SYNCHRONOUS GENERATOR PARAMETER IDENTIFICATION …

SYNCHRONOUS GENERATOR PARAMETER

IDENTIFICATION FROM MEASUREMENT

DATA

A dissertation submitted to The University of Manchester for the degree of

Master of Science in Electrical Power Systems Engineering in the faculty of

Engineering and Physical Sciences

2010

Ali H. Almarhoon

School of Electrical and Electronic Engineering

Synchronous Generator Parameter Identification from Measurement Data 1

LIST OF CONTENTS

List of contents...........................................................................................................................1

List of figures.............................................................................................................................4

List of tables...............................................................................................................................5

List of abbreviations...................................................................................................................6

Abstract....................................................................................................................................10

Declaration...............................................................................................................................11

Intellectual Property Statement................................................................................................12

Acknowledgments....................................................................................................................13

Chapter 1 Introduction and organisation of dissertation...............................................14

1.1 Background and motivation .......................................................................14

1.2 Aims and objectives the project..................................................................15

1.3 Literature review.........................................................................................16

1.3.1 Introduction........... ............................................................................16

1.3.2 On-line Tests......................................................................................16

1.3.2.1 Standstill Frequency Response (SSFR)......................................16

1.3.2.2 Sudden Three Phase Short Circuit Test......................................19

1.3.2.3 Numerical Impulse Method.........................................................27

1.3.3 Parameters Derivation of Power Plant Equipment.............................27

1.3.4 Summary of the Literature Review....................................................29

1.4 Dissertation organisation...….......................................................................31

Chapter 2 Modelling and simulation of synchronous machine.....................................32

2.1 Introduction................................................................................................32

2.2 Synchronous machine representation...........................................................32

2.3 Tow axes models of synchronous machines...............................................33

2.4 Per-unit notation.........................................................................................34

2.5 Park's transformation..................................................................................35

2.6 Simulation of synchronous machine...........................................................37

2.6.1 Simulation in rotor reference frame.....................................................37

2.7 Experimental data during disturbance........................................................40

2.8 Simulated data during a 3-phase short circuit test......................................41

2.9 Adding and filtering the noise.....................................................................41

2.10 Conclusion..................................................................................................43


Chapter 3 Procedures of parameters estimation in MATLAB/SIMULINK................44

3.1 Introduction.................................................................................................44

3.2 Parameter estimation procedures................................................................44

3.2.1 Creating an estimation project...........................................................44

3.2.2 Importing data into GUI....................................................................45

3.2.3 Parameter estimation.........................................................................47

3.2.3.1 Creating an estimation task........................................................47

3.2.3.2 Specifying data for parameter estimation.................................48

3.2.3.3 Specifying parameters for estimation........................................48

3.2.3.4 Starting the estimation...............................................................50

3.2.3.4.1 Specifying and selecting the solver type..............................50

3.2.3.4.2 Specifying and selecting the optimization method.............51

3.2.3.4.2.1 Cost function specification...........................................51

3.2.3.4.2.2 Optimization method specification..............................52

3.3 Parameter Estimation Flowchart.....................................................................54

3.4 Conclusion.......................................................................................................55

Chapter 4 Parameters estimation results.........................................................................56

4.1 Manufacturer data.......................................................................................56

4.2 Calculation of standard machine parameters..............................................57

4.3 Estimation of Parameters for Different Cases............................................58

4.3.1 Case1...................................................................................................58

4.3.1.1 Estimated parameters without including the effect of noise.........58

4.3.1.2 Estimated parameters with including the effect of noise..............59

4.3.2 Case2...................................................................................................59


4.3.3 Case 3..................................................................................................60


4.3.4 Case 4..................................................................................................61


4.3.5 Case 5..................................................................................................62


4.4 Discussion of the Estimated Results.............................................................62

4.5 Conclusion....................................................................................................63


Chapter 5 Project Conclusion and Further Work...........................................................64

5.1 Project Conclusion.......................................................................................64

5.2 Further Work................................................................................................65

References................................................................................................................................66

Appendices...............................................................................................................................70

Appendix A. m-file for complete synchronous machine simulation………….…70

Appendix B. list of the simulated data…………………………………………...71

Appendix C. m-file for adding noise to simulated data……………………….…78

Words (Including footnotes and endnotes): 17,253.


LIST OF FIGURES

Fig (1): (a) General d-axis circuit. (b) Simplified d-axis circuit………......................16

Fig (2): General q-axis equivalent circuit……………………………….....................17

Fig (3): (a) q-axis circuit (XqQ=Xq). (b) Simplified q-axis circuit (XqQ=XQ)……..17

Fig (4): Outputs of the actual system and the identified model………………….......18

Fig (5): Basic procedures………………………………………..…...........................20

Fig (6): Results of simulation for the 190MVA turbogenerator ………….................21

Fig (7): Results of simulation for the 32.6 MVA hydrogenerat…………………….21

Fig (8): Generator model 2.1 with one d-axis and one q-axis damper winding [22]...24

Fig (9): Generator model 2.2 with one d-axis and two q-axis damper winding

Adopted from [22]…………………………………………………………..25

Fig (10): Excitation system model [25]…….………………………………………...28

Fig (11): Schematic diagram of a synchronous generator [1]………………………..33

Fig (12): Generator model 2.1 with one d-axis and one q-axis damper winding [22].33


Adopted from [22]………………………………………………………….34

Fig (14): Block diagram of voltage park transformation.............................................36

Fig (15): Block diagram of current inverse park transformation.................................36

Fig (16): Complete simulink block diagram of synchronous generator.......................40

Fig (17): Experimental data acquisition from synchronous generator terminals [32].40

Fig (18): Filtering configuration adopted from [1]......................................................42

Fig (19): Block diagram of filtering noise....................................................................43

Fig (20): Block diagram of estimator model................................................................44

Fig (21): Control and estimation toolbox manager GUI..............................................45

Fig (22): Importing input data into the control and estimation toolbox manager........46

Fig (23): Importing output data into the control and estimation toolbox manager......46

Fig (24): The estimation task and settings...................................................................47

Fig (25): Selecting the parameters that need to be estimated......................................49

Fig (26): Selecting the parameters and setting up the initial guess.............................49

Fig (27): Different solvers available in optimization toolbox.....................................50

Fig (28): Different optimization methods available in optimization toolbox..............52

Fig (29): Estimated parameters in optimization toolbox.............................................53

Fig (30): Flowchart of parameters estimation processes [22].....................................54


LIST OF TABLES

Table (1): Results for the hydrogenerator …………………………………………...22

Table (2): Results for turbogenerator (identification with two rotor circuits)……….22

Table (3): Results for turbogenerator (identification with three rotor circuits)…..….22

Table (4): Estimated parameters for single d-axis and q-axis damper winding….….26

Table (5): Estimated parameters for damper winding D, G and Q…………………..26

Table (6): Fitted excitation system parameters………………………………………29

Table (7): Derived base quantities……………………………………………………35

Table (8): Typical machine parameters from manufacturer data………………….…56

Table (9): Standard parameters from manufacturer stability study data sheet……….56

Table (10): Formulas of standard parameters………………………………………...57

Case 1

Table (11): Estimated parameters of synchronous machine without including noise..58

Table (12): Manufacturer standard parameters vs the estimated standard

Parameters……………………………………………………………….58

Table (13): Estimated parameters of synchronous machine with including noise…..59


parameters……………………………………………………………….59

Case 2



parameters……………………………………………………………….60

Case 3



parameters……………………………………………………………….61

Case 4



parameters……………………………………………………………….61

Case 5


Table (22): Manufacturer standard parameters vs the estimated standard parameters62


LIST OF ABBREVIATIONS

Stator transformation to zero, direct and quadrature axis parameters

Stator per-phase quantities on conventional a-b-c axis

Damper winding on the direct axis of a synchronous generator

Direct Current

Digital Fault Recorder

Generator internal voltage, leading terminal voltage

Electrical Power Research Institute

Damper winding on the quadrature axis of a synchronous generator

Graphic User Interface

Instantaneous current

Stationary current, proportional to zero sequence current

Vector containing the 0dq currents

Current through stator phase a

Vector containing the abc currents

Current through stator phase b

Stator current base

Current through stator phase c

Current through direct axis

Current through damper winding D

Institute of Electrical and Electronics Engineers

Current through field winding

Current through damper winding G

Neutral Current

Independent Power Producers

Current through quadrature axis

Current through damper winding Q

Stator phase winding a self inductance

Direct axis magnetizing mutual inductance

Quadrature axis magnetizing mutual inductance

Stator inductance base

Stator phase winding b self inductance


Stator phase winding c self inductance

Direct axis leakage inductance

Direct axis leakage inductance

Equivalent direct axis inductance

Field winding leakage inductance

Field winding to damper winding D mutual leakage inductance

Damper winding G leakage inductance

Equivalent neutral inductance

Damper winding Q leakage inductance

Equivalent quadrature axis inductance

MATrix LABoratory, Software package

Maximum Likelihood

Open Circuit Characteristic

Ordinary differential equation

Output Error Estimation

On-Load Frequency Response

Active power

Park's transformation matrix

Reactive power

Stator resistance phase a

Stator resistance phase b

Stator base resistance

Stator resistance phase c

Damper winding D equivalent resistance

Field winding equivalent resistance

Damper winding G equivalent resistance

Root mean square

Damper winding Q equivalent resistance

Stator and rotor MVA base

Supervisory Control and Data Acquisition

Standstill Frequency Response

Time


Instantaneous voltage

Voltage phasor

Zero axis voltage, proportional to zero sequence voltage

Vector of 0dq voltages

Stator phase a voltage

Vector of abc voltages

Stator phase b voltage

Stator base voltage

Stator phase c voltage

Direct axis voltage

Damper winding D voltage

Damper winding Q voltage

Field winding voltage

Damper winding G voltage

Neutral voltage component

Quadrature axis voltage

Synchronous quadrature axis reactance

Vector of simulated output

Vector of experimental data

𝛿 Synchronous machine torque angle in electrical radian

Ψ Instantaneous flux linkage

Flux linkage vector of odq components

Vector of stator flux linkage

Time derivative of flux linkage phase a

Time derivative of flux linkage phase b

Time derivative of flux linkage phase c

Time derivative of flux linkage of damper winding, D

Time derivative of flux linkage of field winding

Time derivative of flux linkage of damper winding, G

Time derivative of flux linkage of damper winding, Q

𝜔 Synchronous angular frequency in radians per second

𝜔 Base synchronous angular frequency in radians per second


𝜔 Rated synchronous angular frequency in radians per second

On-Load Frequency Response

Armature base ohm (impedance), Ω

d-axis synchronous, transient and subtransient reactances

Newly defined open field d-axis subtransient reactance

q-axis synchronous, subtransient reactances

Short circuit d-axis transient and subtransient time constants, S

Open circuit d-axis transient and subtransient time constants, S

d-axis damper winding time constant, S

Open circuit q-axis subtransient time constant, S

2D Two dimension


ABSTRACT

Synchronous machines have still been the most common machines used in

generation since 40 years before. For accurate analysis of a synchronous generator,

its parameters should be identified as precise as possible. These parameters can

generally be determined either by off-line or on-line techniques. On-line test is

preferred due to technical and economical reasons.

The main aim of this project is to develop a model that can be used to estimate the

synchronous generator parameters from on-line data. Non linear least square

method has been implemented for the estimation purpose.

This dissertation is started with literature survey to overview some of the previous

research papers discussing synchronous machine parameter identification. Then, the

developed model is simulated including the effect of noise. Both modeling and

simulation is performed by using MATLAB/SIMULINK package.

The simulation outcomes, in general, show a high accuracy of estimation compared

to the original parameters provided by the manufacture. However, further work

needs to be done in order to limit the significant deviation in estimated Rfd by

considering the effect of saturation. AVR and excitation system parameters can also

be estimated in a future work.


DECLARATION

I, Ali Habib Almarhoon, confirm that no portion of the work referred to in the

dissertation has been submitted in support of an application for another degree or

qualification of this or any university or other institute of learning.


INTELLECTUAL PROPERTY STATEMENT

Certain copyright is owned by the author of this dissertation (including any

appendices and schedules to this dissertation) and has given The University of

Manchester certain rights to use such copyright, including for administrative

purpose.

Copies of this dissertation, either in full or in extracts and whether in hard or

electronic copy, may be made only in accordance with the Copyright, Designs and

Patents Act 1988 (as amended) and regulations issued under it or in accordance

with instructions and licensing agreement given by the author and The University

of Manchester. This page must form part of any such copies made.

The ownership of any Intellectual Property and any reproductions of copyright

works in the dissertation such as graphs and tables which may be described in this

dissertation may not be owned by the author and may be owned by the third parties.

Such Intellectual Property and Reproductions must not be made available for use by

third parties without the prior written permission of the author or the university,

which all terms and conditions of such agreement will be prescribed.

Further information on the conditions under which disclosure, publication and

commercialisation of this dissertation, the copyright and Intellectual Property and

reproductions described in it may take place is available in the university IP policy

and it is also available in the School of Electrical and Electronic Engineering.


ACKNOWLEDGEMENTS

I would like to express my gratitude and thanks to my project academic

supervisor, Professor J. V. Milanovic, who has advised, guided and supported me

throughout the project period. Many thanks for his reading and comments on this

dissertation.

Many thanks should be given to PhD students, Mr.Robin and Mr.Gustavo for their

help. Sincere thanks should be also given to Dr. Soon Yee from Siemens for his

help.

I also wish to thank my parents, wife and son for their encouragement and support

during the entire period of my study.

Moreover, my best friend, Zuhair Alnaser, is to be thanked for his help and support

throughout my academic life in the UK.

Finally, I owe special thanks to the government of Saudi Arabia for sponsoring me

during the entire period of my study.

