GAS TURBINE PERFORMENCE BASED CREEP LIFE ESTIMATION USING

SOFT COMPUTING TECHNIQUE

ALMEHDI MOHAMED ZARTI

A thesis submitted in

fulfillment of the requirement for the award of

the Degree of Master of Mechanical

Engineering

Faculty of Mechanical and Manufacturing Engineering

University Tun Hussein Onn Malaysia

DECEMBER 2012

iv

ABSTRACT

Accurate and simple prediction system has become an urgent need in most

disciplines. Having the accurate prediction system for gas turbine components will

allow the user to produce reliable creep life prediction. Focusing on the turbine

blades and its life, the current method to calculate its creep life is complex and

consumes a lot of time. For this reason, the aim of this research is to use an

alternative performance–based creep life estimation that is able to provide a quick

solution and obtain accurate creep life prediction. By the use of an artificial neural

network to predict creep life, a neural network architecture called Sensor Life Based

(SLB) architecture that produces a direct mapping from gas path sensor to predict the

blade creep life was created by using the gas turbine simulation performance

software. The performance of gas turbine and the effects of multiple operations on

the blade are studied. The result of the study is used to establish the input and output

to train the Sensor Life Based network. The result shows that the Sensor Life-Based

architecture is able to produce accurate creep life predictions yet performing rapid

calculations. The result also shows that the accuracy of prediction depends on the

way, how the gas path sensor is grouped together.

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CONTENTS

TITLE i

DEDICATION ii

ACKNOWLEDGEMENT iii

ABSTRACT iv

CONTENTS v

LIST OF TABLES viii

LIST OF FIGURES ix

LIST OF SYMBOLS AND ABBREVIATIONS xii

CHAPTER 1 INTRODUCTION 1

1.1 Introduction to Gas Turbines

1.2 Problem Statement

1.3 Project Objectives

1.4 Project Scope

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CHAPTER 2 LITERATURE REVIEW 8

2.1 Common Failure in the Gas Turbine Hot Section

2.1.1 Fatigue

2.1.2 High Temperature Corrosion/Oxidation

2.1.3 Creep Deformation

2.2 Creep Life Estimation Approaches

2.1.1 Model-Based Approach

2.1.2 Service-Based Approach

2.1.3 Statistical / Probabilistic Approach

2.1.4 Soft Computing Approach

2.3 Artificial Neural Network

2.3.1 Introduction to Artificial Neural Network

2.3.2 Advantages of Neural Networks

2.3.3 Neuron Model

2.3.4 Multilayer Perceptron Network Architectures

2.4 Some Previous Works

CHAPTER 3 METHODOLOGY 35

3.1 Research Methodology Diagram

3.2 Research Methodology Diagram

3.3 Proposal Model

3.4 Work Setup

3.4.1 Engine Model Construction

3.5 Generate and Train Data

3.5.1 Train Samples

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CHAPTER 4 DATA ANALYSIS AND DISCUSSION 49

