© 2001 by Centrospazio.

J5'h International Symposium on AirBreathing Engines, Bangalore, India, September 200 l.

parameters. ECM is thus in principle not able to predict sudden failures or failures to auxiliary systems, unless they have an impact on some engine parameter. By the simultaneous monitoring of a sufficiently large number of different parameters, ECM methods are also capable of identifying the specific component which is accumulating damage, although this process is still subject to a certain level of uncertainty. The methods more frequently used for ECM purposes in turbomachines are: )> Vibration analysis )> Gas Path Analysis (GPA) )> Trend analysis )> Acoustic analysis )> Lubricant analysis

are linearized by first-order Taylor expansion for a nominal operating point:

Rearranging the above, we can obtain qk -q.o ""!.{ 8

"

i

0,00

..(l,Oi

'a

~

,..J-

0,005

"!'' o.ooo

D(T< l)D(Ptl)D(T!l)D(Pl2)D(TCl)D(PtJ)D(T

"

Fig. 8 Non-dimensional variation of the output values for a 1%, 2% and 3 % compressor deterioration.

I

..{i,010

--0,015

'a

8

z

-0mo

D(Ttl)D(Pt!lD(Tl1)D(l'2)!XfU)D(PtJ)D(Tt4)D(Pt4lD(ft5)D(Pt5) D(m) lXWl

Model ootput l::j."'

0,03

~ ~..,

0,02

g

Fig. 11 Non-dimensional variation of the output values for a 1%, 2% and 3 % second turbine stage deterioration.

1i

-~

O,Ol

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0,00

+---------.....--"'......_"'--_,...._____.......,,..-1

Statistical Infere nee Method (Maximum Likely hood)

'a

~

-0,01

Upon observation of a system, of which we have a mathematical model, a Maximum Likelihood Estimate technique 11 (MLE, based on Bayes' wellknown statistical theorem) can be used in order to derive the joint a posteriori Probability Density Function (PDF) of the state vector. In the statistical characterization of the uncertain quantities, a Gaussian a priori PDF is initially assumed 12 , since it usually simplifies the mathematical problem. A particular set ( g , indicated hereafter as optimal) of states is then obtained which maximizes the likelihood of the observed data with respect to the corresponding model predictions. Equivalently, g minimizes the cost function:

D(T< l)D(Pll )D(fl2)D(P\2)D(Ttl)D(PtJ)D(Tt4)D(P\4)D(TO)D(Pt5) D(m) D(W)

Model output

Fig. 9 Non-dimensional variation of the output values for a 1% ,2% and 3 % combustor deterioration.

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J51h International Symposium on AirBreathing Engines, Bangalore, India, September 200 l.

f[

J {£} =

J:(o)

lo) (t.,_q)T

(t.}

the steepest descent and inverse-Hessian methods, especially convenient when dealing with nonlinear models n.

n=t

where ;:!(o)(t.) and q< l(t.,g_) respectively represent, after normalization, the observed (measured) and computed values of the state variables at the time t. for a given choice of a in the model 13 • For a large number N of samples~ the asymptotic PDF of the uncertain state vector !! is given by: 0

G(g)

Application of the Statistical Inference Method to ECM When applying the MLE technique to the identification of faults in an engine, we assume that N is the number of samples acquired before each identification process and M is the number of outputs of the system (transducers). We also assume that the outputs are affected by a random error (noise) with a null mean and a known standard deviation cri . From the above paragraph, the MLE identification consists in minimizing the following cost function (the so called chi-squared function):

1 [ --1 ( a a~) .:~:;. ( a - a)] ---=exp

21/"·'2

#i'

2- -

- -

where the N0 -dimensional vector g of identified optimal states coincides with the mean value of a and 12:1 is the determinant of the correspondin~ covariance matrix. For N ~ oo the inverse of 2: coincides with Fisher's information matrix 14 and is expressed by:

z

2

2:-1 ::= NN0 0 J(4)

2J(g)

ol

2

(f, £.)

.

t, t, {

~

x / ( n )-0" / (i_, fl_))

r

where #. is the vector of known (measured) inputs. As mentioned in the previous paragraph, the method allows for significance tests to be performed on the identification results, and to test the influence of the number of samples, of the standard deviation in transducer measures and of the possible failure of one of the transducers. In order to provide a well-controlled set of engine data for benchmarking purposes, the turboshaft model detailed above was used also for producing the dataset. The model was run with a predefined fault configuration (determined by the state vector g_ ). A reduced output vector was selected, including the values of Tt2' Pt2' P13' Tt4' P!4 and ~s in order to mimic the non complete measurement capability in normal turbomachine installations. Gaussian noise was then added to each channel of the dataset, with null mean and known standard deviation, D . A set of fault configurations were used in order to test the method (Table 1). Although we will not detail here the important characteristics of the z 2 function, it is interesting to note that the value of the function to which the identification process (optimization) is converging, 2 % (i, g) , is a measure of the significance of the statistical result. As an example, for N 100, M = 6 and N = 4 , we have that the value corresponding to a confidence equal to 1% is:

where N0 is the number of the observed state variables. The important properties of this estimation technique consist, according to the definition of Cramer-Rao lower bound, in being asymptotically unbiased and efficient as well as consistent. It also has to be reminded that the above results for the a posteriori PDF are only valid for globally identifiable systems. For the engine model considered here this condition is indeed satisfied. When applying the MLE technique to the ECM problem, we observe that the availability of G(g) allows for the application of classical statistical methods for testing the significance of the estimated performance parameters of the engine components by comparison with their nominal (not deteriorated) values at the same operating condition. Therefore, the capability of yielding the posterior PDF of the performance parameters (or health factors) has crucial implications in the diagnostic analysis of the engine conditions and represents a very significant asset of the identification method. The optimization of the cost function for the above engine model has been carried out numerically using a Levenberg-Marquardt method, a second order algorithm with performance combining

0

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15°' International Symposium on AirBreathing Engines, Bangalore, India, September 2001.

z~ 99 (I00,6,4) = 657.8871

keeps below 1%, even for a relatively small number of samples (Fig. 13).

When performing the significance tests, we 're therefore accepting an identification with a confidence level equal to p% if for the resulting value of ;r2 the following holds: