IPN ESIME

Gas turbine fault recognitiontrustworthiness

Científica Vol. 10 Núm. 2 pp.65-74© 2006 ESIME-IPN. ISSN 1665-0654. Impreso en México

Gas turbine fault recognitiontrustworthinessIgor Loboda1

Sergiy Yepifanov2

1 Sección de Estudios de Posgrado e Investigación(SEPI),ESIME, "Unidad Culhuacán", Instituto Politécnico Nacional(IPN), Av. Santa Ana 1000, Edif. 2, tercer piso,Col. San Francisco Culhuacán, CP 04430,México, DF.MÉXICO2 National Aerospace University.Chkalov street, 17,KharkovUKRAINE

Tel. (1) 56-56-20-585729-6000 ext 73-254

email: loboda@calmecac.esimecu.ipn.mx

Recibido el 1 de febrero de 2005; aceptado el 8 de noviembre de 2005.

1. Abstract

This paper examines three methods of gas turbine parametricdiagnosing. The functioning methods are simulated in theidentical conditions of gradually developing faults and randommeasurement errors. The objectives are to tune the methods, tocompare them, and to choose the best one on basis ofprobabilistic criteria of class correct and incorrect recognition.So, main focus of the paper is a recognition trustworthinessproblem. A previous research work in this direction is unitedwith new results and they all together are presented in moresystematic form as a common approach. Besides the methodcomparison and selection, other ways to enhance thetrustworthiness are described and the perspectives to realizethe methods in real condition monitoring systems are analyzed.

2. Resumen

Este artículo examina tres métodos del diagnóstico paramétricode turbinas de gas. El funcionamiento de los métodos se simu-

la en condiciones idénticas al desarrollo de fallas y erroresaleatorios de medición. Los objetivos son afinar los métodos,compararlos y escoger el mejor con base en criteriosprobabilísticos del reconocimiento correcto e incorrecto delas clases de fallas. Así, el enfoque principal del artículo es elproblema de autenticidad del reconocimiento. Algunos resul-tados de investigación previa en este campo se unen con nue-vos resultados y todos se presentan en forma más sistemáticacomo un enfoque común. Además de la selección del métodomejor, se describen otros medios para mejorar la autenticidady se analizan perspectivas de la realización de los métodos enlos sistemas actuales de monitoreo.

Key words: Gas turbine fault classification, thermodynamicmodel, diagnosis methods, fault recognition trustworthiness

3. Introduction

Knowledge of machines’ health through condition monitoringcan allow reducing maintenance cost without risk of failureand give industries significant improvements in efficiency.With a new generation of high temperature and high outputgas turbine engines the objectives of attaining a highavailability and limiting degradation is of vital importance[1]. That is why advanced condition monitoring systems forcritical turbomachines and auxiliary equipment are designedand maintained in recent decades.

To examine the wide range of common deteriorationproblems, a comprehensive monitoring system must integratea variety of approaches. In addition to vibration analysis,there are other technologies to be employed such as gaspathanalysis also known as aerothermal [1] or thermodynamicperformance analysis. Gaspath analysis of turbomachinerypresents advanced calculation techniques used to computeand correlate all performance variables of the gaspath. Thistechnology applied in gas turbine monitoring in order tosimulate and detect the failures also provides insight intohow efficiently fuel is being utilized and so favours a fuelsaving.

Gaspath analysis is a multidiscipline incorporating threeinterrelated disciplines [2,3]: common engine statemonitoring, state prognostics, and concerned in this paperdetailed diagnostics (fault localization or identification).

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The faults exert influence on measured and registered gaspathvariables (pressure, temperatures and consumptions of thegas flow, rotation speeds, fuel consumption, and any others).On the other hand, the influence of operational regimechanges is much greater. That is why in the localizationalgorithms, raw measurement data should be subjected to acomplex mathematical treatment to obtain final resultidentified faults of gas turbine modules (compressors,combustors, turbines). Besides the gaspath faults, controlsystem and measurement system malfunctions can be alsodetected analyzing the gaspath variables [4].

However, a lot of negative factors, which are explained inmore detail below, affect the diagnosing process and makedifficulties for the correct detection. So, engine fault detectionpresents a challenging recognition problem.