4.1 Result and Discussion

4.2 Effect of Engine Operation and Health Condition

on the Blade‟s Creep Life

4.2.1 Effect of Altitude

4.2.2 Effect of Engine Rotational Speed

4.2.3 Effect of Ambient Temperature

4.2.4 Effect of Mach Number

4.3 Train Network

4.3.1 The Performance of the Network Using Unseen

Test Samples

4.4 Compare results

CHAPTER 5 CONCLUSION 69

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LIST OF TABLES

2.1 Finalized parameter(s) for each cluster 32

3.1 Test range of selected parameter 42

3.2 Step size for each parameter 43

3.3 Sample parameter 44

3.4 Training parameter 45

3.5 Error ranges used to assess the performance of the trained

Approximates 48

4.1 Summary of the impact analysis 61

4.2 Percentage of samples within the specified range for all

the Tests 62

4.3 Percentage of samples within the specified range for all

the Tests 64

4.4 Percentage of Samples Encompassed Within the

Specified Error Range for RB, FB, SB, SLB 67

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LIST OF FIGURES

1.1 Schematic for a) an Aircraft Jet Engine; and

b) A land-Based Gas Turbine 2

1.2 Cutaway View of the Rolls-Royce Trent

900 Turbofan Engine 3

1.3 Increase of Firing Temperature With Respect

to Turbine Blade Alloys Development 5

2.1 Deformed Turbine Blades under Creep Attack 11

2.2 Life Span of Hot Section Component Due

to Creep Deformation 12

2.3 Single Input Neuron 18

2.4 Some Common Transfer Functions 19

2.5 Multiple-Input Neuron 20

2.6 A Multiple-Input Neuron Using Abbreviated Notation 20

2.7 A Single-Layer Network of S Neurons 21

2.8 Layer of S Neurons, Abbreviated Notation 22

2.9 Three Layer Network 23

2.10 Abbreviated Notation of Multilayer Network 24

2.11 Back Propagation of Error 25

2.12 Weights and Bias Update 25

2.13 SRT Specimen (a) and Its Location in Blades (b) 27

2.14 Failed Stator Blades 28

2.15 Fracture on the Studied Blade 30

2.16 Different Mappings Between the Cluster Parameters 31

2.17 Range-Based Architecture Based on Mapping 1 33

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2.18 Functional-Based Architecture Based on Mapping2 34

2.19 FB Architecture with Degradation Levels and

Operation 35

3.1 Illustration of Research Idea 36

3.2 Research Methodology Diagram 38

3.3 Mappings 4 39

3.4 Simplified Sensor-Based Architecture for Clean Engine

Case 39

3.5 Engine for Helicopter 40

3.6 Engine Model Configuration 41

3.7 The Neural Network Training 45

3.8 Training, Validation and Test Samples During Network

Training 46

3.9 Training, Validation and Test Samples During Network

Training 47

4.1 Plot of TET Against Altitude 50

4.2 Plot of Tm Against Altitude 51

4.3 Plot of Max Stress Against Altitude 51

4.4 Plot of Mass Flow Against Altitude 52

4.5 Plot of Mach No Against Altitude 52

4.6 Plot of PCN Against Altitude 53

4.7 Plot of Creep Life Against Altitude 53

4.8 Plot of TET Against PCN 54

4.9 Plot of Rotor Temp Against PCN 55

4.10 Plot of Max Stress Against PCN 55

4.11 Plot of Mass Flow Against PCN 56

4.12 Plot of Altitude Against PCN 56

4.13 Plot of Mach No Against PCN 57

4.14 Plot of Creep Life Against PCN 57

4.15 Plot of TET Against Ambient Temperature 58

4.16 Plot of TET Against Rotor Temp 58

4.17 Plot of TET Against Max Stress 59

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4.18 Plot of TET Against Mass Flow 59

4.19 Plot of TET Against Creep Life 60

4.20 Enforcement of SLB Neural Creep Life Estimation Architecture 62

4.21 Percentage of Samples Encompassed Within the Specified Error

Range for Test 1 64

4.22 Percentage of Samples Encompassed Within the Specified Error

Range for Test 2 65

4.23 Percentage of Samples Encompassed within the Specified Error

Range for Test 3 65

4.24 Percentage of Samples Encompassed within the Specified Error

Range for Test 4 66

4.25 Percentage of Samples Encompassed Within the Specified Error

Range for RB, FB, SB, SLB 68

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LIST OF SYMBOLS AND ABBREVIATIONS

