research article simulations of transformer inrush current...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 215647 10 pageshttpdxdoiorg1011552013215647

Research ArticleSimulations of Transformer Inrush Current by UsingBDF-Based Numerical Methods

Amir TokiT1 Ivo UglešiT2 and Gorazd Štumberger3

1 Faculty of Electrical Engineering Tuzla University 75000 Tuzla Bosnia and Herzegovina2 Faculty of Electrical Engineering and Computing Zagreb University 11000 Zagreb Croatia3 Faculty of Electrical Engineering and Computer Science Maribor University 2000 Maribor Slovenia

Correspondence should be addressed to Amir Tokic amirtokicuntzba

Received 7 April 2013 Revised 8 July 2013 Accepted 9 July 2013

Academic Editor Evangelos J Sapountzakis

Copyright copy 2013 Amir Tokic et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper describes three different ways of transformer modeling for inrush current simulations The developed transformermodels are not dependent on an integration step thus they can be incorporated in a state-space form of stiff differential equationsystemsThe eigenvalue propagations during simulation time cause very stiff equation systemsThe state-space equation systems aresolved by using A- and L-stable numerical differentiation formulas (NDF2) method This method suppresses spurious numericaloscillations in the transient simulations The comparisons between measured and simulated inrush and steady-state transformercurrents are done for all three of the proposed models The realized nonlinear inductor nonlinear resistor and hysteresis modelcan be incorporated in the EMTP-type programs by using a combination of existing trapezoidal and proposed NDF2 methods

1 Introduction

The transformer represents one of the essential elementsin power systems Proper modeling of the transformer isvery important in different transient and steady-state powersystems simulationsThe nonlinearity of the transformer ironcore is the most important parameter in simulations of low-frequency transients such as transformer inrush currentferroresonance temporary overvoltages during transformerenergizations load rejections and harmonic analysis Allthese transients belong to a frequency range of up to 1 kHz [1]for which it is not necessary to take into account frequencydependencies of the transformer parameters A standardSteinmetz 119879 equivalent circuit model of a two-windingsingle-phase transformer is shown in Figure 1 In additionthere is an analogousΠ-shaped equivalentmodelThismodelis rarely used due to the fact that the detailed transformerconstruction has to be known for determining the parametersof the Π equivalent model

Different types of 119879 transformer models will be usedin this paper in the first place Transformer parameterscan be divided into two basic groups winding and iron-core parameters In Figure 1 119877119901 and 119877119904 represent the serial

winding resistances that generally include the Joule and eddycurrent losses in the windings whilst 119871119901 and 119871 119904 representthe serial leakage inductances divided amongst primary andsecondary windings The shunt branch parameters describeiron core behavior where the resistance 119877119898 represents thecore eddy current losses whilst inductance 119871119898 representsthe saturation or hysteresis phenomena in the transformercore The winding parameters are linear whilst the ironcore parameters describe nonlinear phenomena during theanalysis of low-frequency transformer transients

In general there are three different approaches whenmodeling transformer iron-core nonlinearities

(a) linear resistor 119877119898 and nonlinear inductor 119871119898

(b) nonlinear resistor 119877119898 and nonlinear inductor 119871119898

(c) linear resistor 119877119898 and nonlinear hysteresis inductor119871119898

During the analysis of low-frequency transformer tran-sients such as inrush current and ferroresonance the moreused model is the model (a) because of its simplicityrelatively rare used models are (b) and (c) For example

2 Mathematical Problems in Engineering

t = T0 RpLp

e(t) Rm Lm

Ls RsNp Ns

Figure 1 Standard 119879 equivalent transformer circuit model

120601

120601sN

120601s2

120601s1is1 is2 isN

im

Figure 2 Nonlinear curve of core inductor

[2ndash10] use the (a) based model of transformer [11ndash14] use the(b) but [15ndash18] use the (c) based model of the transformer

Typical procedures for obtaining an instantaneous non-linear magnetizing curve and nonlinear curve of core lossesby the separation technique of losses from the magnetizingcurve are described in [19 20]

The modeling of nonlinear or hysteresis inductor andnonlinear core loss resistor can be done by using the curvefitting procedure for the approximation of nonlinearity or byusing piecewise linear representation of the nonlinear curve