Chapter 1 Introduction and Organisation of Dissertation


Chapter 1: Introduction and Organisation of Dissertation

1.1 Background and Motivation

Electrical machines consist mainly of two parts: stationary part called a stator and

rotating part called a rotor. Depending on type of power fed and produced by the

machines, they are classified into direct current machines (DC machines) and

alternating current machines (AC machines). AC machines can be categorized into

two main types: synchronous machines and asynchronous machines (induction

machines). As they present an enormous impact on system stability study,

synchronous machines have still been the most common machines used in

generation since 40 years before.

The prime motivation of this project is the high need for accurate models of

synchronous generators and an effective utilization of techniques used to estimate

the parameters of synchronous generators. This is needed for the stability study

which is highly affected by the original parameters of the synchronous generator

given by the manufacture. However, the accuracy of these parameters depends on

the age of the machine. In other words, the parameters of synchronous generator are

not fixed throughout the useful life of the machine. They are varying due to change

of the physical characteristics of the machine as its age moves forward. Moreover,

saturation effect causes some parameters like the magnetizing inductances to vary

at different operating points. Furthermore, significant changes can happen in the

generator parameters after the generator is subjected to repair or replacement of

some components. For these reasons, the accuracy of synchronous generator

parameters has been in interest of many past and recent paper and researches.

This project is further motivated by the need of development and modifications of a

synchronous generator model designed by a previous work done in the electrical

engineering department at the University of Manchester in 2008. The developed

model has shown some leakage of accuracy in estimation the parameters of

synchronous machines and hence it is not accurate enough to be used in

synchronous generator dynamic and stability study. Therefore, it is required to

perform some modifications by considering the effect of noise and magnetic

saturation.



According to [1] many methods had been developed in the period between1969

and 1971 to estimate the parameters values of synchronous machines based on

models developed by Dandeno, Suchuzl and Dineley. A direct and quadrature axis

equivalent circuit for round rotor synchronous generators had been developed by

Jackson and Winchester. Simultaneously, Equivalent circuits for field and damper

windings had been developed by Canay. In 1971, Yu and Mosa reported a systemic

procedure applied to estimate the parameters of the synchronous generator [2].

The parameters of synchronous generators can be generally determined either by

off-line or on-line techniques. The off-line methods require interruption of service

during the test. Moreover, they are impractical and inaccurate since they are not

used under normal operating condition and the saturation effect cannot be

considered on these methods. Thus, they may not be preferred especially in case of

large synchronous generators used as base unit. This makes the on-line methods

more attractive from technical and economical point view.

1.2 Aims and Objectives of the Project

The main aims and objectives of this project can be outlined as follows:

To study and understand the behavior of the synchronous machine.

To implement a methodology for identification of machine parameters based on

continuous monitoring (without staged tests and disconnection of the machine

of the network) of machine output variables such as voltage, current, speed, etc.

This methodology has been modified in order to be accurately applied.

To develop a model using a least square algorithm from MATLAB

optimization toolbox.

To simulate the model in order to identify the synchronous generator

parameters and compare it with actual parameters.The modelling and

simulation will be completely done by using MATLAB/SIMULINK package.



1.3 Literature Review

1.3.1 Introduction

The aim of this review is to examine some recent and relevant literature for

synchronous generator parameter identification based on measurement data. The on

line different tests are described and compared with each other.

1.3.2 On line Tests

In general, the methods used to identify parameters of synchronous machines can

be classified as follows.

* Standstill frequency response (SSFR)

* Sudden Short circuit test.

* Numerical impulse method.

1.3.2.1 Standstill Frequency Response

Modelling and identification of the synchronous machine have been conventionally

used in IEEE standards 115, part II [3]. In 1971, a systematic procedure was

reported by [2] in order to find the synchronous machine parameters depending on

simple field test. The general d-axis equivalent circuit of a synchronous machine

shown in Fig (1a) was simplified as shown in Fig (1b). Moreover, the general q-

axis equivalent circuit shown in Fig (2) was simplified as shown in Fig (3).

Fig (1): (a) General d-axis circuit [2]. (b) Simplified d-axis circuit [2].



Fig (2): General q-axis equivalent circuit [2].

Fig (3): (a) q-axis circuit (XqQ=Xq) [2]. (b) Simplified q-axis circuit

(XqQ=XQ) [2].

Mathematical equations were proposed in order to identify the parameters of the

simplified circuits based on field tests. The eight conventional d-axis parameters (

) can be determined from IEEE test code

described in IEEE standards [3]. Nevertheless, an extra test was suggested by [2] to

measure a newly defined parameter ( ) by Dalton and Cameron’s method with

the field winding left open and a damper time constant ( ) can be measured by

varying slip test or decaying current test [2]. At that time, this method was not

applied to large synchronous machine.

In 1981, two large turbo generators named as Nanticoke and Lambton generators

were used to test an identification technique developed by Dandeno, P.L. et al [4].

Those two generators were modelled by using the transfer functions measured

during on-load frequency response (OLFR) and this model was exactly matching

the one obtained by standstill frequency response (SSFR). [4] Proved that the

existence of continues damper winding under continues rotor slot wedges produces

a countable difference between OLFR and SSFR rotor parameters. Furthermore, the

effect of damper winding dynamics is very important especially in sub transient

studies for a single generator. The major difficulty with the damper winding is that

its currents are not available for measurement. Therefore, an algorithm was

proposed by Said, S.A. et al [5] in order to solve this problem. This algorithm

calculates the electric parameter of the stator by implementing the synchronous

machine equations in steady state where the damper currents can be neglected.



Then, the parameters of field and damper windings can be calculated from the

estimated d-axis and q-axis damper currents. A synchronous generator connected to

an infinite busbar electric network and a limited load was used to test the

performance of this algorithm. The results obtained in this test had shown a very

close matching between the identified model and the actual system performance.

See Fig (4).

q-axis currents d-axis currents

Output power Terminal voltage

Fig (4): Outputs of the actual system and the identified model [5].

In 1981, a standstill frequency response test was developed by Coultes, M.E. et al

[6]. The procedure of this test was based on low voltage frequency response

measurements taken from the stator and rotor terminals with fixed rotor. Further

development of these procedures was made by [7] in order to overcome the

disagreements by modifying the model with both stator and rotor values. This

development has also shown a good agreement with [4] regarding the complexity of

modelling the synchronous machines with damper winding.

The maximum likelihood (ML) technique was used in 1989 to estimate the

parameters of the solid rotor linear machine from noise corrupted data. The

technique was applied to the SSFR or time domain test data. Excluding the

saturation effect, ML algorithm was presented to be a very accurate estimation



method involving noise data [8]. In 1994, a direct comparison between the

measured standstill and on-line responses was carried out by [9] on a 5KVA three

phase salient pole synchronous machine and the validation of both the time domain

and the ML estimation was approved. Similar rating machine was used to perform

an on line model identification steps using the ML technique and the small

disturbance responses [10]. Saturation effect was considered in estimating the

mutual inductance. The simplicity of the small disturbance test was shown as there

was no great impact to the interconnected power system during the test.

In 2003, a nonlinear mapping based modelling method was designed to estimate

the parameters of large machine from on-line data [11]. A 460-MVA large steam

turbine was used to test the method. Linear model armature circuit and field

winding parameters were first estimated by using data from small excitation

disturbances. Then, nonlinear mapping functions-based estimators were used to

identify the saturated inductances ( ). Finally, an output error method

(OEM) was implemented to estimate the rotor body parameters. The final

simulation results of this paper have proved that the estimated parameters

outperform data supplied by the manufacturer [11].

1.3.2.2 Sudden three-phase short circuit test

Although it is ideal for process identification, SSFR may not be practical under

some conditions such as the adaptation of linear transfer function parameters for

use in generator models operating rather than standstill condition. Instead, a sudden

three-phase short circuit test is commonly used to estimate the dynamic parameters

of the synchronous machine.

An approach of obtaining synchronous machine d and q axis impedances was

suggested based on the concept of line to line short circuit [12]. The short circuit

test is done by applying line to line short circuit to a machine running at reduced

speed while the rotor is excited to produce line to line short circuit current at

fundamental frequency. In addition to the rotor angle, the line voltage and short

circuit current are recorded in order to compute the operational inductances or

impedances. The major advantage of this technique is that it can be used to

determine the machine characteristics at subsynchronous and supersynchronous

frequencies [12]. However, the application of line to line short circuit may lead the



machine out of the operating limit due to dielectric or mechanical stress. Therefore,

a simple and hazardless test procedure was suggested by de Mello, F.P. et al [13].

In this test, the synchronous machine parameter can be derived by tripping the

breaker of loaded generator. The test involves measurement of voltage and field

current transient deviations under no load condition. The results of the simulation

had shown an important technique of determining the q and d axis supposed to help

the industry in resolving the adequacy of machine modelling methods for system

dynamic studies [13]. The q-axis components identification were considered in

details by a further research done by de Mello, F.P. et al [14] although the effect of

saturation was neglected.

Based on three phase short circuit test, a fully automated software was developed

by Simond, J.J. et al [15] to determine the sub-subtransient, subtranisient and

transient parameters of large synchronous machine. Fig (5) shows the main steps

for the software. First, the characteristics reactance’s and time constants of the

machine are identified based on the phase and excitation currents during the three

phase short circuit test. The corresponding equivalent circuit diagram is also

determined according to the theory of the synchronous machine. Then, this

equivalent circuit is converted to a simulation program. Finally, the same three

phase short circuit test is done by a numerical simulation and the results of the

simulation are compared with the measured values taken in field.

Fig (5): Basic procedures [15].



To test the performance of this software, a large hydrogenerator and turbogenerator

were used. The characteristics quantities and simulation results compared to

measurements for the 190MVA turbogenerator are shown in fig (6). In the other

hand, fig (7) shows the characteristics values and the elements of the equivalent

circuit of the 32.6 MVA, 10.5KV hydrogenerator. A comparison between the

simulated and the measured values is also shown in fig (7).

Fig (6): Results of simulation for the 190MVA turbogenerator [15].

Fig (7): Results of simulation for the 32.6 MVA hydrogenerator [15].



Both tests have shown the effective performance of the developed software. The

main advantage of this method appears in the intrinsic time saving, the higher

accuracy of the results [15].

In another research, Simond, J.J. et al [16] used a 2D Finite Element to determine

the parameters of the same rated large machines based on simulations of no load

sudden three phase short circuit test. The saturation effects and the eddy currents in

the rotor solid iron parts were considered in the simulation. The results obtained for

both machines are shown in the following tables. The measured values are included

for comparison.

Table (1): Results for the hydrogenerator [16]

Simulated Data Measured Data

1 0.3 0.5 0.3 0.15

0.905 1 1 1 1

0.3595 0.4004 0.3777 0.4051 0.4056

0.2403 0.2777 0.2550 0.2722 0.2738

1.1002 1.2914 1.3897 1.5838 1.6504

0.02988 0.03359 0.02724 0.03172 0.03397

Table (2): Results for turbogenerator (identification with two rotor circuits) [16]


1 0.25 0.7 0.5 0.35 0.2

2.09 2.15 2.085 2.15 2.15 2.15

0.246 0.232 0.257 0.250 0.254 0.2160

0.154 0.190 0.179 0.187 0.196 0.215

1.606 1.519 1.299 1.298 1.327 1.415

0.191 0.109 0.151 0.090 0.094 1.160

Table (3): Results for turbogenerator (identification with three rotor circuits) [16]


1 0.25 0.7 0.5 0.35 0.20

2.084 2.15 2.085 2.15 2.15 2.15

0.246 0.232 0.257 0.250 0.254 0.260

0.178 0.220 0.200 0.219 0.222 0.228

0.145 0.186 0.141 0.162 0.170 0.180

1.606 1.519 1.299 1.298 1.327 1.415

0.258 0.296 0.214 0.209 0.185 0.224

0.0764 0.0725 0.0240 0.0258 0.0234 0.0224



According to the tables, an outstanding concordance has been shown with the

measurements in case of the large hydrogenerator. However, the accuracy for the

obtained time constant is not so acceptable in case of the turbogenerator. This

inaccuracy is may be due to the not optimal dimension of the mesh analysis in finite

element [16].

A comparison between results of tests was made by [17] on three large

turbogenerator with different rotor construction. Standstill frequency response

(SSFR), on-line frequency response (OLFR) and three phase short circuit tests were

all implemented. As observed in another research [4], Dandeno, P.L. et al [17]

observed that it is more difficult to obtain the equivalent circuit of d and q axis in

case of more complex rotor construction. He also concluded that the standard

model based on the manufactures data is not enough for simulating the dynamic

responses.

An another comparison between standstill frequency response and three phase short

circuit tests was done by Simond, J.J. et al [18] using 2D finite element design.

Without considering the saturation effect, a good agreement between the two tests

was obtained for laminated salient-pole synchronous machines [18].

Although it has been usually used at no load, three phase short circuit test requires

large equipment and therefore it is expensive and risky especially for voltages

higher than 60% of nominal voltage For these reasons, a DC decay test has been

used as an alternative test as it requires light equipment. This test produces the

characteristic values of synchronous machine in the d and q axes. It is done for the

two intense positions of the rotor two axes [19].

A technique based on least-squares estimation was presented by Kyriakides, E. et al

[20]. An observer was designed to measure the damper currents and use them in the

parameter estimation. Two cases were studied in this paper. A good agreement

between the damper current and the simulation result was shown in the case of d-

axis winding unlike the case of q- axis winding where small difference between the

estimated and the simulated currents was noticed [20]. Further to this paper,

Kyriakides, E. et al [21] used the observer estimator in a Graphic User Interface

(GUI) application. A Visual C++ engine and GUI were both used so that the on-line



measurement can be linked with the estimator. It was shown that the accuracy of

estimation is still acceptable even when multiple parameters are estimated.

Recently, a master thesis about synchronous generator parameters identification has

been written by Nizam, I. [22] for the University of Manchester. The author has

developed a complete Simulink model of synchronous generator including the

damper windings. The parameters have been expressed in per unit system in order

to make it easier to compare between different rated machines. Park’s

transformation has been applied to transform all stator quantities to equivalent dq

quantities. Thus, the generator model can be given as shown in the following matrix

[21].