ANN - Artificial Neural Network

b - Bias

CL - Creep Life

CFD - Computational Fluid Dynamic

CT - Combustion Turbine

DT - Destructive Test

EAbs, - Absolute Prediction Error Percentage

f - Function

FB - Functional-Based

FF - Fuel Flow

HP - High Pressure

HCF - High Cycle Fatigue

LCF - Low Cycle Fatigue

LCMin - Minimum Creep Life

LP - Low Pressure

MSE - Mean Squared Errors

MLP - Multilayer Perceptron

NDT - Non-Destructive Test

P3 - Compressor Outlet Total Pressure

P15 - Power Turbine Outlet Total Pressure

PCN - Relative Compressor Rotational Speed

R - Regressions

RB - Range-Based

SB - Sensor-Based

SLB - Sensor Life-Based

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SP - Shaft Power

Tm - Metal Temperature

T3 - Compressor Outlet Total Temperature

T12 - Power Turbine Inlet Temperature

T15 - Power Turbine Outlet Total Temp

TET - Turbine Entry Temperature

TMF - Thermal-Mechanical Fatigue

W - Weight

Z - Targeted parameter

z^ - Output Parameter

σ - Applied Stress

UTHM - University Tun Hussein Onn Malaysia

Chapter 1

Introduction

1.1 Introduction to Gas Turbines

In the history of energy conversion, however, the gas turbine is relatively new. The

first practical gas turbine used to generate electricity ran at Neuchatel, Switzerland in

1939, and was developed by the Brown Boveri Company. The first gas turbine

powered airplane flight also took place in 1939 in Germany, using the gas turbine

developed by Hans P. von Ohain. In England, the 1930s‟ invention and development

of the aircraft gas turbine by Frank Whittle resulted in a similar British flight in

1941[1].

The name "gas turbine" is somewhat misleading, because to many it implies a

turbine engine that uses gas as its fuel. Actually a gas turbine (as shown

schematically in Figure1.1) has a compressor to draw in and compress gas (most

usually air); a combustor (or burner) to add fuel to heat the compressed air; and a

turbine to extract power from the hot air flow. The gas turbine is an internal

combustion (IC) engine employing a continuous combustion process. This differs

from the intermittent combustion occurring in Diesel and automotive IC engines.

2

Figure 1.1: Schematic for a) an Aircraft Jet Engine; and b) a Land-Based Gas

Turbine.[1]

Because the 1939 origin of the gas turbine lies simultaneously in the electric

power field and in aviation, there have been a profusion of "other names" for the gas

turbine. For electrical power generation and marine applications it is generally called

a gas turbine, also a combustion turbine (CT), a turbo shaft engine, and sometimes a

gas turbine engine. For aviation applications it is usually called a jet engine, and

various other names depending on the particular engine configuration or application,

such as: jet turbine engine; turbojet; turbofan; fanjet; and turboprop or prop jet (if it

is used to drive a propeller). The compressor combustor- turbine part of the gas

turbine is commonly termed the gas generator [1].

The advance of gas turbine engines and the increase in fuel efficiency over

the past 50 years relies on the development of high temperature materials with the

performance for the intended services. The cutaway view of an aero engine is shown

in Figure1.2. During the service of an aero engine, a multitude of material damage

such as foreign object damage, erosion, high cycle fatigue, low cycle fatigue,

3

fretting, hot corrosion/oxidation, creep, and thermo mechanical fatigue will be

induced to the components ranging from fan/compressor sections up front to high

pressure (HP) and low pressure (LP) turbine sections at the rear.

Figure1.2: Cutaway view of the Rolls-Royce Trent 900 Turbofan Engine. [2]

The endurance of the gas turbine engine to high temperature is particularly

marked by the creep resistance of HP turbine blade alloy. Figure 3 shows the trend of

firing temperature and turbine blade alloy capability[2]. Since 1950, turbine bucket

material temperature capability has advanced approximately 850°F/472°C,

approximately 20°F/10°C per year. The importance of this increase can be

appreciated by noting that an increase of 100°F/56°C in turbine firing temperature

can provide a corresponding increase of 8% to 13% in output and 2% to 4%

improvement in simple-cycle efficiency. Advances in alloys and processing, while

expensive and time-consuming, provide significant incentives through increased

power density and improved efficiency.

4

The increases in bucket alloy temperature capability accounted for the

majority of the firing temperature increase until the 1970s, when air cooling was

introduced, which decoupled firing temperature from bucket metal temperature.

Also, as the metal temperatures approached the 1600°F/870°C range, hot corrosion

of buckets became more life-limiting than strength until the introduction of

protective coatings. During the 1980s, emphasis turned toward two major areas:

improved processing to achieve greater bucket alloy capability without sacrificing

alloy corrosion resistance; and advanced highly sophisticated air-cooling technology

to achieve the firing temperature capability required for the new F generation of gas

turbine. The use of steam cooling to further increase combined-cycle efficiencies will

be realized in the 1990s [3].