The well-known curve fitting approximations areobtained by using the polynomial exponential arctg andhyperbolic functions The polynomial approximation wassuggested by Mayergoyz in [21] Exponential approximationwas introduced by Trutt et al in [22] Arctg approximationwas started by Karlqvist in [23] and the hyperbolic functionswere introduced by Takacs in [24] and refined by Takacs in[25]

The piecewise linearization of a nonlinear curve forthe modeling of nonlinear or hysteretic inductor on theother hand is used in [7ndash10] Linearization of a nonlinearcurve has certain advantages and disadvantages Successfulcontrol of the numerical stability properties of the appliednumerical methods is the main advantage [26ndash28] Alsoanother advantage of this linearization is the improved speedcompared to those classical methods used in calculationswithin nonlinear algebraic-differential equation systems Onthe other hand piecewise linearization has disadvantagesrelated to overshooting effects [27]

2 Transformer Iron Core Modeling

Theoriginal way ofmodeling nonlinear transformer iron corein low-frequency transient analysis using three mentionedmodels is derived in this paper

imk

Lm

(a)

imk

S120601k

Lmk i0mk

(b)

Figure 3 (a) Nonlinear inductor and (b) equivalent model

urM

uR

ur2ur1

ir1 ir2 irM

iR

Figure 4 Nonlinear curve of core resistor

21 Modeling of Nonlinear Core Inductor The nonlinear coreinductor can be described using a magnetizing curve mag-netizing current versus magnetic flux as shown in Figure 2

When this curve is piecewise linear the results are theinput vectors magnetizing currents 119868119871

119898

= [1198941199041

1198941199042

119894119904119873

]119879

and magnetic fluxes Φ119871119898

= [1206011199041

1206011199042

120601119904119873

]119879 whereas 119873

is the total number of magnetizing curve regions The mag-netizing current for the 119896th linear region of the magnetizingcurve 119896 = 1 2 (119873 minus 1) is calculated using the equation

119894119898119896

=1

119871119898119896

120601 + sign (120601) 119868119904119896

=1

119871119898119896

120601 + 119878120601119896

(1)

where 119871119898119896

= (120601119904119896+1

minus120601119904119896

)(119894119904119896+1

minus119894119904119896

) and 119868119904119896

= 119894119904119896

minus(1119871119898119896

)120601119904119896

Based on (1) it is possible to create a nonlinear inductorequivalent model using a linear inductor and a correspond-ing current source Figure 3 This model is functionallydependent on the position of the operating point 119896 At thesame time contrary to EMTP-based equivalent models theinductor model is not dependent on the integration step

22 Modeling of Nonlinear Core Resistor Analogous to thenonlinear core inductor the nonlinear core resistor can bedescribed using the curve resistor current versus corre-sponding resistor voltage as shown in Figure 4

In this case the input vectors are the resistor currents119868119877119898

= [1198941199031

1198941199032

119894119903119872

]119879 and resistor voltages 119880119877

119898

= [1199061199031

1199061199032

119906119903119872

]119879 where119872 is the total number of resistor curve

Mathematical Problems in Engineering 3

iRk Rm

(a)

iRk

Srk

Rmk i0Rk

(b)

Figure 5 (a) Nonlinear resistor and (b) equivalent model

regions The resistor current for the 119896th linear region of theresistor curve 119896 = 1 2 (119872 minus 1) is calculated using theequation

119894119877119896

=1

119877119898119896

119906119877 + sign (119906119877) 119868119903119896

=1

119877119898119896

119906119877 + 119878119903119896

(2)

where119877119898119896

= (119906119903119896+1

minus119906119903119896

)(119894119903119896+1

minus119894119903119896

) and 119868119903119896

= 119894119903119896

minus(1119877119898119896

)119906119903119896

In the sameway it is possible to create a nonlinear resistor

equivalent model using a linear resistor and a correspondingcurrent source Figure 5

