Diagnosis of Fuel Pump Measurement Bias In Gas Turbine … · Diagnosis of Fuel Pump ... an engine system can lead to a loss of redundancy ... are employed in the diagnosis of fuel

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Diagnosis of Fuel Pump Measurement Bias

In Gas Turbine Engines

By David Danny Santoso

MSc in Control Systems Engineering August 2001

Supervisor: Professor PJ Fleming, BSc, PhD, CEng, FIEE, FInstMC, MBCS

A dissertation is submitted in partial fulfilment of the requirement for the degree of Master of Science in Control Systems

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EXECUTIVE SUMMARY The ability to robustly diagnose faults within a safety control system is vital from both a safety point of view and from an effective management consideration. The existence of a fault within an engine system can lead to a loss of redundancy in the engine operation which could require that the aircraft abort its planned mission and result in significant additional operational costs. This provides significant motivation to develop capabilities to accurately diagnose the initial stages of fault development. A great deal of research effort in fault diagnosis for gas turbine engines has concentrated on model-based (observer design) and knowledge-based (neural network, expert systems) approaches. In this dissertation, the application of statistical-based techniques, which has been applied widely in the process industries, are employed in the diagnosis of fuel pump measurement bias in gas turbine engines. Aerospace engines and chemical processes share a common characteristic (process non-linearity) that would make statistical techniques applicable, but there are also significant differences (especially differences in system diagnosis) that make this study important. The aim of this dissertation is to diagnose faults within the fuel pump of a gas turbine engine using a statistical technique, an adaptation of Multivariate Statistical Process Control (MSPC), which utilises the theory of Principal Component Analysis (PCA) and Partial Least Squares (PLS) approaches. Furthermore, these techniques are used to categorise the levels of deterioration experienced by the engine in order to improve the diagnosis of any measurement bias. The adaptation of the PCA technique within the PLS Toolbox is required to diagnose faults because the characteristic of an aero engine data is different to chemical process data. Three different adaptation methods were used: Hotelling’s, Normalisation and Quasidimensionless. The latter is a development of the Normalisation Method. These methods were able to identify the existence of sensor errors and the level of engine deterioration. This dissertation highlights that statistical techniques could provide a good indicator for fuel pump measurement bias in a gas turbine engine. A number of useful performance parameter combinations were identified that most clearly categorised the levels of deterioration within the engine. Furthermore, the combinations were implemented within a truth table construction to provide an approach to diagnose sensor failure. Finally, estimation of fuel flow was possible from the validated engine sensors via optimised performance signal combinations. The importance of particular engine sensor signals was highlighted and a physical understanding of why this was for this particular application was outlined. Directions for future research were outlined which included extending the technique for diagnosis of faults other than fuel flow bias. Improvements to the visualisation of the results can be undertaken through development of a GUI in MATLAB.

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ACKNOWLEDGEMENT I would like to express my gratitude to Lord Jesus and Mother Mary for their kindness and blessing in any circumstances, I have been through. Thanks for always staying beside me and encouraging me through people surrounds me. I count the blessing today and I know how grateful I am. Deus Charitas Est. Thanks to my parents, Antonius Soewari and Anastasia Soewari for endlessly love, support and pray. I believe, as my happiness is their happiness too. Thanks to my brothers and sisters, nieces and nephews who kept me notice that their support, pray and love have make me stronger. It has been an honour for myself to get an opportunity to do my master project with Rolls Royce UTC within Department of Automatic Control and Systems Engineering, University of Sheffield. I would like to thank to Prof. Peter J. Fleming, my supervisor for his kind attention, guidance and valuable advise due to the process of consulting and working of this project and informal discussion. Thanks for being so friendly and kind. He makes me comfortable and believes myself even more and it means a lot to me. Thank you. I would like to thank to Dr. Stephen Hargrave, Research Associate to whom I regarded as my co-supervisor. His input, encouragement and discussion due to the simulation of the program, pointing the problems, his critics and challenged have kept me rolling and above all, his patience of my distraction at anytime. Thanks and good luck, Steve. Thanks to all member of B20 Rolls-Royce UTC for kindly and lovely atmosphere and marvellous help during my dissertation period especially Clare, Sog Kyun, Tim Breikin, Shane, Robin and Iban for spending sometime to discuss, also many thanks to Haydn Thompson, Dongik Lee, Jianyong, Ian Griffin, Xiaoxu, James and Daniela. It has been my pleasure to have sometime in B20. I would like to say thanks to Ching-Yi Yang for her patience and support due to my study and writing the dissertation. Thanks also to Mira, Ira, Claire, Noki, Mas Nano, Yanyan, Joseph, Dondys and Muhsin Alsagoutri, Margaritis, Master Course friends, Consortium friends for their friendship and support. I would like to thank to my sponsor Consortium Pertamina-Total Indonesie, especially Mr. Herve Madeo, who gives me opportunity to continue my master study in The University of Sheffield. I hope this knowledge could contribute to TFE in the future. Last but not least, thank to the big family of Indonesian Society in Sheffield, where I feel home. Thanks for gathering and lovely times.

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TABLE OF CONTENTS Cover paper ................................................................................................................................................ i Executive Summary ................................................................................................................................... ii Acknowledgements .................................................................................................................................. iii Table of Contents ..................................................................................................................................... iv Abbreviation and Symbols ....................................................................................................................... vi List of Figures .......................................................................................................................................... vii List of Tables ........................................................................................................................................... ix

Chapter 1: INTRODUCTION 1.1. General Overview .............................................................................................................................. 1 1.2. Aim of Dissertation .......................................................................................................................... 2 1.3. Motivation for Project .................................................................................................................... 2 1.4. Dissertation Structure ........................................................................................................................ 5 Chapter 2: BACKGROUND 2.1. Gas Turbine Engine ........................................................................................................................... 7

2.1.1. Introduction to Gas Turbine Engines ...................................................................................... 7 2.1.2. Basic Concept of How a Gas Turbine Engine Works ............................................................. 7 2.1.3. Type of Jet Propulsion ............................................................................................................. 8 2.1.4. Utilisation of Gas Turbine Engines ....................................................................................... 11

2.2. Fuel Systems .................................................................................................................................... 12 2.2.1. Introduction to Fuel Systems ............................................................................................... 12 2.2.2. Fuel Metering Unit and Fuel Pump Function ...................................................................... 12 2.2.3. Control of the Fuel System .................................................................................................. 12 2.2.4. Types of Fuel Pumps .............................................................................................................. 14

2.2.4.1. Plunger Type Fuel Pump .......................................................................................... 14 2.2.4.2. Gear Type Fuel Pump ............................................................................................. 15

2.3. Summary .......................................................................................................................................... 16 Chapter 3: FDI AND DEGRADATION IN GAS TURBINE ENGINES 3.1. Fault Detection and Isolation ........................................................................................................... 17 3.2. Traditional Model Based Approach .................................................................................................. 18

3.2.1. Off-line Method ................................................................................................................... 18 3.2.2. On-line Method ...................................................................................................................... 19 3.2.3. Unknown Input Observers ..................................................................................................... 20 3.2.4. Eigenstructure Assignment ................................................................................................... 21

3.3. Statistical Approaches ...................................................................................................................... 23 3.3.1. Neural Network ................................................................................................................... 23

3.3.1.1. Supervised Learning ............................................................................................... 23 3.3.1.2. Unsupervised Learning .......................................................................................... 23

3.3.2. Multi Layer Perceptron ........................................................................................................ 24 3.3.3. Radial Basis Function Neural Network ................................................................................. 25

3.4. Degradation in Gas Turbine Systems ............................................................................................... 26 3.4.1. Degradation Mechanisms ..................................................................................................... 26 3.4.2. Component Degradation ........................................................................................................ 27

3.5. Effect of Degradation in Gas Turbine Engines ................................................................................. 28 3.6. Summary .......................................................................................................................................... 28

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Chapter 4: STATISTICAL TECHNIQUES 4.1. Introduction to Statistical Techniques ............................................................................................... 30 4.2. Statistical Process Control ................................................................................................................ 31 4.3. Multivariate Statistical Process Control .......................................................................................... 32

4.3.1. Principal Component Analysis ..............................................................................................34 4.3.1.1. Introduction to PCA ............................................................................................... 34 4.3.1.2. Principal Component ............................................................................................... 35 4.3.1.3. Multi-way PCA ..................................................................................................... 39

4.3.2. Partial Least Squares .............................................................................................................. 40 4.4. Summary .......................................................................................................................................... 41 Chapter 5: DATA ANALYSIS 5.1. Data Information .............................................................................................................................. 42 5.2. Fuel Flow Estimation Analysis ........................................................................................................ 45

5.2.1. Adaptation of the PCA Technique ....................................................................................... 46 5.2.2. Hotelling’s Method ................................................................................................................ 51 5.2.3. Normalisation Method ............................................................................................................ 56

5.2.3.1. WFE vs P30.*N1 1.5 Combination ............................................................................ 61

5.2.3.2. WFE vs P30.*TGT 1.5 Combination ......................................................................... 63

5.2.4. Traditional Quasidimensionless .......................................................................................... 66 5.2.5. Quasidimensionless Adaptation ..............................................................................................71 5.2.6. Performance Comparison ......................................................................................................76

5.3. Deterioration Estimation Analysis ......................................................................................................78 5.4. Test Cases .......................................................................................................................................... 80 5.5. Diagnosis Technique/Consistency Checking .................................................................................... 83 5.6. Summary .......................................................................................................................................... 89 Chapter 6: CONCLUSION AND RECOMMENDATION 6.1. Review of Diagnosis Methods ........................................................................................................... 90 6.2. Dissertation Achievement ............................................................................................................... 93 6.3. Recommendation for Further Investigation .................................................................................... 95 Bibliography ............................................................................................................................................... x Appendix A: Engine Data A.1. Engine Data Generation ..................................................................................................................... a A.2. Engine Data Sample ........................................................................................................................... a Appendix B: Hotelling’s Method B.1. Hotelling’s Information ..................................................................................................................... d B.2. Program Listing ................................................................................................................................ d B.3. Program Results ................................................................................................................................. f Appendix C: Normalisation Method C.1. Normalisation Information ................................................................................................................ h C.2. Program Listing ................................................................................................................................ h C.3. Program Results ................................................................................................................................. j Appendix D: Quasidimensionless Method D.1. Quasidimensionless Information ........................................................................................................ l D.2. Program Listing ................................................................................................................................. l D.3. Program Results and Table ............................................................................................................... p

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ABBREVIATIONS AND SYMBOLS Abbreviation CUSUM Cumulative Sum EEC Electronic Engine Controller EMWA Exponentially Moving Weight Average FDI Fault Detection and Isolation FMU Fuel Metering Unit FMV Fuel Metering Valve FSA Frequency Spectrum Analysis HP High Pressure IP Intermediate Pressure LP Low Pressure MCD Magnetic Chip Detector MLP Multi Layer Perceptron MLR Multiple Linear Regression MN Mach Number MSPC Multivariate Statistical Process Control PC(s) Principal Component(s) PCA Principal Component Analysis PLS Partial Least Squares PCR Principal Component Regression RBFN Radial Basis Function Neural Network SOAP Spectrometric Oil Analysis Program SOV Shut Off Valve SLS Sea Level Static SPC Statistical Process Control TGT Turbine Gas Temperature UIO Unknown Input Observer Symbols ALT Altitude FN Thrust Power IGV Inlet Guide Vanes N1 Low Pressure Shaft Speed N2 Intermediate Pressure Shaft Speed N3 High Pressure Shaft Speed P20 Engine Inlet Pressure at the Fan Face P25 Intermediate Pressure Compressor Exit Pressure P30 High Pressure Compressor Exit Pressure P50 Low Pressure Turbine Exit Pressure P160 By pass duct pressure T20 Low Pressure Compressor Exit Temperature T25 Intermediate Pressure Compressor Exit Temperature T30 High Pressure Compressor Exit Temperature TGT Intermediate Pressure Turbine Entry Temperature WFE Fuel Flow

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LIST OF FIGURES

Motivation for Project Figure 1.1: Fuel Flow Estimation ............................................................................................................... 3 Figure 1.2: Fuel Flow Estimation for Different Engine Data ..................................................................... 4 Figure 1.3: Categorisation of Deterioration Level ...................................................................................... 4 Gas Turbine Engines Figure 2.1: A Whittle-Type Turbo-Jet Engine ............................................................................................ 7 Figure 2.2: Jet Propulsion Mechanism .......................................................................................................8 Figure 2.3: Rolls-Royce Trent 800 Engine ................................................................................................. 9 Figure 2.4: Airflow Systems ...................................................................................................................10 Figure 2.5: The Working Cycle Diagram ............................................................................................... 10 Figure 2.6: Airflow Changing with Altitude .......................................................................................... 13 Figure 2.7: Fuel Flow Changing with Altitude ....................................................................................... 13 Figure 2.8: A Plunger-Type Fuel Pump .................................................................................................. 14 FDI and Degradation in Gas Turbine Engines Figure 3.1: Conceptual Structure of Model Based Fault Diagnosis ........................................................20 Figure 3.2: A Full-Order UIO Structure .................................................................................................. 21 Figure 3.3: Robust Observer-Based Residual Generation ...................................................................... 22 Figure 3.3: Multiple Layers Neurons ........................................................................................................ 24 Figure 3.4: The Structure of Hidden Layer RBFN ................................................................................. 25 Statistical Techniques Figure 4.1: Principal Component Model of 3D Data Set ......................................................................... 38 Data Analysis Figure 5.1: MSPC Model of Gas Turbine Engines ................................................................................. 42 Figure 5.2: Trent 700 Engine Diagram ..................................................................................................... 44 Fuel Flow Estimation Analysis ��Modification of The PCA Technique Figure 5.3: Principal Component Analysis GUI ....................................................................................... 47 Figure 5.4: Eigenvalues vs. Principal Component ................................................................................. 48 Figure 5.5: Plot Scores of PC1 vs. PC2 .................................................................................................. 49 Figure 5.6: Loading Plot of PC1 vs. Variables Number ......................................................................... 50 ��The Hotelling’s Method Figure 5.7: The Hotelling’s Plot of N1 and N2 .......................................................................................... 51 Figure 5.8: The Hotelling’s Plot of N3 and P20 ....................................................................................... 52 Figure 5.9: The Hotelling’s Plot of P25 and P30 ....................................................................................... 53 Figure 5.10: The Hotelling’s Plot of P50 and P160 .................................................................................... 53 Figure 5.11: The Hotelling’s Plot of T20 and T25 .................................................................................... 54 Figure 5.12: The Hotelling’s Plot of T30 and TGT .................................................................................55 ��The Normalisation Method Figure 5.13: The First Four Combinations of WFE vs. N1.*P30

a .............................................................. 57 Figure 5.14: The Last Four Combinations of WFE vs. N1.*P30

a .............................................................. 58 Figure 5.15: Mean and Standard Deviation of the Ideal Data ................................................................. 59 Figure 5.16: Mean and Standard Deviation of 1000 Hours Deterioration Data ....................................... 60 Figure 5.17: Characteristic and Combination Table of WFE vs. P30.*N1

1.5 .............................................62 Figure 5.18: Combination Plot of WFE vs. P30.*N1

1.5 ............................................................................ 62 Figure 5.19: Characteristic and Combination Table of WFE vs. P30.*TGT

1.5 ....................................... 63 Figure 5.20: Combination Plot of WFE vs. P30.*TGT

1.5 ......................................................................... 64

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��Traditional Quasidimensionless Figure 5.21: Quasidimensionless of (TGT.*T20

a) Combination .............................................................. 67 Figure 5.22: Quasidimensionless of (N1.*T20

a) Combination ................................................................. 68 Figure 5.23: Quasidimensionless of (P30.*P20

a) Combination ................................................................. 69 ��Quasidimensionless Adaptation Figure 5.24: Combination of (TGT./T20).*(T25.*T20

a) ............................................................................ 73 Figure 5.25: Combination of (TGTnd).*(N2nd)

a ....................................................................................... 74 Figure 5.26: Combination of (P30./P20).*(TGT./T20)

a ............................................................................... 75 ��Performance Comparison Figure 5.27: First Performance Comparison Plot .................................................................................... 76 Figure 5.28: Performance Comparison Plot of Development ................................................................... 77 Deterioration Estimation Analysis Figure 5.29: Linear Interpolation Plot of Deterioration Estimation Analysis .......................................... 79 Test Cases Figure 5.30: Level of Deterioration Estimation ....................................................................................... 80 Figure 5.31: Calculation of Fuel Flow Estimation .................................................................................... 81 Diagnosis Technique/Consistency Checking Figure 5.32: Diagnosis of P30nd.*N1nd Combination ............................................................................... 85 Figure 5.33: Diagnosis of P30nd.*TGTnd Combination ............................................................................ 86 Figure 5.34: Diagnosis of N1nd.*TGTnd Combination ............................................................................ 87 Review of Diagnosis Methods Figure 6.1: Statistical Approach of The Fuel Flow Estimation ................................................................ 90 Figure 6.2: Fuel Flow Estimation for Different Conditions of Engine Data ............................................. 91 Figure 6.3: Identification of Deterioration Level .................................................................................... 92

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LIST OF TABLES

Table 5.1: Normalisation Method Variables Combinations ....................................................................61 Table 5.2: Traditional Quasidimensionless Combinations ...................................................................... 70 Table 5.3: Combinations of Quasidimensionless Adaptation ................................................................. 71 Table 5.4: Test Data for Unknown Level of Deterioration ...................................................................... 80 Table 5.5: Calculation of Linear Interpolation ....................................................................................... 81 Table 5.6: Fuel Flow Comparison ........................................................................................................... 82 Table 5.7: Diagnosis Technique of Selected Combinations ................................................................... 84

Chapter 1: Introduction – General Overview

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Chapter 1

INTRODUCTION

1.1. General Overview

Fault diagnosis (detection and isolation) in a gas turbine engine’s fuel pump

measurement is a crucial aircraft technology and forms an integral part of aircraft

safety. The ability to diagnose faults in their early stages enables avoidance of

potentially hazardous situations.

A vast number of investigations to diagnose fault in gas turbine engines have been

undertaken and reported in published literature. These have included Model-based

approaches (Merrington et al., 1991; Patel, U.S., 2000; Chen et al., 1999) and Neural

Networks (Zedda et al., 1998; Patel, V.C., 1998).

The Model-based approach has been applied to simplify the complexity of the

problems and focus on optimising the model performance for the ‘real data’ condition

to be simulated, before it is implemented to the real systems, e.g. implementation in

gas turbine engines. This approach is highly dependent on how close the model is to

the real condition it represented and how well the model behaves based with the real

engine data.

The magnificent development of the software technology leads to another method that

has been undertaken using the intelligent techniques, i.e. neural network (MLP,

RBFN). This method creates a learning algorithm, which is able to learn how to adjust

the system to suit the current situation and the ability of the systems to predict the

behaviour of an aero engine for an unexpected condition. The more detail included in

the learning procedure, the better the performance expected.

Diagnosing faults can be carried out through the analysis of the engine data, which

represents the relationship between the inputs and outputs of the system. This

alternative method to those discussed above, which is commonly applied in the

Aim of Dissertation and Motivation for Project

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chemical industry, uses statistical techniques. This technique is known as

chemometric and it has been developed for many years. The technology has reached a

mature level to the extent that a MATLAB Toolbox has been developed by

Eigenvector (PLS Toolbox) to enable its wider application. Chemometric is defined

(Wise et al., 2000) as:

Chemometric is the science of relating measurements made on a

chemical system to the state of the system via application of

mathematical or statistical methods and of designing optimal

experiments for investigating chemical systems.

Established techniques of this technology include the Principal Component Analysis

(PCA) and the Partial Least Squares (PLS) technique. This dissertation utilises the

theory of both techniques and adapts them to the particular requirements of the current

problem.

1.2. Aim of Dissertation

This dissertation will implement a statistical technique based on an adaptation of

Multivariate Statistical Process Control (MSPC), to analyse multivariable engine data

to diagnose faults within the engine. Statistical techniques have been proven to be

able to solve problems of complexity and transform these problems to a smaller

number of variables whilst retaining the underlying process information necessary to

diagnose component futures. By analysing this engine data, diagnosing faults within

the engine can be undertaken and specifically diagnosis/estimation of measurement

bias of the fuel flow within the fuel metering system can be achieved.

1.3. Motivation for Project

Several methods had been applied to diagnose faults (fuel pump measurement bias)

within engine i.e. model-based, neural networks and intelligent agents but none

specifically based on MSPC approaches. This dissertation has been motivated to

assert that statistical techniques have the capability to diagnose faults within gas

turbine engines.

Motivation for Project

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A more descriptive outline of the project is to estimate the measurement bias in the

fuel flow and the proposed approach is based on steady-state engine data. The concept

behind the diagnosis of this measurement bias is most simply explained by observing

the following figures. The first figure illustrates the method for estimating the fuel

flow, and is based on the premise that the performance characteristic of an engine can

be derived and that there is a direct relationship between a performance parameter, the

known engine characteristic and the theoretical fuel flow. This relationship is

illustrated in Figure 1.1.

Figure 1.1: Fuel Flow Estimation

The most convenient method for characterising the performance of an engine is that of

the level of deterioration experienced by the engine. A new engine (ideal) will be

more efficient than that of an engine that has flown for several thousand hours. Figure

1.2 illustrates the changing relationship of a performance parameter to the estimated

theoretical fuel flow for three different levels of engine deterioration.

Therefore, in order to derive a more accurate estimation of the fuel flow into an

engine, it is necessary to determine the level of deterioration experience by the engine.

Motivation for Project

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Figure 1.2: Fuel Flow Estimation for Different Engines Data

It is proposed that this deterioration estimation is undertaken in a similar approach,

but in this case is undertaken from the statistical-based relationship between two

different performance parameters (not fuel flow). This form of categorisation is

illustrated graphically in Figure 1.3, which in this instance shows a level of

deterioration that is partial.

Figure 1.3: Categorisation of Deterioration Level

Dissertation Structure

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Thus, the manner in which the estimate of the fuel flow bias is achieved is based on

the following steps:

1. Estimate engine performance deterioration.

2. Check consistency of performance parameters (to check correctness of engine

signals).

3. Estimate theoretical fuel flow input.

4. Compare theoretical fuel flow input with measured fuel flow to determine bias.

5. Continuously monitor measurement bias to check consistency of this bias

estimate.

1.4. Dissertation Structure

This dissertation is divided into 6 chapters and breakdown of which is on this below:

Chapter 1 introduces the general overview of the project. The brief description of the

aim and the motivation of the dissertation are also stated. The last part of this chapter

contains the structure of the dissertation when the explanation of each chapter is

mentioned in general.

Chapter 2 introduces the gas turbine engines, how it works and some basic concept.

Special topic about fuel metering systems is discussed here related to the dissertation

topic, which is about diagnosis of fuel pump measurement bias in gas turbine engines.

General operation and some parameter within gas turbine engines are explained (i.e.:

N1, N2, N3, P20, P25, P30, P50, P160, T20, T25, T30, TGT, WFE, FN, MN).

Chapter 3 explains about fault and degradation within gas turbine engines, the

previous approach, which has been done, i.e. the Model-based approach and the

Statistical Approach – Neural Network (MLP and RBFN), the type of degradation

within gas turbine engines and the cause.

Chapter 4 discusses new approaches that will be pursued within this dissertation, the

adaptation of Multivariate Statistical Process Control (MSPC) approaches to diagnose

faults within the measurement of Fuel Metering Unit of a gas turbine engine. Theory

about the PCA and the PLS are explained as the analysis method has been developed

Dissertation Structure

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from this theory. Some mathematics principles (Linear Algebra) are briefly stated as

basic theory to develop some methods (Hotelling’s, Euclidean Distance or Statistical

Distance, Normalisation, Quasidimensionless: Traditional and Adaptation, Diagnosis

Technique/Consistency-Checking).