IEEE Standard models 2.1 and 2.2 shown in figure (8) have been used to represent

the synchronous generator modelling.

Fig (8): Generator model 2.1 with one d-axis and one q-axis damper winding [22].




adopted from [22].

Where;

: Stator winding d-axis and q-axis leakage inductances respectively.

: Direct & quadrature axis stator-rotor mutual inductances respectively.

: Field resistance and leakage inductance referred to stator respectively.

: Direct axis damper winding D, resistance and leakage inductance

respectively.

: Quadrature axis damper winding G, resistance and leakage inductance

respectively.

: Quadrature axis damper winding Q, resistance and leakage inductance

respectively.

: Direct axis field –damper mutual leakage inductance.

A balanced load operation has been assumed in the simulation and hence zero

sequence voltages and currents have been ignored. The second order model has

been considered due to a simulation limits and the effect of d-axis field damper

mutual leakage inductance has not been considered. An observer model has been

designed to calculate the damper currents [22].

In this project [22], a 158MVA, 13.8KV and 3600 rpm synchronous machine has

been used to evaluate the developed model. The estimation process has been done

by using the nonlionear least square algorithm from Matlab optimization toolbox.

Table (4) shows the estimation results for single d-axis and q-axis damper winding.



Table (4): Estimated parameters for single d-axis and q-axis damper winding [22]

Deviation (%) Estimated Value (p.u) Initial Value (p.u) Parameter

-17.42 1.3543 1.64

-0.41 1.5536 1.56

0.56 0.160889 0.16

-1.86 0.15702 0.16

76.95 0.0081396 0.0046

1.15 0.00098341 0.0009722

133.81 0.27568 0.11791

Table (4) shows an accurate estimation for while an accounted

difference is obtained for . This high difference is assumed to be due to

neglecting the saturation effect.

The estimation process has been repeated by including the damper windings D,G

and Q. The results are shown in Table (5).

Table (5): Estimated parameters for damper winding D, G and Q[22].

Deviation (%) Estimated Value

(p.u)

Initial Value

(p.u)

Parameter

25.47 2.057700 1.64

-3.26 1.509100 1.56

1.81 0.162894 0.16

-12.30 0.140320 0.16

-16.94 0.003821 0.0046

2.88 0.00099998 0.000972

37.12 0.352410 0.257

Except of , all parameters have shown a very close value to the

initial value. However, the simulation outcomes, in general, are not accurate

enough to be used in synchronous generator dynamic and stability study. Therefore,

it has been concluded that the developed model needs to be modified in order to get

more accurate results [22]. This modification can be done by considering the effect

of noise and magnetic saturation.



1.3.2.3 Numerical Impulse Method

In a numerical impulse method, the obtained parameters describe behaviour of the

machine at the certain operation point. This is an advantage of the numerical

impulse method over the SSFR and the sudden short circuit tests. For this reason,

Olli Makela [23] has chosen the numerical impulse method to estimate two-axis

model parameters for a synchronous machine in his Master degree project where he

used data from linear and nonlinear finite element models to estimate the

parameters. It was noticed that saturation has no big effect when an impulse with

amplitude of 1% of the average RMS value of the line voltages is used.

1.3.3 Parameters Derivation of Power Plant Equipment

In power plant, the generation system is composed of the synchronous generators,

their excitation system and the turbine-governor. Therefore, modeling of

synchronous machine is affected by the modeling of both excitation control system

and turbine-governor control. Similar to the synchronous generator, the excitation

system and turbine-governor control system can be tested either by off-line or on-

line tests.

The excitation system should be tested in conjunction with the commissioning as

the manufactures’ data and manufactures’ representative are available at that time.

Off-line tests are conducted on each part within the excitation system while it is

isolated from field winding and fed by test supplier. On-line tests, in the other hand,

are conducted with the generator synchronized to the network and running at a

range of active and reactive power loadings. The excitation system can also be

tested while the generator is open circuited operating at rated speed and rated

voltage. Automatic voltage regulator (AVR) step response test may be considered

as the most common type of open circuit test. This test is carried out by slightly

changing the AVR reference level for short period. As a result of this change, the

generator terminal voltage will change abruptly and gives a good measure of the

whole response of the excitation system. Another way of open circuit testing is

load rejection with the unit absorbing reactive power. Unlike the AVR step

response, this method doesn’t need any equipment for changing AVR reference

level [24].



Pourbeik, P [25], presented a technique for fitting parameters to power plant

equipment based on off-line tests. The methodology was applied on both brushless

and static exciters. A good agreement between the original data and the measured

values has been shown in most cases. The excitation system model used in this

research is shown in figure (10) below.

Fig (10): Excitation system model [25].

In another paper, Pourbeik, P [26], proposed a novel automated technique for

fitting parameters of power plant equipment based on on-line system disturbances.

A 560MVA and 496MVA large steam-turbine generators in the North American

power system were used to test this method. The test was conducted based on five

loss of generation events occurred due to faults during the period between May to

November 2008. Measurement of speed, field current and field voltage of the

generation system in each event were used to apply the proposed algorithm. After a

number of iterations, the algorithm converged to a very good fit between identified

parameters and the original ones. Table (6) displays identified parameters for some

events in comparison with the original equipment manufacturer [26].



Table (6): Fitted excitation system parameters [26]

Parameter Description OEM Fit

Event 1

Fit

Event 2

Fit

Event 3

Fit 2

Event 2

Transducer Time Constant 0 0.02 0.02 0.02 0.024

AVR Proportional Gain 4.83 4.83 4.83 4.83 4.36

AVR Integral Gain 4.83 4.83 4.83 4.83 3.32

AVR Time Constant 0.01 0.01 0.01 0.01 0.01

Field Voltage Feedback

Gain

0 0 0 0 0

Vfd Feedback Loop P-Gain 1 1 1 1 1

Vfd Feedback Loop I-Gain 0 0 0 0 0

Potential Forcing Gain 6.21 6.21 6.21 6.21 5..74

Forcing Angle 0 0 0 0 0

Current Forcing Gain 0 0 0 0 0

Leakage Reactance 0 0 0 0 0

Communication Loss 0.09 0.09 0.09 0.09 0.05

This research has shown the capability to apply models against real events rather

than against staged tests which were used in the previous work [25].

1.3.4 Summary of the Literature Review

This review has examined some past and recent works and papers published in the

field of synchronous generator parameter identification. The most common on-line

tests such as standstill frequency response (SSFR) and sudden short circuit tests

have been particularly considered and the numerical impulse method has been

briefly described.

The main observations and conclusions can be outlined in the following points:

Since the saturation effect cannot be considered in the off-line test methods and

because of the inaccuracy of this test, the on-line test techniques may be

preferred due to its technical and economical sides.

The existence of continues damper winding under continues rotor slot wedges

produces a countable difference between on-load frequency response (OLFR)

and (SSFR) rotor parameters.

An algorithm that has been used to solve damper winding problem in sub

transient studies for a single generator has an accurate identification results.



Compared to (OLFR), low voltage frequency response, maximum likelihood

and the small disturbance, the nonlinear mapping based modelling method has

shown that the estimated parameters outperform the data supplied by the

manufacturer.

Compared to (SSFR), the sudden three-phase short circuit test is preferred to be

used to estimate the dynamic parameters of the synchronous machine due to its

ability to work in different conditions of the generator operation.

Line to line short circuit technique has an advantage that it is used to determine

the machine characteristics at sub-synchronous and super-synchronous

frequencies.

With no consideration of the effect of the noise and magnetic saturation, the

least square estimation method can show an inaccurate simulation results.

The numerical impulse method has an advantage over (SSFR) and sudden short

circuit tests which describes the behaviour of the machine at the certain

operation point.

In this review, a recent master thesis about synchronous generator parameters

identification written by Nizam, I. [22] has been considered in details. The project

(thesis) has developed a complete Simulink model of synchronous generator

including the damper windings. A 158MVA, 13.8KV and 3600 rpm synchronous

machine has been used to evaluate the developed model. The estimation process

has been done by using the nonlionear least square algorithm from Matlab

optimization toolbox. The results of simulation have shown that the developed

model needs to be modified in order to get more accurate results. This modification

can be done by considering the effect of noise and magnetic saturation.



1.4 Dissertation Organisation

The thesis is structured as follows:

Chapter two describes the modelling and the simulation of synchronous

machine. Development equations of synchronous machine, simulated data

during a 3-phase short circuit and filtering the noise are presented.

Procedures of parameter estimation are covered in chapter three. The

parameters are estimated by using non linear least squares method from

optimization toolbox in MATLAB environment.

In chapter foure, results of parameters estimation are presented.

The results are discussed and evaluated in chapter five. Further work is finally

suggested.

Chapter 2 Modeling and Simulation of Synchronous Machine


Chapter 2: Modeling and Simulation of Synchronous Machine

2.1 Introduction

This chapter presents the representation and models' selection of the synchronous

machines as well as the detailed development of the synchronous machine

equations that will be used in the simulation is presented. The measurements data

during a three-phae short circuit and filtering the noise are also presented.

2.2 Synchronous Machine Representation

A conventional three phase synchronous machine consists of two parts; stationary

part called a stator and rotating part called a rotor. The stator has three-phase

windings that are 120 electrical degrees a part and the rotor has an excitation

winding which DC supply with variable number of damper windings in the direct

and the quadrature axis can be received by the excitation winding. The foundation

of synchronous machine with detailed theory can be found in [27].

The operation of synchronous machine can be represented by the following voltage

equations that can be found as [28]:

(2.1)

(2.2)

(2.3)

(2.4)

(2.5)

(2.6)

(2.7)

Stator and rotor circuit of synchronous generator are shown in figure (11).



Fig (11): Schematic diagram of a synchronous generator [1].

2.3 Two Axes Models of Synchronous Machines

It is necessary to employ a mathematical model in order to formulate the state

estimation equation for a synchronous generator. Depending on the type of study

that is desired to be performed, there are various practical models available for

synchronous generators. The number of rotor circuits in the direct and the

quadarture axes prescribe the order of a synchronous generator model. For stability

studies and representation of various types of generators, lower order models are

often used [1]. Different recommended synchronous generator models are

suggested in IEEE standard, such as models 2.1 and 2.2 and theses models can be

shown in Figs (12) and (13) respectively.

Fig (12): Generator model 2.1 with one d-axis and one q-axis damper winding [22].




adopted from [22].

A balanced load operation has been assumed in the simulation and hence zero

sequence voltages and currents have been ignored. The second order model has

been considered due to a simulation limits and the effect of d-axis field damper

mutual leakage inductance has not been considered and due to its accurate

modeling of quadrature axis [1] [22].

2.4 Per-Unit Notation

The per-unit representation of synchronous machine can be used to normalize the

variables of the machine. In addition, the per-unit system offers computational

simplicity by eliminating units and expressing system quantities as dimensionless

ratios compared to the use of physical units (amperes, volts, ohms, webers, henrys,

etc.) [27]. In this project, the parameters have been expressed in per-unit system in

order to make it easier to compare between different rated machines [29]. In order

to derive the other base quantities that have been defined below in table (7), the

base power Sbase, the base voltage Vbase and synchronous angular frequency

Wbase should be specified. Same base power Sbase is selected in order to calculate

the per-unit quantities for rotor but the voltage and the current are referred to

mutual flux linkage. Moreover, more details about conversion to per-unit quantities

can be found in [29].



Table (7): Derived base quantities

Quantity Formula

Base Current

Base Resistance

Base Inductance

𝜔

Base Flux

2.5 Park's Transformation

The time-varying inductances can be eliminated when the changes of variables are

used in the analysis of ac machines. Changes of variables are also needed in the

analysis of constant parameter power-system components and control systems

associated with electric drives. Indeed, all known real transformations for these

components and controls are contained in the transformation to the arbitrary

reference frame. The same general real transformation can be used for the stator

variables of the induction and synchronous machines and for the rotor variables of

induction machines. One transformation to the arbitrary reference frame can be

formulated which could be applied for all variables [30].

A transformation of the 3-phase variables of stationary circuit elements to the

arbitrary reference frame for a change of variable may be expressed as [30]:

(2.8)

Where:

(2.9)

(2.10)

(2.11)



𝜔

The inverse transformation can be shown in the following equation:

(2.13)

Based on the formulas mentioned above, Simulink models for Voltage and Current

transformations have been designed as shown in figures (14) and (15) respectively.

Fig (14): Block diagram of voltage park transformation (Vabc to Vdqo).

Fig (15): Block diagram of current inverse park transformation (Idqo to Iabc).



2.6 Simulation of Synchronous Machine

The computer simulation for synchronous machine is divided into two types of

simulations. The most common used of simulation derived from the voltage

equations expressed in the rotor reference frame with an arrangement of equations

in the same form of the equations that are used in the induction machine. This kind

of simulation was done by C. H. Thomas [31]. The second type of simulation is that

the stator flux linkages per second are calculated in the arbitrary reference frame

with the rotor flux linkages per second computed in the rotor reference frame [30].

In this dissertation the first type of simulation is only used to be done in

MATLAB/SIMULINK by using the available blocks that have been provided in the

toolbox library.