23 Modeling of Hysteresis Core Inductor By definition themajor loop is the largest possible loop whose end pointsreach the technical saturation Beyond the saturation pointsthe hysteresis is a single-valued function Symmetrical orasymmetrical minor hysteresis loops lie within the majorloop The following is assumed based on experimentalresults for hysteresis modeling (Figure 6) (a) the operatingpoint trajectories lie within the major hysteresis loop (b) inthe direction of the flux increase (decrease) the operatingpoint heads towards the lower (upper) branch of the majorloop and (c) after the flux reversal points are detected (12 3) the operating point trajectories will head towards theprevious reversal point

Therefore following the symmetry the major hysteresisloop is defined by the points of the lower (or upper) half of themajor hysteresis loops that is by input vectors are hysteresiscurrents 119868119871

ℎ

= [119894ℎ1

119894ℎ2

119894ℎ119875

]119879 and magnetic fluxes Φ119871

ℎ

=

[120601ℎ1

120601ℎ2

120601ℎ119875

]119879 where 119875 is the total number of the major

hysteresis loopsThe expression for the hysteresis current of the 119896th piece-

wise region in terms of the main flux 120601119898 119896 = 1 2 (119875minus1)Figure 6 can be stipulated as

119894ℎ119896

=1

119871119898119896

120601119898 + sign (Δ120601119898) 119868ℎ119896

(3)

where Δ120601119898 = 120601119898(119905119895) minus 120601119898(119905119895minus1) 119895 = 1 2 3 119869max 119871119898119896

=

(120601ℎ119896+1

minus 120601ℎ119896

)(119894ℎ119896+1

minus 119894ℎ119896

) and 119868ℎ119896

= 119894ℎ119896

minus (1119871119898119896

)120601ℎ119896

In the general case if it is assumed that the distance

between the operating point and the major loop is a linear

120601

1

2

3

Major loop (120601m)

Minor Loops (120601)

ih

Figure 6 Major and minor hysteresis loops

function of the magnetic flux for both the ascending anddescending trajectory the next equation is valid

119889 = 120583119896120601 + 120578119896 = sign (Δ120601) (120601 minus 120601119898) (4)

The unknown coefficients 120583119896 and 120578119896 are determined oncondition that the two successive reversal points (120601rev1 119889rev1)and (120601rev2 119889rev2) lie on this line

Based on relations (3)-(4) it can be proved that theexpression for the hysteresis current of the 119896th linear regionthat describes the trajectory of the actual operating point is

119894ℎ119896

=1

119871119898119896

minus Δ119871119898119896

120601 + sign (Δ120601) sdot (119868ℎ119896

minus Δ119868119896) (5)

noting that Δ119871119898119896

= (sign(Δ120601)120583119896(sign(Δ120601)120583119896 minus 1))119871119898119896

andΔ119868119896 = (1119871119898

119896

)120578119896In this way the hysteresis inductor 119871ℎ in Figure 3(a) can

be modeled by introducing a linear inductor 119871ℎ119896

in parallelconnection with a current source 119878ℎ

119896

both functionallydependent on the actual linear region 119896 Figure 7 as follows

119871ℎ119896

= 119871119898119896

minus Δ119871119898119896

119878ℎ119896

= sign (Δ120601) sdot (119868ℎ119896

minus Δ119868119896)

(6)

It is noted that all three developed models has a specialroutine for eliminating the possible overshooting effect [26ndash28]

3 Inrush Current Modeling andNumerical Methods

The transformer inrush current is a well-known transientresponse to the energization of a transformer iron coreDuring transformer energization depending on the value ofthe remanent flux the magnetization curve and the breakerswitching instant the iron core magnetic flux can reach atwice higher value than the nominal operating value Inthe case when the remanent flux is nearly as high as thesaturation then the inductance is reduced to near zero


ihkLh

(a)

ihk

Shk

Lhk i0hk

(b)

Figure 7 (a) Hysteresis inductor and (b) equivalent model

The only initial impedance is the small ohmic resistance ofthe primary coil As a result the transformer iron core isdriven into a saturation and inrush current is producedThe transformer inrush current which could be many timeshigher than the operational current may cause irreparabledamages to the circuit without taking into account the designstage

The developedmodels for the nonlinear inductor nonlin-ear resistor and hysteresis inductor are very suitable for for-mulation state-space equations that describe low- frequencytransformer transients such as transformer energizationFigure 8