Chapter 5 presents the MATLAB analysis for adaptation methods. It discusses the

process of engine data as a requirement to create a statistical model and develop an

analysis technique (fuel flow estimation analysis and deterioration estimation

analysis). The MATLAB models are investigated and tested by doing some adaptation

on the statistical calculation. The program has been investigated using different

approaches, each program displays the result and discussion of the performance

analysis for each method is undertaken. The first section is briefly discussed

adaptation of the PCA technique and reason behind. Then the application and

adaptation of the statistical techniques (Hotelling’s Method, Normalisation Method

and Quasidimensionless Method) is presented. Performance comparison is provided

to give perspective about the robustness of this method. To highlight particular faults

in aero engine variables, Diagnosis Technique/Consistency-Checking is discussed.

Chapter 6 contains review of the methods, the achievement and conclusion of the

dissertation. Review all of the undertaken works is briefly explained as objectives of

the motivation in Chapter 1. The achievement using the statistical technique to

diagnose faults problem in fuel pump measurement bias of a gas turbine engines is

explained. Some suggestion for further investigation in the future is provided to

optimise the performance.

Chapter 2: Background - Gas Turbine Engines

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Chapter 2

BACKGROUND

2.1. Gas Turbine Engines

2.1.1. Introduction to Gas Turbine Engines

Modern gas turbine engines are the advanced development of the very first model of

jet engines, which was pioneered by Frank Whittle and Hans Joachim Pabst Von

Ohaion, who worked independently in the early 1930’s. This engine has a significant

speed increase and reliability advantages over the piston engine.

Figure 2.1: A Whittle-Type Turbo Jet Engine ( Rolls-Royce plc, 1986)

Figure 2.1 is the first generation of the jet engine model by Whittle. The outline of the

general process within the turbo jet engine is shown from the flow of air intake

through compressor, combustion chamber, and turbine, where finally the large

quantity of air is trusted backward to produce power to the jet.

2.1.2. Basic Concept of How a Gas Turbine Engine Works

The basic concept of how the aircraft engine works is by its ability to produce thrust

to propel the aircraft forward by thrusting the large mass of air backwards (Rolls-

Royce plc, 1986a). This happens because the engine is designed to accelerate a stream

of air at high velocity to generate thrust to propel an aircraft.

Gas Turbine Engines

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Sir Isaac Newton’s third law states: ’for every force acting on a body, there is an

equal and opposite reaction’. It is based on this law that the propulsive power of an

engine to propel an aircraft forward is achieved. The atmospheric air is injected at the

front of the engine, squeezed, heated and expelled from the rear of the engine to

generate forward thrust to accelerate the aircraft, as shown in Figure 2.2.

Figure 2.2: Jet Propulsion Mechanism ( Rolls-Royce plc, 1986)

The relationship to produce thrust can be stated as: Thrust = m.(Vaircraft – Vjet), where

the jet moves small mass of gas at high velocity (Rolls-Royce Website).

2.1.3. Type of Jet Propulsion

There are four types of jet propulsion, the only difference of these types is in the way

the engine supplies and converts the energy into power for flight.

1. Ram Jet

Ram Jet is commonly called the aero thermodynamic duct (athodyd), which is simple

in design but contain no rotating parts. Generally, this type of engine is used to power

missiles rather than aircraft since it forward motion is required before thrust is

produced.

2. Pulse Jet

It operates on a cyclic fashion requiring two operating stages: the charging and firing.

This engine is used for helicopter rotor propulsion, but not for aircraft propulsive for

reasons of power generation and fuel efficiency.

Gas Turbine Engines

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3. Rocket Jet

The operation is similar to the operation of the Ram Jet except it uses liquid oxygen

for the propellant medium rather than the air. This type of engine is ideal for rocket

where the oxygen is non-existent outside the earth’s atmosphere.

4. Gas Turbine Engines

The gas turbine is the most economical engine to power passenger aircraft. It provides

propulsion at low speed by the inclusion of a turbine driven compressor. This

eliminates the problem of poor efficiency of the rocket engine. A recent example of

which is the Rolls Royce Trent 800 engine, which is depicted in Figure 2.3 and is

used to power the Boeing 777.

Figure 2.3: Rolls-Royce Trent 800 Engine ( Rolls-Royce plc, 2000)

From a control systems point of view, it is possible to divide a gas turbine engine into

three major parts: the bare engine, the accessories and EEC. The bare engine consists

of the compressor combustion chamber, turbine and casings. Figure 2.4 shows the

airflow of the gas turbine engines systems and the component within the systems. The

accessories are located around the bare engines and contain the Fuel Metering Unit,

the starter motor. The Electronic Engine Controller (EEC) provides the interface

between the aircraft systems and the engine to provide thrust control and diagnostic

information about the engine health, related to fault diagnosis within the gas turbine

engines.

Gas Turbine Engines

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Figure 2.4: Airflow Systems ( Rolls-Royce plc, 1986)

Fuel delivery system in the accessories provides regulated fuel flow to the combustion

chamber. It contains the Fuel Metering Unit (FMU) and the fuel pumps, which are

driven by the high-speed shaft gearbox. This Fuel Metering Unit consists of the Fuel

Metering Valve (FMV) and the Shut-Off Valve (SOV). This dissertation will focus on

the discussion of diagnosis of fuel pump measurement bias in gas turbine engines.

Further information regarding the fuel systems itself is described in the next section.

Figure 2.5: The Working Cycle Diagram ( Rolls-Royce plc, 1986)

The gas turbine engine operates on a continuous working cycle as shown on the

Figure 2.5. The continuous process in a gas turbine engine can be divided into four

phases: induction, compression, combustion and exhaust/expansion stages.

Each line represents the working cycle diagram. Line DA, where the air is drawn into

the gas turbine engine, represents the induction stage. While line AB, where the air is

compressed to increase the pressure, represents the compression stage. Line BC, the

fuel is added to the compressed air and light to heat up the air - this represents the

Gas Turbine Engines

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combustion stage, where a mixture of fuel and air within an enclosed placed is burnt

at near constant pressure. Finally, line CD represents the exhaust/expansion stage,

where the heated air is expanded to provide the thrust as it passes the turbines and the

exit nozzle.

2.1.4. Utilisation of Gas Turbine Engines

There are two clear distinctions in the utilisation of gas turbine engines: the aero gas

turbines and the industrial gas turbines. The latter type has been used in many

applications such as the electrical power generation (Scalzo et al., 1996), pipeline

pumps, naval propulsion (Shepard et al., 1995), etc.

The industrial gas turbine engines are not classed as safety critical systems (Patel,

V.C., 1998). However industrial engine have extremely stringent demand on operating

cost, environmental emission and most fuel usage, which have resulted in more

complex fuel metering systems.

In aero gas turbine engines, it is possible to divide their application between military,

civil and rotorcraft applications. Although they have similarities in their configuration

the different demands of their application places different priorities in their design.

Performance, size and weight are significantly more important in military engines

than in civil applications where efficiency and availability are the priorities.

Fuel Systems

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2.2. Fuel Systems

2.2.1. Introduction to Fuel Systems

Fuel systems provide the engine with the necessary fuel to support the combustion

and the flow control such that the required quantity of fuel to enable an easy starting,

acceleration and deceleration in all flight conditions. To achieve this, the fuel pump is

used to send the fuel to the fuel spray nozzles, which injects the fuel in the form of an

atomised spray into the combustion chamber. In this stage, the fuel is mixed with the

air and burnt together to produce hot gas to drive the turbine.

The controlling devices in the fuel systems are fully automatic except on engine

power selection, which is achieved by a manual throttle or power lever, because the

flow rate must vary according to the amount of air passing through the engine, while

maintaining a constant selected engine speed or pressure ratio. To prevent from

exceeding the maximum limitation on the engine gas temperature, compressor

delivery pressure, and the assembly speed, it is necessary to have automatic safety

control. A governor in the fuel system also prevents the over-speeding for safety

reason.

2.2.2. Fuel Metering Unit and Fuel Pump Function

The main function of the FMU and Fuel Pump is to deliver a regulated fuel flow to

the combustor, where the fuel is mixed with the air. The fuel pump consists of a low-

pressure centrifugal pump and a high-pressure gear pump, which are driven by the

high-speed gearbox. The fuel input from the aircraft fuel tanks will flow through an

aircraft fuel pump to the Low-Pressure (LP) pump.

2.2.3. Control of the Fuel System

Regulation of the quantity of fuel injected into the combustion system is important

because it has the most significant effect on the engine behaviour. One of these effects

is on the gas turbine engine control of power or thrust. The relationship between the

airflow through the engine and the fuel supplied is complicated by changes in altitude,

Fuel Systems

13

air temperature and aircraft speed. Varying these factors change the density of the air

within the engine intake and consequently the mass of the air through the engine is

also varied. A typical change of airflow with altitude is shown in the Figure 2.6.

Figure 2.6: Airflow Changing with Altitude ( Rolls-Royce plc, 1986)

It is shown that as the altitude increases to maintain the constant engine speed, the

intake of air decreases almost linearly. To meet this change in airflow, a similar

change in fuel flow is required and this will affect the engine speed from the original

value selected by the throttle lever position. The relationship is shown in the Figure

2.7.

Figure 2.7: Fuel Flow Changing with Altitude ( Rolls-Royce plc, 1986)

Fuel Systems

14

Traditional fuel control systems are achieved by a hydro-mechanical unit (pressure

control and flow control systems). The vast majority of fuel systems use a Multi

plunger type fuel pump. This will be discussed further on the next section.

Recently, there is an electronic system control, which measures and translates

changing engine conditions to automatically adjust the fuel pump output

electronically. Electronic Engine Control system continuously monitors shaft speeds,

temperatures and pressures along the engine to ensure its safe operation. The EEC

commands the FMU to increase or decrease the flow of fuel to the engine to activate

the desired level of thrust.

2.2.4. Type of Fuel Pumps

The two basic types of positive displacement pumps used in engine fuel metering

systems are plunger type and constant delivery gear type. The latter is preferred due to

its lightness.

2.2.4.1. Plunger Type Fuel Pump

Figure 2.8: A Plunger Type Fuel Pump ( Rolls-Royce plc, 1986)

Figure 2.8 illustrates the main components of a plunger type fuel pump. The engine

gear train drives the fuel pump and the output of which is dependent on both the

strokes of the plunger and the rotational speed. Typically, a single unit can deliver

Fuel Systems

15

fuel at a rate of 100 to 2000 gallons per hour at a maximum pressure of 2000 lb/inch2.

This level of performance requires 60 horsepower to drive this pump.

2.2.4.2. Gear Type Fuel Pump

Gear type fuel pumps are shaft driven and the relationship between the output and

shaft speed is approximately proportional. A spill valve, which is pressure sensitive,

controls the amount of spill, while the re-circulating excess fuel delivery controls the

fuel flow to the spray nozzles (Rolls-Royce plc, 1986b).

Specific variation in gravity and its specific energy content has a critical influence on

the changes of fuel to the engine. Changes to lower the quantity of fuel will result in

an increase number of flame luminosity and temperature. This leads to greater

combustor temperature and reduced the combustor and turbine life.

In general, the gas turbine fuel should have the following quality requirement:

• In all operating condition it flows easily

• At all ground conditions it permits engine starting

• Satisfactory flight re-lighting characteristic.

• The combustion is efficient at all condition.

• The calorific value is as high as possible.

• The harmful effect on the fuel system component is minimal.

• There is adequate lubrication for the moving parts of the fuel system.

• The fire hazard is reduced to a minimum.

The maintenance of the fuel can be done by well-planned storage and by making

routine aircraft tank drain checks. The contamination level is reduced to minimum in

practical if the suitable filter, fuel/water separators and selective additives are used.

Avoiding un-dissolved water will prevent serious icing problems, reduce the micro

biological growth and minimise corrosion while reducing the solid matter will prevent

undue wear in the fuel pumps. It will reduce corrosion and lessen the possibility of

blockage occurring within the fuel systems. This prevention should be carried out

Summary of Chapter 2

16

carefully to avoid the existence of fault within the fuel pump. In line to this prevention

step, it is important to diagnose fault within the fuel system as early stage as possible.

2.3. Summary

Whittle and Ohaion pioneered the earliest model of an aero gas turbine engine in the

early 1930’s. The basic concept how a gas turbine works is where some amount of

power is required to thrust the amount of air backward, in order to propel the aircraft

forward. Types of gas turbine engines and the main sections (the bare engines, the

accessories and EEC) are discussed briefly. The operation within gas turbine engines

can be divided into four major continuous cycles (ambient air, compression,

combustion and expansion). The process is very critical and the safety is the highest

priority.

The fuel system is the main concern in order to give an easy starting, acceleration and

stable running on all condition. The behaviour of the fuel consumption to some

parameter (i.e. altitude and air consumption) is important to detect the amount of fuel

consumption in different condition. Fuel pumps responsible to supply the fuel within

the engine. There are two types of fuel pumps: the plunger type and the gear type. The

latter is preferred because of its lightness.

Chapter 3: FDI and Degradation in Gas Turbine Engines

17

Chapter 3

FDI AND DEGRADATION

IN GAS TURBINE ENGINES

3.1. Fault Detection and Isolation

In many industrial applications, especially safety critical systems, fault detect-ability

is a very crucial indicator for the continuity of the process, e.g. aircraft industry,

chemical processes, steel manufacturing, etc. As the complexity of modern control

system and sophisticated development of control algorithm increases, the issues of

availability, cost efficiency, reliability, operating safety and environmental protection

become the major importance. An early indication of the existence of faults can be

used to avoid system breakdown, mission abortion and unnecessary delays that can

result in large financial penalties.

The term fault is defined as an unexpected change within system function, which is

caused deterioration or abnormal operation. Modern fault diagnosis systems attempt

to minimise unscheduled maintenance activities by prevention and isolation. Fault

diagnosis system is a monitoring system to detect fault as early stage as possible and

trace their location in the system to prevent any serious consequences.

The tasks within a monitoring system normally consist of the following items (Kurz et

al., 2001), with the first two tasks are often called Fault Diagnosis.

1. Fault Detection - to detect the condition of the operation.

2. Fault Isolation - to trace and locate the fault.

3. Fault Identification - to identify the size and type of fault.

Fault Diagnosis has a significant contribution for fault tolerance control system, to

retain some system capability in a case of fault, while the necessary decision has been

proceed to make the proper action, to reconfigure according to the faults (e.g. in flight

control system in order to maintain aircraft safety in a fault condition).

Traditional Model-Based Approach

18

The discussion for particular problem in this dissertation will focus on two schemes:

the traditional approach and the statistical approach. It is however appreciated that

there are much wider spectrum of diagnostic technologies and techniques than these

two particular approaches. Isermann et al., 1996, gave excellent reviews of the wide

range of diagnostic technologies. The emphasis is on multivariate analysis technique,

which will be discussed more detail in Chapter 4 and 5.

3.2. Traditional Model-Based Approach

The traditional approach generally divides into two major methods: the off-line

method and the on-line method. Brief information provided here attempts to outline

the basic concepts about traditional approach. For further explanation, it is advisable

to refer to book by Chen et al., 1999a.

3.2.1. Off-line Method

This method of analysis is used after the incident has occurred by detecting faults that

have already occurred. The reason for this is to analyse the gradual degradation before

the failure occurs. The slight deterioration caused by the degradation will grow

bigger, and then leads to eventual fault or even failure to the system, if not detected

early. Slight deterioration is also known as incipient faults, a small fault, which is

within the tolerance but would have to be taken more seriously once it grows to a

more significant size (i.e. potential for fault prognosis). This fault is useful to

highlight particular components for maintenance purposes, which lead to cost

reduction.

The off-line method includes the Frequency Spectrum Analysis (FSA), fault

dictionary, Magnetic Chip Detector (MCD), Spectrometric Oil Analysis Program

(SOAP), performance monitoring, inspection, test equipment, pilot reporting and

usage monitoring. The performance monitoring has been used in many industrial

processes, mainly in chemical plant processes to detect any incipient fault on the

system overtime. Some methods have been developed based on the performance

trending and have been made applicable in wide range of process control system, e.g.

performance monitoring for fault detection in aircraft system (RR Compass System).


19

3.2.2. On-line Method

The online method continuously monitors the deviation from the normal operating

condition of the plant parameters. Some further action will proceed if an abnormal

condition is detected and then decision will be made whether the plant will continue

working while the further investigation is carried on, or the plant should be

reconfigured to accommodate this condition or if it is necessary the plant should be

shut down to prevent further damages.

These online methods include limit checking, fault integration, temporal redundancy,

physical redundancy, and analytical redundancy. Currently, the latter method is

performed using look-up tables indexed against engine parameters to derive what the

output should be, i.e. parameter synthesis.

Model based fault methods diagnose faults during system operation and as such form,

part of the online method since input/output information is generally only available

when the system is operating. It monitors the signal against a given threshold level

and establishes the action. In fact, the consistency checking can provide better

detection and isolation of faults compared to limit checking (Chen et al., 1999b). This

analytical redundancy is achieved through comparison between measured signal and

its mathematical model estimation. The result of this comparison is the residual

quantities. This approach has a major advantage where there is no need to add new

hardware components in order to implement the FDI algorithm. The major drawback

of this approach results from the fact that the mathematical model never exactly

reflects the true process for anything other than the most trivial process.

Some assumptions have to be made to model the uncertainty because there is always a

mismatch between the actual process and the mathematical model. The FDI model has

to be robust (i.e. insensitive to modelling uncertainty), without losing fault sensitivity.

This scheme is known as a robust FDI scheme. Examples of the development of this

type of scheme are the unknown input observer (UIO) and eigenstructure assignment,

which are briefly described in the following section.


20

Figure 3.1: Conceptual Structure of Model Based Fault Diagnosis (Chen et al., 1999.p21)

The crucial task for a robust FDI scheme is to diagnose the incipient faults within a

system before it grows to major problem with its associated costs. This early prompt

can give enough information to take necessary remedial actions.

Figure 3.1 shows the conceptual structure of a model-based fault diagnosis, which is

divided into two stages: the residual generation – to generate the residual from

knowing input and output information as a fault indicator extraction from the system,

and the decision making – to examined the residual to make a decision action. This

fault information then can be used to make a model-based fault diagnosis having

known the input/output to the system.

3.2.3. Unknown Input Observers

In 1982, Watanabe and Himmelblau proposed the original method of Unknown Input

Observer (UIO) approach for designing observers in the presence of inputs

disturbances. This approach makes the state estimation error de-coupled from the

unknown inputs as its principle. There has been much progress in the development of

the approach for designing robust FDI schemes.

One important contribution that had been made in order to avoid some complex

computation is by Chen, Zhang & Patton (1996). They introduced a new full order

UIO structure. This new structure is shown in Figure 3.2. It offers more design

freedom to achieve the performances but the disturbance de-coupling condition must


21

be satisfied. In fault isolation, this extra design of freedom is used to generate

directional residual, to make the state estimation error have minimal variance. It is

essential to know the unknown input distribution matrix, while it is not the case for

the actual unknown input signal.

Figure 3.2: A Full-Order UIO Structure (Chen et al., 1999.p71)

The description of this structure can be stated into two equations:

)()()()( tKytTButFztz ++=� (3.1)

)()()(ˆ tHytztx += (3.2)

where both x̂ and nz ℜ∈ . The x̂ represents the estimated state vector, z represents

the state of full-order observer, while F, T, K and H represent matrices which are

designed to achieve unknown input de-coupling.

3.2.4. Eigenstructure Assignment

The residual is important in model-based FDI because it contains the diagnostic signal

as the required information. Eigenstructure assignment offers an important direct

approach to design robust residual generators by assigning the left eigenvectors of the

observer to be orthogonal to the disturbance distribution direction. This procedure

makes the residual become robust against the disturbances. Chen et all., has

intensively developed this method for industrial monitoring. The assignment of the


22

eigenvector contributes the robust residual generation. Both of the left and right

eigenvectors will affect the relative shape of the system dynamic response.

Figure 3.3: Robust Observer-Based Residual Generation (Chen et al., 1999.p111)

Figure 3.3 shows the relationship of input/output system to the residual generator.

That figure can be describes by the following equation:

)()()()()( 1 tEdtfRtButAxtx +++=� (3.3)

)()()()( 2 tfRtDutCxty ++= (3.4)

where )(),(),(),( tdtutytx and ntf ℜ∈)( . x(t) is the state vector, y(t) is the output

vector, u(t) is the known input vector, d(t) is the disturbances (unknown input) vector,

and f(t) is the fault vector. A, B, C, D represents the matrices and the assumption has

been made for a known matrix E, which has the full column rank. The fault

distribution matrices are represented by R1 and R2.

Statistical Approaches

23

3.3. Statistical Approaches

3.3.1. Neural Network

The implementation of neural networks to analyse the data has been carried out for

many applications. The Multi Layer Perceptron and Radial Basis Functions are two

examples of the networks that have been extensively used. The neural network treats

data as a weight for particular network model due to learning process, hence the

technique need to train the data and find the correlation in between to show the

performance quality of the system.

A simple approach is to view neural network as a black box where the input-output

variables are collected together. The process within the black box will produce some

information base on the input-output relationship. The learning process will make the

system ‘learn’ as so that it has the capability to generalise for future output signals. In

general, the neural network classification for learning process can be divided into two

categories: the supervised leaning and unsupervised learning.

3.3.1.1. Supervised Learning

This type of neural network learns by using the desired inputs as a trainer to the neural

network. The inputs are presented as pattern pairs of an input vector and a desired

output. The neural network task is to establish the relationship between the input and

desired output for each pattern pair. With this procedure, the network is expected to

simplify the unobserved pattern pairs and to produce a reasonable output from the

given previously unobserved input.

3.3.1.2. Unsupervised Learning

This type of neural network learns without requires the availability of the desired

input. This is a self-organising type process, which results in a set of clusters. An

exemplar vector represents each cluster.


24

Neural Networks has been widely implemented in many applications, for example:

NET talk, biomedical engineering (Harrison et al., 2000), economic forecasting,

performance analysis (Artursson et al., 2000), etc. There are some benefits of using

the neural network:

��It tackles problem without the need to have an accurate model of the process.

��It simplifies the previous unobserved data.

��Provides fast speed processing.

��It can create an autonomous decision support.

��It is flexible and ease of maintenance.

The training process needs to be able to generalise so that it gives the ability to make

‘prediction’ for unseen data. It is essential that artificial neural networks are able to

undertaken pattern classification, which involves selecting objects to certain

categories. A brief description of MLP and RBFN type neural networks is given

below.

3.3.2. Multi Layer Perceptron

Multi Layer Perceptron (MLP) is the network that is built by a combination of several

layers. This creates a new multi layer network, although, basically, this type of

network is equivalent to a single layer network from the linearity property

relationship.

Figure 3.3: Multiple Layers Neurons


25

Figure 3.3 illustrates a two-layer network. The input layer actually does no processing

of the input data but acts as a buffer, to forward the data to the active layers.

Multi layer network requires a non-linear activation function to prevent the single

layer equivalence problem, a method of credit assignment to distribute the error at the

output throughout the network, and a differentiable activation function to give more

information about the output error to aids learning.

MLP has been used in wide range of application, e.g. pattern discrimination and

classification, pattern recognition, interpolation, prediction and forecasting, and

process modelling. Typically, it is trained using a supervised training algorithm, back

propagation. It has a limitation in recall mode by having no memory of previous

inputs. To overcome this limitation, it is crucial to include the time history in the

neural network input vector, to model a system that depends upon the time history of

input variables.

3.3.3. Radial Basis Function Neural Network

RBFN is a popular alternative for MLP network because it is easier to train although it

is not suited for larger application (www.brainstorm.co.uk). RBFN is similar to MLP

not only in term of multi layer, feed-forward network but also its wide application

base. One distinction for RBFN is in its hidden layer, which contains a radial basis

function that is a statistical transformation, based on Gaussian distribution function.

Figure 3.4: The Structure of Hidden Layer RBFN [a]

(http://www.brainstorm.co.uk/NCTT/tech/tb6-2.htm)

Degradation in Gas Turbine Systems

26

Further descriptions of the RBFN are given in Moody and Darken, (1989) and

Broomhead and Lowe, (1988). For fault diagnosis using neural networks is given by

Zedda et al., 1998.