To review common works on condition monitoring [1] andfault detection [5] as well as works applied to gas turbines[4,6], it can be stated that a simulation of analyzed systems isan integral part of their diagnostic process. The models fulfillhere two general functions. The first one is to give a gas turbineperformance baseline in order to calculate differences betweenit and current measurements. These differences (or residuals)do not practically depend on operational regime variationsand so serve as good degradation indices. The second functionis related with a fault classification. The models connect mo-dule degradation and the residual changes assisting with afault class’s description.

In the eighties and early nineties, any direct use of complexstatistical recognition methods in an on-line capacity wasprohibitively expensive in time and computer capacity. Itwas therefore often decided to simplify diagnostic techniquesin order to reduce processing requirements. For instance,MacIsaac and Muir [6] used the method based on fault matri-ces where every class (fault signature) is presented byresidual’s signs only. Other example of a simplified techniquecan be found in [7]. To recognise the classes the author applieslinear and non-linear discriminant analysis but he needs tominimize an axis set of the class’s recognition space to redu-ce processing requirements. However, our statisticalsimulations of diagnosing process have shown [8] that thementioned simplifications cause great recognition errors.

Over the last decade there have been significant advances ininstrumentation and computer technology which resultedin more perfect approaches such as [4]. The authors proposesome enhanced diagnosing methods based on non-lineargaspath models, statistical neural networks, and probabilisticfault identification that promise high confidence. However,this work as many other lacks for a numerical estimation of

method’s effectiveness and any comparison with other knownapproaches.

As opposed to the mentioned works, this paper is concentratedon a trustworthiness problem. On basis of proposed earlierprobabilistic indices [9], three methods are optimized andcompared in order to choose the best one and giverecommendations for practical use. The method analysis ispreceded in the paper by a description of developed andapplied models.

4. Development

4.1 Models used

First of all the diagnosing process needs a base-line tocalculate the residuals [5] which may be presented as relativechanges of gaspath variables

0

** −=

Y

YYδ (1)

where Y * is measured value, Y0(U) - base-line value, and U is

the vector of control variables (for example, fuelconsumption) and ambient conditions (ambient air pressureand temperature). So, the vectorial base-line function Y

0(U)

may be interpreted as a model of gas turbine normal behavior.

Such a normal state model may be formed by any abstractfunction as well as a physical model. As an example of an abstractfunction, the second order four arguments full polynomial

Y0(U ) = c

1 + c

2U

1 + c

3U

2 + c

4U

3 + c

5U

4 +

c6U

1U

2 + c

7U

1U

3 + c

8U

1 U

4+

c9U

2U

3 + c

10U

2U

4 + c

11U

3U

4 +

(2)

c12

U1

2 + c13

U2

2 + c14

U3

2 + c15

U4

2

is given which is able to describe correctly an enginebehaviour [10]. To compute the coefficients c

1 –c

15 this model

needs registered data inside a wide range of operationalconditions.

Non-linear thermodynamic model [3], in which every mod-ule is presented by its full manufacture performance map as itis done in [6], demonstrates the option of a physical model.The capacity to reflect the normal behaviour is based onobjective physical principles realized. Since the faults affectthe module performances involving in the calculations thethermodynamic model has additional capacity to simulategas turbine degradation.

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)(

)(0→

→

U

UY

P

PP

P

P

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To have a possibility to displace the module maps ofperformances v (corrected flow parameters or efficiencyparameters) for a fault development introducing into themodel, the correction factors

0j

jj ν

ν=Θ (3)

are introduced as a relative performances, where vj0 is a nomi-

nal value. Fig.1 gives a schematic representation of correctionfactor actions.

So, in the thermodynamic model, the gaspath variables rela-te with the control variables and the correction factors i.e.present a vector function of the view

),(→→→ΘUY (4)

This function is computed as a solution of an algebraicequations system reflecting the conditions of a gas turbinemodules combined work. The software consists ofapproximately 60 subprograms, most of them are universal.Thermodynamic models of more than 20 gas turbine enginesof various schemes were elaborated and applied in healthmonitoring systems [3].

It is known that typical gaspath faults cause relatively small

residuals (4-6%). That is why a linearization of the functional

dependence Y(Θ) is possible and the linear model:

→→Θ= δδ HY (5)

is widely used in diagnostics. It connects the small relativechanges δΘ of correction factors with the relative deviationsδY of gaspath variables by the matrix of influence coefficientsH. The thermodynamic model software is capable to generatethe matrices H for any operational conditions determined bythe vector U.

What is a difference between the simulated deviations δY andthe residuals δY* based on real measurements? Ideally, theyshould be equal however every vector has its own errors.