2.6.1 Simulation in Rotor Reference Frame

The voltage equations expressed in the rotor reference frame are given by [30]:

(2.14)

(2.15)

(2.16)

(2.17)

(2.18)

(2.19)

(2.20)

The equations defining the flux linkages per second are as follows [30]:

(2.21)

(2.22)

(2.23)



(2.24)

(2.25)

(2.26)

(2.27)

The voltage and flux linkage equations can be manipulated in order to obtain

computer simulation. The resulting integral equations are defined as (2.28)-(2.45)

[30]:

(2.28)

(2.29)

(2.30)

(2.31)

(2.32)

(2.33)

(2.34)

Where:

(2.35)

(2.36)

(2.37)

(2.38)

(2.39)



(2.40)

(2.41)

(2.42)

(2.43)

(2.44)

(2.45)

Since the saturation is not taken into account, the torque equation that is used in the

simulation is given by:

(2.46)

And the rotor speed is expressed as:

𝜔

(2.47)

Block diagram showing the computer simulation of the synchronous machine in the

rotor reference frame using Matlab / Simulink is shown in figure (16). The

MATLAB's m-file for the simulation is attached in Appendix A. In general, the

voltages applied to the damper winding are not shown in the block diagram because

the damper windings are always short-circuited and the voltages are zero [30].



Fig (16): Complete simulink block diagram of synchronous generator.

2.7 Experimental Data during Disturbance

Terminal voltages and current of synchronous generator are known as stator

measurements while field winding voltage and current are known as rotor

measurements. Bothe the stator and the rotor measurements can be recorded as

shown in figure (17) [32].

Fig (17): Experimental data acquisition from synchronous generator terminals [32].

Digital Fault Recorder (DFR) can read the experimental data directly from

synchronous generator control panel.



2.8 Simulated Data during a 3-Phase Short Circuit Test

Although it is ideal for process identification, SSFR may not be practical under

some conditions such as the adaptation of linear transfer function parameters for

use in generator models operating rather than standstill condition. Instead, a sudden

three-phase short circuit test is commonly used to estimate the dynamic parameters

of the synchronous machine.

An approach of obtaining synchronous machine d and q axis impedances was

suggested based on the concept of line to line short circuit [12]. The short circuit

test is done by applying line to line short circuit to a machine running at reduced

speed while the rotor is excited to produce line to line short circuit current at

fundamental frequency. In addition to the rotor angle, the line voltage and short

circuit current are recorded in order to compute the operational inductances or

impedances. The major advantage of this technique is that it can be used to

determine the machine characteristics at subsynchronous and supersynchronous

frequencies but the application of line to line short circuit may lead the machine out

of the operating limit due to dielectric or mechanical stress [12]. However, the 3-

phase short circuit was applied across the machine in this project. High load impact

is presented by this sudden short circuit in order to excite the damper windings

[33].

The simulated data were recorded and attached in Appendix B.

2.9 Adding and Filtering the Noise

The noise has been added to the simulated data of the estimator by typing a

MATLAB code in order to make them as realistic data and this code can be shown

in Appendix C. The noise of the data should be filtered and prepared in a form that

can be used by the estimator.

In order to prepare these data, there are many processes that need to be performed

between the data acquisition and the estimator implementation. The filtering of

simulated data to remove inconsistent measurements and noise is the most

fundamental process. In reality, there are different filters that have been developed

and implemented in order to filter the noise. In this project, the digital discrete



filters are considered and the types of theses filters are classified into; the

Butterworth, Chebyshev, Bassel and Moving average filters [1] [34].

A phase shift to the filtered signals is almost introduced. Since this phase shift is

not desired, a zero phase shift filter is needed to provide no phase difference

between the original and filtered signals. The signal in both the forward and the

reserve directions can be filtered by zero phase digital filters [1]. In Reference [35]

more information about zero shift filters can be found.

In this project, a low pass filter is necessary to be employed whose cut off

frequency is selected to maintain the dynamics of the signals in both steady state

and transient conditions. The types of digital discrete filters that have been

mentioned previously can be considered as low pass filters. The fastest digital filter

available and it has good smoothing in time domain is the moving average filter but

it has a slow roll off. Similar characteristics to butterworth filters can be found in

chebyshev and elliptic filters but they have considerable ripple in their passband.

Therefore, the butterworth filter has been used due to its good transient response

and fast roll off [1]. Figure (18) shows the configuration of the filtering while figure

(19) shows the simulink model of filtering the noise.

Fig (18): Filtering configuration adopted from [1].



Fig (19): Block diagram of filtering noise.

2.10 Conclusion

Various recommended IEEE models and development equations for synchronous

machines are presented in this chapter. A Simulink model for synchronous

generator has been designed based on the development equations. Main parts of

modeling and simulation of synchronous machine have been individually

considered. The function of each part has been built. By the end of this chapter, the

synchronous generator model is ready for simulation.

Chapter 3 Procedures of Parameters Estimation in MATLAB/SIMULINK


Chapter 3: Procedures of Parameters Estimation in MATLAB/SIMULINK

3.1 Introduction

The main objective of this project is to develop a method & a model that can be

used to estimate the synchronous generator parameters fitting with the

measurements parameters. In order to achieve this objective, the method and the

model will be done in MATLAB/SIMULINK package.

3.2 Parameters Estimation Procedures

The parameters estimation process of the proposed model of the synchronous

generator has been done by using complete model of Synchronous Machine which

is given in figure (16), estimator model that have been built up to be used for the

estimation as shown in figure (20) and Optimization Toolbox (GUI) in MATLAB

by implementing the steps below.

Fig (20): Block diagram of estimator model.

3.2.1 Creating an Estimation Project

Before we start importing data, an estimation project must be created and set up by

configuring the appropriate parameters, solvers, and the cost functions. A Graphical

User Interface (GUI) is provided by Simulink Optimization Software that makes

setting up the estimation project quick and easy.



An estimation project can be created by the following steps:

Open synchronous generator and estimator models.

Open the Control and Estimation Tools Manager GUI by selecting Tools ˃

Parameter Estimation in the simulink model window [36].

The Control and Estimation Tools Manager GUI is depicted in figure (21).

Fig (21): Control and Estimation Tools Manager GUI.

3.2.2 Importing Data into the GUI

After creating an estimation project, the estimation data can be imported. In order

to import transient (measured) data for the estimator model, many steps need to be

implemented as follows:

In the Control and Estimation Tools Manager, select Transient Data under the

Estimation Task node of the Workspace tree.

Right-click Transient and select New to create New Data node.

Select New Data node under the Transient Data node.

Import input and output data from data import dialog box.

Select Time / Ts cell from dialog box.



Import the time vector for the input and output data [36].

Importing input and output data into the Control and Estimation Tools Manager are

shown in figures (22) and (23) respectively.

Fig (22): Importing input data into the Control and Estimation Tools Manager GUI.

Fig (23): Importing output data into the Control and Estimation Tools Manager

GUI.



3.2.3 Parameter Estimation

When the model parameters are estimated, the measured data are compared with

the data generated using a simulink model by Simulink Design Optimization

Software. By using optimization techniques, the parameters and initial conditions of

states are estimated by the software in order to minimize a user-selected cost

function. The cost function typically calculates a least-square error between the

empirical and model data signal [36].

After importing and processing the estimation data, the following steps should be

followed to estimate model parameters:

3.2.3.1 Creating an Estimation Task

An estimation task is created and the estimation settings are configured as follows:

In the Control and Estimation Tools Manager, right-click the estimation node in

the Workspace tree and select New.

Select the New Estimation node [36].

Figure (24) shows the estimation task and settings.

Fig (24): The estimation task and settings.



3.2.3.2 Specifying Data for Parameter Estimation

To specify a data set for estimation, the data must be imported in the GUI and an

Estimation Task must be created.

The steps of specifying data are as follows:

Select the selected check box to the right of the New Data of data set.

Specify the weight of each output from the model by setting the weight column

in the output data weights table.

Use less weight when an output is noisy.

Use more weight when an output strongly affects parameters [36].

3.2.3.3 Specifying Parameters for Estimation

Simulink Design Optimization software lets you estimate scalar, vector and matrix

parameters. Estimating model parameter is an iterative process. Usually, it is more

practical to estimate a small group of parameters and use the final estimated values

as a starting point for further estimation of parameters that are more difficult.

However, if a large number of parameters need to be estimated, the parameters that

influence the output should be estimated [36].

The parameters for estimation in the GUI are specified by implementing the

following steps:

In the Control and Estimation Tools Manger, select the Variables node in the

Workspace tree to open the estimated parameters part.

In the estimated parameters pane, click Add to open the select parameters

dialog box.

Select the parameters that need to be estimated and then click OK as shown in

figure (25).

In the New Estimation node of the Control and Estimation Tools Manager GUI,

select the parameters tab and select the parameters that need to be estimated.



Enter the initial values for the parameters in the Initial Guess column.

Specifying the Upper/Lower bounds [36].

Selecting the parameters and setting up the initial guess and upper/lower bounds are

shown in figure (26).

Fig (25): Selecting the parameters that need to be estimated.

Fig (26): Selecting the parameters and setting up the initial guess and upper/lower

bounds.



3.2.3.4 Starting the Estimation

Before starting the estimation, the estimation options in the new estimation node of

the Control and Estimation Tolls Manager GUI should be specified as follows:

3.2.3.4.1 Specifying and selecting the solver type

In the simulation options of the estimation options, the types of solvers are divided

into two major types according to [36].

Variable-Step which the error can be kept within the specified tolerance by

adjusting the step size of solver uses when it is used.

Fixed-Step which a constant step size can be used.

The variable-step solver is selected due to its ability to keep the error within the

specified tolerance and for faster simulation. For each type of the solvers that have

been mentioned beforehand, different solvers are available for differential equations

in Optimization Toolbox as shown in figure (27). Indeed, the most famous methods

of solving different equations incorporated with Ode 23 and Ode 45 are

implemented in MATLAB package. A solution for a simple second and third order

model can be provided by Ode 23. However, a solution for fourth and fifth order

can be provided by Ode 45 with a higher accuracy. Finally, the Ode 23 is used due

to its economical computation side and fast convergence for Lower-Order model

with less data [36].

Fig (27): Different solvers available in Optimization Toolbox.



3.2.3.4.2 Specifying and Selecting the Optimization Method

3.2.3.4.2.1 Cost Function Specification

In order to set up the optimization method the cost function should be specified.

Ideally, the most common methods that can be used in this project to minimize the

deviations of the estimated measurements from the actual measurements are as

follows:

1. The weighted least-squares method

It is used to minimize the sum of the squares of the weighted deviations of the

estimated data from actual data and it can give the best linear unbiased estimate for

any distribution with finite variance [1].

2. The maximum likelihood method

It can be used to maximize the probability of estimating the state variable [22].

3. The maximum variance method

The anticipated value of the sum of the squares of the error between the estimated

components of the state variable vector and the actual components of the state

variable vector can be minimized by using this method [22].

In this project, the least-squares method will be used due to its ability to give the

best minimization of the sum of the squares of the difference between the estimated

output and experimental data compared to other methods [1].

The error signal is given by:

(3.1)

Where:

the deviation between the simulated model outputs.

the current set of model parameters.

the experimental measurements.



Since the negative and positive deviations need to be taken into account, the cost

function is given by [22]:

(3.2)

3.2.3.4.2.2 Optimization Method Specification

In order to minimize the objective function , the nonlinear least squares method

is selected as shown in figure (28), and it is selected due to its ability to be used for

discontinuous and highly nonlinear functions, and also for the functions that have

unreliable and undefined derivatives.

Fig (28): Different optimization methods available in Optimization Toolbox.

From figure (28), it can be seen that there are different methods available in

MATLAB's Optimization Toolbox.

Pattern Search Method which can be used to compute the first approximation

of parameters as initial simulation.

Gradient Descent Method which can be used to optimize the response signal

subject to the constraints.

Simplex Search Method which can use a direct search method to optimize the

response and it is the most useful for simple problems [36].

Nevertheless, Nonlinear Least Squares Method with lower and upper bounds is

used to achieve accurate parameters representing models with nonlinear equations.

Finally, after setting up the simulation and optimization options, the estimation can

be started and the estimated parameters will be appeared as shown in figure (29).



Fig (29): Estimated parameters in Optimization Toolbox.



3.3 Parameter Estimation Flowchart

The estimation processes of the parameters that have been applied into the

Optimization Toolbox in Simulink can be outlined in the following flowchart:

Fig (30): Flowchart of parameters estimation processes [22].

Obtain Va, Vb, Vc, Wr, Efd, Ifd, Ia, Ib & Ic

(Experimental Measurements)

Priori System Knowledge

Complete Synchronous Generator Model

(MATLAB/SIMULINK)

Simulated Output: Ifd, Ia, Ib & Ic

℮ ˂ ε

Estimated Parameters

Parameter Adjustment

Algorithm

Va, Vb, Vc, Efd, & Wr

Ifd, Ia, Ib & Ic

Yes

No

Add Noise to Experimental

Measurements

Filter the Noise of Experimental

Measurements



3.4 Conclusion

A Simulink model for the estimator has been designed in this chapter. The

estimator has been designed based on the obtained measurements data from on-line

test. These measurements have been used in the estimator as inputs and outputs

data. Furthermore, the procedures of parameters estimation have been described

based on non-linear least squares method from Optimization Toolbox in Matlab.

Main steps of parameters estimation have been individually considered.

Chapter 4 Parameters Estimation Results


Chapter 4: Parameters Estimation Results

4.1 Manufacturer Data

In this project, a 158MVA, 13.8KV and 3600 rpm synchronous machine has been

used to evaluate the developed model. The typical machine parameters that have

been obtained from manufacturer data are listed in table (8). These parameters have

been used as priori knowledge and were subjected to changes during optimization

process at each iteration. Moreover, the available parameters from manufacturer

stability study data sheet are given by [1] and shown in table (9) and the

experimental data from 3-ph short circuit test that have been used for the simulation

and estimation are attached in Appendix B.

Table (8): Typical machine parameters from manufacturer data

Parameters Original Values (p.u.)

1.64

1.56

0.11791

0.16

0.0009722

0.0046

Table (9): Standard parameters from manufacturer stability study data sheet

Parameters Original Value

1.8 p.u.

0.27 p.u.

0.1971 p.u.

1.72 p.u.

0.49 p.u.

0.1793 p.u.