3.4. Degradation in Gas Turbine Systems

The output of a gas turbine is the result of the fined-tuned co-operation of many

different components within the engines, which creates a system including the engines

driven equipment (compressor or pump). Engine components are subject to tear and

wear over time. The degradation of the gas turbine components will affect the

performance of the system although this degradation principally only on particular

components of the gas turbine engines system. Mainly, the degradation is caused by

the engine mechanism problems, such as: changes in blade surfaces due to erosion of

fouling, effect on the blade aerodynamics, changes in seal geometries and clearances,

effect on parasitic flows, etc.

In particular, the aerodynamic components such as the engine compressor, the

turbines, the shaft driven pumps have to operate in an environment that will invariably

degrade their performance. By analysing the data for each particular component, it is

anticipated that the effects of degradation and those caused by faults can be separated

and a robust diagnosis made of the engine behaviour.

The following section will briefly describe both the degradation mechanism and the

engine components, which are normally affected.

3.4.1. Degradation Mechanism

Degradation mechanism is the type of degradation observers in engines, which is

causes a reduction in the efficiency of the engine performance. These degradation

mechanisms includes:

��Fouling is caused by the adherence of particles to airfoils and annulus surfaces.

The adherence is caused by oil or water mists. Cleaning can normally eliminate it.

It is common for a ‘compressor wash’ to be undertaken periodically to remove

this type of fouling.

Degradation in Gas Turbine Systems

27

��Erosion is the abrasive removal of material from the flow path by hard particles

impinging on flow surfaces. This is more of a problem for aero engine application

and in particular shaft driven compressors and pumps.

��Large foreign objects striking the gas path components often cause damage.

��Abrasion is caused when rotating surface rubs on a stationary surface, it will

increase seal or tip gaps to increase.

3.4.2. Component Degradation

Degradation happen on particular component within the engine, including:

��Airfoils. Erosion, deposits, or other damages can change the geometric shape of

airfoils. This will always reduce its performance, which is caused by significantly

increased boundary layer growth, premature transition to turbulent flow, and

premature flow separation. All these influences will create higher losses. Erosion

has a significantly affect the location and extent of the transition from laminar to

turbulent boundary layers.

��Clearances. Clearances between rotating blades and the stationary casing have a

tendency to open-up during the ageing process of the components. On driven

equipment such as compressors and pumps, increased clearances in inter-stage

labyrinth seals or balance piston seals, or should seals will reduce the overall

efficiency of the engine.

��Compressor. There are three major effects that determine the performance

deterioration of the compressor. The ones that typically lead to non-recoverable

degradation are increased tip clearances and changes in airfoil geometry. While

the effect to the changes in airfoil surface quality can be at least partially reversed

by washing the compressor.

��Turbine. The flow capacity may be reduced by thicker boundary layers on the

blades and sidewalls, especially near choking conditions. If the degradation of the

turbine section leads to material removal, especially in the nozzle area, the

positive affect is the flow capacity increases for any given pressure ratio.

��Combustor. The efficiency of the combustion will usually not decrease

significantly to cause performance deterioration, although it could lead to a

variation in the combustor exit temperature profile. However, these distorted exit

temperature distributions will arise the problems: local temperature peaks can

Effect of Degradation in Gas Turbine Engines

28

damage the turbine section, the altered temperature profile will increase secondary

flow activity, which will reduce the turbine efficiency.

��Driven Equipment. The driver and the driven equipment will show the

performance degradation over time. Solid and liquid contaminants in the process

gas can damage the impellers or form a dirt layer on the aerodynamic surfaces,

which can increase friction losses.

3.5. Effect of Degradation in Gas Turbine Engines

The amount of degradation depends on the loss of compressor efficiency, compressor

pressure ratio, compressor airflow, gas generator turbine efficiency, power turbine

efficiency and the control mode. Degradation of the compressor and the gas generator

turbine effectively move the match point to lower ambient temperatures. For the same

amount of degradation, the relative loss in efficiency is significantly lower than for an

engine at full load.

Compressor deterioration will generally cause higher power losses rather than losses

in heat rate, because higher compressor exit temperature will result due to lower

efficiency of the compressor for a fixed combustion temperature.

The affect of degradation is typically more severe for two shaft engines than for single

shaft engines since compressor degradation on two shafts engine leads to a larger drop

in gas generator speed (Tarabrin et al., 1998).

Readjusting the control system can accommodate these losses. The reduction of gas

turbine efficiency will lead to an under firing of the engine. Furthermore, it will also

affect the optimum power turbine speed.


29

3.6. Summary

As complexity of the system increases, the possibility for the fault to happen becomes

higher and the systems continuity is threaten. It is very crucial to detect and allocate

the fault when it is in the earliest stage. Undetectable fault is known as incipient fault,

the small amount, which possible grows bigger and degrades the performance of the

system. If this condition is undetectable, it can cause the failure and catastrophe

situation. Not only the systems could broke but also the losses in cost and possible

dangerous situation may happen.

Fault Detection and Isolation (FDI) is the method to detect the availability of fault in

the early stage and locate the place to do isolation, so the fault will not spread to

bigger area. There are several methods of doing FDI, i.e. traditional model based (off

line and on line method – UIO and Eigenstructure Assignment) and statistical model

approach (neural network – MLP, RBFN).

In gas turbine engines, the degradation will lead to the failure. Since aero industry is

safety critical systems, the degradation needs to be detected and repair in the early

stage. There are two types of degradation: the mechanism and components. This

degradation will affect the performances of the gas turbine engines. Doing analysis of

the engine data, the degradation can be located and necessary action can be taken.

Chapter 4: Statistical Techniques - Introduction

30

Chapter 4

STATISTICAL TECHNIQUES

4.1. Introduction to Statistical Techniques

Statistical techniques have been used for many years in a wide range of applications

to analyse and solve problems using modelling, data analysis, prediction and

forecasting. This includes applications to problems in modern gas turbine engine

control. However, the majority of the application of this technique is in the field of

chemical process control and is known as chemometric. This statistical approach

introduces a modelling technique to describe the processes in statistical term, which is

based on data and the analysis is then performed on a time-series analysis of the

process data.

Due to the uncertainties that surround some non-linear dynamics processes, which

varies heavily, the statistical approach proved suitable for the purpose of data analysis

of multi variables parameters. The statistical theory of decision systems and

information is given based on the statistical performance. Correlation model is utilised

to quantify the degree of similarity between two variables and monitor their variation

during the process.

Statistical approach has been used to monitor the quality performance in many

industrial processes for reasons of simplicity and data complexity. The previous

method of analysing the fault detection in gas turbines has been carried out using the

technique of model-based fault detection and neural network. The developments in

industrial processes have lead to design of new statistical techniques to overcome

problems in analysing the complex data. One such method is Statistical Process

Control (SPC).

Statistical Process Control

31

4.2. Statistical Process Control

Statistical Process Control (SPC) has been widely applied in manufacturing industries

for more than 30 years and just recently been implemented in process industries. It

uses the statistical model and procedures to adapt the process control in order to

improve product quality with a reduced operating cost. Cost is a very important issue

in industrial processes and it becomes the main motivator to dense new methods.

The aim of SPC is to keep the process in a state of statistical control although natural

variations remain in the process. If the number of qualities being monitored is small,

univariate SPC such as Shewhart charts, CUSUM (cumulative sum) plots and EWMA

(Exponentially Weighted Moving Average) chart are typically used (Tham, 1999a). It

works by comparing the current process performance against process behaviour when

the product in progress has to satisfy the specification limits.

Univariate SPC detects disturbances effectively and provides information about the

interaction between variables in complex processes. It addresses the analysis of

individual variables and based upon the magnitude of the variable deviations and any

directional information resulting from variable interaction is ignored.

Process malfunctions can lead to many problems. It reduces product quality, reduces

production rates results in plant shutdowns, increases re-working, increases

environmental impact and gives a low return on plant assets (Tham, 1999b). It is

important to robustly detect the indicator of error that can cause process malfunction

at the earliest possible stage.

A process is said to be in statistical control if certain process or product variables

remain close to their desired values and the only source of variation is common. We

need to verify that the process remains in the state of statistical control over time.

In fact, the huge quantity of variables involved and the complexity in the industrial

processes, results in standard univariate SPC procedures being no longer effective

since it only addresses one variable at a time to find whether it is within the control

Multivariate Statistical Process Control

32

limit or not. Thus, with the majority of the control processes being multivariate in

nature, the needs for an improved technique are required.

4.3. Multivariate Statistical Process Control

As the demand for better performance increases, so the complexity of the process also

increases and which in turn places new demands of the methods of analysis. One

important aspect that we need to consider is the number of variables used in the

process since it creates demands on the data analysis. By monitoring their variation,

we can quantify the degree of similarity to find the correlation models between

variables. This correlation is important to analyse the behaviour of the systems for

particular purpose e.g. for early fault detection in this case, to build the reference

model as the filter or performance indicator for future engine data.

It is claimed that multivariate method will take SPC to a new level and promise to

revolutionise quality control and fault diagnosis by using computing tools to analyse

problem. MSPC potentially provides the process information where effective

monitoring of the process performance with greater reliability and flexibility is

required. Furthermore, MSPC has the potential to provide early warning of process

malfunctions and abnormalities, reduce raw material and energy usage, reduce in re-

work and waste and ensure the manufacture of a consistent production with improved

quality (Tham, 1999c). These advantages will give a major contribution in reducing

the cost of maintenance; improve availability and fuel consumption for a fleet of

aircraft engines.

MSPC has the potential to provide the necessary key for process information that is

required to effectively monitor the process to achieve the stated benefits above. In a

gas turbine engine, there are lots of variables that affect its performance. By applying

the MSPC method, it is expected that the reduction of the variable numbers will lead

to simplification of the problem analysis.

MSPC monitoring technique requires Principal Component Analysis (PCA) and

Projection to Latent Structures, which is also known as Partial Least Squares (PLS)

Multivariate Statistical Process Control

33

method. The techniques form the basis of monitoring schemes to overcome the

problem of variable dimensionality for both on-line and off-line monitoring.

This dissertation will discuss the implementation of Multivariate Statistical Process

Control (MSPC) technique for monitoring and design of a robust fault detection

scheme for a fuel pump system within a gas turbine engine. The application in this

case is to discover whether the measurement in the fuel pump is bias or unbiased

through some combinations of variables within the engine, i.e. pressure, temperature,

shaft speed, altitude, Mach number and thrust power and to be able to relate this to the

fuel consumption. Accurate measurement of the fuel consumption is very important in

aerospace applications, since it enables the aircraft to fly with the minimum necessary

fuel quantity for safe operation, which makes a considerable cost and weight saving.

Any inaccuracies in the fuel measurement need to be accommodated for by the

addition of extra fuel to the aircraft to ensure safe operation.

PCA and PLS techniques offer a better analysis compared to univariate SPC. PCA

and PLS Technique will be explained in the following section prior to their

applications to the specific aerospace related problem.

Principal Component Analysis

34

4.3.1. Principal Components Analysis

4.3.1.1. Introduction to PCA

One major disadvantage in industrial process control generally which includes in civil

and military aerospace engine is its inability to monitor the quality of multiple

independent variables. Large number of the variables makes the task of analysing the

problem complicated and time consuming. An alternative solution to solve these types

of problem is to find the relationship between these variables and hence reduced the

number of variables to make analyse more efficiently and simpler (i.e. eliminate

redundant information).

When the number of data variables is greater than two, it is appropriate to use a

technique called Principal Component Analysis (PCA) to analyse the data. PCA can

monitor the system process through a single block of information, i.e. the data process

from each variable, which represent the process performance. In this case, PCA

method reduces the number of the monitored variables whilst maintaining monitoring

of the entire process. This technique has been used in wide spectrum of application in

science, geology, psychology, engineering, etc.

The reduction in the data dimensionality is not the only benefits provided by PCA,

because it can also be applied to classified variables, outliers detection (where the data

which is not within tolerance could alarm the process) and recent applications of PCA

are to fault diagnosis of industrial processes.

PCA is helpful for analysing the covariance structure of the data set, by finding the

directions of small or large variability. Theoretically, this is important due to the

relationship between elliptical distributions and standard distance, where the data is

analysed based on a Euclidian method by finding the centre and maintaining its

distance within a specific tolerance. PCA provides a means of defining which

variables are potentially related to the quality or production variables, especially when

the variable is redundant. PCA is an effective method of dissecting the

interrelationships among a group of variables.


35

4.3.1.2. Principal Component

PCA transforms the correlated variables into uncorrelated ones for the reason of

reducing the dimensionality of the data, with minimal loss of information. This makes

the data easier to understand. This reduction of the dimensionality will make the

analysis process simpler and faster to undertake since the amount of data to be

analysed is reduced. The advantage of this data compression is it still gives the

information of the overall process performance through information extraction

(Jackson, 1991), which describes the major trends in a data set.

Principal Components (PCs) or factors are a set of linear combination of the variables

that redefines the existing variable space. The original variables are transformed to the

new uncorrelated variables and upon which the analysis is performed. These PCs will

represent all known variables within the process. By analysing the performance of the

PCs, we can still extract the information for the whole system.

This method consists of finding an orthogonal transformation of the original variables

to a new set of uncorrelated variables (principal component), which are described in

decreasing order of importance. The first PC is the combination of variables that

explains the greatest amount of variation while the second PC defines the second

largest amount of variation. Both PCs are independent of each other.

PCA utilises an eigenvector decomposition of the covariance or correlation matrix of

the process variables. Consider matrix X with m rows and n columns, where the rows

correspond to samples, while columns correspond to variables. For a given set process

data, the covariance of matrix Xmxn is defined as:

1)cov(

−=

m

XXX

T

(4.1)

An assumption is made that the columns of X have been adjusted to have zero mean

by subtracting the mean of each column, to make it mean centred or auto scaled. Auto

scaled means that the columns of X have been adjusted to zero mean and unit variance


36

by dividing each column with its standard deviation. This will gives the correlation

matrix of X.

PCA decomposes the data matrix X as the sum of the outer product of vectors ti and pi

plus a residual matrix E by the approach, which transforms the measurement matrix

from the process variables X into a matrix of mutually uncorrelated variables ti (where

i=1-n) and a set of orthogonal loading vectors pi. A new terminology is introduced at

this point termed scores, which is defined as the individual value of the PCs. The

equation for the transformation is written as follows:

∑=

+=],min[

1

nm

k

Tkk EptX (4.2)

In general, the equation above can be written in the form of:

ETPEptptptX TTkk

TT +=++++= ...2211 (4.3)

where k must be less than or equal to the smaller dimension of X, ],min[ nmk ≤ . The

scores, ti vector contains information on how the samples relate to each other, while

the pi vectors are eigenvectors of the covariance matrix and known as loadings and

contain information on how variables relate to each other. The relationship for each pi

is given by:

iii ppX λ=)cov( (4.4)

where iλ is the eigenvalue associated with the eigenvector pi. For X and any ti,pi pair,

the relationship can be stated as:

ii tXp = (4.5)

In this case, ti forms an orthogonal set, i.e. 0=jTi tt for ji ≠ while pi also forms an

orthonormal set, i.e. 0=jTi pp for ji ≠ , 1=j

Ti pp for ji = . The score vector ti is the


37

linear combination of the original data X defined by pi. The order of pairs ti,pi are

arranged in descending order according to the associated iλ , which measures the

amount of variance described by those pairs. In this sense, we can think of variance as

information. This arrangement idea is to make the first pair captures the largest

amount of variation in the data with a linear factor and the last pair captures the least.

The subsequent pairs capture the greatest possible variance remaining at each step. In

this way, it is possible find the smallest number of Principal Components ],min[ nm to

represent the required amount of power in covariance matrix, when the remaining

power constitutes the error term E. When the above equation is applied to a single

vector of new measurements, XT, the resulting term E is called the prediction error.

These components are linear combinations of the original variables. The first few

components will account for most of the variation in the original data, so that the

effective dimensionality of the data can then be reduced to simplify the subsequent

analysis. By analysing these variations, the interaction among variables in the process

is understood and is required to minimise the amount of variation. Variation among

these variables indicates deviation from normal, which highlights the potential

abnormal process behaviour resulting from a process fault.

The concept of principal components (PCs) is shown in the Figure 4.1 where the 3D

data is lying on the single plane with the Q and T2 outliers. It is observed that the

description of the data is well represented by only two principal components. The first

eigenvector aligns with the greatest variation in the data; the second aligns with the

next greatest amount of variation but is orthogonal to the first PC. In practise, two or

three principal components are frequently sufficient and the data can be adequately

described. The number of principal components is much less than original variables

dimension (k<<n). For understandable fault diagnosis, lower order components are

often advantageous.

Since most of the process industry is non-linear, this approach of analysis should

consider the relationship of the variables within the process control. This method is

applicable in condition that the relationship among variables is linear. The solution


38

available is to develop non-linear counterparts to PCA or to attempt to linearise the

problem, which has proved more effective.

Figure 4.1: Principal Component Model of 3D Data Set (Redrawn from Wise et al., 2000)

Conventional PCA is linear procedure, to analyse the performance of the PCA model

on how it captures the amount of variation in each sample, we can calculate Q and T2

for each sample.

The Q statistic indicates how well each sample conforms to the PCA model. It

measures the amount of variation in each sample not captured by the k principal

components retained in the model. Q is simply the sum of squares of each sample row

of E e.g. for the jth sample in X, xj is shown as:

Tj

Tkkj

Tjjj xPPIxeeQ )( −== (4.6)

where ej is the jth row of E, Pk is the matrix of the first k loading vectors, each vector

is a column of Pk and I is the identity matrix (nxn). A confidence limit should be

placed on Q conditioned for each data vector.


39

Hotelling’s T2 statistic measures the variation within the PCA model. It is simply the

sum of normalised squared scores, given by:

Tj

Tj

Tjjj xPPxttT 112 −− == λλ (4.7)

where tj refers to the jth row of Tk, the matrix of k score vectors from the PCA model.

1−λ is given by the diagonal matrix containing the inverse eigenvalues associated with

the k eigenvectors (principal components) retained in the model. A high value of T2

indicates a poor density of training data of the measured vector.

It is suggested (Goulding et al., 1999) to:

��Monitor Q using EWMA (exponentially weighted moving average) with time

constraints appropriate to anticipated faults.

��Monitor T2 to validate the Q statistic.

This statistical limit can be developed for Q and T2.

We can use the PCA model, once the model has been developed (mean, variance

scaling vectors, eigenvalues, loadings, statistical limits on the scores, Q and T2) and

used with new process data to detect changes in the physical process. For detecting

systems faults, it was chosen to primarily use Q and T2 where Q is a measure of the

variation of the data outside of the PCA model. A feature of the PCA is that it is often

the smaller component that describes the noise in the data is necessary to be excluded

on the process analysis.

4.3.1.3. Multi-way PCA

The concept of the multi-way PCA is a relatively straightforward extension of the

PCA approach and is applied to continuous systems, but it considers the mean

trajectories rather than steady state. This method considers collecting the data in a

sequential manner as a block before it is processed in a multi-way matrix.

This technique creates an unfolded data matrix and then develops a PCA model i.e.

the coefficient is calculated for every measured process variable at each sampling

Partial Least Squares

40

instant. Most of the application of the multi-way PCA is to analyse the batch process

data (Wise et al., 1997) or the measurement of data with more than two parameters

(e.g. the technique of PCA and PLS to handle missing data).

4.3.2. Partial Least Squares

In industrial processes, Multiple Linear Regression (MLR) is an important technique

used to identify the relationship between the quality information and the process

variables. The reduction of time spent on analysis, the prediction of the final product

becomes easier to undertake.

Unfortunately, for multidimensional system where the correlations among variables

are typically high, this technique is no longer appropriate because the model becomes

numerically unstable even to small perturbations. Alternatively the Principal

Component Regression, the PCR may be implemented to overcome this problem. It

has been shown to effectively handle both singularity and dimensionality, however

the approach treats the variables as independent. Partial Least Squares or commonly

identified as PLS, overcomes this limitation and enables the information to be treated

as dependent, whilst it handles the singularity and dimensionality aspects of the

problem.

The concept of PLS is similar to PCA. It is a statistical method, which relates the

multivariate descriptor data sets X to multivariate response data sets Y and finds the

highest correlation.

The PLS relationship can be explained by given a set of information on m process

variables, X, and k quality variables, the outer relations between X and Y data sets are

then evaluated by the factor ti and ui as follow:

iit ϖΧ= (4.8)

and

ii cu Υ= (4.9)


41

The factor weights, ti and ui are the most correlated to one another. To find the inner

relation between X and Y blocks, we need to perform a linear regression between the

first pairs of factors,

ε+= 111 tbu (4.10)

while the final stage of the algorithm requires the regression of X on t1 and Y on u1

with its residual E and F.

Ept T +=Χ 11 (4.11)

and

Fqu T +=Υ 11 (4.12)

It is possible to repeat these steps with residual E replacing X, and residual F

replacing Y and derived up to m factors. The first few pairs contain the most important

information of the process operation in exactly the same manner as the PCA worked.

The monitoring approach that is commonly suggested (Goulding et al., 2000) for the

PLS model is:

• Monitor Q of the inputs for the validity of the regression model.

• Monitor T2 the process exhibits a highly non-linear behaviour.

• Monitor R2 for the regularised T2 statistic, where R2 = λT2 + (1-λ) Q and λ will be

a function of the data and is obtained from non-parametric approaches.

4.4. Summary

MSPC is a further development of SPC to overcome the increasing complexity of the

problems. It has the potential to provide early warning of process malfunctions and

abnormalities, reduce raw material and energy usage, reduce in re-work and waste and

ensure the manufacture of a consistent production with improved quality Utilising

MSPC can be undertaken by the PCA and the PLS analysis. The PLS is used to

predict the process outputs from inputs, this prediction is then analysed using PCA

approach.

Data Information

42

Chapter 5

DATA ANALYSIS

5.1. Data Information

The data that requires analysis consist of parameters from an aero gas turbine engines

process. These parameters are measured variables from the engine sensors. These data

can be classified into four major variable types: the engine shaft speeds (N), the

temperatures (T), the pressures (P), and the fuel flow (WFE). These engine variables

are affected by the altitude position (ALT), the thrust power (FN) and the Mach

Number (MN).

Figure 5.1: MSPC Model of Gas Turbine Engines

Figure 5.1 shows the relationship between inputs and outputs of the process in aero

gas turbine engines. Knowing both input and output data, the main task is to find the

relationship between them, i.e. the MSPC model. This model will describe, which

variables have significant effects on the performance of aero gas turbine engines,

(especially to able to relate this to the fuel flow). In this way it will be possible to

detect measurement bias in the fuel meter and consequently to reduce operating costs.

The statistical analysis is done through the integration of the PLS Toolbox functions

and user developed MATLAB routines.

An essential output of the analysis is in the form of a scatter plot surrounded by an

ellipse, which will describe the performance of components in the aero gas turbine

engine. The analysis requires engine data to be proceeding in order to find out

whether the engine operates under normal condition with no deterioration (which

means there is no problem within the engine) or there is a signal disturbance (which

could result from engine deterioration in particular location), or fault is present. In

Data Information

43

finding out this information, it is possible to undertake the necessary actions to isolate

the problem and avoid further costs.

The shaft speed, pressure and temperature outputs are spread across different location

in gas turbine engines. The high pressure, intermediate pressure and low-pressure

regions relate to the locations along the length of the engine at which the respective

three engine shafts reside.

The engine shaft speeds (N) comprise three variables:

��N1 - Low Pressure (LP) shaft speed

��N2 - Intermediate Pressure (IP) pressure shaft speed

��N3 - High Pressure (HP) shaft speed

The pressures (P) comprise five variables:

��P20 - the engine inlet pressure at the fan face

��P25 - pressure of the exit of IP compressor

��P30 - high pressure (HP) compressor delivery pressure

��P50 - hot nozzle pressure

��P160 - by pass pressure

The temperatures (T) comprise four variables:

��T20 - Low Pressure compressor exit temperature (engine inlet)

��T25 - exit of Intermediate Pressure (IP) compressor

��T30 - exit of High Pressure (HP) compressor

��TGT - the temperature of gas stream entering the HP turbine from the combustion

chamber.