As described before, either the non-linear model (4) or thelinear one (5) are capable to simulate the fault developmentand for this reason can be classified as diagnostic models.However, fault modeling accuracy presents a separate problemand this is not an object of current study. In this paper, thehypothesis is accepted that the diagnostic models adequatelydescribe the mechanisms of gaspath deterioration;consequently, the vector δY does not contain errors. In section4.6, some arguments are given to support the hypothesis.

As regards the vector δY*, its errors occur due to measurementerrors in Y and U as well as possible inherent inaccuracy of thefunction Y

0(U). It is supposed that a systematic component of

these errors does not depend on a deterioration developmentand a random component is normally distributed due to variousfactors affecting the accuracy of the residuals. As a consequence,the residuals δY* can be presented as a sum of the deviations δYand the standard normal distribution vector ε

n multiplied by

the diagonal matrix ΣY of maximal dispersions. In this way,

→→→

Σ+= nYYY εδδ * (6)

Inside the following approach to gas turbine diagnostics andmethods’ comparison in basis of trustworthiness criteria, thedescribed models aid to form a gas path fault classification.

4.2 Recognition trustworthiness: common approach

The pattern recognition theory supposes three principlestages of total recognition process: a classification forming,a recognizing itself, and a trustworthiness estimating. Thesestages are concretized below in application to a gas turbinediagnostics.

Fig. 1. Correction factor effects.

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P

PP

PP

P

PP

P

PP PP P

P PP

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4.2.1 Classification

Many gaspath faults are known from scientific literature. Forinstance, Meher-Homji et al. [11] give an excellent discussionof performance degradation mechanisms. Existent fault varietyis too great to distinguish all possible gas turbine degradationstates, moreover a maintenance personnel does not need sucha detailed diagnosing. That is why the degradation states shouldbe divided into limited number of classes.

There are many difficulties to form a representativeclassification based on real fault appearances only. The faultsappear rarely and their displays depend on a fault severity,engine type and operational conditions. A few degradationmodes only, for instance, a compressor contamination ofstationary gas turbines and compressor erosion of helicopterengines, can be interpreted as a permanent problem.

As a result, the real classification could be theoretically madeup only for a great engine fleet maintained over a long periodof time and model-based classifications are widely used indiagnostics [2,6].

In this paper, applying the models (4) and (5) the classificationis made up in the multidimensional space of the correctedresiduals

Y

YYZ

i

iii

−= (0

σ (7)

and a diagnostic decision about the actual state is taken in thesame space. Here σY

i is a maximal random error amplitude of the

deviation [Yi −

Y

0i(U)] and m is a number of measured variables.

So, the residual vector Z corresponds to the vector Y. The vectorZ corresponding to the measure Y* is formed in the same way.

Fig. 2. Single and multiple fault classes.

The hypothesis is accepted that an object (engine) state Dcan belong to one of q determined beforehand classes

D1,D

2,...,D

q (8)

only, as it is often supposed in the pattern recognition theory.Consequently,

1

1

=∑=

q

jjD and 0)/( =

≠ ljlj

DDP

We consider two types of classes: single and multiple.

The single type class has one independent parameter of faultseverity, for example, one correction factor or some correctionfactors changed proportionally. This type is convenient todescribe any known permanent fault of variable severity. InFig. 2, the class D

1 demonstrates this type. The point O

corresponds here to an engine normal state. The figure Ω1

presented by the line O-L1 reflects theoretical changes of the

residuals Z while fault severity increasing to a limiting enginestate in the point L

1. The figure Ω

1

∗ presents the residuals Zincorporating their random components induced bymeasurement errors. To complete class forming, any residualssample or distribution function inside the region Ω

1

∗ isrequired.

In contrast to the single type class, the multiple type classhas more than one independent parameter, for example, somecorrection factors to be changed independently. This classmay be useful to combine some faults when their owndisplays and descriptions are uncertain. The class D

2 in Fig. 2

is formed by independent changes of two correction factors,that is why the region Ω

2 presents a surface and the region Ω

2

∗

has more complex form.

4.2.2 Recognition

A nomenclature of possible diagnosis

d1,d

2,...,d

q (9)

corresponds with the accepted classification D1,D

2,...,D

q .

To make a diagnosis d a method dependent criterion

Rj = R(Z*, D

j) (10)

is introduced as a closeness measure between current re-s idual vector Z* (pat tern) and every i tem of theclassification (8) and a decision rule

miY

U

i

i −=→

1 ,)(

PPPP

PP

P

P

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d = dl if R

l = max (R

1,R

2,...,R

q) (11)

is established.