The state-of-the-art turbine blade alloys are single crystal Ni-base super

alloys, which are composed of intermetallic γ‟ (Ni3Al) precipitates in a solution-

strengthened γ matrix, solidified in the [100] crystallographic direction. Turbine disc

alloys are also mostly polycrystalline Ni-base super alloys, produced by wrought or

powder metallurgy processes. Compressor materials can range from steels to

titanium alloys, depending on the cost or weight-saving concerns in land and aero

applications. Coatings are often applied to offer additional protection from thermal,

erosive and corrosive attacks. In general, the advances in gas turbine materials are

often made through thermo mechanical treatments and/or compositional changes to

suppress the failure modes found in previous services, since these materials

inevitably incur service-induced degradation, given the hostile (hot and corrosive)

operating environment.

5

Figure1.3: Increase of Firing Temperature with Respect to Turbine Blade Alloys

Development. [3]

Therefore, the potential failure mechanisms and lifetimes of gas turbine

materials are of great concern to the designers, and the hot-section components are

mostly considered to be critical components from either safety or maintenance points

of view. Because of its importance, the methodology of life prediction has been

under development for many decades. [3]

As the firing temperatures are increased and the operating cycles become

more complicated, the traditional approaches are too costly and time-consuming to

keep up with the fast pace of product turn-around for commercial competition. The

challenges in life prediction for gas turbine components indeed arise due to their

severe operating conditions:

High mechanical loads and temperatures in a high-speed corrosive

Erosive gaseous environment.

The combination of thermo mechanical loads and a hostile environment may

induce a multitude of material damages including low-cycle fatigue, creep,

fretting and oxidation.

6

Gas turbine designers need analytical methods to extrapolate the limited

material.[2].

The attempt to signified the traditional process have been done by Abdul

Ghafir[4] via soft computing technique to develop an alternative performance-based

creep life estimation method that is able to provide a quick solution to creep life

prediction while at the same time maintaining the achieved accuracy and reliability

as that of the model-based method. Using an artificial neural network, the existing

creep life prediction sub processes and secondary inputs are „absorbed‟ into simple

parallel computing units that are able to create direct mapping between various gas

turbine operating and health conditions or gas path sensors and creep life. As his

work has shown promising outcomes, it is the intention of this project to complete

his work and look for any opportunity to improvement.

1.2 Problem Statement

In depth blade life estimation takes place in the design stage, current model-based

creep life estimation methods have become more and more complicated and

therefore demand huge amounts of work and significant amount of computational

time [10, 11, 12,13]. For this reason, there are used to find an alternative solution that

can reduce complexity of the estimation creep life, to be able to perform rapid

computation and more accurate, also reduce the cost of design.

1.3 Project Objectives

Its measurable objectives are as follows:

a) To study the effects of different operating conditions on component creep life

consumption for a selected turbine engine.

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b) To construct an alternative neural-based creep life estimation model that

enables a direct link between the gas turbine operating conditions and the

components‟ corresponding life.

1.4 Project Scope

This project is primarily concerned with the creep life estimation, the scopes of this

project are:

a) Creep deformation is only considered in the research component of the life

assessment.

b) The component that‟s they studied is high pressure turbine blade.

c) ANN is used to construct the alternative creep life estimation model where

the reduction of the complexity will be done at a macro level.

d) A turbo shaft engine performance model is used to show the application of

the alternative creep life estimation method.

Chapter 2

LITERATURE REVIEW

2.1 Common Failure in the Gas Turbine Hot Section

The failure of metal parts is a complex phenomenon that depends on material,

temperature, deformation, and the rate at which strain is applied. When a metal

component breaks, two major questions need to be answered; what are the modes of

the failure and what is the origin of the damage; hence examination of the „how‟ is

essential to understand the deterioration phenomena [4].