An arbitrary numerical method could be applied in sucha state-space form In the EMTP-type of programs theelements are strongly dependent on the integration step thisfact becomes apparent when a trapezoidal numerical methodis applied to the relevant branch The general algorithmprocedure for generating state-space matrixes is shown in[27] According to this procedure a state-space equationis developed that describes transformer transients duringinrush current analysis

119889119883 (119905)

119889119905= 119860119895119896119883 (119905) + 119861119895119896119880 (119905) = 119865 (119883 119905) (7)

where the state-space vector and input vector are

119883 (119905) = [119894119871119901

120601]119879

119880 (119905) = [119890 (119905) 119878120601119896

119878119903119895]119879

(8)

System matrixes are respectively as follows

119860119895119896 =

[[[[[[

[

minus

119877119901 + 119877119898119895

119871119901

119877119898119895

119871119901119871119898119896

119877119898119895

minus

119877119898119895

119871119898119896

]]]]]]

]

119861119895119896 =

[[[[[[[[[

[

1

119871119901

0

119877119898119895

119871119901

minus119877119898119895

119877119898119895

119871119901

minus119877119898119895

]]]]]]]]]

]

119879

(9)

It is very important to discover whether the demonstratedsystem is a stiff system Stiff systems are categorized asthose different components of solutions which evolve on verydifferent time scales occurring simultaneously that is therates of change of the various components of the solutionswhich differ markedly Stiffness is a property of system withstrong implications for its practical solution using numericalmethods The stiffness of system is defined by two factorsstiffness ratio 120585119895119896 and stiffness index 120577119895119896 for all state matrixes119860119895119896

120585119895119896 =

max119894100381610038161003816100381610038161003816Re (120582(119894)

119895119896)100381610038161003816100381610038161003816

min119894100381610038161003816100381610038161003816Re (120582(119894)

119895119896)100381610038161003816100381610038161003816

120577119895119896 = max119894

10038161003816100381610038161003816Re (120582(119894)

119895119896)10038161003816100381610038161003816

(10)

where 120582(119894)119895119896 119894 = 1 2 dim(119860119895119896) are eigenvalues of all state

matrixes 119860119895119896 In the case for 120585119895119896 ≫ 1 it is a stiff system andin the case for 120577119895119896 rarr infin it is a very stiff system

The state equations (7) are traditionally solved using theimplicit 119860-stable second-order trapezoidal method [7]

119883119899+1 = 119883119899 +Δ119905

2[119865 (119883119899 119905119899) + 119865 (119883119899+1 119905119899+1)]

119899 = 1 2 3

(11)

The trapezoidal method is suitable for nonstiff andmoderate stiff systems however for a very stiff system thismethod could be inaccurate and other techniques shouldbe used [26ndash30] Indeed the trapezoidal method is not 119871-stable such that for state matrixes 119860119895119896 with eigenvaluescontaining large negative real parts this method producesunwanted numerical oscillations In addition the classicalexplicit numerical methods are not applicable for stiff or verystiff systems due to finite regions of numerical stability

On the other hand the Backward Differentiation Formu-las (BDF119901) methods of 119901th order 119901 le 5 as introduced byGear in 1971 have the following form [29 30]

119901

sum

119898=1

1

119898nabla119898119883119899+1 = Δ119905119865 (119883119899+1 119905119899+1) (12)

The order of the truncation errors for the BDF119901 methodsis Δ119905119901+1 [29 30] The BDF119901 methods are 119860(120572)- and 119871-stable with stability angles ranging between 90∘ 90∘ 86∘73∘ and 51∘ [25] The 119871-stability properties of the BDF119901methods lead to the suppression of numerical oscillations


e(t)

t = T0

Rp Lp

iLp

Rmj Srjum Lmk Shk

Ls RsNp Ns

Figure 8 Equivalent transformer model for inrush current simulations

minus8 minus6 minus4 minus2 0 2 4 6 8minus2

minus15

minus1

minus05

0

05

1

15

2

Magnetizing current (A)

Mag

netic

flux

(Vs)

Magnetizing curveHysteresis loop

Figure 9 Magnetizing curve and hysteresis loop

whereas the trapezoidal method is not free from numericaloscillations [29 30] In addition Numerical DifferentiationFormulas (NDF119901) as suggested by Klopfenstein in 1971 area modification of the BDF119901rsquos methods with the next relations[31 32]