4.2.3 Trustworthiness

Various negative factors affect the diagnosing process andthe final recognition. So, to ensure the diagnosis d it needs tobe accompanied by any trustworthiness assessment.Unfortunately, few probabilistic recognition methods onlyare capable to compute a confidence parameter for a currentdiagnostic action. On the other hand, such a particularparameter depending on the current measure Y* can not serveas a criterion of average method reliability, enginecontrollability, and general diagnosing effectiveness.

For this reason mean trustworthiness characteristics aredetermined for every analyzed method by a statistical testingprocedure. This procedure repeats numerously cycles of amethod action. In every cycle, the procedure generates randomnumbers of the current class, fault severity, and measurementerrors according chosen distribution laws, then computesactual pattern Z, and finally takes a diagnostic decision dcorresponding to this vector. A square diagnosis matrix Dd(see Table 1) accumulates diagnosing results according therule

,1ljlj DdDd += (d=d1) (12)

All simulated patterns Z compose a testing sample Z*t. Itsvolume Nt corresponds to a total cycle number. After a testingcycle’s termination and diagnosis accumulation, the matrixDd is transformed into a diagnosis probability matrix Pd ofthe same format by a normalization rule

∑=

=q

lljlj DdPd

1 Dd

lj (13)

The diagonal elements Pdll forming a probability vector Pt of

true diagnosis represent indices of distinguishing possibilitiesof the classes. Mean number of these elements - scalar Pt -characterizes the total controllability of the engine with itsmeasurement system. No diagonal elements give probabilitiesof false diagnosis and their great values help to identify thecauses of bad class distinguishability. These elements makeup probabilities of false diagnosis

PtPe jj −= and 1 (14)

also applied in comparative analysis.

The number Nc that determines computational precision ofthe described indices is chosen as a result of compromisebetween a time T to execute the procedure and diagnosingaccuracy requirements. In any case, uncertainty in theprobabilities should be less than studied effects of changesof the method or diagnosing conditions.

4.2.4 Methods

Three recognition techniques which present differentapproaches in a recognition theory have been chosen fordiagnosing. The first technique is based on the Bayesianapproach [12], the second operates with the Euclidiandistance to recognize gas turbine fault classes, and the thirdapplies the neural networks which present a fast growingcomputing technique expanding through many commonfields of applications. The techniques have been adapted forthe diagnosing and statistically tested by the aboveprocedure. While the testing the settings of these diagnosingmethods were adjusted and the methods were compared inequal conditions.

4.2.5 Comparison conditions

Fixed conditions in which the diagnosing methods aresimulated and compared are described below.

A. Gas turbine operational conditions determined by thevector U are: a maximal gas turbine regime established bythe compressor rotation speed variable and standard ambientconditions.

B. Measured parameters structure and accuracy correspondto a gas turbine regular measurement system that includes 6gaspath parameters to be included in the vector Y. It isassumed that fluctuations of the residuals (7) mainly inducedby measurement errors are normally distributed.

C. Classification parameters. Two classification variants areconsidered.

Table 1. Diagnosis matrix.

P

P

PP

P

P

P

69

tPeP −= 1

P

Diagnosis

d1

d2

...d

q

D1

Dd11

Dd21

...Dd

q1

D2

Dd12

Dd22

...Dd

q2

...

...

...

...

...

Dq

Dd1q

Dd2q

...Dd

qq

Classes

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()( if j dDD ∧≡)( ldd ≡ljDd

)( if jDD ∧≡

)( ldd ≡ljDd

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The first incorporates nine single classes and every one isconstituted by a variation δΘ

j of one correction factor

(inlet pressure losses factor increase and flow andefficiency factors decrease for four principle engine mo-dules: compressor, combustion chamber, compressorturbine, power turbine).

The second includes four multiple classes corresponding tothe principle modules. Every multiple class is formed byindependent variations of two correction factors of the sameengine module and describes possible faults of the module.This classification corresponds to maintenance needs – toknow engine condition to every module to be able to repairfaulty one.

All variations δΘj which present here fault severities are

uniformly distributed within the interval [0,5%]. The classesalso have a uniform distribution, so every class is equallyprobable.

D. Testing sample volume. Analyzing precision of theaveraged probabilities, the sample volume was establishedas a function of class number Nt=1 000 q.