For this analysis, the engine data is sampled order into four different conditions of

engine deterioration: ideal (i.e. new engine data), conditions representative of 1000

hours of deterioration, conditions representative of 2000 hours deterioration and

finally, conditions representative of 3000 hours deterioration. The ideal data with no

deterioration is used to build the reference model as an indicator or filter for future

analysis. This ideal model will be used to analyse and locate faults within aero gas

turbine engines for engines with 1000 hours, 2000 hours and 3000 hours deterioration.

Data Information

44

This analysis is applied to the entire engine system to identify any discrepancies that

might have happened. Figure 5.2 shows the location of each variable within a typical

3-shafts high bypass turbofan engine (Trent 700 in this instance).

Figure 5.2: Trent 700 Engine Diagram

Steady state engine data was derived from the engine model at varying flight

conditions around cruise. Cruise conditions are defined from speed, altitude and

thrust. Mach Number is a speed measurement and is the ratio of the speed of a body

to the local speed of sound. Mach 1.0 represents a speed equal to the local speed of

sound. As the aircraft speed increases, the fuel consumption will increase, and the

thrust will decrease in direct proportion. The engine data around cruise comprised of

three different Mach Numbers (0.81, 0.83, 0.85).

Altitude is the measurement of height above SLS and the effect of increasing altitude

is to reduce ambient air pressure and temperature. The reduced pressure affects the

engine speed, which cause the thrust to fall. The altitude around cruise conditions

used to generate the engine data is varied from 34000 to 37000 feet in increments of

1000 feet.

Finally, Thrust is a measurement of the engine power and around cruise conditions for

a Trent 800 engine would be in the region of 17580 lbf. Engine data was generated at

thrust conditions of ± 2% ~ ± 4% of this nominal value.

Fuel Flow Estimation Analysis

45

The various combinations of thrust, altitude and speed variations around cruise

conditions resulted in a data set of 60 conditions and were used in the MSPC analysis.

All above engine data is the ideal data. It is used to build a model with no error, which

will be the performance indicator for future data analysis. The following analysis

(Fuel Flow Estimation and Deterioration Estimation) is built within MATLAB using

statistical technique to diagnose faults.

5.2. Fuel Flow Estimation Analysis

MATLAB is used to analyse the given engine data set. The approach is to build a fault

detection model based on the performance of the ideal data, which could then be used

as a model reference for future data analysis, i.e. to detect the deterioration within the

engine from the new examined engine data.

Since the engine data comes with different units and scales, it is convenient to

minimise and if possible eliminate these differences by undertaking normalisation of

the engine data. The reason for this elimination is to put all the engine data into non-

dimensions terms by removing the effect of Mach number, thrust power and altitude

variations and to effectively relate the engine data to the sea level static conditions.

Unfortunately, for some combinations, this is not possible. The best possibility for this

data is to achieve a semi dimensionless state (often known as Quasidimensionless).

This normalisation still has dimensions but achieves simplification of the parameter

variation complexity (Walsh et al., 1998). This normalisation can be achieved by

multiplying combinations of parameters together.

The methods of analysis utilised within the chapter consist of Hotelling’s method and

multivariate statistical method (i.e. the PCA and the PLS analysis). Some

combinations in statistical techniques have been adapted for the reasons of improved

clarity and adaptation to this fault diagnosis problem. The reason underlying the

necessary adaptation of traditional technique will become clear as the simulation and

modelling is developed throughout this chapter.

Adaptation of the PCA Technique

46

5.2.1. Adaptation of the PCA Technique

The implementation of the Principal Component Analysis technique to this

application, which is to diagnose fuel pump measurement bias in gas turbine engine,

requires some adaptation. However, this adaptation is remains in line with the

statistical techniques in which the PCA was originally used. There are some

limitations with respect to the PCA techniques that require to be effectively utilised in

this application. The specific drawbacks of the algorithms are:

��Each parameter of the engine data is independent. The MN, ALT and FN cause

direct effects to the engine data. No variation exists within the engine data.

�� Elimination of the rest existence of parameters based on the PCA technique is not

possible because of the independency of each parameter. All the parameters

either are contributed to or are affected by fuel flow changing by differing

percentage amounts.

�� The chosen Principal Component does not exactly represent the whole condition

of the engine performance.

�� The combinations of parameters in a non-linear relationship are not possible and

some adaptation of the PCA technique is needed to accommodate this

requirement in order to find the relationship to the fuel flow.

To overcome these issues, the adaptation of the MATLAB PCA implementation was

undertaken using simple mathematics and statistical theory. The Hotelling’s method

was based on the same theory where it requires the calculation of the statistical

distances between data. The Normalisation method was developed to suit the needs of

data plotting, as a development to more clearly explain the analysis of the Hotelling’s

method. Both Hotelling’s and Normalisation method utilise standard statistical theory,

whilst the Quasidimensionless approach is based on gas turbine theory (Walsh et al.,

1998) in order to equate any flight condition to reference SLS.

Figure 5.3 illustrates a screen-shot of the PCA analysis based on the ideal engine data.

The mean centred scaling is chosen from the alternatives of: no scaling and auto

scaling. Both mean centred and auto scaling demonstrated relatively close results and

highlights the dominance of the first and second principal components of N3 and N2

data. N1 appears as the third PC when the auto scaling is chosen. Based on this PLS


47

Toolbox on PCA technique; these PCs represent overall performance parameters

within gas turbine engine, which is not exactly faithful to the real condition. The

elimination of the remaining parameters will affect the reliability of the analysis since

it is not a reliable fact that the only parameters that has an effect to the fuel flow is the

shaft speed component variation.

Figure 5.3: Principal Component Analysis GUI

The engine data is given as the matrix: (60x16), the variable is represented by matrix

(16x4), in the sequence of: N3, N2, N1, WFE, P30, TGT, T20, P20, P50, P25, P160, T25,

T30, ALT, MN, FN.

The first and second PCs in total represent 99.55% of the engine data performance

and accommodate the variance among engine data. Ideally, from the PCA technique

point of view, the number of the PC should not be greater than 4. When two PCs

cover more than 85%, this is a very good representative for data analysis. The

simplification of the variability within engine data is quite radical in this instance, but

when further analysis is carried out, significant biases are noted. In fact, the selection

of the PC does not accommodate the engine data structure. Through the eigenvalue


48

analysis, the number of the PCs can be chosen based on knowledge on the given

engine data (Wise et al., 2000). These eigenvalues approach captures as much of the

original variables variability. Figure 5.4 shows the condition where the first PC

captures 13e+05 variance of the original variables, while the second PC captures

1.8e+05 variance. The combined retained and represent 99.55% of the variation

within the engine data. It raised a question whether this theoretical data compression

can be achieved in practise for the engine data application in condition when

deterioration and fault factors are included.

Figure 5.4: Eigenvalues vs. Principal Component

The requirement for adaptation of the PCA implementation in this application is

clearly demonstrated from the score plot of the relationship between sampled data.

Examining the two PCs scenario, the representative score plot is shown in the Figure

5.5. The scores plot can be displayed based on 5 different compositions for each axis

(i.e. sample, PC1, PC2, Q-Res and T^2).

Q-Res explains the measurement of the variation of the data outside of the PCs

defined, included in the PCA representation, whilst the T^2 explains the unusual

variation inside the PCA model. The 95% limit is used to bind the samples for certain

area. The sample below is taken to show how the analysis is difficult to perform since

the characteristics of the data are spread around the circle area. The correlation


49

between the first two PCs, which covered the most variation within the gas turbine

engine, does not appear to be clear. It is a difficult task to uncover the correlation

between the parameter behaviour to fuel flow and degradation with this

implementation of the technique.

Figure 5.5: Plot Scores of PC1 vs. PC2

The fact that the samples are spread in four areas within the circle with a 95% limit

made this representation of the engine data performance has no direct relationship to

the changing of the fuel flow. This shows 0% variation (e.g. the fourth PC), which

will make diagnosis of fuel flow bias troublesome.

The corresponding loading plot of the plot scores is shown on Figure 5.6, where there

exist a significant correlation between the altitude (ALT) and the fuel flow (WFE)

signals. The altitude is loaded positively; the fuel flow is loaded negatively, while

there exist small variations of the loading characteristic among the remains

parameters. This result matches with Figure 2.7 where the fuel flow changes with

altitude. Increasing altitude results in lesser fuel flow consumption. However the

correlation between the remaining parameters to the fuel flow variation has yet to be

explained in such a physical interpretation.


50

There is little fluctuation among other parameters. N3 and N2 show a small degree of

fluctuation, but there is virtually a straight line through the remainder. It is clear from

this situation that these individual engine parameters provide limited correlation to the

fuel flow. However it would be useful to identify whether combinations of the engine

parameters would provide a more informative correlation to the fuel flow

measurement. It is common in gas turbine theory to combine engine parameter to

provide dimensionless parameters that describe the engine behaviour more abstractly.

Figure 5.6: Loading Plot of PC1 vs. Variable Number

The modification is aimed at finding the relationship between the parameters of the

engine data to the fuel flow. The statistical theory is involved to find the variance of

the engine data, finding the right combination of the parameter, which has the must

direct effect with changing values of the fuel flow.

Further improvements have been achieved by applying: Hotelling’s, Normalisation

and Quasidimensionless (Traditional and Adaptation) methods. The entire method is

covered in the section 5.2.6: Performance Comparison to highlight the ability of each

approach. Diagnosis Technique/Consistency Checking in section 5.5, where the

location of the faults is traced, represents the implementation of these methods.

Hotelling’s Method

51

5.2.2. Hotelling’s Method

The first technique is build using Hotelling’s T2 as a multivariate SPC procedure. This

approach was developed by Hotelling’s to analyse the bombsite data. Second similar

approach is that of Mahalanobis. The difference between the two techniques reduces

to only on a constant term (Tham, 1999).

The Hotelling’s T2 utilised standardised statistical distance between two points and

eliminates the effect of varying scales and units. This contributes to simplification of

the analysis of the data plotting and discovering the relationship between parameters.

Figure 5.7 shows the Hotelling’s plot of N1 and N2 against the WFE. The ideal engine

data and deteriorated engine data are plotted in the Figure below. Figure 5.7a depicts

WFE vs. N1 plot. All the engine data (ideal, 1000 hours, 2000 hours, 3000 hours

deterioration) are on top of each other, which make the analysis locating the fault

within the engine difficult. Figure 5.7b illustrates WFE vs. N2 plot and demonstrates a

slight variation in location resulting from 2000 hours of engine deterioration.

Figure 5.7: The Hotelling’s Plot of N1 and N2

Given the condition when the value of N1 has been deliberately changed in order to

represent particular fault condition, it would be expected that the first graph would


52

show a significant changes. The fact that there is no observable change proves that

Hotelling’s is not suitable in this application to detect the changes in certain

parameters to detect fault conditions. It was anticipated that a significant shift from

the ideal engine data would be apparent.

In general, the overall plot is quite scattered although for each combination, the y-axis

scale varies considerably. This variation represents the density of the plot related to

the correlation between parameter and fuel flow. The smaller the y-axis scale, the

greater density the combination has and the concentration of the samples will be close

to the point of origin (0,0). It represents a good correlation between this parameter to

the fuel flow.

Identical patterns of graph are shown on Figure 5.8 ~ Figure 5.11, for combinations

of: N2, N3, P20, P25, P30, P50, P160, T20, T25 and T30 against the fuel flow.

Figure 5.8: The Hotelling’s Plot of N3 and P20

Shaft speeds (N1, N2, N3) and temperatures (T30 and TGT) achieve the highest density

combinations whilst other parameters are more widely scattered. This Hotelling’s

method does not provide a good correlation for the given variations in the engine data

and consequently the analysis is difficult to undertake because the data is not


53

concentrated at a particular point. Figure 5.9 and Figure 5.10 are the result of plotting

the engine data for different conditions and result with the data sitting on top of each

other.

Figure 5.9: The Hotelling’s Plot of P25 and P30

Figure 5.10: The Hotelling’s Plot of P50 and P160


54

Figure 5.11: The Hotelling’s Plot of T20 and T25

The distinct pattern to this observation apart for WFE vs. N2 plot is achieved for the

combination of WFE vs. TGT and is illustrated in Figure 5.11b. TGT demonstrates a

difference between the ideal engine data and the deterioration engine data.

It is interesting to note that the correlated data (i.e. data concentrated at the point of

origin (0,0)) of the WFE vs. TGT combination behaviour generally is similar to other

combinations. However further away from the origin, the data scatter of deteriorated

to ideal increases.

It was noted from Figure 5.12 that the correlation between TGT and WFE is tighter

when comparing to other combinations examined. This can be explained physically,

since TGT is the first measurement of temperature after the combustion stage and the

correlation to the fuel flow is closest.

The fact that little data scatter is apparent for different levels of engine deterioration

with the examined parameter combination. This will make diagnosing fuel pump from

deterioration a non-trivial task (and subject to considerable error) with this approach.

The root cause of this is that the method uses the mean and variance value for

calculating the statistical distance.


55

Figure 5.12: The Hotelling’s Plot of T30 and TGT

Thus, for the engine data generated at different engine condition (60 data samples),

which was effectively normalised with the mean and variance values, results in the

classification of the data to identical ratio. This is manifested as the statistical

distances being identical no matter what value of engine deterioration was present.

Due to this problem, it was proposed to analyse the engine data with engine parameter

combinations based on a Normalisation Method.

Normalisation Method

56

5.2.3. Normalisation Method

This method is used to identify good combinations of engine parameters and in which

parameters are raised to specific powers to avoid straight ratios and also to exhibit

closeness to dimensionless units in gas turbine theory. Combinations of parameters

were raised to the specific powers of [-2.0 –1.5 –1.0 –0.5 +0.5 +1.0 +1.5 +2.0] and

evaluated to identify the most informative parameter relationships to enable detection

of fuel flow faults.

The range of chosen parameter powers and parameter permutations for the 12

different measurement signals resulted in 1056 different combinations. The

combinational explosion that would result from increasing the power range and more

than dual combination of parameters necessitated the restricted range to be examined.

Although all 1056 combinations were examined, the automatic approach adapted to

the analysis only stores the top ranking combinations. The adopted approach is easily

extended to cover a wider range of power values or parameter combinations. Only the

top ranking solutions will be used in the subsequent analysis of the engine behaviour

to diagnosis fault within the Fuel Metering Unit.

Statistical techniques (mean and standard deviation) are used to find the correlation

between samples for the given engine data. The variables are plotted against fuel flow

(WFE) to find the relationship between combined variables and fuel flow. This

information is an important indicator in the analysis of the engine performance with

respect to fuel pump measurement faults and selecting the most appropriate

combinations of variable among the possible 1056 combinations is vital. The criteria

used to assess the appropriateness of parameter combinations are that which has the

most significant effect with changing values of fuel flow.

The method proposed here is for ranking and storing of the most effective parameter

combinations is based on selection of the best ten results obtained from varying the

power variable for each possible engine parameter. The ranking is based on how well

the samples correlated to each other. In so doing for each of the 12 available

parameters: N1, N2, N3, T20, T25, T30, TGT, P20, P25, P30, P50 and P160, a set of 10

optimal combinations is stored. Thus, a total of 120 best combinations are stored. It


57

was chosen to store in this manner so that a wide variety of engine variables are

available for the subsequent fault diagnosis (storing only the global optimal values

results in a dominance of a single engine variable).

With the aid of PLS Toolbox 2.1 functions and some modification in multivariate

PCA and PLS techniques to suit to this fault diagnosis purposes, an ellipse can be

drawn around the data sample to classify outliers (i.e. the samples which are outside a

predefined tolerance). For the four sets of data samples (ideal, 1000 hours, 2000 hours

and 3000 hours of deterioration) these elliptic categories were defined around the data

samples in order to observe the scatter characteristics. In so doing, these elliptic

regions defined the operating deterioration of the engine. Therefore, if these regions

do not intersect, it should be possible to use the data sample to classify the level of

engine deterioration and thus help in the diagnosis of fuel pump failure. The more

highly these elliptic classification can be made and the greater scattered between

elliptic regions of different deterioration, the greater use this calculated data sample is

for diagnostic purposes.

Figure 5.13 illustrates examples of the possible 1056 combinations of engine variables

for the data samples of ideal and 3000 hours deterioration. In particular Figure 5.13

illustrates the combinations of WFE vs. N1.*P30a where a=[-2.0 –1.5 –1.0 –0.5 +0.5 +1.0

+1.5 +2.0]. It is interesting to note that varying the power associated with P30 results in

a different characteristic of the elliptic classifier. Furthermore, the effect of engine

deterioration is to shift the data sample to the right, which can be explained physical

by the fact that to achieve a certain degree of N1.*P30a a greater flow of fuel is

necessary due to the drop in the engine efficiency resulting from the deterioration.

Figure 5.13: The first four combinations of WFE vs N1.*P30

a


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The overlapping area between the ideal and deteriorated data sets is relatively large

while the percentage of the outliers is approximately 35% of total samples. This gives

a good indicator on how this model could separate those different samples into two

categories of outlier and samples within tolerance. It is these attributes that can be

used a performance measure to assess the ranking of how good each particular

variable combination is. A further interesting feature is the direction of the ellipse.

They are tilted such that it has a negative gradient, which also means there is a

negative correlation between the combination and the fuel flow. This is an unusual

correlation since the most majority of parameter combinations have a positive

correlation with fuel flow.

The significant overlap between the elliptic regions of ideal and 3000 hours

deterioration data sets makes robustly classifying unknown data within the overlap

impossible. Clearly, any overlap should be minimal and if possible the distance

between elliptic regions were made as large as possible to aid the robustness of

classification. Figure 5.14 illustrates the response with an alternative variable

combinations of N1.*P30+0.5 ~ N1.*P30

+2.0 and clearly demonstrates an improved

performance at classifying deterioration over that of Figure 5.13.

Figure 5.14: The last four combinations of WFE vs N1.*P30

a

The overlapping samples are minimised to less than 10% of the total samples for

N1.*P30+1.5 and N1.*P30

+2.0, and the ellipses cover nearly 100% of all samples within

the given engine data. Furthermore for each engine data, these parameters

combinations give samples that are well dense to the semi major line across the

ellipse. Figure 5.14 (a) and (b) demonstrates two separate elliptic regions with a


59

degree of separate distance between ideal and deteriorated data sets. This means these

combinations are well suited to diagnose and analyse the performance behaviour of

the different condition of the engine data.

The further distance between two ellipses, the better the performance at robustly

classifying the engine data. The ellipses are tilted upward, which means there is a

positive correlation between the combinations and the fuel flow.

The mean and standard deviation values are used to describe the distance of each

sample point to the line across the ellipse (semi major). It is preferable to minimise

this distance to create density along the line, which represents a high correlation

among the samples (i.e. close to linear equation).

Combinations Mean Stdev

WFE vs N1.*P30^-2.0 0.1308 0.0934

WFE vs N1.*P30^-1.5 0.1395 0.0990

WFE vs N1.*P30^-1.0 0.1583 0.1123

WFE vs N1.*P30^-0.5 0.2172 0.1543

WFE vs N1.*P30^0.5 0.0333 0.0270

WFE vs N1.*P30^1.0 0.0299 0.0220

WFE vs N1.*P30^1.5 0.0510 0.0370

WFE vs N1.*P30^2.0 0.0615 0.0453

Figure 5.15: Mean and Standard Deviation of the Ideal Data

Figure 5.15 illustrates in graphical and tabular form the mean and the standard

deviation of the possible combination of WFE vs. N1.*P30a for the ideal data. The

optimal combination is that of N1.*P30 1.0 which has a mean value of 0.0299 and a

standard deviation of 0.0220. The stem plot made this point obvious by showing the

smallest value on the graph.

By way of a comparison, the mean and standard deviation tables and plots for the

N1.*P30a combinations for the 1000 hours deterioration data set is illustrated in Figure

5.16. The smallest mean and standard deviation is also achieved with a combination


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of N1.*P30 1.0 and has values of 0.0277 and 0.0204 respectively. This combination

gives a good performance in term of correlation between samples.

The normalisation method continues form identifying the most appropriate parameter

combinations by performing the normalisation of the combination absolute value to a

range of 0 to 1. This normalisation minimises the effect of different unit values. The

chosen range is purely to simplify the normalisation method.

Combinations Mean Stdev

WFE vs N1.*P30^-2.0 0.1271 0.0909

WFE vs N1.*P30^-1.5 0.1356 0.0963

WFE vs N1.*P30^-1.0 0.1540 0.1093

WFE vs N1.*P30^-0.5 0.2120 0.1504

WFE vs N1.*P30^0.5 0.0343 0.0272

WFE vs N1.*P30^1.0 0.0277 0.0204

WFE vs N1.*P30^1.5 0.0484 0.0350

WFE vs N1.*P30^2.0 0.0587 0.0431

Figure 5.16: Mean and Standard Deviation of 1000 Hours Deterioration Data

Table 5.2 illustrates a range of optimal parameter combination based on the

normalisation method outlined above and get maintain a wide spectrum of parameter

content. It is noted that most of the 20 variables combination involves P30. It is very

dominant for every variables combination on the table. A physical explanation of this

is that the HP compressor pressure is physically located close to the combustion zone

and is therefore directly affected by changes in the fuel flow.

In a similar vain to P30, the TGT and N1 are also dominant on the Table 5.2. These

three variables have a direct effect to the fuel flow measurement and proved

importantly the engine power that provides a sound basis for their use to classify fuel

system faults. The mean and standard deviation are the indicators of how well the

samples data are correlated to each other for particular combinations. The

combination of WFE vs. P30.*N1 1.5 and WFE vs. P30.*TGT 1.5 represent the best

combination to diagnose fault within the gas turbine engine. The samples data are


61

well correlated and separated by a distinct distance, which enables a robust

classification of engine system faults from new data. The next section will discuss in

detail the two best variables combinations obtained by the normalisation method.

Combination Mean Standard Deviation ’WFE vs P30.*N1 ^1.5’ 0.0046 0.0035 ’WFE vs P30.*TGT ^1.5’ 0.0147 0.0089 ’WFE vs N2.*P30 ^0.5’ 0.0187 0.0138 ’WFE vs P30.*N2 ^2.0’ 0.0193 0.0143 ’WFE vs TGT.*P50 ^0.5’ 0.0198 0.0148 ’WFE vs P50.*TGT ^2.0’ 0.0209 0.0164 ’WFE vs N1.*P50 ^0.5’ 0.0211 0.0176 ’WFE vs N3.*P30 ^0.5’ 0.0223 0.0170 ’WFE vs P50.*N1 ^2.0’ 0.0226 0.0156 ’WFE vs P30.*N3 ^2.0’ 0.0228 0.0180 ’WFE vs P30.*TGT ^1.0’ 0.0249 0.0181 ’WFE vs TGT.*P30 ^1.0’ 0.0249 0.0181 ’WFE vs N1.*P30 ^1.0’ 0.0299 0.0220 ’WFE vs P30.*N1 ^1.0’ 0.0299 0.0220 ’WFE vs TGT.*P25 ^0.5’ 0.0317 0.0230 ’WFE vs P30.*N1 ^2.0’ 0.0328 0.0249 ’WFE vs T30.*P30 ^0.5’ 0.0330 0.0240 ’WFE vs P30.*P25 ^-0.5’ 0.0331 0.0187 ’WFE vs N1.*P30 ^0.5’ 0.0333 0.0270 ’WFE vs P25.*TGT ^2.0’ 0.0337 0.0247

Table 5.1: Normalisation Method Variables Combinations

5.2.3.1. P30.*N1 1.5 Combination

This combination resulted in the highest samples correlation and corresponding the

lowest the mean values and the standard deviation values of all examined

combinations. The performance of P30.*N1 a combination is illustrated in Figure 5.17

and clearly this optimal P30.*N1 1.5 combination is sited at position seven and shows

significant performance improvements over its closest neighbours.