119901

sum

119898=1

1

119898nabla119898119883119899+1 = Δ119905119865 (119883119899+1 119905119899+1)

+ 120581120574119901 (119883119899+1 minus 119883[0]

119899+1)

(13)

where the coefficients are 120574119901 = sum119901

119898=1(1119898) the predicted

value is119883[0]119899+1

= sum119901

119898=0nabla119898119883119899 and parameter 120581was introduced

by Shampine and Reichelt [32] which is a compromise madein balancing efficiency in step size and stability angle ofthe NDFp These methods are also 119860(120572)- and 119871-stable [32]Compared with the BDFprsquos the NDFp methods achievethe same accuracy as BDFp methods with a bigger step-size percents 26 26 26 12 and 0 respectivelyThis implies improvements in the efficiency of the methodsHowever the percent changes in the stability angle are 00 minus7 minus10 and 0 respectively It can be concludedthe NDFp methods are more precise than the BDFp butnot more stable The other modifications of original BDFp

0 0025 005 0075 01 0125 015 0175 020

100

200

300

400

500

600

700

800

Resistor current (A)

Volta

ge (V

)

Linear resistorNonlinear resistor

Figure 10 Nonlinear core loss resistor curve

Figure 11 Measured transformer inrush current

methods are the extended BDFmethods as proposed by Cashin [33] the generalized 119860-BDF methods as introduced byFredebeul in [34] the diagonalizable extended BDFmethodsas suggested by Frank and vanderHouwen in [35] a predictormodification to the extended BDF methods as introduced


Figure 12 Measured transformer steady-state current

Figure 13 Harmonic content of measured inrush current

0 002 004 006 008 01 012 014

0

2

4

6

8

10

Time (s)

Curr

ent (

A)

Inrush current (simulation)

005 0055 006minus05

0

05

Figure 14 Inrush current simulation Model 3 trapezoidal methodΔ119905 = 100 120583sec

by Alberdi and Anza in [36] and the block BDF methodsdeveloped by Yatim et al in [37]

The main aim of this paper was to explore the types ofdifferential equation systems describing the energization ofa transformer using appropriate numerical method for thecorrect simulations of equations In general for simulatingelectrical systems the priority is that the numericalmethod is119860-stable has a high accuracy and successfully solves a (very)stiff system

In regard to these requirements the conclusion is thatmethods BDF1 and -2 and NDF1 and -2 are 119860- and 119871-stableThey are generally well suited for simulations of nonlinearelectrical systems This paper realizes an algorithm for thesolution of (7) based on the usages of the NDF2 or BDF2methods These methods were selected because of the factthat they have the same truncation errors of order Δ1199053 likea trapezoidal method and they are 119860- and 119871-stable Onthe other hand it is shown that it is possible to combinethese methods using the widely used trapezoidal method forsimulating electrical systems

4 Inrush CurrentMeasurements and Simulations

Three developed single-phase transformer models weretested on an example of a transformer inrush current tran-sient

The system and transformer parameters are(i) system voltage 119890(119905) = 328 cos(120596119905 minus 39∘)(ii) nominal transformer power 119878tr = 300VA(iii) primarysecondary voltage 119880119901 = 22024V(iv) short circuit voltage 119906119896 = 545(v) primary winding resistance 119877119901 = 375Ω(vi) primary winding leakage inductance 119871119901 = 254mH(vii) core loss resistance 119877119898 = 3826ΩThe nonlinear magnetizing curve with a major hysteresis

loop is shown in Figure 9 A nonlinear curve of an iron corelosses resistor is shown in Figure 10

Figures 11 and 12 present the results of measurementof transformer inrush and steady-state current In additionharmonic analysis of the transformer inrush current for thefirst four harmonic components is realized Figure 13

Figure 14 shows the simulation results of the developedalgorithm for Model 3 realized using the classical trapezoidalmethod with integration step Δ119905 = 100 120583sec

Figure 14 clearly shows the existence of numerical oscil-lations during the trapezoidal methodrsquos usage Otherwiseany simulations of the developed models (Model 1 Model 2or Model 3) using trapezoidal method clearly showed theexistence of unwanted numerical oscillations