When gas turbine hot section components are being operated at extreme

operating conditions, several damage mechanisms such as fatigue, high temperature

corrosion/oxidation, and creep deformation will inevitably emerge. The presence of

such mechanisms will cause the component to lose its ability to sustain its intended

function, increase its life consumption rate, and to some extent, will cause the

component to fail prematurely [4].

9

2.1.1 Fatigue

Fatigue, broadly speaking, is caused by repetitive loads that produce fluctuations in

the components‟ stress, hence if large enough, will cause the component to fail even

though the . Is much lower than that required for failure on single load application.

Consequently, it will lead to crack initiation and propagation which ends with a

fracture.

Fatigue, in this context, can be either mechanical or thermal-mechanical

fatigue (TMF). Mechanical fatigue is a failure occurring under cyclic loading which

is, for example, caused by vibration. on turbine blades during gas turbine start-stop

cycle and power change. Mechanical fatigue can be further divided into two: high

cycle fatigue (HCF), and low cycle fatigue (LCF). The distinction between them is

where the repetitive application of load is taking place. HCF is categorised by high

frequency and low amplitude elastic strain. An example of HCF will be when the

turbine or the compressor blade is subjected to repeated bending, such as when the

blade passes behind a stator vane, hence emerges into the gas path which will bend

the blade due to high velocity gas pressure. This will force the blades to vibrate and

the excitation at some point will match the blade‟s resonant frequency causing the

amplitude of vibration to increase significantly LCF on the other hand is categorized

by low frequency and high amplitude plastic strain. When dealing with LCF, the

yield limit of the material is often exceeded and the material becomes plastic;

therefore, repetitive plastic deformation is the main cause of LCF. Although there is

no distinct border between the two types of failure, the traditional approach is to

classify failures as HCF and those occurring below that value as LCF , TMF on the

other hand occurs when the component is not only exposed to cyclic loads but is also

experiencing variations in temperature gradient, resulting in significant thermal

expansion and contraction. According to Jacobsson, turbine blade cooling which is

used to lower the turbine blade‟s temperature will induce high temperature gradients

between the blades high and cold regions thus generating σ, and during service, the

effect of variation results in TMF [4]

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2.1.2 High Temperature Corrosion/Oxidation

Both turbine and compressor are exposed to aggressive corrosive and oxidizing

conditions that may be caused by several factors [14]:

a. Ingested air which contains sodium and chloride in the form of salt from the sea or

from runaway de-icing treatment or marine environments.

b. Atmospheric contaminants resulting from pollutions from industry or forest fires

which usually contain sulphur and sodium.

c. Volcanic activity which can generate significant levels of pollutants particularly

sulphur.Gaseous combustion products which contain elements such as sulphur,

vanadium or even lead and bromine from fuel at higher temperatures.

Although the blades have a protective coating, corrosive and oxidation

attacks are unavoidable, especially when the blades are exposed to sufficiently

contaminated surroundings or too much harmful fuel element [4].

2.1.3 Creep Deformation

Creep is a time dependent, thermally assisted deformation caused by prolonged high

operating temperatures coupled with constant mechanical loading (below the yield

stress of the material). Creep will cause the gas turbine hot section components to

„stretch‟ or elongate. Taking turbine blades for example, in the event of severe creep

deformation, the physical shape of the blades will change and hence can no longer

function properly. In addition, the elongation will cause the blades to be in contact

with the casing, causing the blades to fracture and finally lead to engine failure.

Figure 2.1 depicts several deformed turbine blades under creep attack. Note that the

blades have already lost their original features at the tip, indicating a severe creep

attack.

11

Figure 2.1: Deformed Turbine Blades Under Creep Attack. [4]

Although different materials have different strengths to resist creep

deformation, it can generally be said that creep becomes significant when the

homologous temperature (ratio between the material temperature and its melting

temperature) is more than 0.5 but it can be in the range of 0.4 to 0.6.

2.2 CREEP LIFE ESTIMATION APPROACHES

When a hot section component is put into service and operates at a creep regime, its

life will be consumed progressively as shown in Figure 2.2. This is due to the fact

that both the deformation and fracture are becoming time-dependent. The rate of

useful life consumption will depend on the ability of the material to resist creep

deformation and also the gas turbine operating condition. The more volatile the

operating condition, the faster the material will be degraded thus the quicker the

useful life will be consumed.