The fact that the correlation between WFE and P30.*N1 1.5 is so tight, it would be

expected that the elliptic boundary around the data set would be very thin (and close

to a straight line). Thus, creating an almost one to one correlation between each axis.

Normalisation method needs the best-selected variables combinations to create a

reliable model as a benchmark for diagnosing new engine data. In order to


62

demonstrate that the combination has a good performance, a plot of the combination

is illustrated in Figure 5.18.

Figure 5.17: Characteristic and Combination Table of WFE vs P30.*N1 1.5

Figure 5.18 shows the plots of P30.*N1 1.0 and P30.*N1

1.5 and an interesting feature of

which is the significant correlation performance decline observed by the small change

in parameter combination. The samples become closer to each other across the

diagonal line. This highly correlated samples shape an ellipse to cover the whole

samples within the given engine data.

Figure 5.18: Combination Plot of WFE vs P30.*N1 1.5

Combination Mean Std dev

’WFE vs P30.*N1 ^-2’ 0.2147 0.1529

’WFE vs P30.*N1 ^-1.5’ 0.1848 0.1318

’WFE vs P30.*N1 ^-1’ 0.1545 0.1104

’WFE vs P30.*N1 ^-0.5’ 0.1239 0.0888

’WFE vs P30.*N1 ^0.5’ 0.0616 0.0445

’WFE vs P30.*N1 ^1’ 0.0299 0.0220

’WFE vs P30.*N1 ^1.5’ 0.0046 0.0035

’WFE vs P30.*N1 ^2’ 0.0328 0.0249


63

Figure 5.18(b) demonstrates a clear separation of different deterioration conditions of

the engine data. Indeed, all data (ideal and three deterioration conditions) are distinct

with the P30.*N1 1.5 combination. It clearly visible that the data set with 3000 hours

deterioration has dragged the ideal data along x-axis (WFE) and caused an effective

bias in the measurement of the fuel flow for every point of the sample for particular

distance and which is a result of the decreased efficiency of the engine due to the

deterioration.

The clear separation of ellipses represents the ability of this model to identify levels of

deterioration for a given set of engine data. Each data set is well covered by the

elliptic boundary, which means it is well correlated.

5.2.3.2. P30.*TGT 1.5 Combination

This combination gives the second best level of performance based on the

normalisation method, only second to P30.*N1 1.5. In a similar vain to the previous

combination, Figure 5.19 illustrates in graphical and tabular form the correlation

performance of the combination of P30.*TGT a. The seventh combination appears to

have the smallest value as shown on the stem plot.

Figure 5.19: Characteristic and Combination Table of WFE vs P30.*TGT 1.5

Figure 5.20 illustrates the scatter plots of the ideal and deterioration engine data sets

for P30.*TGT a combination. The plots illustrate the relative performance of the

combination at categorising the engine data. Each data set is well covered by the

elliptic boundary and line is a small distance between boundaries of different

Combination Mean Std dev

’WFE vs P30.*TGT ^-2’ 0.2530 0.1779

’WFE vs P30.*TGT ^-1.5’ 0.2103 0.1483

’WFE vs P30.*TGT ^-1’ 0.1695 0.1199

’WFE vs P30.*TGT ^-0.5’ 0.1304 0.0928

’WFE vs P30.*TGT ^0.5’ 0.0575 0.0416

’WFE vs P30.*TGT ^1’ 0.0249 0.0181

’WFE vs P30.*TGT ^1.5’ 0.0147 0.0089

’WFE vs P30.*TGT ^2’ 0.0364 0.0315


64

deteriorations. The correlation of data sets to fuel flow is good and is observed from

the width of the elliptic boundaries.

From the x-axis consideration, there is a shifting along x-axis (WFE) for the chosen y-

axis value. If the new engine data is exactly on the top of the ideal data, then there is

clear indication that no fault is diagnosed. But if the data is well separated as shown

on the Figure 5.20(a) then it is obvious that the fault existed. The small intersection

outliers normally show the insipient fault (i.e. the fault in the earliest stage). This fault

has to be diagnosed soon to avoid further fault that may lead to failure within the

systems. This fuel pump measurement bias is drifting bigger as the deterioration is

getting longer.

Figure 5.20: Combination Plot of WFE vs P30.*TGT 1.5

Comparison of the performance of P30.*TGT 1.5 with that of P30.*N1

1.5 is possible due

to the normalisation of the combination sample to the range 0 to 1. It is observed that

P30.*N1 1.5 gives a better performance than that of P30.*TGT 1.5 which is exemplified

by the fact that the data set is highly correlated across thin ellipse, closely shape a

straight line for the given different conditions of engine data (1000 hours, 2000 hours,

and 3000 hours deterioration). While from Figure 5.20, the scatter data although well

dense along semi major ellipse but the semi minor is slightly bigger. This means the

correlation between data is not as good as the combination of P30.*N1 1.5.


65

Both combination of P30.*TGT 1.5 and combination of P30.*N1 1.5 have shown good

performance and are highly prevalent in Table 5.1 of top ranking solution. This good

performance was achieved with the normalisation and will also be shown to have

good performance characteristic with the Quasidimensionless method outline in the

subsequent sections. The physical explanation of this lies in the physical proximity of

these measurements to the combustion chamber where fuel flow consumption/burning

is undertaken.

The fact that the normalisation approach required the normalising of the engine data

calculated from the various combinations to a fixed range of 0 to 1 means that a

comparison of a the relative abilities of the different combinations was straight

forward. Direct comparison of mean and standard deviation of data sets was possible.

However, normalisation of few samples (with unknown fault/deteriorations) will

result in their parameter variation being normalised to a new relative range, which

would make vertical shifts in the parameter values not possible. Vertical shifts in the

parameter magnitudes would provide a useful mechanism to detect fault conditions

and thus, an absolute rather than relative parameter range is necessary. It is this

concept that drove the implementation of a Quasidimensionless method outlined in

the follow section.

The advantage of this approach is the calculation has simplified the complexity or the

random data and with simple statistical approach to find mean value and standard

deviation, we can differ the sample, whether it is within or outside tolerance with the

aid of plotting ellipse. There is no need to create a physical model to diagnose fault

and training the model. Mathematically, the calculation concept is not too complex, in

fact is quite simple and it is commonly used in many multivariate techniques (i.e.

scaling, linear equations, ellipse, linear algebra, etc).

Traditional Quasidimensionless

66

5.2.4. Traditional Quasidimensionless

The following method examines the optimisation of Quasidimensionless unit groups

for faults diagnosis. Quasidimensionless minimises the effect of different unit scales

by combining parameters, which have different unit scales of measurement. It is often

termed semi-dimensional because the unit still has a dimension. The equations are

chosen for reasons of minimising the existence of likely error (i.e. second order

effects upon engine matching). This method will correct all sets of data to the standard

condition so the comparison between values is more absolute. For further information

about the Quasidimensionless, it is advised to refer to Walsh, P.P. and Fletcher, P.

(1998).

Essentially the effect of using Quasidimensionless unit is to equate the behaviour of

the engine at any flight condition to that of a standard condition (typically SLS). This

makes comparison of engine data over the flight envelope simpler. The way in which

engine measurements are redefined in Quasidimensionless units depends on the type

of engine signal. The approach adopted in this implementation is based upon:

��Fuel flow (WFE) is divided by the multiplication of P20 and 20T .

2020 *. TP

WFE (5.2.4.1)

��Temperatures (T25, T30 and TGT) are defined by dividing by the intake

temperature, thus:

20T

Tall (5.2.4.2)

��Shaft speeds (N1, N2 and N3) are redefining by dividing by square root of

intake temperature, thus:

20T

Nall ; (5.2.4.3)

��Pressures (P25, P30, P50 and P160) are redefined by divided by the engine intake

pressure P20, thus:

20P

Pall (5.2.4.4)


67

The first implementation of Quasidimensionless method is to the temperature

variables of engine data. Elimination of the dimension for both axes results in a more

absolute value plot.

TGT demonstrates significant improvement and better performance compare to T25

and T30 against the chosen criteria to assess its fault diagnosis capability. The ideal

data was more clearly differentiable from that of an engine suffering 3000 hours of

deterioration with TGT parameter and is illustrated in Figure 5.21. The performance

of the Traditional Quasidimensionless TGT parameter (TGT.*T20 -1.0) clearly

outperforms that of its closeness neighbour TGT.*T20 –0.5 combination.

Figure 5.21: Quasidimensionless of (TGT.*T20 a) Combination

Furthermore, Figure 5.21(a) shows better performance compare to Figure 5.21(b) in

terms of correlation between samples, which closely shape a straight line and density

of samples along semi major of the ellipse. The smaller the semi minor of the ellipse

represents the higher correlation between samples of the engine data. The ellipse

represents the grouping factor of the correlated data.


68

This performance of the TGT parameter can be explained physically because TGT is

the first measurement point of temperature after the combustion stage and hence bears

the closest relationship to the fuel flow.

Non-dimensional shaft speed is undertaken by dividing the shaft speed by the square

root of the temperature intake. These elimination of dimension results in N1.*T20 –0.5

combination shows in Figure 5.22.

Figure 5.22: Quasidimensionless of (N1.*T20 a)Combination

The scatter plots of ideal and deteriorated data results in an intersection more than

70% of data in Figure 5.22(a). While in Figure 5.22(b), which displays the Traditional

Quasidimensionless data is clearly capable of distinguising ideal and deteriorated

data. The density along semi major axis demonstrates a straight line relationship and

is a good indicator to explain how well the data correlated. The elliptic boundary

clearly cover the data samples into two distincts.


69

Quasidimensionless pressures consist of four different variables: P25, P30, P50 and P160.

Each Quasidimensionless pressure variable is achieved by dividing by inlet pressure

(P20).

Figure 5.23 shows P30.*P20 -1.0 combination with 4 different conditions of engine data

(ideal, 1000 hours, 2000 hours and 3000 hours deterioration). The straight-line

relationship between Quasidimensionless P30 and fuel flow highlights the fact that the

data is well correlated. Furthermore, the semi major axis of each ellipse is small,

which enables each sample group to be clearly distinct and this enables categorising

of engine deterioration.

Figure 5.23: Quasidimensionless of (P30.*P20^ a) Combination

The greater the deterioration exists, the greater the shifting of the data along x-axis

(the fuel flow, WFE).

All the Traditional Quasidimensionless combinations are summarised in Table 5.2,

where the mean values and the standard deviation values show the correlation

between samples (i.e. how well sample correlated to each other). If the values of mean


70

and standard deviation are smaller, it means the correlation of that combination is

higher.

Combination Name Mean Value Std Value

WFE/(P20.*sqrt T20) vs TGT.*T20^-1.0 0.0168 0.0111

WFE/(P20.*sqrt T20) vs T25.*T20^-1.0 0.0171 0.0098

WFE/(P20.*sqrt T20) vs T30.*T20^-1.0 0.0198 0.0131

WFE/(P20.*sqrt T20) vs N1.*T20^-0.5 0.0091 0.0052

WFE/(P20.*sqrt T20) vs N2.*T20^-0.5 0.0188 0.0090

WFE/(P20.*sqrt T20) vs N3.*T20^-0.5 0.0289 0.0151

WFE/(P20.*sqrt T20) vs P30.*P20^-1.0 0.0095 0.0058

WFE/(P20.*sqrt T20) vs P25.*P20^-1.0 0.0126 0.0065

WFE/(P20.*sqrt T20) vs P160.*P20^-1.0 0.0131 0.0064

WFE/(P20.*sqrt T20) vs P50.*P20^-1.0 0.0145 0.0106

Table 5.2: Traditional Quasidimensionless Combinations

Quasidimensionless Adaptation

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5.2.5. Quasidimensionless Adaptation

Quasidimensionless Adaptation approach is a continuation of the Normalisation

approach but with the application of Quasidimensionless performance parameters.

This approach combines the Normalisation Method and Quasidimensionless theory

(Traditional Quasidimensionless) to explore a variety of combinations in non-

dimensional term for fault diagnosis purposes.

The ten non-dimensional performance variables are multiplied by another non-

dimensional variables, which is in turn raised to power range of [-2.0 –1.5 –1.0 –0.5

+0.5 +1.0 +1.5 +2.0] that create a set of 800 possible dimensionless combinations. The

dimensionless performance combinations are correlated against non-dimensional fuel

flow. A simple statistical analysis of this correlation, which was outlined in the

normalisation approach was undertaken to identify the set of 20 best combinations

and illustrated in Table 5.3.

Combination Mean Distance S-minor Ratio

’WFE/(P20.*sqrt T20) vs (P160./P20).*(P160./P20 )^-1’ 0 -5.63E-18 0 0

’WFE/(P20.*sqrt T20) vs (TGT./T20).*(T25/T20 )^-2’ 3.40E-04 -3.18E-04 7.97E-04 1.83E+01

’WFE/(P20.*sqrt T20) vs (T25./T20).*(TGT./T20 )^-0.5’ 9.36E-05 8.78E-05 2.19E-04 1.82E+01

’WFE/(P20.*sqrt T20) vs (TGT./T20).*(N2./sqrt(T20))^-2’ 1.30E-08 -8.29E-09 4.11E-08 1.77E+01

’WFE/(P20.*sqrt T20) vs (N2./(sqrt(T20))).*(TGT./T20 )^-0.5’ 3.00E-02 1.92E-02 9.47E-02 1.72E+01

’WFE/(P20.*sqrt T20) vs (N3./(sqrt(T20))).*(N2./sqrt(T20))^-1.5’ 1.01E-05 -7.63E-05 2.08E-05 1.61E+01

’WFE/(P20.*sqrt T20) vs (TGT./T20).*(T30./T20 )^-1.5’ 2.04E-04 -1.46E-04 6.04E-04 1.61E+01

’WFE/(P20.*sqrt T20) vs (TGT./T20).*(N3./sqrt(T20))^-1.5’ 1.48E-07 3.42E-07 3.86E-07 1.49E+01

’WFE/(P20.*sqrt T20) vs (T25./T20).*(N2./sqrt(T20))^-1’ 5.59E-07 1.07E-07 1.67E-06 1.49E+01

’WFE/(P20.*sqrt T20) vs (N2./(sqrt(T20))).*(T25/T20 )^-1’ 2.31E-02 -4.42E-03 6.88E-02 1.48E+01

’WFE/(P20.*sqrt T20) vs (TGT./T20).*(P30./P20 )^-0.5’ 3.95E-04 -4.59E-04 1.11E-03 1.39E+01

’WFE/(P20.*sqrt T20) vs (P30./P20).*(TGT./T20 )^-2’ 8.36E-04 1.01E-03 2.36E-03 1.37E+01

’WFE/(P20.*sqrt T20) vs (P30./P20).*(TGT./T20 )^-1.5’ 3.04E-03 6.73E-03 5.67E-03 1.18E+01

’WFE/(P20.*sqrt T20) vs (TGT./T20).*(N3./sqrt(T20))^-2’ 7.13E-09 -1.71E-09 2.44E-08 1.18E+01

’WFE/(P20.*sqrt T20) vs (N3./(sqrt(T20))).*(TGT./T20 )^-0.5’ 5.52E-02 1.33E-02 1.89E-01 1.15E+01

’WFE/(P20.*sqrt T20) vs (TGT./T20).*(N2./sqrt(T20))^-1.5’ 3.46E-07 5.35E-07 1.07E-06 1.13E+01

’WFE/(P20.*sqrt T20) vs (P160./P20).*(P30./P20 )^1’ 3.06E-02 6.13E-01 8.63E-02 1.12E+01

’WFE/(P20.*sqrt T20) vs (P30./P20).*(P160./P20 )^1’ 3.06E-02 6.13E-01 8.63E-02 1.12E+01

’WFE/(P20.*sqrt T20) vs (P30./P20).*(TGT./T20 )^-1’ 8.87E-03 2.61E-02 1.25E-02 1.09E+01

’WFE/(P20.*sqrt T20) vs (T25./T20).*(N2./sqrt(T20))^-1.5’ 1.26E-07 -2.29E-07 1.87E-07 1.07E+01

Table 5.3: Combinations of Quasidimensionless Adaptation


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It is interesting to note that the first combination on Table 5.3 represents elimination

of non-dimensional variables where non-dimensional fuel flow is plotted against a

constant value of 1. It is desired that the variation of variables should create a rich set

of non-dimensional combinations of engine data. However the first combination in

Table 5.3 does not provide the unique correlation between performance combination

and fuel flow and thus can be ignored.

Variations within dimensionless engine variable combinations illustrate the

characteristic of non-linear engine data and how they are related to changes in fuel

flow. A significant distance between different conditions of engine data (ideal, 1000

hours, 2000 hours and 3000 hours deterioration) is a desirable feature to aid

classification of how well the combinations identify faults. Assessment of the

distinction of engine deterioration was achieved by the ratio between the distance of

engine conditions and the semi minor ellipse of ideal data.

The second combination of TGTn.d..*(T25n.d.) -2.0 in Table 5.3 has the mean value of

dispersion of 3.40E-04 and ratio of 18.3 ellipse widths between deterioration

categories. This compare to the fourth combination of TGTn.d..*(N2n.d.) -2.0, which has

a mean value of dispersion of 1.30E-08 and ratio of 17.7. In terms of samples density

(i.e. how well engine data samples correlate to each other), the fourth combination has

a better correlation than that of the second. However, in considering the separation

distance, which classifies the different engine conditions as a ratio of the ellipse

boundary width and as an absolute distance, the second combination outperforms that

of the fourth.

The distance between two different conditions of engine data is calculated using

simple linear algebra by measuring the distance of an orthogonal line between ellipse

centres of deterioration (ideal and 3000 hours) and the semi major ellipse of ideal

data. By dividing the separation distance by that of the semi minor ellipse for ideal

data gives a ratio of separation for all possible performance combinations. This

classification filter selects the best dimensionless combinations, and improves the

ability to diagnose faults in more precise manner.


73

Figure 5.24 illustrates the best combination of Quasidimensionless Adaptation where

(TGT./T20)..*(T25./T20)-2.0 combination is plotted in four different engine data

conditions along with (TGT./T20)..*(T25./T20)-1.5 combination.

Figure 5.24: Combination of (TGT./T20)..*(T25./T20)â

Clearly, it appears that each engine data condition is well covered by the ellipse

boundaries. Figure 5.24(a) demonstrates thinner ellipses and greater distances

between ellipses, which makes for a bigger than that of Figure 5.24(b). An interesting

point is the direction of ellipses, which demonstrates both a positive and negative

correlation with fuel flow. These directions explain the relationship between

dimensionless fuel flow consumption and dimensionless combinations.

A further important criteria is that greater deterioration results in increased distance of

this data category away from that of ideal engine. This represents the operational

condition on a gas turbine engine.

Figure 5.25 illustrates TGTn.d..*(N2n.d.)-2.0 and TGTn.d..*(N2n.d.)

-1.5 combinations. The

relationship between TGTn.d. and N2n.d. performs the fourth best overall (Table 5.3).

The different dimensionless combinations demonstrate similar level of performance.


74

Figure 5.25: Combination of TGTn.d..*(N2n.d.)a

A key feature of the results of optimal non-dimensional parameter combinations in

Table 5.3 is the rich variation in performance parameters, which achieved good

performance. This looks well for development of a truth table analysis for diagnosis

technique (consistency checking) of sensor signals.

The results of the previous approaches (Normalisation Method and Traditional

Quasidimensionless), demonstrates that the TGT, P30 and N1 have shown good

performance. The optimal Quasidimensionless Adaptation combinations have tended

to employ these variables as part of the best 30. This demonstrates that

Quasidimensionless Adaptation holds these combinations and makes potential

improvements of fault diagnosis performance.

Figure 5.26 shows (P30./P20).*(TGT./T20)â combinations. The fact that ellipse in

Figure 5.26(b) tends to shape a straight line and is well-correlated data along its semi

major demonstrates the improved performance of this combination. The different

levels of deterioration are clearly visible and classification of data within the elliptic

boundaries is achieved with no overlapping.


75

Figure 5.26: Combination of (P30./P20).*(TGT./T20) a

The tightness of the elliptic boundaries enables accurate correlation of fuel flow to

performance parameter combination and vice versa. This would make estimation of

fuel flow to a reasonable degree of accuracy achievable.

Performance Comparison

76

5.2.6. Performance Comparison

Diagnosis of fuel pump measurement bias in gas turbine engine utilised statistical

mathematics, an adaptation of MSPC technique (PCA and PLS analysis) resulted in

development of three methods: Hotelling’s, Normalisation and Quasidimensionless

(Traditional and Adaptation). The latter accommodated factors required to robustly

estimate fuel flow given an estimate of engine deterioration, which had been

problematic for the previous methods.

Hotelling’s method has a limitation as a ‘filter model’ to diagnose faults when it could

not clearly highlight the given different conditions of engine data. The sample data are

on top of each other and scatter in wide area makes the analysis is difficult to carry

out. This method utilised dimensional variables, where the engine variables were

plotted as functions of fuel flow. The scatter data was an accumulation result of

different unit scale, where there was a big gap between the smallest and the highest

sample to plot together.

Figure 5.27: First Performance Comparison Plot

The Normalisation method was devised to overcome the limitation of the Hotelling’s

method to accommodate different engine deterioration conditions. Selection of

optimal parameter combinations through mean and standard deviation of the scatter

data resulted in 20 best combinations. The deterioration of engine data captured as

shifting patterns of the elliptic boundary along the x-axis (WFE). This method clearly

Performance Comparison

77

highlighted degradation of engine data along x-axis (WFE) but could not manage to

diagnose the movement along y-axis (i.e. to find exact value of the performance

parameter) since the Normalisation approach converted y-axis to a predefined spread.

Figure 5.27 shows the comparison of Hotelling’s, Normalisation and Traditional

Quasidimensionless. It is obvious the stage changes from each method to reach a

better performance.

The Quasidimensionless method built on the techniques devised in the Normalisation

method but made use of dimensionless engine performance variables. Elliptic

boundaries were drawn around the engine data of different deterioration to enable

classification and estimation of fuel flow. Optimal combinations were identified.

Figure 5.28: Performance Comparison Plot of Development

Figure 5.28 illustrates a comparison of the optimal combinations of the Normalisation

and Quasidimensionless Methods. It is clearly appeared that Figure 5.28(b) could

accommodate four different conditions of engine data with an exact separation

distance while Figure 5.28(a) only able to accommodate three different conditions of

engine data. Both figures showed exact separations to diagnose fault within the engine

data, the main differences is the axes values.

Deterioration Estimation Analysis

78

5.3. Deterioration Estimation Analysis

Real engine sensor data provides information about the engine behaviour through

engine measurements in a gas turbine engine system. Although all measurements

provide information, each has it own level of reliability. WFE is the least reliable

variable due to inaccuracies of the sensor and difficulty of measurement. An accurate

sensor is very expensive; consequently an estimate of the fuel flow is desirable to

identify inaccuracies in the current flow sensing devise.

Pressures and temperatures measurements are less reliable than that of shaft speed.

The measurement of pressures varies according to the sensor position, and the

dynamic of the sensor has an influence on the measurement. Temperatures

measurements are influenced by a thermal inertia caused by a thermocouple being

embedded within the engine casings. Shaft speed measurement is most reliable

variables. It is measured using 3 encoders and the majority-voting scheme is used to

identify any faults.

The ability to accurately estimate the flow of fuel into the engine from the previous

analysis requires knowledge of the level of deterioration inherent within the engine.

Therefore a method to estimate this deterioration is outlined. The technique is based

on the same statistical approach used estimate fuel flow. However, in this instance the

two performance parameter are plotted against each other. One of these performance

parameters was chosen to be HP shaft speed, since this is well correlated with fuel

flow, which was shown to be able to categorise deterioration levels.