In order to investigate the causes of numerical oscillationsduring simulation of maximum and minimum eigenvaluesstiffness ratios and stiffness indexes were calculated at eachintegration step Figure 15 shows the results of the calculatedmaximum and minimum eigenvalues during the simulationtimes


0 014minus16

minus155

minus15Min eigenvalue

Time (s)

Model 1

0 014minus45

minus25


Time (s)

Model 2

0 014minus49

minus47


Time (s)

Model 3

Time (s)

Model 1

Time (s)

Model 2

Time (s)

Model 3

0 014minus100

minus50

0Max eigenvalue

0 014minus100

minus50

0Max eigenvalue

0 014minus100

minus50

0Max eigenvalue

times106 times106 times106

Figure 15 Eigenvalue propagation during the simulation time all three models

0 002 004 006 008 01 012 014

0

2

4

6

8

10

Time (s)

Curr

ent (

A)

Inrush current (measurement and simulation)

MeasuredModel 1

Model 2Model 3

Figure 16 Measured and simulated transformer inrush currentNDF2 method Δ119905 = 100 120583sec

Table 1 shows the borders of the analyzed stiffness param-eters It is obvious that they are very stiff systems for all threecases

It is interesting that the extreme values of parameters120585119895119896 rose in the unsaturated regions of the magnetizing(hysteresis) curve and the extreme values of parameter 120577119895119896

01 011 012 013 014 015 016minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time (s)

Curr

ent (

A)

Steady-state current (measurement and simulation)

MeasuredModel 1

Model 2Model 3

Figure 17 Measured and simulated transformer steady-state cur-rent NDF2 method Δ119905 = 100 120583sec

rose in the saturated region of this curve Also it is interestingthat the time function of parameter 120577119895119896 is an analog to theinrush current waveform in all three cases

In view of the mentioned reasons the BDF2 and NDF2methods were used for simulation of transformer inrushcurrent Figures 16 and 17 show the results when comparing


1 2 3 4 5 60

2

4

6

8

10

Peak number

Relat

ive e

rror

()

Model 1Model 2Model 3

Figure 18 Relative error of peak values of inrush current

0 002 004 006 008 01 012 0140

05

1

15

2

25

3

35

Time (s)

1st h

arm

onic

(A)

MeasuredModel 1

Model 2Model 3

MeasuredModel 1

Model 2Model 3

0 002 004 006 008 01 012 0140

05

1

15

2

25

3

35

Time (s)

2nd

harm

onic

(A)

0 002 004 006 008 01 012 0140

05

1

15

2

25

3

35

Time (s)

3rd

harm

onic

(A)

0 002 004 006 008 01 012 0140

05

1

15

2

25

3

35

Time (s)

4th

harm

onic

(A)

Figure 19 Harmonic content of measured and simulated inrush currents


Table 1 Stiffness ratios and stiffness indexes of the system

Model 1 1508 sdot 106le 120577119895119896 le 1596 sdot 10

6

1965 sdot 104le 120585119895119896 le 4949 sdot 10

4


6

1256 sdot 104le 120585119895119896 le 1183 sdot 10

7


6

4622 sdot 104le 120585119895119896 le 1566 sdot 10

7

Table 2 Peak error of steady-state current

Model Model 1 Model 2 Model 3Peak error [] 1318 1198 456

the measured and simulated inrush current and steady-statetransformer current by usage of the NDF2 method with anintegration step of Δ119905 = 100 120583sec

Figure 18 shows the peak values for the inrush currentserrors for all three models It is necessary to note that themaximum error when usingModels 1 and 2 was 529 whilstwhen using Model 3 the maximum error was 430

Table 2 shows the error in peak value for the transformersteady-state current Minimum error was in Model 3

In addition a comparison of the harmonic contentsbetween the measured and simulated inrush current isrealized Figure 19This figure shows that the best results werefor Model 3 whilst Models 1 and 2 provided mostly identicalsimulation results

On the basis of all the aforementioned it is possible tosuggest the hybrid numerical method as a linear combinationof the traditional trapezoidalmethod and the proposed BDF2(NDF2) method In [38] a hybrid method was proposedby Tadeusiewicz and Halgas in 2005 that presented a linearcombination of the trapezoidal and BDF2method in the form