4.7963 S

0.049 S

0.49 S

0.059 S

0.1085 S



4.2 Calculation of Standard Machine Parameters from Estimated Derived

Parameters

The available machine parameters in table (9) are referred as standard machine

parameters and the direct and quadrature axis reactances and their transient and

subtransient components are included as well as the transient and subtransient time

constants. Indeed, the estimated parameters that have been obtained through the

developed model derived from the standard parameters. Wherefore, it is necessary

to calculate the standard parameters obtained through the selected algorithm [1].

The formulas that are needed to perform the conversion from derived parameters to

standard parameters are outlined in table (10) [28].

Table (10): Formulas of standard parameters

Parameters Formula

Note: all time constants are in p.u. and they are divided by 2*π*f in order to get the

values in seconds.



4.3 Estimation of Parameters for Different Cases

The parameters have been estimated in the Optimization Toolbox by using

nonlinear least square method with setting up the initial guesses of parameters

values that have been provided table (8) with different cases as follows:

4.3.1 Case 1

In this case, the initial guesses in the estimation process have set up to be matched

with the initial values of the original parameters.

4.3.1.1 Estimated Parameter without Including the Effect of Noise

The parameters of the developed synchronous machine model were estimated

without including the effect of the noise to the measurements data. The results are

shown in table (11) and the estimated standard parameters are compared with

manufacturer standard parameters as shown in table (12).

Table (11): Estimated parameters of synchronous machine without including noise

Parameters Original

Values (p.u.)

Initial Guess

(p.u.)

Estimated Values

(p.u.)

Deviation (%)

1.64 1.64 1.6407 -0.0426

1.56 1.56 1.5645 -0.288

0.11791 0.11791 0.11786 0.0424

0.16 0.16 0.16001 -0.000625

0.0009722 0.0009722 0.00075286 22.56

0.0046 0.0046 0.0047643 -3.57

Table (12): Manufacturer standard parameters vs the estimated standard parameters

Deviation (%) Estimated Values Original Values Parameters

0 0.27 (p.u.) 0.27 (p.u.)

-0.0507 0.1972 (p.u.) 0.1971 (p.u.)

-0.0408 0.4902 (p.u.) 0.49 (p.u.)

-0.1673 0.1796 (p.u.) 0.1793 (p.u.)

-29.18 6.196 (s) 4.7963 (s)

0 0.049 (s) 0.049 (s)

-0.22 0.4911 (s) 0.49 (s)

0 0.059 (s) 0.059 (s)

0.34 0.1048 (s) 0.1085 (s)

Note: the deviation has been calculated by using the following equation:



4.3.1.2 Estimated Parameters with Including the Effect of Noise

With including the effect of noise, the parameters estimation process has been

repeated. The results are listed in table (13) and the estimated standard parameters

are depicted in table (14).

Table (13): Estimated parameters of synchronous machine with including noise

Parameters Original

Values (p.u.)

Initial Guess

(p.u.)

Estimated Values

(p.u.)

Deviation (%)

1.64 1.64 1.6399 0.006

1.56 1.56 1.538 1.41

0.11791 0.11791 0.1179 0.008

0.16 0.16 0.16003 -0.018

0.0009722 0.0009722 0.00098936 1.32

0.0046 0.0046 0.0046037 -0.08



0 0.27 (p.u.) 0.27 (p.u.)

-0.1014 0.1973 (p.u.) 0.1971 (p.u.)

0.183 0.4891 (p.u.) 0.49 (p.u.)

0.055 0.1792 (p.u.) 0.1793 (p.u.)

-1.332 4.8602 (s) 4.7963 (s)

0 0.049 (s) 0.049 (s)

1.06 0.4848 (s) 0.49 (s)

0.338 0.0588 (s) 0.059 (s)

0.092 0.1084 (s) 0.1085 (s)

4.3.2 Case 2

The initial guesses of the original parameters in the estimation process have been

set up by taking 80% of (Xmd, Xmq, Xlfd, Xls, rfd and rs) initial values.


The estimated parameters are shown in table (15) and the estimated standard

parameters are compared with manufacturer standard parameters as can be shown

in table (16).




Parameters Original

Values (p.u.)

Initial Guess

(p.u.)

Estimated Values

(p.u.)

Deviation (%)

1.64 1.312 1.6404 -0.024

1.56 1.248 1.5646 -0.29

0.11791 0.094328 0.11788 0.025

0.16 0.128 0.15999 0.0063

0.0009722 0.00077776 0.00060011 38.27

0.0046 0.00368 0.0047631 -3.54



0 0.27 (p.u.) 0.27 (p.u.)

-0.0507 0.1972 (p.u.) 0.1971 (p.u.)

-0.041 0.4902 (p.u.) 0.49 (p.u.)

0.055 0.1792 (p.u.) 0.1793 (p.u.)

-62 7.7719 (s) 4.7963 (s)

0 0.049 (s) 0.049 (s)

-0.244 0.4912 (s) 0.49 (s)

0 0.059 (s) 0.059 (s)

3.4 0.1084 (s) 0.1085 (s)

4.3.3 Case 3

The initial guesses of the original parameters in the estimation process have been

set up by taking 120% of (Xmd, Xmq, Xlfd, Xls, rfd and rs) initial values.


The estimated parameters are shown in table (17) and the estimated standard

parameters are compared with manufacturer standard parameters as can be shown

in table (18).


Parameters Original

Values (p.u.)

Initial Guess

(p.u.)

Estimated Values

(p.u.)

Deviation (%)

1.64 1.968 1.6409 -0.054

1.56 1.872 1.5645 -0.288

0.11791 0.141492 0.11786 0.0424

0.16 0.192 0.16002 -0.0125

0.0009722 0.00116664 0.00066337 31.766

0.0046 0.00552 0.0047659 -3.61





0 0.27 (p.u.) 0.27 (p.u.)

-0.0507 0.1972 (p.u.) 0.1971 (p.u.)

-0.041 0.4902 (p.u.) 0.49 (p.u.)

0.055 0.1792 (p.u.) 0.1793 (p.u.)

-46.6 7.0327 (s) 4.7963 (s)

0 0.049 (s) 0.049 (s)

-0.002 0.4911 (s) 0.49 (s)

0 0.059 (s) 0.059 (s)

3.4 0.1048 (s) 0.1085 (s)

4.3.4 Case 4

The initial guesses of the original parameters in the estimation process have set up

by taking 80% of (Xmd, Xmq & Xlfd) and 120% of (Xls, rfd & rs) values.


Table (19) shows the estimated parameters and table (20) shows the estimated

standard parameters.


Parameters Original

Values (p.u.)

Initial Guess

(p.u.)

Estimated Values

(p.u.)

Deviation (%)

1.64 1.312 1.641 -0.06

1.56 1.248 1.5644 -0.28

0.11791 0.094328 0.11785 0.051

0.16 0.192 0.16003 -0.01875

0.0009722 0.00116664 0.0006 38.28

0.0046 0.00552 0.0047662 -3.6



0 0.27 (p.u.) 0.27 (p.u.)

-0.0507 0.1972 (p.u.) 0.1971 (p.u.)

-0.041 0.4902 (p.u.) 0.49 (p.u.)

0.055 0.1792 (p.u.) 0.1793 (p.u.)

-62.12 7.7758 (s) 4.7963 (s)

0 0.049 (s) 0.049 (s)

-0.002 0.4911 (s) 0.49 (s)

0 0.059 (s) 0.059 (s)

3.4 0.1048 (s) 0.1085 (s)



4.3.5 Case 5

The initial guesses of the original parameters in the estimation process have set up

by taking 80% of (Xmd, Xlfd & rfd) and 120% of (Xmq, Xls & rs) values.


The estimated parameters can be seen in table (21) and the estimated standard

parameters can be found in table (22).


Parameters Original

Values (p.u.)

Initial Guess

(p.u.)

Estimated Values

(p.u.)

Deviation (%)

1.64 1.312 1.641 -0.06

1.56 1.872 1.5644 -0.28

0.11791 0.094328 0.11785 0.051

0.16 0.192 0.16003 -0.01875

0.0009722 0.00077776 0.0006 38.28

0.0046 0.00552 0.0047662 -3.6



0 0.27 (p.u.) 0.27 (p.u.)

-0.0507 0.1972 (p.u.) 0.1971 (p.u.)

-0.041 0.4902 (p.u.) 0.49 (p.u.)

0.055 0.1792 (p.u.) 0.1793 (p.u.)

-62.12 7.7758 (s) 4.7963 (s)

0 0.049 (s) 0.049 (s)

-0.002 0.4911 (s) 0.49 (s)

0 0.059 (s) 0.059(s)

3.4 0.1048 (s) 0.1085 (s)

4.4 Discussion of the Estimated Results

The estimated parameters of the synchronous generator without including the effect

of noise with different cases of setting up the initial guesses of the original values

are shown in tables 11, 15, 17, 19 & 21. All parameters confirm a high accuracy of

estimation compared to the original values except rs and rfd. These high deviations

may be due to:

Their small p.u. values.

Imprecision in the modeling of the synchronous generator.



Estimating more than one parameter at the same time.

Inaccurate adjustment of the function tolerance in algorithm.

Nevertheless, various tests have been tried in order to reduce these high deviations

but the results have shown the same high deviations. These various tests are as

follows:

Converting their values from p.u. to Ohms and re-estimating them.

Changing in certain features of algorithm.

Converting all p.u. values to their absolute values.

Moreover, tables 12, 16, 18, 20 & 22 show the calculated standard parameters

without considering the effect of noise. All parameters emphasize the high accuracy

of the estimation except . This is because it is related with rfd.

However, the estimated parameters and the calculated standard parameters with

including the effect of noise are depicted in tables 13 & 14 respectively. All

parameters show a high accuracy compared to the original values.

4.5 Conclusion

The simulation has been performed on a 158 MVA, 13.8 KV and 3600 rpm

synchronous machine in order to evaluate the proposed model. Parameters

estimation outcomes have been presented and analyzed in this chapter based on

non-linear least squares method. The main observations can be outlined as follows:

High accuracy of the proposed model and method has been proved for

parameters estimation.

All estimated parameters without considering the effect of noise have shown

high accuracy estimation except rs and rfd.

High deviation found in estimated rfd may be due to its small p.u. value and

estimating more than one parameter at the same time in addition to inaccurate

adjustment of the function tolerance in algorithm.

High accuracy of all estimated parameters with including the effect of noise

has been emphasized compared to original values.

Chapter 5 Project Conclusion and Further Work


Chapter 5: Project Conclusion and Further Work

5.1 Project Conclusion

The purpose of this project was to develop a model and modify a methodology that

can be used to estimate the synchronous generator parameters from on-line data.

The model has been developed by using MATLAB/SIMULINK package while the

methodology has been implemented and modified from Optimization Toolbox in

MATLAB.

The work has been started by reviewing some of the research papers written and

experimental works done on synchronous machine parameters identification. The

parameters of synchronous machine can generally be determined either by off-line

or on-line techniques. The on-line techniques are mainly considered in this work

due to technical and economical reasons.

Various recommended IEEE models and development equations for synchronous

machine are presented in chapter three first for the purpose of modeling. Then, a

Simulink model for synchronous generator has been designed based on the

development equations. After that, the theory of main parts of modeling and

simulation of synchronous machine has been individually explained. In addition,

the function of each part has been described and its Simulink model has been built.

By the end of chapter three, the proposed model has been ready for simulation.

After it is used for obtaining the simulated data from on-line test, a Simulink model

for the estimator has been designed and built in chapter four first in order to be used

for parameters estimation. Then, the procedures of parameters estimation have been

described based on non-linear least squares method from Optimization Toolbox in

MATLAB environment. Furthermore, main steps of parameters estimation have

been individually explained.

The simulation has been performed on a 158 MVA, 13.8 KV and 3600 rpm

synchronous machine in order to evaluate the proposed model. Parameters

estimation outcomes have been presented and analyzed. High accuracy of the

proposed model and method has been proved for parameters estimation. Compared

to original values, high accuracy of all estimated parameters with including the

effect of noise has been emphasized. However, imprecision has been noticed in

Chapter 5 Project Conclusion and Further Work


estimation the parameters rs and rfd when the effect of noise is ignored. The

significant deviation that can be found in rs and rfd may be justified by their small

p.u. values and estimating more than one parameter at the same time in addition to

inaccurate adjustment of the function tolerance in algorithm. Therefore further

work is suggested.

5.2 Further Work

Further work needs to be done in order to limit the significant deviation in

estimated rs and rfd by including the effect of saturation and adjusting the function

tolerance of the proposed method in addition to adjusting the max/min bounds of

the parameters. Other models in power plant, such as AVR and excitation systems,

are linked to the synchronous generators and it is worth to estimate their parameters

in a future work.