An example of a near optimal combination of performance parameters to classify

engine deterioration is illustrated in Figure 5.29. In this example 4 data points are

plotted within a level of deterioration between that of 1000 and 2000 hours. These real

engine data are represented by ‘+’ sign. A simple linear interpolation is undertaken to

estimate the exact level of deterioration experienced by the engine from where these

measurements were taken. The interpolation is based on simple mathematic formula:

Deterioration={(x1/x2)(2000-1000)}+1000 (5.3.1)

Deterioration Estimation Analysis

79

Figure 5.29: Linear Interpolation Plot of Deterioration Estimation Analysis

Where x1 represents the distance of calculation point to 1000 hours deterioration level

and x2 represents the distance between 1000 hours and 2000 hours deterioration along

the calculation point.

This procedure produces a list of four deterioration levels for each real engine sensor

data, which are summarised in Table 5.5. Taking mean value to find the average value

of real data, results in 1535.63 while the exact deterioration level is 1500 or there is an

estimated error of 2.3% due to manual calculation. This estimation error represents

reliable approach to the automatic calculation in term of numerical precision.

Finding this information is crucial to identify the exact deterioration level. By

analysing this value inline with the previous methods, the calculation of fuel pump

measurement bias between the WFEgiven and the WFEestimation can be founded. This

time, the analysis is carried on for the combination of performance parameters and

fuel flow, which will be discuss in the next section.

Test Cases

80

5.4. Test Cases

The follow section presents a numerical example of the diagnosis procedure to

estimate the fuel flow bias within the fuel metering system. Four sets of engine data

were provided from an engine, which had suffered an unknown level of deterioration.

The fuel flow measurement from this engine data had been seeded with flow biases

between 0 and 10%. The test data are illustrated in Table 5.4, to accommodate the

selected combination as an example.

Case WFE N2 N3 P20 T20 TGT 1 7.1075e+03 6.0620e+03 9.0928e+03 5.5821 2.4981e+02 1.3613e+03

2 7.8583e+03 6.1581e+03 9.2281e+03 5.4327 2.4900e+02 1.4006e+03

3 7.9847e+03 6.1153e+03 9.1683e+03 5.8148 2.5274e+02 1.3833e+03

4 8.1958e+03 6.1893e+03 9.2734e+03 5.0388 2.4801e+02 1.4135e+03

Table 5.4: Test Data for Unknown Level of Deterioration

The initial step in the diagnosis procedure is to estimate the level of deterioration

experienced by the engine. This is illustrated graphically in Figure 5.30.

Figure 5.30: Level of Deterioration Estimation

Test Cases

81

Simple linear interpolation of the data from each test case was undertaken and the

estimate of the deterioration is given in Table 5.5.

Case Estimated Deterioration

1 1642.86

2 1500

3 1466.67

4 1533

Table 5.5: Calculation of Linear Interpolation

Thus to approximate level of deterioration experienced by the engine was 1535.63

hours. This ties is very well with the engine data, which had had a deterioration of

1500 hours implanted. The detail procedure of this stage is covered in the section 5.3:

Deterioration Estimation Analysis.

The second stage of the diagnosis is to utilise this known level of deterioration to

estimate the fuel flow. This is illustrated in Figure 5.31, where the fuel flow

estimations are gained in such a way as directions shown by red arrows.

Figure 5.31: Calculation of Fuel Flow Estimation

Test Cases

82

These non-dimensional fuel flow estimates are converted manually to dimensional

units to enable comparison to measured flows and are depicted in Table 5.6.

Case WFExn WFEx WFEgiven Estimated Bias True Bias

1 81.818 7.2186e+003 7.1075e+003 1.5631% 0%

2 89.881 7.7052e+003 7.8583e+003 1.9483% 2%

3 81.964 7.5770e+003 7.9847e+003 5.1060% 5%

4 93.296 7.4032e+003 8.1958e+003 9.6708% 10%

Table 5.6: Fuel Flow Comparison

WFExn represents the normalisation fuel flow (fuel flow estimation from Figure 5.30

in dimensionless), while WFEx represents the non-normalise fuel flow from WFExn.

This is undertaken by multiplying WFExn to P20.*sqrt(T20).

By non-normalised the normalised WFE, it is expected to find the estimation value of

the original WFE to compare with the given WFE in order to clarify the existence of

bias within fuel pump measurement.

The true biases added to the flow were 0%, 2%, 5% and 10% respectively. Table 5.6

shows a set of comparison values where the amount of bias between estimated and

true bias is very well similar, apart from the first case. This can be understood since

the approach was undertaken manually. Overall, this approach proves an accurate

estimation of fuel flow measurement to diagnose the biases within a fuel pump gas

turbine engine.

Diagnosis Technique/Consistency-Checking

83

5.5. Diagnosis Technique/Consistency-Checking

The previous analysis of the performance data to identify deterioration level and

estimate fuel flow bias had assumed that the measurements of the engine variables

were correct. Any inaccuracies in the measurement of engine signals will result in an

inaccurate estimate of the fuel flow and consequently also its bias. However, the

statistical approaches identified a number of optimal parameter combinations with

which the fuel flow or deterioration level could be estimated. Therefore, excluding

these signals from the estimation procedures can accommodate for inaccuracies in

known signals.

The approach proposed to diagnoses sensor faults (and hence exclude from fuel flow

estimation) is based on a consistency-checking approach. It can be assumed that

intake pressure and temperature measurements are correct since measurement can be

taken from the aircraft and engines and with which to compare and isolate faults.

Furthermore, shaft speed measurements are highly reliable and the simple majority-

voting scheme is capable if violating faults robustly. Thus the consistency-checking

approach is aimed at diagnosing faults within the remaining pressure and temperatures

signals along the length of the engine.

This approach is based on the assumption that deterioration levels within the engine

remain essentially static over short period of time. This estimation of the deterioration

level can be undertaken with a wide range of engine sensor combinations. It is

reasonable to assume that the estimate of deterioration with the diverse range of

sensors signals should be approximately consistent. Any significant inconsistency

between deterioration level estimation could be due to measurement accuracies of the

signals used for that particular estimation. Thus a truth table construction can be used

to identify inconsistent measurement signals from the rich variety of performance

combination available.

This technique is easily understood through the following illustration on Table 5.7.

The sample is selected randomly from the best combinations in previous methods

(Hotelling’s and Normalisation) but also perform well on Quasidimensionless


84

(Traditional and Adaptation) Method. Analysing could be performed with different

method but some adjustment is needed to suit the method.

Combination P30 nd N1 nd TGT nd P30 nd.*N1 nd 1 1 0

P30 nd.*TGT nd 1 0 1

N1 nd.*TGT nd 0 1 1

Table 5.7: Diagnosis Technique of Selected Combinations

These combinations on Table 5.7 are plotted for ideal engine data (as perfect model)

and for different conditions of deterioration engine data (1000 hours, 2000 hours and

3000 hours deterioration) to collect degradation information or any discrepancies that

might happen on engine performance. Detection and isolation fault is undertaken

through performance comparison of new engine data plot to deterioration engine data

plot. Any mismatch pattern will be highlighted for identification, to specify particular

component, the diagnosis technique is carried out.

There are two significant patterns of new plot: scatter condition when sample data are

spread out in wide area and shifting condition when each ellipse moving in group to

different conditions from the perfect model (ideal engine data plot). The first case is

relatively easy to detect in order to diagnose faults condition. Normalisation and

Quasidimensionless (Traditional and Adaptation) Methods manage to diagnose this

phenomenon very well. The second case could not diagnose in Normalisation since it

normalises y-axis, which makes shifting to up and down direction undetectable. In

order to overcome this, it needs new approach, which utilises an absolute value

without any normalisation procedure for the axes. Both Quasidimensionless methods

fulfil this requirement and beneficial to analyse the correlation in dimensionless

variables and absolute values.

Hotelling’s, Normalisation and Quasidimensionless Method are capable to diagnose

faults fuel pump measurement bias in a reasonably good performance, but

Quasidimensionless Adaptation has proven the best to overcome this task in a set

contains a wide range dimensionless variables combinations. For this particular type

of steady state engine data, Quasidimensionless Adaptation is a well-suited method.


85

Normalisation capable to handle most of the tasks to diagnose faults although not as

precise as Quasidimensionless, but since characteristic of engine data is non-linear, it

is recommended to utilise Quasidimensionless. Hotelling’s method is less likely

recommended for this particular objective (fault detection of fuel pump measurement

bias), for the reason of limitation on visual analysis (plot characteristic), which makes

it difficult to perform. It is typically applicable for chemical processes data when the

variability makes flexible analysis for better performance.

Knowing advantages of each method, diagnosis technique adapts these advantages to

trace the location of faults by comparing combinations performance. Zero (0)

represents no error condition while one (1) represents a variable error condition. In

this example three dimensionless variable: P30nd, N1nd and TGTnd are used. It is

expected that diagnosis technique could highlight the variable error.

Figure 5.32: Diagnosis of P30nd.*N1nd Combination

The second column in Table 5.4 illustrates an error within P30nd.*N1nd combination

and N1nd.*TGTnd combination, but no bias within P30nd.*TGTnd combination. It is

noticed that each variable is involved within the combination pairs. Knowing that

P30nd and TGTnd contain no variable error, investigation lead to N1nd. To prove this


86

analysis, plotting engine data for different conditions is required. Comparing

performance combinations will highlight an error variable. The engine data for

deterioration 2000 hours is added 5% to highlight particular variable to represent error

condition. The investigation is carried out to prove that this Diagnosis technique is

able to find this variable error.

Figure 5.31 illustrates P30nd.*N1nd combination with two conditions: plotting all

engine data conditions as shown by Figure 5.31(a) and plotting all engine data

condition but for deterioration 2000 hours data utilise engine data which is added 5%

value (to simulate variable error).

Normally, the combination should behave as shown by figure (a) on left side where

greater degradation drags ellipse plot along WFEnd, by shifting the ellipse to the right

In fact degradation 2000 hours data shifts the ellipse further away than degradation

3000 hours data, and tends to move it down. This unusual behaviour highlights that

this combination contains error in one of its variable combinations. To find whether

an error is occurred in variable P30 or N1, further investigation is required.

Figure 5.33: Diagnosis of P30nd.*TGTnd Combination


87

Plotting by the same procedure for P30nd.*TGTnd combination results in Figure 5.32.

Both figures are identical. Although deterioration 2000 hours is added 5% value for

particular variable, it has no effect on plot analysis. The normal degradation is clearly

seen without any radical behaviour. Ellipse shifting is as predicted and follows the

pattern as deterioration becomes greater.

This combination is success of diagnosing faults within aero gas turbine by certain

distance of different conditions of engine data and thin ellipse shape, closely shapes a

straight line to represent high correlation between data samples. This means that both

variable P30 and TGT clearly contain no error condition.

The third combination on Table 5.6 is N1nd.*TGTnd, where one of the variable

contains error condition. The facts that negative correlation exist for this combination

does not affect the ability of this combination to diagnose faults within aero gas

turbine. A different condition of engine data is well covered by ellipse boundary in

certain distance. Although semi minor ellipse is not closely shape a straight line but

the correlation between sample data is reasonably good. It means this combination

could easily recognise any degradation of engine performances.

Figure 5.34: Diagnosis of N1nd.*TGTnd Combination


88

Contrast to the figure on its left side; Figure 5.33(b) illustrates a radical shifting when

deterioration 2000 hours moves up away from its original place. Normally it should

lie between ideal engine data and deterioration 3000 hours engine data as shown on

Figure 5.33(a).

Conclusion can be withdrawn from this behaviour that one of the combination

variable must contain error condition, which cause the combination misbehaved.

Considering three combinations made up from three variables in two pair

combinations, it is clearly highlight N1 as an error variable which cause all this

misbehave condition on P30nd.*N1nd and N1nd.*TGTnd combination.

Thus, the diagnosis process would work by looking for inconsistencies relating to

combinations of any single measurement signal compared to all the other engine

signals. If all the estimates gave a consistent deterioration level, whether that the

ideal, 1000 hours, 2000 hours or 3000 hours, then it would be assumed that the

measurement signal were correct.

It is feasible to extent this consistency-based approach to include a temporal element,

where it would be assumed that the deterioration level estimated would not change

significantly in a short period of time. Any sudden deterioration change would most

likely be caused by a measurement error.


89

5.6. Summary

The PCA technique required adaptation to overcome diagnosing problem in gas

turbine engine. This adaptation raised three different methods (Hotelling’s,

Normalisation and Quasidimensionless), which had been applied to diagnose faults

within fuel pump in aero gas turbine engine. These methods utilised statistical theory

and simple linear algebra to accommodate non-linear behaviour of engine data.

Hotelling’s method is less likely appropriate to diagnose faults, it is predicted that the

characteristic of engine data is not suitable for this method, which require more

variation within data as its originally applied in chemical processes. Normalisation

and Quasidimensionless (Traditional and Adaptation) came as a development of

previous method limitation. Both methods proved their capability to diagnose faults

from different conditions of engine data. Since Normalisation normalised y-axis

value, it could not manage to detect any discrepancies of performance combinations

along y-axis, although it performs well to diagnose any degradation along x-axis.

Quasidimensionless covered all limitation of the previous methods by combining non-

dimensional variables into dimensionless combinations, and create a robust detector

and isolator of faults existence. Further development from its traditional approach

(Walsh, et all., 1998) leads to Quasidimensionless Adaptation which overcome any

variation within engine data behaviour effectively.

Diagnosis Technique/Consistency-Checking took the idea of Normalisation and

Quasidimensionless (Traditional and Adaptation) Method to detect and isolate faults

within aero gas turbine engine through simple comparison of variable combinations.

This approach highlight particular variable, which cause error to combinations.

Chapter 6: Conclusions and Further Work - Review

90

Chapter 6

CONCLUSIONS AND FURTHER WORK

6.1. Review of Diagnosis Methods

The motivation of the project was to apply a statistical based technique to diagnose

measurement bias within a fuel metering system of a gas turbine engine (see Figure

1.1~1.3). It has been shown that it is feasible to achieve this diagnosis by the

application and adaptation of MSPC (PCA and PLS) approaches to several methods

(Hotelling’s, Normalisation and Quasidimensionless).

In adapting MSPC approaches, it was revealed that it is possible to statistically

correlate the fuel flow measurement to a combination of engine sensor signals. This is

illustrated in Figure 6.1 as which (TGT./T20).*(N2./sqrt(T20))-1.5 combination is well

correlated to normalised fuel flow (Quasidimensionless Adaptation Method). This

demonstrates that it is possible to relate engine sensor measurements to an estimated

of fuel flow. Comparison of the estimated and measured fuel flow signals provides a

mechanism to detect any measurement bias in fuel metering unit.

Figure 6.1: Statistical Approach of The Fuel Flow Estimation

Review of Diagnosis Methods

91

However, the relationship between performance signal combination and fuel flow are

dependent on the deterioration level of the engine. This deterioration level varies

depending upon the wear that has taken place within the engine. This feature is

illustrated in Figure 6.2, where an engine with four different levels of deterioration is

plotted for a performance combination of (P30./P20).*(TGT./T20)-1.5 against

dimensionless fuel flow, (WFE./(P20.*sqrt(T20))).

It should be noted that for the different levels of deterioration, different estimates of

the fuel flow are derived (marked a, b and c). Thus, to achieve a robust estimate of the

fuel flow, it is also necessary to determine the exact level of deterioration within the

engine.

Figure 6.2: Fuel Flow Estimation for Different Conditions of Engine Data

Identification of the deterioration level is achieved through correlation of two sets of

performance parameters (not fuel flow) and an example of which, is illustrated in

Figure 6.3. The four levels of engine deterioration are clearly distinguishable from

each other through classification of ellipses with separation.

Review of Diagnosis Methods

92

Figure 6.3 plots a set of real engine sensor data points (marked 1 to 4) from an

operational engine with unknown level of deterioration but which can be estimated by

a linear interpolation to have a deterioration level of approximately 1500 hours. It is

then possible to use this estimate of the engine deterioration to determine the

estimated fuel flow from Figure 6.2.

Figure 6.3: Identification of Deterioration Level

Dissertation Achievement

93

6.2. Dissertation Achievement

Fuel flow measurement has an important role in a gas turbine engine, not only from

an operational point of view, but also in the economical management point of view.

Safety regulations dictate that an aircraft carries a reserve of fuel necessary to account

for any inaccuracies in the measurement of the fuel flow consumed within the aircraft

engines. This, therefore, motivates the desire to identify measurement biases in the

fuel flow sensors in order to minimise the additional reserve fuel need to be carried,

which adds to operational costs of the aircraft.

Investigations to diagnose the fuel flow measurement bias have been undertaken with

a range of methods: model-based approach, neural networks, and knowledge based.

This dissertation examines the potential of statistical techniques to solve this

diagnostic problem. Statistical techniques have been successfully applied to

diagnostic problems in the chemical industry and it would therefore be anticipated

that the technology should be applicable to the gas turbine problem.

The variability and characteristics of the engine data, which is significantly different

to chemical process data, will require some adaptation of the statistical techniques

(MSPC approaches) to diagnose bias in the fuel pump measurement. This adaptation

combines statistical theory and mathematics approaches to establish classification

methods (Hotelling’s, Normalisation and Quasidimensionless) that are capable of

diagnosing sensors faults within an engine based on steady state performance data.

Hotelling’s uses a statistical distance between sample points in order to find possible

correlations within the engine data. However, the approach has a major limitation on

performance analysis, where the scatter samples make the plot difficult to analyse.

Normalisation approach performed well on classifying different conditions of engine

data. The elliptic boundaries clearly demonstrated separation between engine of

different levels of deterioration and good correlation to fuel flow data.

Unfortunately, normalisation of the performance parameters impaired some of its

diagnosis capability. The Quasidimensionless approach overcomes this limitation

through use of dimensionless variables.

Dissertation Achievement

94

The Quasidimensionless Adaptation approach combines the normalisation approach

with dimensionless performance variables, which both improve the accuracy of the

deterioration classification and fuel flow bias estimation. The rich combinations of

performance parameters that are well correlated to the engine behaviour provide a

method to estimate fuel flow consumption but also a mechanism to check the validity

of performance signals through a consistency-based approach.

In summary, the specific achievements of this dissertation are:

�� Development of a framework to diagnose fuel flow measurement bias and sensor

failure based on statistical techniques

�� Diagnose sensor faults based on a statistical analysis of steady state performance

data through consistency checking.

�� Estimate Level of deterioration through statistical comparison of different

performance parameters.

�� Estimate fuel flow from correlated performance combinations against fuel flow

and estimate of deterioration level. A comparison of this fuel flow estimate

against the measured signal provides a mechanism to identify any measurement

bias.

Recommendation for Further Work

95

6.3 Recommendation for Further Work

There are a number of areas where further work could usefully be undertaken. This

preliminary study has shown that there is potential for the application of MSPC

approaches to diagnose sensor faults within a gas turbine engine and a broad area for

further work would be to extend the current work to examine other statistical

techniques.

More specific recommendation would be:

��Automate the current diagnostic process to estimate fuel flow measurement bias.

��Extend consistency-checking process to include a temporal element to improve

diagnosis of sensor faults.

��Include a simple knowledge base in estimation of deterioration to identify sudden

changes in the deterioration level (deterioration levels should only increase in time

in a smooth fashion, sudden changes should be treated carefully).

It is suggested to expand the investigation method of analysis by optimising GUI

within MATLAB environment integrated to programming language. Such integration

will overcome the limitation in the PLS Toolbox to accommodate different approach

to analyse the engine data characteristic.

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Appendices

xiii

APPENDICES

All the programs written during the simulation processes of this dissertation are available at:

http://www.geocities.com/david_antory/links/dissertation/programs

Appendix A: Engine Data

a

APPENDIX A: ENGINE DATA

A.1. Engine Data Generation

The aero engine data is generated in steady-state format. The data is taken in four

different conditions based on the position of Mach number, Altitude and Thrust

Power. The generated data are: the ideal data (that contains no error as the perfect

model to diagnose the existence of faults by comparing to the rest of engine data

conditions), deterioration 1000 hours, deterioration 2000 hours and deterioration 3000

hours data.

The parameter of ALT, MN and FN affect the performance of the rest engine

parameters (N1, N2, N3, T20, T25, T30, TGT, P20, P25, P30, P50, P160, WFE). Specifically,

the generated data is influenced by:

��The Altitude is generated within the range of 34000 ~ 37000 with increment of

1000 feet.

��The Mach number is generated within the range of 0.81, 0.83 and 0.85.

��The Thrust Power condition is in the region of 17580 lbf within %4~%2 ±± of

nominal value.

In contrast to steady state data, the real engine sensor data is taken from the

operational condition. This is used to test the performance behaviour of the model,

which was built using statistical techniques through the implementation of real data

condition to represent operational condition within the gas turbine engine systems.

A.2. Engine Data Sample

The generated data is in the form of matrix data, which contains 60 samples for each

engine performance parameter. The performance parameter is in the following

sequence, respectively:

Appendix A: Engine Data

b

The engine sensor then took four different conditions of engine data by setting the

value of ALT, MN and FN. The sample below is taken for data with 1000 hours

deterioration level. The same format is applied for different conditions. The order is as

shown on the performance parameter.

These data have to be formatted in a form, which suit to the requirement. MATLAB

provides sets of function to do matrix manipulation to take the data for each

performance parameter to suit the statistical techniques model. Each parameter will

have 60 samples data, which represent the parameter performance for that aero engine

condition.

Appendix B: Hotelling’s Method

d

APPENDIX B: HOTELLING’S METHOD

B.1. Hotelling’s Information

Hotelling’s Method is developed using the statistical approach in the form of

Statistical Distance (SD) to find the distance between sample points. Mathematically,

it is simply formulated as:

SD2=2

2

222

21

211 )()(

σµ

σµ −

+− xx

(B.1.1)

By finding the mean value (µ) and the variance value (σ) from the data, the

requirement calculation of the statistical distance (Ming Tam, 1999) can be proceed to

find the correlation within engine data.

The correlation is higher if the scatter samples are concentrated on the point of origin

(0,0). Some boundaries have to be made for acceptable toleration limit, which is

normally 5% of the y-axis maximum value. In the PLS Toolbox 2.1, sub section PCA

GUI, it is known as limit 95%.