119883119899+1 = (1 +120579

3)119883119899 minus

120579

3119883119899minus1 + (1 minus 120579)

Δ119905

2119865 (119883119899 119905119899)

+ (1 +120579

3)Δ119905

2119865 (119883119899+1 119905119899+1)

(14)

In expression (14) 120579 = 0 leads to the trapezoidal methodwhereas 120579 = 1 leads to the BDF2 method

In the same way we have proposed a hybrid method thatrepresents a linear combination of the trapezoidal and NDF2methods in the form

119883119899+1 = (1 +120579

5)119883119899 minus

3120579

10119883119899minus1 + (1 minus 120579)

Δ119905

2119865 (119883119899 119905119899)

+ (1 +120579

5)Δ119905

2119865 (119883119899+1 119905119899+1) +

120579

10119883[0]

119899+1

(15)

In the expression (15) 120579 = 0 leads to the trapezoidalmethod whereas 120579 = 1 leads to the NDF2 method Thismethod overcomes all the eventual problems during theusage of the standard trapezoidal method

5 Conclusions

This paper investigates the different models and numericalmethods that could be implemented for the simulations ofinrush current in a single-phase transformer Various coremodels of a transformer are developed in the paper a modelwith a nonlinear inductor and a linear resistor a model witha nonlinear inductor and a nonlinear resistor and a modelwith a hysteresis inductor and a linear resistor In addi-tion the paper investigates eigenvalues propagations duringsimulation time for all three models The defined stiffnessparameters the stiffness ratios and the stiffness indexesindicated that very stiff systems should be solved for all threedeveloped models The state-space equation system is solvedby using proposed 119860- and 119871-stable numerical differentiationformulas (NDF2) (BDF2) method The proposed methodhas the same order as a trapezoidal method however thismethod overcomes the main drawback of the trapezoidalmethod (numerical oscillations) In the case where thenumerical oscillations are generated by use of the trapezoidalmethod the NDF2 (BDF2) method efficiently suppressesthem Comparisons between the measured and simulatedinrush and steady-state currents of a transformer are con-ducted for all three proposed models The best simulationresults were obtained by using the developed Model 3 whichincorporates hysteresis inductor The developed Models 1and 2 gave almost identical simulation results The realizednonlinear inductor nonlinear resistor and hysteresis modelcan be incorporated within the EMTP-type of programsby using a combination of the existing trapezoidal andproposed NDF (BDF) method of order 2 In this way thesuggested numerical method could be used in the simulationof electrical networks with nonlinear elements The futurework will investigate the propagation of eigenvalues of asystem in simulations of inrush currents in a three-phasepower transformer

References

[1] CIGRE Working Group SC 3302 ldquoGuidelines for representa-tion of network elements when calculating transientsrdquo CIGREBrochure vol 39 pp 1ndash29 1990

[2] D A N Jacobson P W Lehn and R W Menzies ldquoStabilitydomain calculations of period-1 ferroresonance in a nonlinearresonant circuitrdquo IEEE Transactions on Power Delivery vol 17no 3 pp 865ndash871 2002

[3] G Stumberger S Seme B Stumberger B Polajzer and DDolinar ldquoDetermining magnetically nonlinear characteristicsof transformers and iron core inductors by differential evolu-tionrdquo IEEE Transactions on Magnetics vol 44 no 6 pp 1570ndash1573 2008

[4] C Perez-Rojas ldquoFitting saturation and hysteresis via arctangentfunctionsrdquo IEEE Power Engineering Review vol 20 no 11 pp55ndash57 2000

[5] X Zhang ldquoAn efficient algorithm for transient calculation ofthe electric networks including magnetizing branchesrdquo Math-ematical Problems in Engineering vol 2011 Article ID 47962611 pages 2011

[6] A Ben-Tal V Kirk and G Wake ldquoBanded chaos in powersystemsrdquo IEEE Transactions on Power Delivery vol 16 no 1 pp105ndash110 2001


[7] H W Dommel Electromagnetic Transients Program ReferenceManual (EMTP Theory Book) Bonneville Power Administra-tion Portland Ore USA 1986