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APPENDICES


APPENDICES

Appendix A. M-file for complete synchronous machine simulation

w=1; % angular speed in p.u we=2*pi*60; wb=2*pi*60; Po=2; % number of poles % Parameters of the machine rs=0.0046; %r Xls=0.16; % lq and ld Xq=1.72; % LAQ+0.16 Xd=1.80; % LAD+0.16 rfd=9.722E-4; % rF Xlfd=0.11791; % lF rQ2=0.01632; % rQ (0 for salient pole machine) rD=0.0125; % rD XlQ2=0.033; % lQ (inf for salient pole machine) XlD=0.121; % lD XlQ1=0.4185; % lG rQ1=0.01071; % rG Xmq=Xq-Xls; Xmd=Xd-Xls; % Calculate Initial Conditions of the machine

Im=1; Vm=1; lang=-acos(0.85); Ea=1+(rs+j*Xq)*Im*(cos(lang)+j*sin(lang)); eang=angle(Ea); t=0; c=wb*t+eang; Vabc=[Vm*cos(wb*t);Vm*cos(wb*t-2*pi/3);Vm*cos(wb*t+2*pi/3)]; Iabc=[Im*cos(wb*t+lang);Im*cos(wb*t-

2*pi/3+lang);Im*cos(wb*t+2*pi/3+lang)]; P=2/3*[1/2 1/2 1/2; cos(c) cos(c-120*pi/180) cos(c+120*pi/180);

sin(c) sin(c-120*pi/180) sin(c+120*pi/180)]; Voqd=P*Vabc; Ioqd=P*Iabc; % Steady State Conditions (p 218) vd=Voqd(3); vq=Voqd(2); io=Ioqd(1); iq=Ioqd(2); id=Ioqd(3); Exfd=abs(Ea)+(Xd-Xq)*id; vfd=Exfd*rfd/Xmd; ifd=Exfd/Xmd; fq=-Xls*iq+Xmq*(-iq); % state 1 --> x(1) fd=-Xls*id+Xmd*(-id+ifd); % state 2 --> x(2) fo=-Xls*io; % state 3 --> x(3) fQ1=Xmq*(-iq); % state 4 --> x(4) fQ2=Xmq*(-iq); % state 5 --> x(5) ffd=Xlfd*ifd+Xmd*(-id+ifd); % state 6 --> x(6) fD=Xmd*(-id+ifd); % state 7 --> x(7) Te=fd*iq-fq*id; % Initial Electrical Torque wb=2*pi*60; Tm=Te; H=5.6; dt=0.0004;