B.2. Program Listing

% ------------------------------------------------------------------------------------- % Program: Hotelling’s Method % Finding the Statistical Distance between 2 parameter % Relationship one to one between WFE and parameter % ------------------------------------------------------------------------------------- % By David Danny Santoso % Last update: 27 June 2001 % ------------------------------------------------------------------------------------- sWFE=WFE(1:60)-mean(WFE); % x-axis is WFE kuadrat1=(sWFE.*sWFE); % y-axis is parameters var1=(var(WFE))^2; pos1=kuadrat1/var1; dN1=N1(1:60)-mean(N1); % Parameter N1 k2N1=(dN1.*dN1); v2N1=(var(N1))^2; p2N1=k2N1/v2N1; set(figure(1),’Color’,[1 1 1]); % Set background plot(pos1,p2N1,’ko’,’MarkerFaceColor’,’g’);


e

xlabel(’Fuel Flow’,’Color’,’b’); ylabel(’Statistical Distance’,’Color’,’b’); title(’WFE vs N1’,’Color’,’r’); grid on; dN2=N2(1:60)-mean(N2); % Parameter N2 k2N2=(dN2.*dN2); v2N2=(var(N2))^2; p2N2=k2N2/v2N2; set(figure(2),’Color’,[1 1 1]); % Set background plot(pos1,p2N2,’ko’,’MarkerFaceColor’,’g’); xlabel(’Fuel Flow’,’Color’,’b’); ylabel(’Statistical Distance’,’Color’,’b’); title(’WFE vs N2’,’Color’,’r’); grid on; dN3=N3(1:60)-mean(N3); % Parameter N3 k2N3=(dN3.*dN3); v2N3=(var(N3))^2; p2N3=k2N3/v2N3; set(figure(3),’Color’,[1 1 1]); % Set background plot(pos1,p2N3,’ko’,’MarkerFaceColor’,’g’); xlabel(’Fuel Flow’,’Color’,’b’); ylabel(’Statistical Distance’,’Color’,’b’); title(’WFE vs N3’,’Color’,’r’); grid on; dP20=P20(1:60)-mean(P20); % Parameter P20 k2P20=(dP20.*dP20); v2P20=(var(P20))^2; p2P20=k2P20/v2P20; set(figure(4),’Color’,[1 1 1]); % Set background plot(pos1,p2P20,’ko’,’MarkerFaceColor’,’g’); xlabel(’Fuel Flow’,’Color’,’b’); ylabel(’Statistical Distance’,’Color’,’b’); title(’WFE vs P20’,’Color’,’r’); grid on; dP25=P25(1:60)-mean(P25); % Parameter P25 k2P25=(dP25.*dP25); v2P25=(var(P25))^2; p2P25=k2P25/v2P25; set(figure(5),’Color’,[1 1 1]); % Set background plot(pos1,p2P25,’ko’,’MarkerFaceColor’,’g’); xlabel(’Fuel Flow’,’Color’,’b’); ylabel(’Statistical Distance’,’Color’,’b’); title(’WFE vs P25’,’Color’,’r’); grid on; dP30=P30(1:60)-mean(P30); % Parameter P30 k2P30=(dP30.*dP30); v2P30=(var(P30))^2; p2P30=k2P30/v2P30; set(figure(6),’Color’,[1 1 1]); % Set background plot(pos1,p2P30,’ko’,’MarkerFaceColor’,’g’); xlabel(’Fuel Flow’,’Color’,’b’); ylabel(’Statistical Distance’,’Color’,’b’); title(’WFE vs P30’,’Color’,’r’); grid on; dP50=P50(1:60)-mean(P50); % Parameter P50 k2P50=(dP50.*dP50); v2P50=(var(P50))^2; p2P50=k2P50/v2P50; set(figure(7),’Color’,[1 1 1]); % Set background plot(pos1,p2P50,’ko’,’MarkerFaceColor’,’g’); xlabel(’Fuel Flow’,’Color’,’b’); ylabel(’Statistical Distance’,’Color’,’b’); title(’WFE vs P50’,’Color’,’r’); grid on; dP160=P160(1:60)-mean(P160); % Parameter P160 k2P160=(dP160.*dP160); v2P160=(var(P160))^2;


f

p2P160=k2P160/v2P160; set(figure(8),’Color’,[1 1 1]); % Set background plot(pos1,p2P160,’ko’,’MarkerFaceColor’,’g’); xlabel(’Fuel Flow’,’Color’,’b’); ylabel(’Statistical Distance’,’Color’,’b’); title(’WFE vs P160’,’Color’,’r’); grid on; dT20=T20(1:60)-mean(T20); % Parameter T20 k2T20=(dT20.*dT20); v2T20=(var(T20))^2; p2T20=k2T20/v2T20; set(figure(9),’Color’,[1 1 1]); % Set background plot(pos1,p2T20,’ko’,’MarkerFaceColor’,’g’); xlabel(’Fuel Flow’,’Color’,’b’); ylabel(’Statistical Distance’,’Color’,’b’); title(’WFE vs T20’,’Color’,’r’); grid on; dT25=T25(1:60)-mean(T25); % Parameter T25 k2T25=(dT25.*dT25); v2T25=(var(T25))^2; p2T25=k2T25/v2T25; set(figure(10),’Color’,[1 1 1]); % Set background plot(pos1,p2T25,’ko’,’MarkerFaceColor’,’g’); xlabel(’Fuel Flow’,’Color’,’b’); ylabel(’Statistical Distance’,’Color’,’b’); title(’WFE vs T25’,’Color’,’r’); grid on; dT30=T30(1:60)-mean(T30); % Parameter T30 k2T30=(dT30.*dT30); v2T30=(var(T30))^2; p2T30=k2T30/v2T30; set(figure(11),’Color’,[1 1 1]); % Set background plot(pos1,p2T30,’ko’,’MarkerFaceColor’,’g’); xlabel(’Fuel Flow’,’Color’,’b’); ylabel(’Statistical Distance’,’Color’,’b’); title(’WFE vs T30’,’Color’,’r’); grid on; dTGT=TGT(1:60)-mean(TGT); % Parameter TGT k2TGT=(dTGT.*dTGT); v2TGT=(var(TGT))^2; p2TGT=k2TGT/v2TGT; set(figure(12),’Color’,[1 1 1]); % Set background plot(pos1,p2TGT,’ko’,’MarkerFaceColor’,’g’); xlabel(’Fuel Flow’,’Color’,’b’); ylabel(’Statistical Distance’,’Color’,’b’); title(’WFE vs TGT’,’Color’,’r’); grid on;

B.3. Program Results

Hotelling’s Method creates twelve different plots. Each plot is the combination of

WFE against parameter. The plot is the result of the statistical between both axes. It

aims to find the relationship between the performance parameter and the fuel flow.

Figure B.3.1 shows a sample of data plot when the ideal data (green circle) is plotted

together with the deterioration 3000 hours data (yellow square). Although the

concentration of most of plot lies on the point of origin (0,0) but the remaining


g

samples are widely scatter and this makes the analysis to find the exact pattern of the

plot behaviour is difficult to perform. It is expected that there exist and exact

separation between two different conditions of engine data in order to identify faults

within the engine. Both engine data is plotted on top of each other for most of the

combinations with clear distinction only appears on the combination of WFE vs. TGT

and WFE vs. N2 in Figure B.3.2.

Figure B.3.1: Hotelling’s Plot

Considering this situation, analysing each combination performance is not an easy

task since the y-axis varies for different correlation. The combination of WFE vs.

TGT is more dense compare to other combination but it is not clearly diagnose the

existence of faults. The sample is sited on top of each other which make visual

analysis is difficult to undertake.

Figure B.3.2: Performance Hotelling’s Plot of N2 and TGT

Appendix C: Normalisation Method

h

APPENDIX C: NORMALISATION METHOD

C.1. Normalisation Information

Normalisation is made to overcome the limitation of the Hotelling’s Method. The aim

is to accommodate all data samples through mathematics manipulation using a

statistical distance in a normalise range to simplify the plot analysis and to clearly

identify the existence of faults given different conditions of engine data.

The y-axis is normalised within the range of 0~1 to accommodate different variation

in y-axis. This represents the correlation in 2D space between the fuel flow

consumption and the performance parameter. The non-linearity characteristic within

the engine data is shown by the parameters, which is influenced by the power in the

range of [–2.0 –1.5 –1.0 –0.5 +0.5 +1.0 +1.5 +2.0].

It is expected that there exist certain classification of engine data for different

condition. This is important to identify faults within the engine, which represent faults

in fuel pump measurement since the correlation is created between the performance

parameter and the fuel flow. This measurement bias could highlight particular

parameter to have an attention.

C.2. Program Listing

% ------------------------------------------------------------------------ % Program: Normalisation Method % Finding the highest correlation through Normalisation % ------------------------------------------------------------------------ % WFE vs (N1 .* All^p) -- extendable for other parameter % where p=[-2.0 -1.5 -1.0 -0.5 0.5 1.0 1.5 2.0] % ------------------------------------------------------------------------ % By David Danny Santoso % Last update: 12 June 2001 % ------------------------------------------------------------------------ p=[-2.0 -1.5 -1.0 -0.5 +0.5 +1.0 +1.5 +2.0]; k=[N2 N3 P20 P25 P30 P50 P160 T20 T25 T30 TGT]; nk=[’N2 ’;’N3 ’;’P20 ’;’P25 ’;’P30 ’; ’P50 ’;’P160’;’T20 ’;’T25 ’;’T30 ’;’TGT ’]; for j=1:11 for i=1:8


i

y1=N1.*(k(:,j).^p(i)); % Normalisation for all y-axis bb=max(y1)-min(y1); aa=y1-min(y1); normy1=aa./bb; % Put to the range of 0-1 dta=[WFE normy1]; % Ellipse orientation dts=sortrows(dta,[1 2]); xa1=dts(1,1); xb2=dts(60,1); y1=dts(1,2); y2=dts(60,2); if y1>y2 ya1=1; yb2=0; % Downward Direction else ya1=0; yb2=1; % Upward Direction end dis=sqrt((xb2-xa1).^2+(yb2-ya1).^2); % Distance between two points m=(yb2-ya1)/(xb2-xa1); % y = mx + c ; m is the gradient c=ya1-(m.*xa1); % Constant value xp1=WFE; yp1=normy1; atas=(m.*xp1)-yp1+c; bawah=sqrt((m*m)+1); dpo=abs(atas./bawah); % Distance between point to a line emax=dis./2; % Semi major ellipse emin=max(dpo); % Semi minor ellipse ang=atan(m); % Angular ellipse cty=(emax.*sin(ang)); % Centre of ellipse: (ctx, cty) ctx=min(WFE)+emax.*cos(ang); set(figure(j),’Color’,[1 1 1]); subplot(2,4,i); % Plotting the samples plot(WFE,normy1,’ks’,’MarkerFaceColor’,’y’); hold on; ellps([ctx,0.5],[emax,emin],’-r’,ang); grid on; xlabel(’Fuel Flow - WFE’); ylabel(’Normalisation distance’); title([’WFE vs N1.*’,nk(j,:),’̂ ’,num2str(p(i))],’Color’,’b’); meandpo(i)=mean(dpo); stddpo(i)=std(dpo); % Mean & Std between points to line set(figure(20+j),’Color’,[1 1 1]); subplot(2,1,1); % Steam plot for mean & std stem(i,meandpo(i),’:r’,’fill’); hold on; ylabel(’mean distance’); title([’mean distance: WFE vs N1.*’,nk(j,:),’̂ a’],’Color’,’k’); subplot(2,1,2);stem(i,stddpo(i),’:b’,’fill’); hold on; xlabel(’combination number’); ylabel(’std distance’); title([’stdev distance: WFE vs N1.*’,nk(j,:),’̂ a’],’Color’,’k’); co(i).name = [’WFE vs N1.*’,nk(j,:),’̂ ’,num2str(p(i))]; % Display the data information mc(i)=meandpo(i); % Display the mean value sc(i)=stddpo(i); % Display the standard deviation end cn=[{co(1).name};{co(2).name};{co(3).name}; {co(4).name}; {co(5).name}; {co(6).name};{co(7).name}; {co(8).name}]; akhir=8.*j; awal=akhir-7; hu=awal:1:akhir; nu=hu’; cm=mc’; cs=sc’; fprintf(’ - comb# - mean - stdev : %g.\n’,j) a=[nu cm cs]; eval(sprintf(’a%d=a’,j)); fprintf(1,’ Combination names :\n’) cn; eval(sprintf(’cn%d=cn’,j)); end


j

dtall=[a1; a2; a3; a4; a5; a6; a7; a8; a9; a10; a11]; % Display all the data nmall=[cn1;cn2;cn3;cn4;cn5;cn6;cn7;cn8;cn9;cn10;cn11]; % Combination names dame=dtall(:,2); % Mean values dasd=dtall(:,3); % Std values

C.3. Program Results

This method creates 1056 combinations for 12 parameter with 8 possible power range

plotted against fuel flow. The selection is made through the criteria of how well the

combination capable to identify faults conditions from different conditions of engine

data (exact separation on ellipse classification) and how well each samples correlate

to each other within the engine data condition (small semi minor ellipse closely shape

a straight line, analogue to a linear equation approach).

.

Figure C.2.1: Normalisation Plot - Intersection

This sample is chosen from the combination of WFE vs. P30.*TGTâ, where a is

within the range of [–2.0 –1.5 –1.0 –0.5 +0.5 +1.0 +1.5 +2.0]. The power represents

the non-linear characteristic within the engine data. This combination is chosen

because P30 and TGT have a closer relationship to the fuel flow consumption, WFE

compare to other parameters.

The first four combinations in Figure C.2.1 show a condition where there exists an

intersection between the ideal data (green circle) and deterioration 3000 hours data

(yellow square). This means some of the error is assumed within tolerance.

Identification of faults condition should be made clear without any intersection.


k

Figure C.2.2: Normalisation Plot – Clear Separation

It is expected to minimise the number of samples in the intersection area to clearly

identify the existence of fault. Figure C.2.2 shows the exact separation between these

two different conditions of engine data. Clearly, the justification of the engine

performance can be given.

These combinations performance are summarized on the Figure C.2.3 in term of the

mean value and the standard deviation value, which shown by a stem plot of the

combinations. The mean and standard deviation values represent the correlation

between samples as a categorisation procedure to select best combination.

Figure C.2.3: Table and Stem Plot

Unfortunately, normalisation method creates a limitation to identify fault, which lies

in y-axis. The normalisation normalises this condition and made this phenomenon

unidentified as faults.

Appendix D: Quasidimensionless Method

l

APPENDIX D: QUASIDIMENSIONLESS METHOD

D.1. Quasidimensionless Information

Quasidimensionless is the approach to analyse the engine data performance through

the graph plots of dimensionless parameter combinations in both axes. This approach

extends the ability of the statistical technique whilst overcoming the limitation in

Normalisation method. This new combination of both methods (Normalisation and

Quasidimensionless) results in Quasidimensionless Adaptation, which capable to

diagnose faults in the earlier stage and capable to perform the measurement bias in

fuel pump.

Quasidimensionless Adaptation uses the combination of dimensionless parameter

suggested by Walsh, P.P. and the statistical technique (extension of Normalisation

method) to eliminate different effect caused by different unit scale of the parameters.

It is expected that this method will retains the ability of the Normalisation approach to

identify faults through certain classification of engine conditions but also capable to

diagnose any characteristic behaviour of faults. Eliminating normalisation in y-axis

and changed to dimensionless unit scale should clearly highlight the characteristic of

engine performance related to the justification of the fuel pump measurement bias in

gas turbine engines.

D.2. Program Listing

% ------------------------------------------------------------------------- % Program: Quasidimensionless Method (TGT) % Best combination plotting, no normalisation % Extendable to other parameter % ------------------------------------------------------------------------- % WFE /(P20.*sqrt T20) vs ((Pall)./P20).*(All_nd)â % where a=[-2.0 -1.5 -1.0 -0.5 0.5 1.0 1.5 2.0]; % ------------------------------------------------------------------------- % By David Danny Santoso % Last update: 07 August 2001 at 17.21 am % ------------------------------------------------------------------------- p=[-2.0 -1.5 -1.0 -0.5 +0.5 +1.0 +1.5 +2.0]; k=[N1./sqrt(T20) N2./sqrt(T20) N3./sqrt(T20) P25./P20 P30./P20 P50./P20 P160./P20 T25./T20 T30./T20 TGT./T20]; nk=[’N1./sqrt(T20)’;’N2./sqrt(T20)’;’N3./sqrt(T20)’;’P25./P20 ’;’P30./P20 ’; ’P50./P20 ’;’P160./P20 ’;’T25/T20 ’;’T30./T20 ’;’TGT./T20 ’];


m

sbx=WFE./(P20.*(sqrt(T20))); % Non-dimensionless fuel flow for j=1:10 % For all combinations for i=1:8 % Power to represent non-linearity data y1=(TGT./(T20)).*(k(:,j).^p(i)); % Non-dimensionless y-axis dta=[sbx y1]; % Ellipse orientation dts=sortrows(dta,[1 2]); % Sorting procedure xa1=dts(1,1); xb2=dts(60,1); % Non-dimensionless WFE ya1=dts(1,2); yb2=dts(60,2); % Non-dimensionless y-axis if ya1>yb2 ya1=max(y1); yb2=min(y1); % Downward direction else ya1=min(y1); yb2=max(y1); % Upward direction end dis=sqrt((xb2-xa1).^2+(yb2-ya1).^2); % Distance between two points m=(yb2-ya1)/(xb2-xa1); % Gradient of y=mx+c c=ya1-(m.*xa1); % Constant Value xp1=sbx; % x-axis yp1=y1; % y-axis atas=(m.*xp1)-yp1+c; bawah=sqrt((m*m)+1); dpo=abs(atas./bawah); % Distance between point to a line emax=dis./2; % Semi major ellipse emin=max(dpo); % Semi minor ellipse ang=atan(m); % Angular ellipse cty=((max(y1)+min(y1))/2); % Center of ellipse ctx=min(xp1)+emax.*cos(ang); %---------------------------------------------------------------------------------- set(figure(j),’Color’,[1 1 1]); subplot(2,4,i); plot(sbx,y1,’ko’,’MarkerFaceColor’,’g’); hold on; ellps([ctx,cty],[emax,emin],’-k’,ang); grid on; xlabel(’WFE/(P20.*sqrt T20)’,’Color’,’b’); ylabel([’(TGT./T20).*(’,nk(j,:),’)^’,num2str(p(i))],’Color’,’b’); meandpo(i)=mean(dpo); stddpo(i)=std(dpo); grad(i)=m; cons(i)=c; centy(i)=cty; centx(i)=ctx; btas(i)=emin; format short e % ---------------------------------------------------------------------------------- % Display the data information co(i).name = [’WFE/(P20.*sqrt T20) vs (TGT./T20).*(’,nk(j,:),’)^’,num2str(p(i))]; mc(i)=meandpo(i); sc(i)=stddpo(i); gra(i)=grad(i); con(i)=cons(i); ceny(i)=centy(i); cenx(i)=centx(i); batas(i)=btas(i); end cn=[{co(1).name};{co(2).name};{co(3).name};{co(4).name}; {co(5).name};{co(6).name};{co(7).name};{co(8).name}]; akhir=8.*j; awal=akhir-7; hu=awal:1:akhir;


n

nu=hu’; cm=mc’; cs=sc’; gr=gra’; ct=con’; % gradient and constant value tx=cenx’;ty=ceny’; % centre of ellipse (tx,ty) bt=batas’; fprintf(’ - comb# - mean - stdev : %g.\n’,j) a=[nu cm cs]; eval(sprintf(’a%d=a’,j)); fprintf(’ mean grad const mid-x mid-y : %g.\n’,j) b=[bt gr ct tx ty];eval(sprintf(’b%d=b’,j)); fprintf(1,’ Combination names :\n’) cn; eval(sprintf(’cn%d=cn’,j)); end dtall=[a1; a2; a3; a4; a5; a6; a7; a8; a9; a10;]; urutall=[b1; b2; b3; b4; b5; b6; b7; b8; b9; b10]; nmall=[cn1;cn2;cn3;cn4;cn5;cn6;cn7;cn8;cn9;cn10]; % --------------------------------------------------------------------- % Program: Quasidimensionless Deterioration (TGT) % Performance Parameter – Deterioration Level % Extendable for different combination % --------------------------------------------------------------------- % N3./sqrt(T20) vs. (All_nd).*(All_nd)â % where a=[-2.0 -1.5 -1.0 -0.5 0.5 1.0 1.5 2.0] % --------------------------------------------------------------------- % By David Danny Santoso % Last update: 10 August 2001 % --------------------------------------------------------------------- p=[-2.0 -1.5 -1.0 -0.5 +0.5 +1.0 +1.5 +2.0]; k=[N1./sqrt(T20) N2./sqrt(T20) N3./sqrt(T20) P25./P20 P30./P20 P50./P20 P160./P20 T25./T20 T30./T20 TGT./T20]; nk=['N1./sqrt(T20)';'N2./sqrt(T20)';'N3./sqrt(T20)';'P25./P20 ';'P30./P20 '; 'P50./P20 ';'P160./P20 ';'T25/T20 ';'T30./T20 ';'TGT./T20 ']; sbx=N3./(sqrt(T20)); % Non-dimensional N3 for j=1:10 % For all combinations for i=1:8 % Power to represent non-linearity data y1=(TGT./T20).*(k(:,j).^p(i)); % Non-dimensionless y-axis dta=[sbx y1]; % Ellipse orientation dts=sortrows(dta,[1 2]); % Sorting procedure xa1=dts(1,1); xb2=dts(60,1); % Non-dimensionless WFE ya1=dts(1,2); yb2=dts(60,2); % Non-dimensionless y-axis if ya1>yb2 ya1=max(y1); yb2=min(y1); % Downward direction else ya1=min(y1); yb2=max(y1); % Upward direction end dis=sqrt((xb2-xa1).^2+(yb2-ya1).^2); % Distance between two points m=(yb2-ya1)/(xb2-xa1); % Gradient of y=mx+c c=ya1-(m.*xa1); % Constant Value xp1=sbx; % x-axis yp1=y1; % y-axis atas=(m.*xp1)-yp1+c; bawah=sqrt((m*m)+1); dpo=abs(atas./bawah); % Distance between point to a line


o

emax=dis./2; % Semi major ellipse emin=max(dpo); % Semi minor ellipse ang=atan(m); % Angular ellipse cty=((max(y1)+min(y1))/2); % Centre of ellipse ctx=min(xp1)+emax.*cos(ang); %---------------------------------------------------------------------------------- set(figure(j),’Color’,[1 1 1]); subplot(2,4,i); plot(sbx,y1,’kd’,’MarkerFaceColor’,’c’); hold on; ellps([ctx,cty],[emax,emin],’-b’,ang); grid on; xlabel(’N3./sqrt(T20)’,’Color’,’b’); ylabel([’(TGT./T20).*(’,nk(j,:),’)^’,num2str(p(i))],’Color’,’b’); meandpo(i)=mean(dpo); stddpo(i)=std(dpo); grad(i)=m; cons(i)=c; centy(i)=cty; centx(i)=ctx; btas(i)=emin; format short e %---------------------------------------------------------------------------------- % Display the data information co(i).name = [’N3./(sqrt(T20)) vs (TGT./T20).*(’,nk(j,:),’)^’,num2str(p(i))]; mc(i)=meandpo(i); sc(i)=stddpo(i); gra(i)=grad(i); con(i)=cons(i); ceny(i)=centy(i); cenx(i)=centx(i); batas(i)=btas(i); end cn=[{co(1).name};{co(2).name};{co(3).name};{co(4).name}; {co(5).name};{co(6).name};{co(7).name};{co(8).name}]; akhir=8.*j; awal=akhir-7; hu=awal:1:akhir; nu=hu’; cm=mc’; cs=sc’; gr=gra’; ct=con’; % gradient and constant value tx=cenx’;ty=ceny’; % centre of ellipse (tx,ty) bt=batas’; fprintf(’ - comb# - mean - stdev : %g.\n’,j) a=[nu cm cs]; eval(sprintf(’a%d=a’,j)); fprintf(’ mean grad const mid-x mid-y : %g.\n’,j) b=[bt gr ct tx ty];eval(sprintf(’b%d=b’,j)); fprintf(1,’ Combination names :\n’) cn; eval(sprintf(’cn%d=cn’,j)); end dtall=[a1; a2; a3; a4; a5; a6; a7; a8; a9; a10;]; urutall=[b1; b2; b3; b4; b5; b6; b7; b8; b9; b10]; nmall=[cn1;cn2;cn3;cn4;cn5;cn6;cn7;cn8;cn9;cn10];


p

D.2. Program Results

Quasidimensionless Adaptation creates 800 combinations by taking out P20 and T20

from the list of parameters. The remaining 10 parameters are combined with 8

possible powers to establish new combinations. These dimensionless performance

parameters are plotted against dimensionless fuel flow.

Figure D.2.1: P30nd.*TGTnd Combination

Figure D.2.1 illustrates set of Quasidimensionless Adaptation plots for the

combination of WFEnd vs. P30nd.*TGTnd. This combination is selected from the best

500 possible combinations. It appears how the characteristic of the engine data varies.

Inclusive group classification for different conditions of engine data shows that this

combination well diagnoses the existence of faults. Exact separation and certain

distance, along with tight ellipse shows how well the samples correlate to each other.

The relationship between the fuel flow and performance parameter in dimensionless

term is analogue to a linear equation.

Quasidimensionless Adaptation creates rich sets of new dimensionless combinations.

These combinations manage to show the ability to diagnose biases/faults within the

fuel pump measurement of a gas turbine engine.

Figure D.2.2 shows P30nd.*N1nd Combination, where the identification of faults for

different conditions of engine data is clearly seen. The classification of the data is well

covered by ellipse boundaries and analogue to the straight line, a linear equation.


q

Figure D.2.2: P30nd.*N1nd Combination

There are two basic criteria should be undertaken to select certain combinations to

further investigate the combination’s performance analysis. These criteria are:

��Thin (small semi minor) ellipse represents well-correlated data.