12

As the material degrades progressively, micro-cracks will start to initiate on

the surface and will propagate. According to Betten et al., the influence of micro-

cracks on creep behavior begins even at the primary stage and the cracks become

visible at the tertiary creep stage when the linkage of blunted micro-cracks into

macro-crack occurs. The macro-cracks will propagate before the final creep fracture

takes place [4].

Figure 2.2: Life span of hot section component due to creep deformation. [4]

There are several methods for estimating creep life. In general, these methods

can be classified into four broad approaches [4]:

a. Model-Based approach

b. Service-Based approach,

c. Statistical/Probabilistic-Based approach.

d. Soft Computing approach

It is important to note that in later sections of this thesis, some of the works related to

the life estimation of turbine blades will be given for each approach. Nevertheless

some works relating to the life estimation of other components are given as well.

13

2.2.1 Model-Based Approach

The model-based approach involves empirical, analytical or numerical model

creation in order to calculate the component‟s creep life. Depending on how the

model is created, external sources are needed to accompany the estimation such as

the component‟s geometrical, material data, and engine operating and health

conditions [4]. According to Wood [15], non-destructive test (NDT) and destructive

test (DT) should be carried out using conventional techniques when the life fraction

is greater than 0.5 (50%).

The model-based approach can further be classified into two sub-approaches

which are the total life approach and the damage tolerance approach. In the total life

approach, the life span of the component is predicted prior to the failure of the

component or at the time failure occurs. On the other hand, the damage tolerance

approach focuses on modeling the crack initiation and propagation processes. It is

important to note that in this section; only the total life approach will be discussed

since the development of the author‟s integrated creep life prediction model is based

on this approach [4].

Creep life estimation using the total life approach can be done by using either

a life-based model, a strain-based model, or a damage-based model.

2.2.2 Service-Based Approach

The service-based approach involves damage evaluation and remaining life

assessment of the service exposed component, which requires direct access to the

components. The present status of the component‟s material is positioned within the

standard scatter band either by measuring its properties, hence providing a refined

prediction or direct assessment of the extent of the damage experienced by the

component as a result of actual service exposed [4].

14

The main methods of assessing the remaining life will involve both

nondestructive tests (NDTs) and destructive tests (DTs).

2.2.3 Statistical / Probabilistic Approach

Although a statistical approach has been partially used in some of the approaches,

most of the approaches used statistics in order to assist the model development which

in general is deterministic in nature.

In a statistical/probabilistic approach, statistical/probabilistic theory is used as

the main lifing method. This approach is used either as a means to substitute the

existing model, or as a means to account for uncertainties in the influencing

parameters or variables that will affect the creep life behavior, or to provide the basis

of performing failure risk analysis [4]

2.2.4 Soft Computing Approach

Although the model-based and service-based approaches have been successfully used

over the years, limitations and complexity of the approaches have driven researchers

to use another form of creep life estimation approach. For this reason, soft computing

techniques such as ANN, fuzzy logic and evolutionary algorithm have been used.

Among these soft computing approaches, ANN has been predominantly used. In this

section some of the works which involve the application of ANN in creep modeling

and life prediction will be given [4].

15

2.3 Artificial Neural Network

2.3.1 Introduction to Artificial Neural Network

In recent years, neural computing has emerged as a practical technology with

successful applications in many field [17] such as in image processing and computer

vision, signal processing, medicine, military systems, financial systems, power

systems [18], material science etc. ANN has evolved from a biologically inspired

layer network connected by neurons into various categories of computational models

with different algorithms and a large variety of architectural design [19].

A Neural Network is a powerful data-modelling tool that is able to capture and

represent complex input/output relationships. The motivation for the development of

neural network technology stemmed from the desire to develop an artificial system

that could perform "intelligent" tasks similar to those performed by the human brain.

Neural networks resemble the human brain in the following two ways:

A neural network acquires knowledge through learning.