APPENDICES


Appendix B. List of the experimental data that have been used for the simulation

and estimation

Simulated Data from Synchronous Generator Model

Time Va Vb Vc Wr Efd Ia Ib Ic Ifd

0 -0.738 -0.738 -0.74 -0.68 -0.74 -0.74 -0.74 -0.738 -0.74

0.0004 -0.569 -0.601 -0.6 7.195 -0.54 -0.57 -0.61 -0.59 -0.56

0.0008 -0.385 -0.491 -0.5 26.09 -0.28 -0.4 -0.52 -0.458 -0.35

0.0012 -0.2 -0.403 -0.43 52.61 0.006 -0.22 -0.46 -0.35 -0.13

0.0016 -0.041 -0.326 -0.37 83.95 0.311 -0.04 -0.42 -0.278 0.09

0.002 0.0816 -0.253 -0.31 117.8 0.618 0.155 -0.39 -0.257 0.318

0.0024 0.1697 -0.187 -0.26 152.4 0.915 0.376 -0.37 -0.301 0.554

0.0028 0.2287 -0.13 -0.21 186.3 1.193 0.623 -0.33 -0.423 0.794

0.0032 0.2651 -0.082 -0.17 218.4 1.449 0.89 -0.27 -0.625 1.038

0.0036 0.2871 -0.044 -0.13 248.2 1.677 1.17 -0.18 -0.905 1.29

0.004 0.299 -0.013 -0.1 275.1 1.877 1.455 -0.05 -1.259 1.55

0.0044 0.2981 0.0132 -0.07 299 2.049 1.736 0.133 -1.678 1.818

0.0048 0.2891 0.034 -0.05 319.6 2.196 2 0.367 -2.153 2.096

0.0052 0.2751 0.0483 -0.03 337.2 2.316 2.236 0.657 -2.668 2.381

0.0056 0.2476 0.0773 -0.02 351.7 2.412 2.437 0.997 -3.205 2.671

0.006 0.2029 0.131 -0.03 363.6 2.487 2.602 1.365 -3.741 2.957

0.0064 0.1418 0.2008 -0.06 373 2.543 2.734 1.74 -4.257 3.227

0.0068 0.066 0.2794 -0.08 380.1 2.583 2.833 2.108 -4.739 3.474

0.0072 -0.02 0.3593 -0.1 385.4 2.61 2.902 2.458 -5.174 3.69

0.0076 -0.112 0.4349 -0.11 389.1 2.625 2.944 2.782 -5.558 3.866

0.008 -0.208 0.5022 -0.11 391.4 2.634 2.961 3.076 -5.889 3.996

0.0084 -0.306 0.5562 -0.09 392.6 2.636 2.958 3.338 -6.168 4.077

0.0088 -0.404 0.5952 -0.06 392.9 2.635 2.936 3.569 -6.394 4.103

0.0092 -0.499 0.6181 -0.01 392.6 2.627 2.895 3.768 -6.566 4.075

0.0096 -0.584 0.6232 0.043 391.8 2.614 2.834 3.935 -6.689 3.993

0.01 -0.658 0.6097 0.112 390.7 2.6 2.758 4.072 -6.762 3.86

0.0104 -0.719 0.5774 0.192 389.4 2.584 2.667 4.176 -6.784 3.676

0.0108 -0.765 0.525 0.279 387.9 2.566 2.566 4.246 -6.763 3.443

0.0112 -0.794 0.4533 0.371 386.5 2.55 2.458 4.281 -6.706 3.167

0.0116 -0.807 0.3656 0.463 385 2.536 2.351 4.281 -6.614 2.852

0.012 -0.803 0.2655 0.552 383.7 2.524 2.251 4.249 -6.491 2.509

0.0124 -0.78 0.1571 0.631 382.4 2.514 2.164 4.185 -6.343 2.144

0.0128 -0.739 0.0422 0.698 381.3 2.505 2.096 4.089 -6.178 1.766

0.0132 -0.682 -0.076 0.754 380.3 2.496 2.05 3.961 -6.005 1.385

0.0136 -0.611 -0.194 0.798 379.5 2.488 2.031 3.807 -5.833 1.011

0.014 -0.527 -0.309 0.829 378.8 2.481 2.041 3.632 -5.672 0.65

0.0144 -0.429 -0.418 0.841 378.3 2.476 2.083 3.442 -5.526 0.31

0.0148 -0.318 -0.521 0.834 377.8 2.474 2.159 3.245 -5.402 0.001

0.0152 -0.198 -0.616 0.807 377.5 2.475 2.265 3.047 -5.309 -0.27

0.0156 -0.072 -0.698 0.76 377.2 2.475 2.4 2.854 -5.249 -0.5

0.016 0.0562 -0.764 0.695 377 2.475 2.557 2.675 -5.227 -0.68

0.0164 0.1825 -0.813 0.615 376.9 2.476 2.726 2.519 -5.242 -0.81

0.0168 0.3053 -0.844 0.52 376.8 2.477 2.9 2.393 -5.292 -0.89

0.0172 0.4222 -0.856 0.415 376.8 2.477 3.07 2.3 -5.375 -0.91

0.0176 0.5305 -0.849 0.301 376.8 2.476 3.229 2.245 -5.486 -0.88

0.018 0.6263 -0.824 0.181 376.8 2.476 3.371 2.227 -5.617 -0.78

APPENDICES


0.0184 0.7073 -0.78 0.054 376.8 2.477 3.492 2.243 -5.759 -0.64

0.0188 0.7724 -0.717 -0.07 376.8 2.477 3.589 2.292 -5.905 -0.45

0.0192 0.8206 -0.637 -0.2 376.9 2.48 3.657 2.37 -6.051 -0.21

0.0196 0.8516 -0.541 -0.32 376.9 2.484 3.695 2.47 -6.19 0.061

0.02 0.8645 -0.432 -0.43 376.9 2.486 3.702 2.588 -6.315 0.365

0.0204 0.8568 -0.311 -0.54 376.9 2.491 3.683 2.716 -6.422 0.691

0.0208 0.8282 -0.182 -0.64 376.9 2.494 3.641 2.848 -6.506 1.035

0.0212 0.779 -0.048 -0.72 376.9 2.496 3.577 2.978 -6.57 1.389

0.0216 0.7101 0.0874 -0.78 376.8 2.496 3.496 3.103 -6.611 1.745

0.022 0.6246 0.2193 -0.83 376.8 2.496 3.403 3.218 -6.632 2.093

0.0224 0.527 0.3456 -0.86 376.7 2.496 3.301 3.322 -6.632 2.426

0.0228 0.4179 0.464 -0.86 376.7 2.496 3.198 3.413 -6.615 2.74

0.0232 0.2989 0.5723 -0.85 376.6 2.496 3.096 3.491 -6.584 3.026

0.0236 0.1732 0.6653 -0.82 376.6 2.496 2.995 3.559 -6.543 3.277

0.024 0.0444 0.7404 -0.77 376.5 2.496 2.896 3.618 -6.496 3.486

0.0244 -0.086 0.7972 -0.71 376.5 2.495 2.799 3.668 -6.442 3.648

0.0248 -0.215 0.8368 -0.63 376.5 2.49 2.703 3.711 -6.385 3.761

0.0252 -0.339 0.8604 -0.53 376.4 2.485 2.609 3.748 -6.327 3.824

0.0256 -0.454 0.8659 -0.42 376.4 2.481 2.516 3.78 -6.268 3.835

0.026 -0.557 0.8526 -0.3 376.4 2.477 2.42 3.808 -6.203 3.795

0.0264 -0.646 0.821 -0.18 376.4 2.472 2.321 3.831 -6.129 3.706

0.0268 -0.724 0.7699 -0.05 376.4 2.467 2.222 3.847 -6.045 3.569

0.0272 -0.787 0.7008 0.082 376.4 2.462 2.119 3.854 -5.952 3.388

0.0276 -0.834 0.6153 0.21 376.4 2.458 2.012 3.851 -5.845 3.163

0.028 -0.86 0.517 0.333 376.4 2.454 1.906 3.834 -5.725 2.904

0.0284 -0.865 0.4078 0.449 376.5 2.451 1.803 3.798 -5.592 2.617

0.0288 -0.85 0.2906 0.552 376.5 2.449 1.71 3.741 -5.449 2.308

0.0292 -0.815 0.1668 0.642 376.6 2.448 1.633 3.659 -5.298 1.981

0.0296 -0.762 0.0387 0.718 376.6 2.449 1.574 3.552 -5.142 1.646

0.03 -0.691 -0.091 0.777 376.7 2.452 1.538 3.424 -4.988 1.313

0.0304 -0.608 -0.221 0.818 376.7 2.456 1.53 3.276 -4.84 0.989

0.0308 -0.512 -0.348 0.842 376.8 2.458 1.553 3.109 -4.703 0.681

0.0312 -0.404 -0.466 0.847 376.9 2.459 1.606 2.931 -4.583 0.393

0.0316 -0.286 -0.573 0.834 376.9 2.46 1.689 2.748 -4.487 0.132

0.032 -0.161 -0.668 0.802 377 2.464 1.798 2.568 -4.418 -0.1

0.0324 -0.031 -0.748 0.752 377.1 2.468 1.929 2.398 -4.379 -0.29

0.0328 0.1021 -0.811 0.685 377.1 2.472 2.079 2.244 -4.373 -0.43

0.0332 0.2325 -0.855 0.601 377.1 2.476 2.241 2.113 -4.4 -0.53

0.0336 0.359 -0.879 0.504 377.2 2.478 2.408 2.008 -4.46 -0.59

0.034 0.4769 -0.882 0.394 377.2 2.478 2.574 1.929 -4.55 -0.59

0.0344 0.5833 -0.864 0.274 377.2 2.476 2.734 1.883 -4.661 -0.55

0.0348 0.6773 -0.826 0.148 377.2 2.472 2.88 1.87 -4.789 -0.46

0.0352 0.7555 -0.77 0.015 377.2 2.471 3.006 1.888 -4.926 -0.33

0.0356 0.8159 -0.698 -0.12 377.2 2.471 3.108 1.936 -5.066 -0.16

0.036 0.8562 -0.613 -0.25 377.1 2.474 3.18 2.009 -5.204 0.053

0.0364 0.8751 -0.513 -0.37 377.1 2.479 3.223 2.102 -5.334 0.297

0.0368 0.874 -0.402 -0.48 377 2.48 3.24 2.211 -5.45 0.566

0.0372 0.8545 -0.28 -0.59 377 2.481 3.23 2.328 -5.549 0.854

0.0376 0.8191 -0.15 -0.68 376.9 2.485 3.198 2.452 -5.63 1.154

0.038 0.7658 -0.018 -0.76 376.8 2.487 3.148 2.574 -5.693 1.463

0.0384 0.6942 0.1156 -0.82 376.8 2.489 3.085 2.688 -5.739 1.772

0.0388 0.6065 0.247 -0.86 376.7 2.49 3.01 2.794 -5.769 2.072

APPENDICES


0.0392 0.5026 0.3728 -0.89 376.6 2.492 2.928 2.891 -5.782 2.357

0.0396 0.3859 0.4899 -0.89 376.6 2.494 2.84 2.977 -5.783 2.622

0.04 0.259 0.5959 -0.87 376.5 2.495 2.748 3.054 -5.771 2.862

0.0404 0.1271 0.6869 -0.83 376.4 2.496 2.654 3.123 -5.75 3.069

0.0408 -0.007 0.7619 -0.77 376.4 2.495 2.56 3.184 -5.72 3.241

0.0412 -0.14 0.8192 -0.7 376.3 2.494 2.469 3.236 -5.683 3.374

0.0416 -0.27 0.8566 -0.61 376.3 2.493 2.376 3.284 -5.641 3.465

0.042 -0.394 0.8745 -0.51 376.3 2.493 2.281 3.326 -5.593 3.509

0.0424 -0.508 0.8727 -0.4 376.2 2.493 2.186 3.364 -5.539 3.507

0.0428 -0.61 0.851 -0.28 376.2 2.492 2.09 3.399 -5.475 3.461

0.0432 -0.697 0.8109 -0.15 376.2 2.489 1.99 3.428 -5.401 3.372

0.0436 -0.767 0.754 -0.01 376.2 2.486 1.889 3.448 -5.319 3.243

0.044 -0.818 0.6804 0.119 376.2 2.483 1.784 3.46 -5.226 3.079

0.0444 -0.855 0.5912 0.247 376.3 2.481 1.678 3.459 -5.119 2.882

0.0448 -0.874 0.4883 0.368 376.3 2.482 1.573 3.44 -4.997 2.654

0.0452 -0.873 0.3742 0.479 376.3 2.483 1.476 3.401 -4.859 2.4

0.0456 -0.852 0.2521 0.579 376.4 2.482 1.39 3.339 -4.712 2.126

0.046 -0.811 0.1237 0.665 376.4 2.48 1.319 3.254 -4.559 1.84

0.0464 -0.751 -0.008 0.735 376.5 2.478 1.268 3.15 -4.402 1.548

0.0468 -0.673 -0.138 0.789 376.5 2.476 1.241 3.026 -4.249 1.26

0.0472 -0.581 -0.265 0.825 376.6 2.475 1.24 2.883 -4.104 0.979

0.0476 -0.478 -0.384 0.841 376.7 2.477 1.265 2.724 -3.973 0.712

0.048 -0.365 -0.494 0.839 376.7 2.48 1.319 2.558 -3.861 0.465

0.0484 -0.241 -0.596 0.818 376.8 2.482 1.399 2.387 -3.772 0.245

0.0488 -0.109 -0.685 0.781 376.8 2.485 1.506 2.218 -3.711 0.056

0.0492 0.0267 -0.76 0.725 376.9 2.489 1.635 2.058 -3.682 -0.1

0.0496 0.1605 -0.814 0.651 376.9 2.493 1.78 1.913 -3.683 -0.21

0.05 0.2884 -0.85 0.559 377 2.498 1.933 1.786 -3.713 -0.29

0.0504 0.4089 -0.865 0.451 377 2.502 2.091 1.684 -3.771 -0.33

0.0508 0.5209 -0.86 0.336 377 2.506 2.246 1.613 -3.855 -0.32

0.0512 0.6215 -0.836 0.216 377 2.508 2.394 1.571 -3.96 -0.27

0.0516 0.7082 -0.793 0.091 377 2.508 2.526 1.559 -4.081 -0.17

0.052 0.777 -0.731 -0.04 377 2.508 2.64 1.576 -4.208 -0.04

0.0524 0.8266 -0.653 -0.17 377 2.507 2.732 1.621 -4.341 0.121

0.0528 0.8572 -0.557 -0.29 377 2.504 2.801 1.69 -4.473 0.313

0.0532 0.867 -0.449 -0.41 376.9 2.503 2.843 1.777 -4.598 0.531

0.0536 0.8578 -0.334 -0.52 376.9 2.503 2.86 1.877 -4.71 0.769

0.054 0.831 -0.214 -0.62 376.8 2.505 2.854 1.985 -4.809 1.025

0.0544 0.7863 -0.091 -0.71 376.8 2.507 2.825 2.096 -4.893 1.291

0.0548 0.7228 0.035 -0.78 376.7 2.509 2.778 2.206 -4.958 1.559

0.0552 0.6438 0.1608 -0.83 376.7 2.51 2.723 2.314 -5.007 1.823

0.0556 0.5517 0.2837 -0.87 376.6 2.514 2.658 2.417 -5.04 2.078

0.056 0.4469 0.4016 -0.88 376.5 2.519 2.584 2.511 -5.061 2.316

0.0564 0.3316 0.5118 -0.87 376.5 2.52 2.504 2.596 -5.07 2.534

0.0568 0.2084 0.6117 -0.84 376.4 2.519 2.421 2.675 -5.068 2.728

0.0572 0.0783 0.6999 -0.8 376.4 2.517 2.335 2.747 -5.054 2.894

0.0576 -0.055 0.7739 -0.74 376.3 2.513 2.246 2.811 -5.03 3.031

0.058 -0.187 0.8286 -0.66 376.3 2.508 2.158 2.87 -5 3.132

0.0584 -0.316 0.8603 -0.57 376.2 2.505 2.067 2.925 -4.964 3.199

0.0588 -0.439 0.8717 -0.46 376.2 2.505 1.972 2.975 -4.921 3.225

0.0592 -0.552 0.8625 -0.34 376.2 2.505 1.873 3.018 -4.868 3.212

0.0596 -0.654 0.833 -0.22 376.2 2.505 1.771 3.053 -4.805 3.16

APPENDICES


0.06 -0.74 0.7855 -0.09 376.2 2.504 1.667 3.081 -4.73 3.07

0.0604 -0.807 0.723 0.045 376.2 2.503 1.564 3.1 -4.644 2.946

0.0608 -0.856 0.6455 0.177 376.2 2.504 1.46 3.107 -4.545 2.792

0.0612 -0.885 0.5521 0.306 376.2 2.505 1.354 3.1 -4.434 2.61

0.0616 -0.895 0.4458 0.426 376.3 2.505 1.253 3.076 -4.312 2.404

0.062 -0.884 0.3295 0.534 376.3 2.503 1.161 3.035 -4.18 2.181

0.0624 -0.853 0.2057 0.63 376.3 2.502 1.082 2.973 -4.039 1.944

0.0628 -0.804 0.0764 0.711 376.4 2.501 1.021 2.89 -3.891 1.696

0.0632 -0.737 -0.056 0.776 376.4 2.503 0.981 2.786 -3.743 1.445

0.0636 -0.652 -0.188 0.823 376.5 2.506 0.963 2.664 -3.597 1.198

0.064 -0.554 -0.316 0.854 376.5 2.507 0.97 2.526 -3.46 0.959

0.0644 -0.445 -0.439 0.865 376.6 2.506 1.001 2.375 -3.338 0.733

0.0648 -0.327 -0.549 0.857 376.6 2.506 1.061 2.212 -3.238 0.525

0.0652 -0.204 -0.646 0.826 376.7 2.506 1.145 2.048 -3.161 0.341

0.0656 -0.073 -0.729 0.775 376.7 2.504 1.252 1.888 -3.112 0.184

0.066 0.0623 -0.794 0.706 376.8 2.504 1.379 1.737 -3.092 0.056

0.0664 0.1969 -0.841 0.62 376.8 2.505 1.518 1.602 -3.101 -0.04

0.0668 0.3248 -0.868 0.521 376.8 2.507 1.665 1.488 -3.135 -0.1

0.0672 0.4428 -0.875 0.414 376.9 2.506 1.816 1.397 -3.197 -0.12

0.0676 0.5498 -0.862 0.3 376.9 2.505 1.962 1.334 -3.282 -0.11

0.068 0.6453 -0.829 0.179 376.9 2.503 2.099 1.297 -3.385 -0.06

0.0684 0.7254 -0.778 0.053 376.9 2.501 2.225 1.288 -3.501 0.025

0.0688 0.7859 -0.71 -0.07 376.9 2.497 2.333 1.307 -3.628 0.139

0.0692 0.8265 -0.629 -0.2 376.8 2.493 2.42 1.351 -3.759 0.284

0.0696 0.8492 -0.534 -0.32 376.8 2.49 2.486 1.415 -3.891 0.456

0.07 0.8548 -0.427 -0.43 376.8 2.486 2.53 1.494 -4.016 0.649

0.0704 0.8427 -0.311 -0.54 376.7 2.484 2.551 1.585 -4.131 0.858

0.0708 0.812 -0.186 -0.63 376.7 2.484 2.548 1.686 -4.233 1.081

0.0712 0.7631 -0.056 -0.71 376.7 2.485 2.523 1.792 -4.318 1.312

0.0716 0.6966 0.0756 -0.78 376.6 2.486 2.482 1.899 -4.389 1.547

0.072 0.6148 0.2061 -0.83 376.6 2.486 2.428 2.003 -4.441 1.779

0.0724 0.5181 0.3323 -0.86 376.5 2.488 2.366 2.103 -4.474 2.001

0.0728 0.4089 0.4507 -0.86 376.5 2.491 2.298 2.2 -4.493 2.208

0.0732 0.2899 0.5584 -0.86 376.4 2.493 2.225 2.291 -4.501 2.399

0.0736 0.1635 0.6543 -0.83 376.4 2.494 2.147 2.375 -4.496 2.567