In that way, section 4.2 embraces explanations of the approachinvolved (formation of the fault classification, faultrecognition rule, and diagnosis trustworthiness indices) aswell as gives common conditions to compare the methods.So, we have all necessary general information to beginpresentation of every method. In the following three sections,the methods and their adjustment are described in moredetails and some trustworthiness characteristics computed inthe described conditions are given.

Table 3. Trustworthiness indices (case of multiple class type)

4.3 Method 1: Bayesian recognition

For actual measurement Y* and corresponding Z the Bayesformula permits to determine a posteriori probabilities:

)/(

1

*

qjj ZDP−=

→

= (15)

whereP(D

j) is a priory probability of the class D

j and

f(Z*/Dj) is its pattern density function

Density function assessment is a principle problem of

statistics. To simplify it the function f(Z*/Dj) was presented

by elemental distributions f(Z/Dj) and f(Z*/Z) .

∫Ω

=→

j

j fDZf )/( *

(16)

and the following assumptions were taken: 1) adequacy of

the linear model (5) applied to simulate faults, 2) uniform

distribution f(Z/Dj) of the model values Z with a different

fault severity, 3) normal distribution f(Z*/Z) of residual errors.

Pointed assumptions considerably simplified the calculationof the sought function: for single fault classes, an analyticalformula to take the integral (16) has been obtained as well asa simple numerical algorithm for multiple ones.

According to the Bayesian rule the recognition decision dl is

taken when P(Dl /Z*)is maximal in the set P(D

j /Z*), j=1−q

Table 2. Trustworthiness indices (case of single class type).

.861

.0

.014

.0

.0

.063

.001

.004

.057

.001

.759

.002

.096

.030

.057

.014

.040

.001

.012

.001

.871

.0

.003

.089

.008

.012

.004

.0

.139

.002

.745

.033

.028

.023

.030

.0

Pd.0.025.006.016.856.056.017.024.0

.008

.004

.036

.004

.017

.828

.017

.059

.027

.001

.020

.0087

.005

.014

.084

.834

.032

.003

.002

.009

.020

.004

.014

.139

.020

.788

.004

.059

.0

.010

.0

.0

.099

.001

.013

.818

Pt .861 .759 .871 .745 .856 .828 .834 .788 .818

Pt = 0.8178

P

P

Pt .891 .784 .940 .883

Pt = 0.8744

P

Pd.891.045.050.014

.062

.784

.038

.116

.016

.016

.940

.028

.003

.072

.042

.883

/(

/(

*

1

*

q

l

DZf

DZf→

=

→

Σ )()/

)()/

*ll

jj

DPD

DPD→

P

PP P P

PP P

P

P P

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→→

j

ZZf )/( *

Θ→→→

j

j dDZfZZf )/()/( *

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that corresponds with the general criterion (10) and rule (11)

if we put Rj(D

j/Z*) .

To assess average trustworthiness of this method (method 1),the diagnosing algorithm based on Bayesian recognition hasbeen elaborated and inserted inside the testing proceduredescribed above. The resulting probabilities (see section4.2.3) corresponding to a single type classification are placedin Table 2 and the same data for a multiple type classificationare included in Table 3.

From Table 2 it can be seen that the classes D2 and D

4 have

the lowest distinguishability according Pt and the elevatedmagnitudes Pd

42 and Pd

24 explain the cause – a great mutual

intersection of these classes.

Comparison of Table 2 and Table 3 shows individual (forevery class by the vector Pt) and total (by Pt) trustworthinessgrowth for the multiple classification that is a result of twoopposite tendencies. On one hand, the replacement of a sin-gle type by a multiple one generally leads to more close classintersection and lower trustworthiness. On the other hand,the significant reduction of class total quantity (from nine tofour) has a contrary influence. In our case, the second tendencydominates. Of course, another diagnosing method will changethe presented probabilities and, probably, the conclusions.

This is an advantage of the method 1 that every diagnosismade for actual measurement may be accompanied by aconfidence probabilistic estimate and, on average, suchestimates will be maximal.

However, the method is not without its difficulties. It seemsto be too complicated to restore density functions of a gene-ral form in a multi-dimensional recognition space utilizingreal measurements (patterns). Therefore, simple type classesbased on the linear model and ordinary theoreticaldistributions may be described only.

That is why a class representation directly by pattern sets isconsidered too as well as the methods 2 and 3 capable to treatthem.