A neural network's knowledge is stored within inter-neuron connection

strengths known as synaptic weights [16].

An artificial neural network is a massively parallel distributed processor made up

of simple processing units (neurons), which has the ability to learn functional

dependencies from data.

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2.3.2 Advantages of Neural Networks

Adaptive learning: An ability to learn how to do tasks based on the data

given for training or initial experience.

Self-Organization: An ANN can create its own organization or

representation of the information it receives during learning time.

Real Time Operation: ANN computations may be carried out in parallel,

and special hardware devices are being designed and manufactured which

take advantage of this capability.

Fault Tolerance via Redundant Information Coding: Partial destruction of

a network leads to the corresponding degradation of performance. However,

some network capabilities may be retained even with major network damage

The procedure used to perform the learning process is called a learning

algorithm, the function of which is to modify the synaptic weights of the network in

an orderly fashion to attain a desired design objective.

Each neuron is a simple processing unit which receives some weighted data,

sums them with a bias and calculates an output to be passed on the function that the

neuron uses to calculate the output is called the activation function. Typically,

activation functions are generally non-linear having a "squashing" effect. Linear

functions are limited because the output is simply proportional to the input.

The manner in which the neurons of a neural network are structured is

intimately linked with the learning algorithm used to train the network. The most

common architecture is the multilayer perceptron (MLP). These networks are a feed

forward network where the neurons are structured in one or more hidden layers. Each

17

perceptron in one layer is connected to every perceptron on the next layer; hence

information is constantly "fed forward" from one layer to the next.

By varying the number of nodes in the hidden layer, the number of layers,

and the number of input and output nodes, one can classify points in arbitrary

dimensional space into an arbitrary number of groups.

2.3.3 Neuron Model

The multilayer perception neural network is designed from simple components.

Initially, we will describe a single input neuron which is then extended to include

multiple inputs. Next, these neurons will be stack together to produce a layer [16].

Finally, a number of layers are cascaded together thus forming a multilayer network.

Single-input neuron: A single-input neuron is shown in Figure 4.22. The

scalar input p is multiplied by the scalar weight "W" to form "Wp", one of the terms

that are sent to the summer. The other input, "1", is multiplied by a bias "b" and then

passed to the summer. The summer output "n" often referred to as the net input, goes

into a transfer function "f" which produces the scalar neuron output (sometimes

"activation function" is used rather than transfer function and offset rather than bias).

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Figure 2.3: Single Input Neuron. [16]

From Figure 2.3, both "w" and "b" are both adjustable scalar parameters of

the neuron. Typically the transfer function is chosen by the designer and then the

parameters "w" and "b" will be adjusted by some learning rule so that the neuron

input/output relationship meet some specific goal. The transfer function in Figure

4.22 may be a linear or nonlinear function of n. A particular transfer function is

chosen to satisfy some specification of the problem that the neuron is attempting to

solve. Figure 2.4 shows some common transfer functions.

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Figure 2.4: Some Common Transfer Functions.[16]

Multiple-input neuron: Typically, a neuron has more than one input. A

neuron with R inputs is shown in figure 2.5. The individual inputs 1 2p , p p g are

each weighted by corresponding elements 1,1 1,2 1,RW 燱 , .W of the weight matrix

"W". The neuron has a bias "b", which is summed with the weight inputs to form the

net input "n":

1,1 1 1,2 2 1,R Rn =W p +W p +... +W p + b (2.1)

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This expression can be written in matrix form as:

n = Wp + b (2.2)

Figure 2.5: Multiple-Input Neuron. [16]

A particular convention in assigning the indices of the elements of the weight

matrix has been adopted [16]. The first index indicates the particular neuron

destination for the weight. The second index indicates the source of the signal fed to

the neuron. Thus, the indices in "1,1W " say that this weight represents the connection

to the first (and only) neuron from the second source [16]. A multiple-input neuron

using abbreviated notation is shown in Figure 2.5.