0.074 0.0339 0.7351 -0.78 376.3 2.493 2.063 2.451 -4.481 2.71

0.0744 -0.095 0.7982 -0.71 376.3 2.493 1.976 2.519 -4.458 2.823

0.0748 -0.223 0.8448 -0.62 376.3 2.494 1.888 2.582 -4.425 2.903

0.0752 -0.345 0.8741 -0.52 376.2 2.497 1.794 2.638 -4.386 2.951

0.0756 -0.458 0.8829 -0.41 376.2 2.497 1.696 2.69 -4.34 2.966

0.076 -0.562 0.8711 -0.29 376.2 2.496 1.596 2.736 -4.284 2.946

0.0764 -0.653 0.8392 -0.16 376.2 2.495 1.492 2.776 -4.217 2.895

0.0768 -0.731 0.7882 -0.03 376.2 2.493 1.385 2.808 -4.141 2.815

0.0772 -0.793 0.7208 0.102 376.2 2.491 1.278 2.83 -4.053 2.706

0.0776 -0.837 0.6359 0.232 376.2 2.488 1.169 2.837 -3.955 2.572

0.078 -0.862 0.5361 0.356 376.2 2.484 1.062 2.825 -3.846 2.412

0.0784 -0.866 0.4264 0.47 376.2 2.48 0.96 2.792 -3.724 2.231

0.0788 -0.849 0.3075 0.574 376.3 2.476 0.869 2.739 -3.592 2.033

0.0792 -0.812 0.1829 0.665 376.3 2.473 0.793 2.666 -3.454 1.823

0.0796 -0.757 0.0548 0.741 376.4 2.471 0.734 2.573 -3.314 1.607

0.08 -0.685 -0.077 0.8 376.4 2.47 0.696 2.464 -3.175 1.391

0.0804 -0.598 -0.21 0.84 376.5 2.469 0.679 2.339 -3.041 1.18

APPENDICES


0.0808 -0.498 -0.339 0.861 376.5 2.469 0.69 2.2 -2.917 0.978

0.0812 -0.387 -0.459 0.864 376.6 2.469 0.728 2.051 -2.809 0.789

0.0816 -0.266 -0.567 0.849 376.6 2.472 0.792 1.897 -2.721 0.614

0.082 -0.139 -0.66 0.816 376.7 2.475 0.881 1.744 -2.654 0.46

0.0824 -0.009 -0.739 0.765 376.7 2.478 0.991 1.594 -2.613 0.331

0.0828 0.1223 -0.8 0.699 376.7 2.481 1.117 1.454 -2.597 0.23

0.0832 0.2498 -0.845 0.617 376.8 2.483 1.254 1.33 -2.606 0.159

0.0836 0.3708 -0.872 0.52 376.8 2.482 1.401 1.225 -2.642 0.119

0.084 0.4836 -0.879 0.409 376.8 2.478 1.547 1.144 -2.703 0.11

0.0844 0.5852 -0.865 0.289 376.8 2.471 1.687 1.088 -2.785 0.133

0.0848 0.6754 -0.831 0.161 376.8 2.465 1.818 1.057 -2.885 0.185

0.0852 0.7508 -0.778 0.029 376.8 2.46 1.935 1.052 -2.997 0.266

0.0856 0.8081 -0.707 -0.1 376.8 2.456 2.036 1.071 -3.117 0.372

0.086 0.8463 -0.621 -0.23 376.8 2.453 2.117 1.113 -3.24 0.503

0.0864 0.8643 -0.521 -0.35 376.8 2.451 2.18 1.172 -3.36 0.653

0.0868 0.8624 -0.409 -0.47 376.7 2.449 2.221 1.245 -3.475 0.821

0.0872 0.8416 -0.286 -0.57 376.7 2.449 2.242 1.332 -3.581 1.002

0.0876 0.8039 -0.156 -0.66 376.7 2.452 2.242 1.427 -3.675 1.193

0.088 0.7491 -0.022 -0.74 376.6 2.457 2.223 1.528 -3.755 1.388

0.0884 0.6759 0.113 -0.8 376.6 2.464 2.189 1.629 -3.82 1.586

0.0888 0.5884 0.245 -0.84 376.5 2.47 2.141 1.73 -3.871 1.782

0.0892 0.4899 0.3718 -0.86 376.5 2.475 2.081 1.828 -3.911 1.972

0.0896 0.3801 0.4907 -0.86 376.5 2.478 2.014 1.922 -3.938 2.149

0.09 0.2591 0.5985 -0.84 376.4 2.48 1.941 2.011 -3.954 2.31

0.0904 0.1305 0.6933 -0.8 376.4 2.479 1.865 2.094 -3.959 2.452

0.0908 -0.001 0.7719 -0.74 376.3 2.477 1.785 2.17 -3.954 2.573

0.0912 -0.133 0.8316 -0.66 376.3 2.477 1.704 2.24 -3.938 2.668

0.0916 -0.261 0.8732 -0.57 376.3 2.476 1.618 2.303 -3.911 2.734

0.092 -0.383 0.8947 -0.47 376.3 2.476 1.526 2.358 -3.874 2.773

0.0924 -0.495 0.8947 -0.35 376.2 2.477 1.431 2.41 -3.828 2.781

0.0928 -0.598 0.8748 -0.23 376.2 2.48 1.333 2.456 -3.774 2.76

0.0932 -0.688 0.8356 -0.1 376.2 2.484 1.234 2.494 -3.71 2.709

0.0936 -0.764 0.7786 0.036 376.2 2.484 1.134 2.524 -3.633 2.629

0.094 -0.823 0.7055 0.168 376.2 2.483 1.032 2.543 -3.543 2.523

0.0944 -0.863 0.6167 0.295 376.2 2.482 0.929 2.549 -3.442 2.396

0.0948 -0.883 0.5126 0.414 376.3 2.482 0.83 2.538 -3.331 2.25

0.0952 -0.886 0.3967 0.521 376.3 2.482 0.739 2.507 -3.209 2.087

0.0956 -0.87 0.2709 0.616 376.3 2.483 0.658 2.457 -3.079 1.912

0.096 -0.831 0.1386 0.696 376.4 2.485 0.59 2.389 -2.947 1.728

0.0964 -0.773 0.0049 0.763 376.4 2.487 0.538 2.304 -2.814 1.54

0.0968 -0.697 -0.128 0.814 376.4 2.487 0.506 2.201 -2.686 1.352

0.0972 -0.604 -0.256 0.849 376.5 2.487 0.498 2.08 -2.566 1.171

0.0976 -0.497 -0.376 0.868 376.5 2.486 0.515 1.945 -2.457 0.995

0.098 -0.379 -0.485 0.868 376.5 2.485 0.557 1.801 -2.363 0.829

0.0984 -0.255 -0.583 0.847 376.6 2.486 0.622 1.652 -2.284 0.678

0.0988 -0.126 -0.669 0.808 376.6 2.487 0.709 1.505 -2.226 0.545

0.0992 0.0074 -0.738 0.75 376.7 2.488 0.815 1.364 -2.19 0.434

0.0996 0.1414 -0.793 0.674 376.7 2.488 0.937 1.234 -2.176 0.348

0.1 0.2717 -0.832 0.582 376.7 2.488 1.071 1.119 -2.188 0.291

0.1004 0.3943 -0.85 0.476 376.7 2.488 1.209 1.022 -2.226 0.262

0.1008 0.507 -0.847 0.359 376.7 2.486 1.348 0.945 -2.287 0.261

0.1012 0.6083 -0.825 0.234 376.7 2.483 1.483 0.889 -2.368 0.286

APPENDICES


0.1016 0.6961 -0.785 0.103 376.7 2.482 1.606 0.856 -2.466 0.339

0.102 0.7677 -0.727 -0.03 376.7 2.482 1.716 0.85 -2.577 0.416

0.1024 0.821 -0.651 -0.16 376.7 2.483 1.811 0.867 -2.692 0.515

0.1028 0.8576 -0.562 -0.29 376.7 2.486 1.888 0.902 -2.81 0.635

0.1032 0.8733 -0.459 -0.41 376.7 2.487 1.945 0.956 -2.923 0.772

0.1036 0.8695 -0.346 -0.52 376.7 2.488 1.984 1.025 -3.03 0.922

0.104 0.8478 -0.225 -0.62 376.7 2.487 2.004 1.104 -3.132 1.084

0.1044 0.8073 -0.099 -0.71 376.6 2.487 2.007 1.193 -3.223 1.252

0.1048 0.7466 0.028 -0.78 376.6 2.487 1.991 1.288 -3.301 1.424

0.1052 0.6669 0.1519 -0.84 376.6 2.487 1.961 1.386 -3.366 1.599

0.1056 0.5715 0.2713 -0.87 376.5 2.487 1.919 1.485 -3.42 1.77

0.106 0.4638 0.3848 -0.87 376.5 2.489 1.867 1.581 -3.461 1.929

0.1064 0.3446 0.4895 -0.86 376.5 2.492 1.805 1.673 -3.492 2.077

0.1068 0.2175 0.5824 -0.84 376.4 2.496 1.734 1.763 -3.511 2.21

0.1072 0.0852 0.663 -0.79 376.4 2.498 1.659 1.848 -3.519 2.325

0.1076 -0.049 0.7286 -0.72 376.4 2.497 1.58 1.929 -3.518 2.42

0.108 -0.18 0.778 -0.65 376.4 2.496 1.496 2.003 -3.506 2.493

0.1084 -0.307 0.8116 -0.55 376.3 2.495 1.405 2.071 -3.482 2.542

0.1088 -0.424 0.8276 -0.45 376.3 2.491 1.309 2.132 -3.447 2.565

0.1092 -0.531 0.8246 -0.33 376.3 2.486 1.209 2.186 -3.401 2.562

0.1096 -0.626 0.8021 -0.21 376.3 2.481 1.107 2.232 -3.345 2.533

0.11 -0.708 0.7591 -0.08 376.3 2.477 1.006 2.269 -3.279 2.481

0.1104 -0.776 0.6974 0.057 376.3 2.476 0.903 2.294 -3.201 2.408

0.1108 -0.827 0.6209 0.187 376.3 2.475 0.803 2.305 -3.11 2.313

0.1112 -0.86 0.5321 0.312 376.3 2.473 0.708 2.303 -3.005 2.199

0.1116 -0.872 0.4309 0.43 376.3 2.472 0.619 2.286 -2.893 2.069

0.112 -0.864 0.3167 0.541 376.3 2.47 0.538 2.253 -2.772 1.927

0.1124 -0.836 0.1935 0.639 376.4 2.469 0.466 2.2 -2.646 1.776

0.1128 -0.787 0.0662 0.722 376.4 2.469 0.408 2.127 -2.515 1.618

0.1132 -0.721 -0.063 0.789 376.4 2.469 0.369 2.041 -2.385 1.455

0.1136 -0.637 -0.192 0.838 376.5 2.47 0.353 1.939 -2.261 1.293

0.114 -0.537 -0.316 0.869 376.5 2.469 0.357 1.822 -2.144 1.136

0.1144 -0.426 -0.435 0.881 376.5 2.465 0.381 1.693 -2.039 0.987

0.1148 -0.306 -0.545 0.874 376.6 2.461 0.427 1.558 -1.949 0.848

0.1152 -0.182 -0.644 0.852 376.6 2.459 0.495 1.421 -1.878 0.721

0.1156 -0.052 -0.73 0.811 376.6 2.46 0.582 1.283 -1.825 0.613

0.116 0.0777 -0.798 0.747 376.7 2.461 0.683 1.15 -1.792 0.526

0.1164 0.2047 -0.847 0.664 376.7 2.463 0.798 1.027 -1.784 0.46

0.1168 0.3256 -0.877 0.565 376.7 2.464 0.922 0.919 -1.802 0.419

0.1172 0.4397 -0.886 0.452 376.7 2.466 1.053 0.83 -1.844 0.401

0.1176 0.5464 -0.874 0.328 376.7 2.468 1.184 0.764 -1.908 0.405

0.118 0.6417 -0.841 0.197 376.7 2.47 1.312 0.719 -1.99 0.431

0.1184 0.7225 -0.79 0.065 376.7 2.47 1.43 0.696 -2.086 0.479

0.1188 0.7844 -0.721 -0.07 376.7 2.471 1.536 0.692 -2.192 0.547

0.1192 0.828 -0.634 -0.2 376.7 2.47 1.628 0.71 -2.304 0.635

0.1196 0.8519 -0.531 -0.32 376.7 2.47 1.703 0.747 -2.421 0.744

0.12 0.8537 -0.415 -0.44 376.7 2.472 1.759 0.802 -2.534 0.87

0.1204 0.8363 -0.289 -0.54 376.7 2.476 1.795 0.87 -2.64 1.006

0.1208 0.8008 -0.157 -0.64 376.7 2.482 1.813 0.95 -2.739 1.149

0.1212 0.7484 -0.025 -0.71 376.7 2.487 1.813 1.037 -2.831 1.297

0.1216 0.681 0.1049 -0.77 376.6 2.49 1.799 1.129 -2.911 1.447

0.122 0.6 0.2317 -0.82 376.6 2.489 1.774 1.226 -2.978 1.597

APPENDICES


0.1224 0.505 0.3525 -0.85 376.6 2.487 1.737 1.323 -3.035 1.743

0.1228 0.3981 0.4662 -0.85 376.5 2.486 1.687 1.418 -3.081 1.883

0.1232 0.2822 0.5697 -0.84 376.5 2.485 1.626 1.508 -3.112 2.013

0.1236 0.1602 0.6597 -0.81 376.5 2.485 1.556 1.594 -3.129 2.127

0.124 0.0349 0.7346 -0.76 376.4 2.484 1.477 1.677 -3.137 2.225

0.1244 -0.091 0.7934 -0.7 376.4 2.484 1.392 1.755 -3.136 2.304

0.1248 -0.216 0.8355 -0.61 376.4 2.483 1.301 1.827 -3.125 2.362

0.1252 -0.337 0.8588 -0.51 376.4 2.482 1.206 1.897 -3.102 2.4

0.1256 -0.45 0.8628 -0.4 376.4 2.483 1.108 1.96 -3.066 2.418

0.126 -0.555 0.8483 -0.28 376.4 2.483 1.007 2.014 -3.02 2.412

0.1264 -0.651 0.8162 -0.16 376.4 2.484 0.906 2.057 -2.961 2.383

0.1268 -0.732 0.7663 -0.03 376.4 2.487 0.806 2.087 -2.892 2.333

0.1272 -0.798 0.6983 0.104 376.4 2.49 0.706 2.105 -2.81 2.263

0.1276 -0.844 0.6129 0.233 376.4 2.491 0.61 2.112 -2.717 2.176

0.128 -0.869 0.5115 0.358 376.4 2.492 0.519 2.103 -2.613 2.075

0.1284 -0.875 0.3984 0.476 376.4 2.493 0.436 2.077 -2.501 1.96

0.1288 -0.862 0.2769 0.584 376.4 2.493 0.362 2.034 -2.381 1.835

0.1292 -0.829 0.1499 0.677 376.4 2.493 0.3 1.971 -2.257 1.702

0.1296 -0.777 0.0191 0.754 376.5 2.491 0.252 1.888 -2.131 1.565

0.13 -0.707 -0.111 0.812 376.5 2.488 0.218 1.788 -2.004 1.424

0.1304 -0.62 -0.236 0.85 376.5 2.489 0.206 1.675 -1.88 1.283

0.1308 -0.52 -0.356 0.87 376.5 2.489 0.216 1.551 -1.767 1.146

0.1312 -0.409 -0.469 0.873 376.6 2.488 0.246 1.418 -1.668 1.017

0.1316 -0.291 -0.571 0.858 376.6 2.486 0.295 1.279 -1.587 0.898

0.132 -0.166 -0.661 0.825 376.6 2.484 0.364 1.14 -1.526 0.79

0.1324 -0.036 -0.735 0.774 376.7 2.481 0.45 1.005 -1.485 0.698

0.1328 0.0932 -0.794 0.704 376.7 2.479 0.549 0.879 -1.464 0.624

0.1332 0.2198 -0.838 0.615 376.7 2.478 0.661 0.766 -1.468 0.567

0.1336 0.3405 -0.865 0.513 376.7 2.476 0.783 0.668 -1.494 0.527

0.134 0.4532 -0.873 0.4 376.8 2.474 0.91 0.589 -1.544 0.51

0.1344 0.5559 -0.861 0.278 376.8 2.471 1.036 0.53 -1.611 0.515

0.1348 0.6485 -0.828 0.148 376.8 2.467 1.159 0.49 -1.695 0.54

0.1352 0.729 -0.777 0.015 376.8 2.463 1.273 0.473 -1.791 0.581

0.1356 0.7903 -0.708 -0.12 376.8 2.458 1.374 0.478 -1.895 0.642

0.136 0.832 -0.623 -0.25 376.8 2.454 1.459 0.503 -2.004 0.722

0.1364 0.8535 -0.521 -0.37 376.8 2.45 1.528 0.549 -2.115 0.818

0.1368 0.8546 -0.408 -0.49 376.7 2.448 1.581 0.613 -2.226 0.929

0.1372 0.8367 -0.284 -0.59 376.7 2.445 1.616 0.691 -2.334 1.051

0.1376 0.8008 -0.154 -0.68 376.7 2.444 1.634 0.781 -2.435 1.181

0.138 0.7469 -0.021 -0.75 376.7 2.444 1.636 0.879 -2.525 1.315

0.1384 0.6771 0.1123 -0.81 376.7 2.445 1.622 0.981 -2.606 1.453

0.1388 0.592 0.2434 -0.85 376.6 2.447 1.594 1.085 -2.674 1.59

0.1392 0.4926 0.3678 -0.87 376.6 2.449 1.549 1.187 -2.728 1.72

0.1396 0.3824 0.4832 -0.88 376.6 2.45 1.491 1.284 -2.77 1.842

0.14 0.2633 0.5871 -0.86 376.6 2.45 1.425 1.378 -2.801 1.952

0.1404 0.1385 0.6761 -0.82 376.5 2.449 1.353 1.468 -2.818 2.048

0.1408 0.0106 0.7481 -0.77 376.5 2.449 1.275 1.553 -2.82 2.127

0.1412 -0.119 0.8039 -0.7 376.5 2.447 1.192 1.63 -2.81 2.19

0.1416 -0.249 0.8417 -0.61 376.5 2.446 1.106 1.701 -2.789 2.236

0.142 -0.372 0.8599 -0.51 376.5 2.445 1.015 1.763 -2.757 2.262

0.1424 -0.485 0.8576 -0.39 376.5 2.442 0.917 1.817 -2.716 2.268

0.1428 -0.588 0.8353 -0.27 376.5 2.44 0.816 1.861 -2.664 2.255

APPENDICES


0.1432 -0.677 0.7912 -0.14 376.5 2.443 0.714 1.893 -2.601 2.224

0.1436 -0.751 0.7271 -0.02 376.5 2.45 0.613 1.915 -2.527 2.177

0.144 -0.808 0.6461 0.113 376.5 2.457 0.513 1.924 -2.444 2.114

0.1444 -0.847 0.5504 0.241 376.5 2.463 0.416 1.922 -2.352 2.036

0.1448 -0.868 0.443 0.365 376.5 2.467 0.324 1.906 -2.25 1.945

0.1452 -0.866 0.3266 0.478 376.5 2.471 0.242 1.875 -2.139 1.841

Appendix C. M-file for adding noise to the simulated data

% Adding noise to the simulated data Van=Va+0.05.*randn(length(Va),1); Vbn=Vb+0.05.*randn(length(Vb),1); Vcn=Vc+0.05.*randn(length(Vc),1); Efdn=Efd+0.05.*randn(length(Efd),1); Wrn=Wr+0.05.*randn(length(Wr),1); Ian=Ia+0.05.*randn(length(Ia),1); Ibn=Ib+0.05.*randn(length(Ib),1); Icn=Ic+0.05.*randn(length(Ic),1); Ifdn=Ifd+0.05.*randn(length(Ifd),1);

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