4.4 Method 2: Euclidian distance

Such a simplification as density functions replacement onthe pattern sets permits simulating a fault severity growth bythe more exact nonlinear thermodynamic model and formingcomplex multiple classes described by three and morecorrection factors. Furthermore, this permits forming real databased classes of general type without any model assistance

and consequently without negative influence of model propererrors. Fig. 3 demonstrates new class representation; patternsets of four classes are given here in the three-dimensionalspace Z.

The recognition space Z of residuals can be classified asuniform since all residuals have the same dispersion accordingto the transformation (7). That is why it will be correctly tointroduce a geometrical measure of closeness between thecurrent vector Z and the class D

j to be used as the recognition

criterion Rj (10). For concerned method, this measure is based

on the Euclidian distance between two points in amultidimensional space.

The computational algorithm has been realized inside thetesting procedure and incorporates the following items.Firstly, outside the testing procedure, a reference sample Zrof the volume Nr incorporating the pattern sets Zr

j for all

classes is composed. Secondly, inside the testing procedure,the criteria R

j are calculated for actual Z* of the testing sample

and every Zrj* of the reference set. Thirdly, the diagnosing

decision dl is accepted by the general rule (11). The

trustworthiness indices (13) and (14) are calculated thenaccording to the general scheme (see section 4.2.3).

To select the best criterion type, the following variants wereconsidered:

- variant 1 - mean inverse distance M(1/di) between a testing

sample point and reference points;

Fig. 3. Classes representation by the pattern sets

P

P

P

P

P

P

P

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- variant 2 - mean inverse quadratic distance M(1/di2) between

the testing point and the reference points;- variant 3 - mean distance M(d

i) between the testing point

and the reference points;- variant 4 - distance between the testing point and a referencesample gravity center.

In preliminary statistical testing, the variant 2 ensured the bestclass distinguishability and has been selected for further use.

Because the reference sample volume Nr influences thecomputational accuracy and executing time in the same modeas it does Nt, the value Nr = Nt = 1000q has been accepted tocarry out next calculations.

4.5 Method 3: neural networks

Neural networks consist of simple parallel elements calledneurons. According to the common scheme of supervisedlearning networks are trained on the known pairs of inputand output (target) vectors. The connections (weights)between the neurons change in such a manner that ensuresdecreasing a mean difference (error) e between the target andnetwork output. Many such input/target pairs are used toadapt a network to a particular function.

Above an input layer and output layer of neurons a networkmay incorporate one or more hidden layers of computationnodes when high network flexibility is necessary. To solve

difficult pattern recognition problems multilayer feedforwardnetworks or multilayer perceptrons are successfully applied[13] since a back-propagation algorithm had been proposedto train them. So, a back-propagation network promisesreliable fault recognition and we also included it in thecomparative analysis.

The network realized in the method 3 has the nextcomposition depending on the measurement system and faultclassification structures.

The input layer vector includes 6 elements since networkinputs are residuals. The output vector presents the concernedclasses and therefore incorporates 9 elements for the singletype classification and 4 elements for the multiple one.Network complexity and resolution capability are in closerelation with hidden layers quantity and their nodes numbersand, to a first approximation, the network has one hiddenlayer of 12 nodes.

Differentiable layer transfer functions are of sigmoid type. Inthe hidden layer, a tan-sigmoid function is applied; it variesfrom -1 to 1 and is typical for internal layers of a back-propagation network. A log-sigmoid function operates in theoutput layer; it varies from 0 to 1 and is convenient to solverecognition problems.

To put the compared methods under equal conditions thesame reference and testing samples as used before are appliednow as data sources in network training and verifyingprocesses. Firstly, the training algorithm is performed on thesample Zr*. Secondly, the trained network passes a

Table 4. Training algorithms. "Cycles" refers to the totalcycle number of training process. Pt and Pt'' meanprobabilities of truthful diagnosis obtained on theteaching and testing samples correspondingly.

Table 5. False diagnosis probabilities (single typeclassification).

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72

e

0.0300

0.0290

0.0285

0.0280

0.0275

Algorithm

123123123123123

Cycles

38365

11148595

114598118122836149134

1616212178

Pt0.81740.81320.81740.82110.81680.82130.82340.81960.82090.82640.82420.82340.82760.82690.8267

Time, s

1103770

1274471

1584975

2015680

37070

100

Pt’’0.81320.80900.81720.81570.81260.81810.81730.81490.81800.81970.81590.82140.81940.81980.8228

P P

P P

Indices Methods

d1

d2

d3

d4

d5

d6

d7

d8

d9

Pe

Pe

10.1390.2410.1290.2550.1440.1720.1660.2120.182

0.1822

20.2660.2480.2190.3900.2240.0150.2120.2430.215

0.2258

30.1700.2520.1370.2500.1450.1700.1430.1730.160

0.1772

P

P

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verification stage to compute the probabilistictrustworthiness indices (see section 4.2.3) corresponding tothe samples Zr* and Zt*.