Multiple input neurons

Figure 2.6: A Multiple-Input Neuron Using Abbreviated Notation. [16]

21

As shown in Figure 2.6, the input vector "p" is represented by the solid

vertical bar at left. The dimensions of "p" are displayed below the variable as "Rx1",

indicating that the input is a single vector of "R" elements. These inputs go to the

weight matrix "W", which has "R" columns but only one row in this single neuron

case.

A constant "1" enters the neuron as an input and is multiplied by a scalar bias

"b". The net input to the transfer function “f” is "n", which is the sum of the bias "b"

and the product "Wp". The neuron's output is a scalar in this case. If there exit more

than one neuron, the network output would be a vector [16].

2.3.4 Multilayer Perceptron Network Architectures

Generally in the most application one neuron, even with many inputs, may not be

sufficient. We might need five or ten, operating parallel, in what we will call a

“layer". This concept of a layer is discussed below.

A layer of neurons: A single-layer network of "S" neurons is shown in

Figure 2.7. Note that each of the "R" inputs is connected to each of the neurons and

that the weight matrix now has s rows.

Figure 2.7: A Single-Layer Network of S Neurons. [16]

22

The layer includes the weight matrix, the summers, the bias vector " b ", the

transfer function boxes and the output vector " a ". Each element of the input vector "

p " is connected to each neuron through the weight matrix "W". Each neuron has a

bias ib , a summer, a transfer function f and an output " ia ". Taken together, the

outputs form the output vector "a ". It is common for the number of inputs to a layer

to be different from the number of neurons (i.e. R S). The input vector elements

enter the network through the weight matrix "W":

1,1 1,

,1 ,

R

S S R

w w

w

w w

(2.3)

The row indices of the elements of matrix "W" indicate the destination

neuron associated with that weight, while the column indices indicate the source of

the input for that weight. Thus, the indices in "3,2W " say that this weight represents

the connection to the third neuron from the second source. The S-neuron, R-input,

one-layer network also can be drawn in abbreviated notation as shown in Figure 2.8.

Figure 2.8: Layer of S Neurons, Abbreviated Notation. [16]

Here again, the symbols below the variables tell that for this layer, "p" is a

vector of length "R", "W" is an "S*R" matrix and " a " and " b " are vectors of length

"S". As defined previously, the layer includes the weight matrix, the summation and

23

multiplication operations, the bias vector " b ", the transfer function boxes and the

output vector.

Multilayer perceptron (MLP): Considering a network with several layers,

each layer has its own weight matrix "W", its own bias vector "b", a net input vector

"n" and an output vector " 2a ". Some additional notation should be introduced to

distinguish between these layers. Superscripts are used to identify these layers. The

number of the layer as a superscript is appended to the names for each of these

variables. Thus, the weight matrix for the second layer is written as 2w . This

notation is used in the two-layer network shown in Figure 2.9.

There are "R" inputs, " 1S " neurons in the first layer, " 2S " neurons in the

second layer, etc. The output of layers one and two are the inputs for layers two and

three. Thus layer 2 can be viewed as a one-layer network with R= 1S inputs, S= 2S

neurons and an 1S * 2S weight matrix 2W . The input to layer 2 is 1a and the output is

2a . A layer whose output is the network output is called an output layer. The other

layers are called hidden layers [16].

Figure 2.9: Three Layer Network. [16]

24

The network shown in Figure 2.9 has an output layer (layer3) and two hidden

layers (layers 1 and 2). Figure 2.10 shows abbreviated notation of multilayer

network.

Figure 2.10: Abbreviated Notation of Multilayer Network. [16]

The network learns about the input through an interactive process of adjusting

the weights and the bias. This process is called supervised learning and the algorithm

used is the learning algorithm. One of the most common is the error back

propagation algorithm. This algorithm is based on the error-correction learning rule,

based on gradient descent in the error surface.

Basically, a set of cases, with the corresponding targets, is given to the

network. The input data is entered into the network via the input layer and is

processed through the layers - forward pass. Then, the output is compared to the

expected output (the targets) for that particular input. This results in an error value.

This error value is back propagated thought the network, against the direction of the

weights. The weights and bias are adjusted to make the actual response of the

network move closer to the desired response in a statistical sense, figure 2.11 shows

Back propagation of error.

71

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