��Exact ellipse separation in particular distance represents well identification of

faults through inclusive classification.

Figure D.2.3: TGTnd.*T25nd Combination


r

Figure D.2.3 illustrates chosen combinations, which capable to identify inclusive

classification for four different conditions of engine data. The ideal as the model

highlights the level of deterioration and explain the behaviour of deterioration data in

implicitly. The bias could be calculated and the further analysis could be performed.

This method is the backbone of the diagnosis technique/consistency checking, where

the measurement bias is identified through comparison between combinations.

The diagnosis (detection and isolation) performs tracing the location of the fault and

highlights the particular component through the comparison performances when

different engine data are plotted together. Comparison between the ideal data and new

given engine data will highlight the degradation. By comparing those plots, we can

point the indication of the error on particular component.

Table D.1 illustrates set of lists of combinations for Quasidimensionless Adaptation.

Combination Distance SMinor Ratio

’WFE/(P20.*sqrt T20) vs (P160./P20).*(P160./P20 )^-1’ 4.4409E-16 0 0

’WFE/(P20.*sqrt T20) vs (TGT./T20).*(T25/T20 )^-2’ 0.02920258 7.97E-04 1.83E+01

’WFE/(P20.*sqrt T20) vs (T25./T20).*(TGT./T20 )^-0.5’ 0.00796553 2.19E-04 1.82E+01

’WFE/(P20.*sqrt T20) vs (TGT./T20).*(N2./sqrt(T20))^-2’ 1.4525E-06 4.11E-08 1.77E+01

’WFE/(P20.*sqrt T20) vs (N2./(sqrt(T20))).*(TGT./T20 )^-0.5’ 3.25551442 9.47E-02 1.72E+01

’WFE/(P20.*sqrt T20) vs (N3./(sqrt(T20))).*(N2./sqrt(T20))^-1.5’ 0.00066937 2.08E-05 1.61E+01

’WFE/(P20.*sqrt T20) vs (TGT./T20).*(T30./T20 )^-1.5’ 0.01944482 6.04E-04 1.61E+01

’WFE/(P20.*sqrt T20) vs (TGT./T20).*(N3./sqrt(T20))^-1.5’ 1.1537E-05 3.86E-07 1.49E+01

’WFE/(P20.*sqrt T20) vs (T25./T20).*(N2./sqrt(T20))^-1’ 4.9742E-05 1.67E-06 1.49E+01

’WFE/(P20.*sqrt T20) vs (N2./(sqrt(T20))).*(T25/T20 )^-1’ 2.03195325 6.88E-02 1.48E+01

’WFE/(P20.*sqrt T20) vs (TGT./T20).*(P30./P20 )^-0.5’ 0.03082395 1.11E-03 1.39E+01

’WFE/(P20.*sqrt T20) vs (P30./P20).*(TGT./T20 )^-2’ 0.06474217 2.36E-03 1.37E+01

’WFE/(P20.*sqrt T20) vs (P30./P20).*(TGT./T20 )^-1.5’ 0.13410201 5.67E-03 1.18E+01




’WFE/(P20.*sqrt T20) vs (P160./P20).*(P30./P20 )^1’ 1.94117224 8.63E-02 1.12E+01



’WFE/(P20.*sqrt T20) vs (T25./T20).*(N2./sqrt(T20))^-1.5’ 4.0081E-06 1.87E-07 1.07E+01

’WFE/(P20.*sqrt T20) vs (TGT./T20).*(T30./T20 )^-1’ 0.03195836 1.52E-03 1.05E+01

’WFE/(P20.*sqrt T20) vs (P25./P20).*(P160./P20 )^1.5’ 0.54207505 2.66E-02 1.02E+01


’WFE/(P20.*sqrt T20) vs (N2./(sqrt(T20))).*(P160./P20 )^2’ 3.77852426 1.90E-01 9.93E+00


’WFE/(P20.*sqrt T20) vs (P30./P20).*(N1./sqrt(T20))^1.5’ 3.42456043 1.80E-01 9.52E+00

’WFE/(P20.*sqrt T20) vs (N2./(sqrt(T20))).*(T30./T20 )^-0.5’ 2.51021541 1.33E-01 9.42E+00


s

’WFE/(P20.*sqrt T20) vs (P30./P20).*(N2./sqrt(T20))^2’ 4.76957492 2.60E-01 9.16E+00

’WFE/(P20.*sqrt T20) vs (N1./(sqrt(T20))).*(P25./P20 )^1.5’ 2.71303005 1.49E-01 9.09E+00

’WFE/(P20.*sqrt T20) vs (T30./T20).*(TGT./T20 )^-1’ 0.00902578 5.00E-04 9.03E+00

’WFE/(P20.*sqrt T20) vs (N3./(sqrt(T20))).*(T25/T20 )^-1.5’ 1.29444469 7.24E-02 8.94E+00










’WFE/(P20.*sqrt T20) vs (N2./(sqrt(T20))).*(N1./sqrt(T20))^2’ 3.36740926 1.99E-01 8.48E+00


’WFE/(P20.*sqrt T20) vs (TGT./T20).*(N3./sqrt(T20))^-1’ 0.00023494 1.41E-05 8.30E+00













’WFE/(P20.*sqrt T20) vs (N3./(sqrt(T20))).*(N2./sqrt(T20))^-2’ 5.6937E-05 3.97E-06 7.17E+00





’WFE/(P20.*sqrt T20) vs (TGT./T20).*(P160./P20 )^-1’ 0.09584803 6.82E-03 7.03E+00


’WFE/(P20.*sqrt T20) vs (T25./T20).*(P160./P20 )^2’ 0.12843779 9.20E-03 6.98E+00


’WFE/(P20.*sqrt T20) vs (TGT./T20).*(T25/T20 )^-1.5’ 0.03836618 2.78E-03 6.90E+00



’WFE/(P20.*sqrt T20) vs (N2./(sqrt(T20))).*(N1./sqrt(T20))^1.5’ 3.61395875 2.64E-01 6.84E+00



’WFE/(P20.*sqrt T20) vs (N3./(sqrt(T20))).*(TGT./T20 )^-1’ 2.51195905 1.88E-01 6.69E+00


’WFE/(P20.*sqrt T20) vs (T25./T20).*(P30./P20 )^1.5’ 3.53574048 2.66E-01 6.64E+00







t


’WFE/(P20.*sqrt T20) vs (P30./P20).*(T30./T20 )^-1’ 0.32822592 2.53E-02 6.49E+00

’WFE/(P20.*sqrt T20) vs (P30./P20).*(T30./T20 )^-1.5’ 0.18503448 1.43E-02 6.45E+00




































’WFE/(P20.*sqrt T20) vs (P30./P20).*(T25/T20 )^2’ 2.51507939 2.26E-01 5.57E+00










’WFE/(P20.*sqrt T20) vs (P30./P20).*(T25/T20 )^-2’ 0.24052205 2.29E-02 5.26E+00




u



’WFE/(P20.*sqrt T20) vs (N2./(sqrt(T20))).*(N1./sqrt(T20))^-1’ 0.01399559 1.34E-03 5.24E+00








’WFE/(P20.*sqrt T20) vs (P160./P20).*(N2./sqrt(T20))^-2’ 1.2569E-07 1.23E-08 5.11E+00


’WFE/(P20.*sqrt T20) vs (P30./P20).*(T25/T20 )^1.5’ 2.2169784 2.19E-01 5.07E+00

’WFE/(P20.*sqrt T20) vs (N2./(sqrt(T20))).*(P160./P20 )^-0.5’ 1.83096729 1.81E-01 5.07E+00













’WFE/(P20.*sqrt T20) vs (T25./T20).*(N1./sqrt(T20))^2’ 1.95580912 2.03E-01 4.83E+00

’WFE/(P20.*sqrt T20) vs (P30./P20).*(P25./P20 )^-1.5’ 0.00976772 1.01E-03 4.82E+00

’WFE/(P20.*sqrt T20) vs (P30./P20).*(T25/T20 )^-1.5’ 0.34595183 3.61E-02 4.79E+00









’WFE/(P20.*sqrt T20) vs (N3./(sqrt(T20))).*(T30./T20 )^-1’ 1.45661909 1.56E-01 4.65E+00












’WFE/(P20.*sqrt T20) vs (TGT./T20).*(T30./T20 )^-2’ 0.01179416 1.31E-03 4.52E+00



v

’WFE/(P20.*sqrt T20) vs (T30./T20).*(P30./P20 )^-0.5’ 0.00747219 8.36E-04 4.47E+00











’WFE/(P20.*sqrt T20) vs (TGT./T20).*(T25/T20 )^-1’ 0.05036355 5.80E-03 4.34E+00




’WFE/(P20.*sqrt T20) vs (TGT./T20).*(N3./sqrt(T20))^-0.5’ 0.00461663 5.37E-04 4.30E+00




’WFE/(P20.*sqrt T20) vs (P30./P20).*(N3./sqrt(T20))^-0.5’ 0.03651059 4.27E-03 4.28E+00


’WFE/(P20.*sqrt T20) vs (P30./P20).*(P25./P20 )^-1’ 0.06238665 7.36E-03 4.24E+00

’WFE/(P20.*sqrt T20) vs (TGT./T20).*(TGT./T20 )^2’ 4.02298263 4.79E-01 4.20E+00



’WFE/(P20.*sqrt T20) vs (P30./P20).*(N3./sqrt(T20))^-1.5’ 4.1352E-05 4.94E-06 4.18E+00

’WFE/(P20.*sqrt T20) vs (P30./P20).*(N3./sqrt(T20))^-1’ 0.00125384 1.50E-04 4.18E+00




’WFE/(P20.*sqrt T20) vs (P30./P20).*(T30./T20 )^0.5’ 1.62262102 1.99E-01 4.08E+00







’WFE/(P20.*sqrt T20) vs (TGT./T20).*(T30./T20 )^-0.5’ 0.05255503 6.56E-03 4.00E+00




’WFE/(P20.*sqrt T20) vs (P30./P20).*(T30./T20 )^2’ 2.64822258 3.34E-01 3.97E+00











w




























’WFE/(P20.*sqrt T20) vs (T25./T20).*(N2./sqrt(T20))^1.5’ 4.33171877 5.95E-01 3.64E+00








’WFE/(P20.*sqrt T20) vs (TGT./T20).*(TGT./T20 )^1.5’ 2.43419705 3.38E-01 3.60E+00















’WFE/(P20.*sqrt T20) vs (N2./(sqrt(T20))).*(T25/T20 )^0.5’ 3.91113706 5.67E-01 3.45E+00


x






’WFE/(P20.*sqrt T20) vs (N1./(sqrt(T20))).*(TGT./T20 )^2’ 2.68953515 3.97E-01 3.39E+00



















’WFE/(P20.*sqrt T20) vs (TGT./T20).*(T25/T20 )^-0.5’ 0.06594596 1.04E-02 3.16E+00







’WFE/(P20.*sqrt T20) vs (TGT./T20).*(TGT./T20 )^1’ 0.94756319 1.51E-01 3.14E+00


’WFE/(P20.*sqrt T20) vs (T25./T20).*(P160./P20 )^-1’ 0.01172667 1.90E-03 3.08E+00



’WFE/(P20.*sqrt T20) vs (P30./P20).*(TGT./T20 )^0.5’ 1.48142801 2.41E-01 3.07E+00









’WFE/(P20.*sqrt T20) vs (N2./(sqrt(T20))).*(T25/T20 )^1’ 3.57966427 6.07E-01 2.95E+00







y








’WFE/(P20.*sqrt T20) vs (T25./T20).*(TGT./T20 )^2’ 1.55806357 2.72E-01 2.87E+00







’WFE/(P20.*sqrt T20) vs (TGT./T20).*(P30./P20 )^1.5’ 2.09070068 3.73E-01 2.81E+00








’WFE/(P20.*sqrt T20) vs (TGT./T20).*(TGT./T20 )^0.5’ 0.30562438 5.50E-02 2.78E+00

















’WFE/(P20.*sqrt T20) vs (P160./P20).*(TGT./T20 )^2’ 0.86585832 1.63E-01 2.66E+00



’WFE/(P20.*sqrt T20) vs (T25./T20).*(N3./sqrt(T20))^-0.5’ 0.00020372 3.87E-05 2.63E+00



’WFE/(P20.*sqrt T20) vs (N2./(sqrt(T20))).*(T30./T20 )^0.5’ 3.77417776 7.19E-01 2.63E+00







z


’WFE/(P20.*sqrt T20) vs (N3./(sqrt(T20))).*(N3./sqrt(T20))^1’ 5.20307188 1.01E+00 2.57E+00


’WFE/(P20.*sqrt T20) vs (N1./(sqrt(T20))).*(TGT./T20 )^1.5’ 2.24269054 4.38E-01 2.56E+00


’WFE/(P20.*sqrt T20) vs (T30./T20).*(TGT./T20 )^2 2.0867683 4.10E-01 2.54E+00






’WFE/(P20.*sqrt T20) vs (T30./T20).*(T30./T20 )^-1’ 5.5511E-16 1.11E-16 2.50E+00


’WFE/(P20.*sqrt T20) vs (P30./P20).*(P30./P20 )^-1’ 5.5511E-16 1.11E-16 2.50E+00

’WFE/(P20.*sqrt T20) vs (T25./T20).*(T25/T20 )^-1’ 5.5511E-16 1.11E-16 2.50E+00



’WFE/(P20.*sqrt T20) vs (TGT./T20).*(TGT./T20 )^-1’ 5.5511E-16 1.11E-16 2.50E+00






’WFE/(P20.*sqrt T20) vs (N3./(sqrt(T20))).*(N3./sqrt(T20))^0.5’ 5.21162138 1.05E+00 2.47E+00













’WFE/(P20.*sqrt T20) vs (N1./(sqrt(T20))).*(P160./P20 )^-1’ 0.62263146 1.30E-01 2.39E+00


’WFE/(P20.*sqrt T20) vs (TGT./T20).*(P30./P20 )^2’ 2.48887208 5.21E-01 2.39E+00












’WFE/(P20.*sqrt T20) vs (T25./T20).*(TGT./T20 )^1.5’ 0.53389474 1.16E-01 2.30E+00


aa




’WFE/(P20.*sqrt T20) vs (TGT./T20).*(TGT./T20 )^-0.5’ 0.01817834 4.03E-03 2.25E+00



























’WFE/(P20.*sqrt T20) vs (TGT./T20).*(N1./sqrt(T20))^0.5’ 0.86200484 2.07E-01 2.08E+00







’WFE/(P20.*sqrt T20) vs (TGT./T20).*(T25/T20 )^0.5’ 0.11184069 2.73E-02 2.05E+00















bb



’WFE/(P20.*sqrt T20) vs (N2./(sqrt(T20))).*(T30./T20 )^1’ 3.13095811 7.88E-01 1.99E+00



















’WFE/(P20.*sqrt T20) vs (TGT./T20).*(TGT./T20 )^-1.5’ 0.00323605 8.59E-04 1.88E+00













’WFE/(P20.*sqrt T20) vs (TGT./T20).*(T30./T20 )^0.5’ 0.14071087 3.87E-02 1.82E+00








’WFE/(P20.*sqrt T20) vs (T25./T20).*(T30./T20 )^-0.5’ 0.00134523 3.79E-04 1.77E+00







’WFE/(P20.*sqrt T20) vs (TGT./T20).*(N1./sqrt(T20))^1’ 1.55966621 4.48E-01 1.74E+00



cc


’WFE/(P20.*sqrt T20) vs (TGT./T20).*(T25/T20 )^1’ 0.14424839 4.16E-02 1.74E+00

’WFE/(P20.*sqrt T20) vs (TGT./T20).*(TGT./T20 )^-2’ 0.00271952 7.87E-04 1.73E+00




































’WFE/(P20.*sqrt T20) vs (T30./T20).*(T25/T20 )^-2’ 0.00193124 6.33E-04 1.53E+00


’WFE/(P20.*sqrt T20) vs (TGT./T20).*(T25/T20 )^1.5’ 0.18621549 6.15E-02 1.51E+00












dd





’WFE/(P20.*sqrt T20) vs (TGT./T20).*(T30./T20 )^1’ 0.22858807 7.94E-02 1.44E+00






















’WFE/(P20.*sqrt T20) vs (TGT./T20).*(T25/T20 )^2’ 0.23692309 8.85E-02 1.34E+00

’WFE/(P20.*sqrt T20) vs (T30./T20).*(T25/T20 )^-1.5’ 0.00151723 5.68E-04 1.34E+00

























ee



’WFE/(P20.*sqrt T20) vs (TGT./T20).*(T30./T20 )^1.5’ 0.36748404 1.53E-01 1.20E+00




















’WFE/(P20.*sqrt T20) vs (N1./(sqrt(T20))).*(P50./P20 )^-1.5’ 2.17171691 1.02E+00 1.06E+00





’WFE/(P20.*sqrt T20) vs (TGT./T20).*(T30./T20 )^2’ 0.56236052 2.72E-01 1.04E+00





’WFE/(P20.*sqrt T20) vs (N1./(sqrt(T20))).*(P50./P20 )^-2’ 2.41839126 1.21E+00 9.99E-01

’WFE/(P20.*sqrt T20) vs (P50./P20).*(P25./P20 )^-0.5’ 0.00457709 2.31E-03 9.90E-01

’WFE/(P20.*sqrt T20) vs (P25./P20).*(P160./P20 )^-1’ 0.0243293 1.23E-02 9.87E-01

’WFE/(P20.*sqrt T20) vs (N1./(sqrt(T20))).*(P50./P20 )^-1’ 1.64404445 8.36E-01 9.83E-01

’WFE/(P20.*sqrt T20) vs (P25./P20).*(N2./sqrt(T20))^-1’ 9.4634E-05 4.81E-05 9.83E-01

’WFE/(P20.*sqrt T20) vs (P30./P20).*(TGT./T20 )^1.5’ 0.63570929 3.25E-01 9.79E-01


’WFE/(P20.*sqrt T20) vs (T25./T20).*(T25/T20 )^2’ 0.04392815 2.32E-02 9.49E-01

’WFE/(P20.*sqrt T20) vs (N2./(sqrt(T20))).*(P30./P20 )^-0.5’ 0.26866113 1.42E-01 9.48E-01

’WFE/(P20.*sqrt T20) vs (N3./(sqrt(T20))).*(P50./P20 )^-1.5’ 2.32317002 1.24E+00 9.35E-01


’WFE/(P20.*sqrt T20) vs (T30./T20).*(N3./sqrt(T20))^0.5’ 0.43859631 2.35E-01 9.32E-01

’WFE/(P20.*sqrt T20) vs (N2./(sqrt(T20))).*(T30./T20 )^-2’ 0.37723052 2.07E-01 9.09E-01



’WFE/(P20.*sqrt T20) vs (TGT./T20).*(P25./P20 )^0.5’ 0.11015121 6.21E-02 8.86E-01

’WFE/(P20.*sqrt T20) vs (T30./T20).*(N1./sqrt(T20))^0.5’ 0.17673687 1.00E-01 8.81E-01


’WFE/(P20.*sqrt T20) vs (T25./T20).*(T25/T20 )^1.5’ 0.02633037 1.50E-02 8.80E-01


ff

’WFE/(P20.*sqrt T20) vs (N1./(sqrt(T20))).*(T25/T20 )^-1.5’ 0.17220873 9.86E-02 8.73E-01



’WFE/(P20.*sqrt T20) vs (T30./T20).*(N3./sqrt(T20))^-0.5’ 0.00035981 2.09E-04 8.62E-01






’WFE/(P20.*sqrt T20) vs (P50./P20).*(N1./sqrt(T20))^-1.5’ 5.4998E-06 3.28E-06 8.37E-01




’WFE/(P20.*sqrt T20) vs (N2./(sqrt(T20))).*(TGT./T20 )^0.5’ 1.15891925 7.19E-01 8.06E-01




’WFE/(P20.*sqrt T20) vs (T30./T20).*(TGT./T20 )^0.5’ 0.04253683 2.70E-02 7.88E-01

’WFE/(P20.*sqrt T20) vs (TGT./T20).*(P50./P20 )^2’ 0.17691632 1.13E-01 7.86E-01





’WFE/(P20.*sqrt T20) vs (N3./(sqrt(T20))).*(TGT./T20 )^1’ 1.18502408 8.44E-01 7.02E-01

’WFE/(P20.*sqrt T20) vs (TGT./T20).*(N3./sqrt(T20))^1’ 1.18502408 8.44E-01 7.02E-01

’WFE/(P20.*sqrt T20) vs (N3./(sqrt(T20))).*(N1./sqrt(T20))^-2’ 5.1461E-05 3.67E-05 7.02E-01



’WFE/(P20.*sqrt T20) vs (P50./P20).*(TGT./T20 )^2’ 0.41590842 2.98E-01 6.98E-01



’WFE/(P20.*sqrt T20) vs (T25./T20).*(T25/T20 )^-0.5’ 0.00135639 1.04E-03 6.55E-01








’WFE/(P20.*sqrt T20) vs (T25./T20).*(T25/T20 )^-2’ 0.00107078 9.03E-04 5.93E-01



’WFE/(P20.*sqrt T20) vs (T25./T20).*(T30./T20 )^0.5’ 0.00964574 8.46E-03 5.70E-01




’WFE/(P20.*sqrt T20) vs (N2./(sqrt(T20))).*(N3./sqrt(T20))^-1.5’ 4.8887E-05 4.67E-05 5.23E-01

’WFE/(P20.*sqrt T20) vs (T25./T20).*(T30./T20 )^2’ 0.0899792 8.60E-02 5.23E-01





gg





’WFE/(P20.*sqrt T20) vs (N1./(sqrt(T20))).*(T30./T20 )^-1.5’ 0.1114181 1.16E-01 4.80E-01



’WFE/(P20.*sqrt T20) vs (N2./(sqrt(T20))).*(TGT./T20 )^1’ 0.64243554 7.08E-01 4.54E-01








’WFE/(P20.*sqrt T20) vs (N1./(sqrt(T20))).*(T25/T20 )^-2’ 0.08148148 1.07E-01 3.81E-01




’WFE/(P20.*sqrt T20) vs (TGT./T20).*(N2./sqrt(T20))^1.5’ 0.41975798 6.40E-01 3.28E-01



’WFE/(P20.*sqrt T20) vs (T30./T20).*(T30./T20 )^-0.5’ 0.00137026 2.34E-03 2.93E-01






’WFE/(P20.*sqrt T20) vs (T30./T20).*(T30./T20 )^-2’ 0.00056835 1.06E-03 2.67E-01










’WFE/(P20.*sqrt T20) vs (TGT./T20).*(N3./sqrt(T20))^1.5’ 0.27010516 8.14E-01 1.66E-01

’WFE/(P20.*sqrt T20) vs (T30./T20).*(T25/T20 )^-1’ 0.00063358 1.96E-03 1.62E-01


’WFE/(P20.*sqrt T20) vs (N3./(sqrt(T20))).*(N1./sqrt(T20))^-1.5’ 8.9904E-05 3.05E-04 1.47E-01







’WFE/(P20.*sqrt T20) vs (N1./(sqrt(T20))).*(N2./sqrt(T20))^-0.5’ 0.00107822 5.82E-03 9.26E-02




hh



’WFE/(P20.*sqrt T20) vs (N2./(sqrt(T20))).*(N1./sqrt(T20))^-2’ 3.2926E-06 2.67E-05 6.16E-02





’WFE/(P20.*sqrt T20) vs (N1./(sqrt(T20))).*(P160./P20 )^-0.5’ 5.1298E-05 5.52E-02 4.65E-04

Table D.1: Combinations of Quasidimensionless Adaptation

These combinations are selected through a comparison between the distance (between

ideal data and deterioration level data) and semi minor ellipse of ideal data. It is

expected that in order to avoid any intersection between these ellipses (to clearly

identify the existence of faults), the distance should be bigger than two times of semi

minor ellipse. The results are collected in a ratio column. To fulfil this condition, the

ratio should be positive, which means there exist a certain distance between ideal data

and data with level of deterioration (1000 hours, 2000 hours and 3000 hours).

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