There are a number of variations on the basic back-propagation training algorithm. In order to choose the bestone under concrete conditions of fault recognition, twelvevariations were tested under fixed given accuracy e (meandiscrepancy between all targets and network outputs) andcompared by execution time. Three more perspectivealgorithms - variable (adaptive) learning rate algorithm(algorithm 1), resilient back-propagation (algorithm 2), andscaled conjugate gradient algorithm (algorithm 3) - wereverified additionally for the different accuracy levels e = 0.03-0.0275, where the value 0.0275 is close to the final obtainableaccuracy. The results given in Table 4 show that the algorithm1 obviously loses the competition, the algorithm 2 seems tobe a little more rapid than the algorithm 3 while the last ismore reliable on the testing sample. Taking into account ourpriority - recognition trustworthiness - the scaled conjugategradient algorithm is selected for next calculations.

An influence of the hidden layer node number was examinedtoo. Above the chosen number 12, the numbers 8, 16, 20were also verified for options of the chosen back-propagationalgorithm and fixed cycle number 200. The reduction of thenode number from 12 to 8 has demonstrated visible changesfor the worse of the obtainable accuracy Äe=0.00103 and theprobabilities Pt’=−0.0068 and Pt’’=−0.0068. On the otherhand, the augmentation to 16 and 20 nodes has not improvedthe algorithm’s characteristics. That is why the node number12 remains for further calculations.

4.6 Methods comparison

To compare all three methods, the statistical testing of themethods 2 and 3 preliminarily checked and adjusted has beenperformed under the conditions of the method 1 testing and forthe same two classification configurations. The resultingtrustworthiness characteristics - the probabilities of false diag-nosis (14) – are placed in Table 5 (single type classification) andTable 6 (multiple type classification). For the method 1 theseprobabilities correspond to the data of Table 2 and Table 3.

Firstly, the methods 2 and 3 are compared. They operate withthe same input and output data representation and, from thispoint of view, do not have any significant advantages/limitationsone against the other. So, the trustworthiness level and executiontime are only arguments to choose the best method.

The indices presented in Tables 5 and 6 demonstratesignificantly lower probabilities of false diagnosing by the

method 3. Since calculating accuracy for the probability Peworks out at 0.01, the effect of trustworthiness enhancement isnot a result of statistical simulation errors. The comparativecalculations repeated for other gas turbine operating conditionsprove the conclusion about superiority of the method 3.

As to the methods 1 and 3, the differences between them indiagnosis trustworthiness are lower. As it can be seen in Tables5 and 6, these methods differ in the probability by 0.0037-0.0050. As the differences are opposite by the sign and smallerthen the calculating inaccuracy, the trustworthiness levels ofthe methods 1 and 3 are considered as equal.

However in contrast to the methods 2 and 3, the method 1applying the Bayesian approach has an advantage toaccompany every diagnosis by its confidence estimation.Therefore the diagnostic method based on the Bayesianapproach may be recommended for practical applicationalways when we are able to describe the classes by densityfunctions; for example, in the case of model-basedclassification concerned in this paper. Otherwise, thetechniques like neural networks should be used.

4.7 Perspectives of practical application

Above the current work focused on a method comparing,numerous investigations were fulfilled by means of adiagnosing process statistical testing and trustworthinessindices analysis in order to study other factors that also affectthe gas diagnosing trustworthiness.

The variety of these factors includes- measurement system structure;- measured parameters accuracy;- number and structure of gas turbine operational regimes

including dynamic regimes;- classification structure and type;

Indices Methods

d1

d2

d3

d4

Pe

Pe

10.1090.2160.0600.117

0.1256

20.2370.3730.0510.051

0.1790

30.1040.2140.0720.217

0.1293

P

Table 6. False diagnosis probabilities (multiple typeclassification)

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- joint recognition of gaspath faults and measurementsystem proper defects.

Presented trustworthiness analysis may be classified as model-based since gas turbine models describe a normal behavior andfault influence. That is why a reasonable question appears howwe could ensure that the obtained results be correct in practice.

To guarantee the results of model-based calculations thestatistical testing is carried out for a wide range of possiblediagnosing conditions and the conclusions are generalized.In addition, important results are confirmed on real data, ifany. For instance, a practical mode has been proposed [10] toenhance the base-line function Y

0(U) and reduce the deviation

errors σYi by maintenance data analysis; the method based on

Bayesian approach was adapted for recognition of physicallysimulated gas turbine faults and a real compressorcontamination and has demonstrated a satisfactory accuracyof the previous model-based realization of the method.

The idea appears of a combined classification: to start a healthmonitoring system development and operation with model-based classes and later, along with maintenance informationaccumulation, to introduce one by one the classes formed byreal fault description.

5. Conclusions

In this paper, we discuss a statistical testing of gas turbinediagnosing process in order to determine and elevaterecognition trustworthiness indices which are averagedprobabilities of true/false diagnosis. A thermodynamic modelserves to simulate gas turbine degradation modes and form afaults classification. To conduct the trustworthiness analysisin more general form, two types of gas turbine fault classescalled single and multiple are considered.

Three diagnosing methods functioning inside the statisticaltesting procedure are adjusted, verified, and finally compared.Two of them – the method utilizing Bayesian rule and themethod applying neural networks - have demonstrated anequally high trustworthiness level and are recommended forthe use in condition monitoring systems.

The presented investigation is a part of total work focused onthe trustworthiness growth of gas turbine diagnosing; theother investigation lines including an analysis of maintenancedata are noted.

AcknowledgmentsThe work has been carried out with the support of the NationalPolytechnic Institute of Mexico (project 20050709).

6. References

[1] Rao, B.K.N., 1996. «Handbook of Condition Monitoring».Elsevier Advanced Technology, Oxford.

[2] Tsalavoutas A., Mathioudakis K., Aretakis N., «StamatisA., 2000. Combined advanced data analysis methodfor the constitution of an integrated gas turbinecondition monitoring and diagnostic system». In: Proc.ASME Turbo Expo Conf., Germany, 8p.

[3] Yepifanov, S.V., Kuznetsov, B.I., Bogaenko, I.M., 1998.«Design of Gas Turbine Engine Control and Diagnosingsystems». Tehnika, Kiev, Ukraine.

[4] Roemer, M.J., Kacprzynski G.J., 2000. «Advanceddiagnostics and prognostics for gas turbine engine riskassessment». In: Proc. ASME Turbo Expo Conf.,Germany, 10p.

[5] Basseville, M., 2003. «Model-based statistical signalprocessing and decision theoretic approaches tomonitoring». In: Proc. Fifth IFAC Symposium on FaultDetection, Supervision and Safety of Technical Process.Washington, D.C., pp.1-12.

[6] MacIsaac, B.D., Muir, D.E., 1991. «Lessons learned inGas Turbine Performance Analysis». In: Proc.Canadian Gas Association Symposium on IndustrialApplication of Gas Turbine, Banff, Canada, 27p.

[7] Pipe, K., 1987. «Application of advanced patternrecognition techniques in machinery failure prognosisfor Turbomachinery». In: Proc. Condition Monitoring1987 Int. Conf., British Hydraulic ResearchAssociation, UK, pp.73-89.

[8] Yepifanov, S., Loboda, I., 1995. «Controllability analysisof gas turbine engines by parametrical methods.Aerospace Techniques and Technology»: Journal ofKharkov Aviation Institute, Ukraine, pp.73-79.

[9] Loboda, I., 2003. «Trustworthiness problem of gas turbineparametric diagnosing». In: Proc. Fifth IFACSymposium on Fault Detection, Supervision and Safetyof Technical Process. Washington, D.C., 6p.

[10] Loboda, I., Yepifanov, S., Feldshteyn, Y., 2004. «Deviationproblem in gas turbine health monitoring». In: Proc.IASTED International Conference on Power and EnergySystems, Clearwater Beach, Florida, USA, 6p.

[11] Meher-Homji, C.B., Chaker, M.A., Motivwala, H.M.,2001. «Gas turbine performance deterioration». In: Proc.Thirtieth Turbomachinery Symposium, Texas, pp.139-175.

[12] Duda, R.O., Hart P.E., Stork D.G., 2001. PatternClassification. Wiley-Interscience, New York.

[13] Haykin, S., 1994. Neural Networks. Macmillan CollegePublishing Company, New York.

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