[8] C E Lin C L Cheng C L Huang and J C Yeh ldquoInvestigationofmagnetizing inrush current in transformersmdashpart I numeri-cal simulationrdquo IEEE Transactions on Power Delivery vol 8 no1 pp 246ndash254 1993

[9] E H Badawy andRD Youssef ldquoRepresentation of transformersaturationrdquo Electric Power Systems Research vol 6 no 4 pp301ndash304 1983

[10] S Seker T C Akinci and S Taskin ldquoSpectral and statisticalanalysis for ferroresonance phenomenon in electric powersystemsrdquo Electrical Engineering vol 94 no 2 pp 117ndash124 2012

[11] I Claverıa M Garcıa-Gracia M A Garcıa and L MontanesldquoA time domain small transformer model under sinusoidaland non-sinusoidal supply voltagerdquo European Transactions onElectrical Power vol 15 no 4 pp 311ndash323 2005

[12] J R Lucas P G McLaren W W L Keerthipala and RP Jayasinghe ldquoImproved simulation models for current andvoltage transformers in relay studiesrdquo IEEE Transactions onPower Delivery vol 7 no 1 pp 152ndash159 1992

[13] Y Baghzouz and X D Gong ldquoVoltage-dependent model forteaching transformer core nonlinearityrdquo IEEE Transactions onPower Systems vol 8 no 2 pp 746ndash752 1993

[14] T Tran-Quoc and L Pierrat ldquoEfficient non linear transformermodel and its application to ferroresonance studyrdquo IEEE Trans-actions on Magnetics vol 31 no 3 pp 2060ndash2063 1995

[15] A D Theocharis J Milias-Argitis and T Zacharias ldquoSingle-phase transformer model including magnetic hysteresis andeddy currentsrdquoElectrical Engineering vol 90 no 3 pp 229ndash2412008

[16] J Faiz and S Saffari ldquoInrush current modeling in a single-phasetransformerrdquo IEEE Transactions onMagnetics vol 46 no 2 pp578ndash581 2010

[17] H Akcay and D G Ece ldquoModeling of hysteresis and powerlosses in transformer laminationsrdquo IEEE Transactions on PowerDelivery vol 18 no 2 pp 487ndash492 2003

[18] S-R Huang S C Chung B-N Chen and Y-H Chen ldquoAharmonic model for the nonlinearities of single-phase trans-former with describing functionsrdquo IEEE Transactions on PowerDelivery vol 18 no 3 pp 815ndash820 2003

[19] W L A Neves and H W Dommel ldquoOn modelling iron corenonlinearitiesrdquo IEEE Transactions on Power Systems vol 8 no2 pp 417ndash422 1993

[20] A Tokic I Uglesic V Milardic and G Stumberger ldquoCon-version of RMS to instantaneous saturation curve inrushcurrent and ferroresonance casesrdquo in Proceedings of the 14thInternational IGTE Symposium on Numerical Field Calculationin Electrical Engineering pp 1ndash4 Graz Austria 2010

[21] I D Mayergoyz Mathematical Models of Hysteresis and TheirApplications Academic Press Elsevier New York NY USA2003

[22] F C Trutt E A Erdelyi and R E Hopkins ldquoRepresentation ofthe magnetization characteristic of DC machines for computeruserdquo IEEE Transactions on Power Apparatus and Systems vol87 no 3 pp 665ndash669 1968

[23] O Karlqvist Calculation of the Magnetic Field in the Ferro-magnetic Layer of a Magnetic Drum vol 86 of Transactions ofthe Royal Institute of Technology Elanders boktr StockholmSweden 1954

[24] J Takacs ldquoA phenomenological mathematical model of hystere-sisrdquo COMPEL The International Journal for Computation andMathematics in Electrical and Electronic Engineering vol 20 no4 pp 1002ndash1014 2001

[25] J Takacs ldquoThe Everett integral and its analytical approxima-tionrdquo in Advanced Magnetic Materials pp 203ndash230 IntechPublication 2012

[26] A Tokic VMadzarevic and I Uglesic ldquoNumerical calculationsof three-phase transformer transientsrdquo IEEE Transactions onPower Delivery vol 20 no 4 pp 2493ndash2500 2005

[27] A Tokic and I Uglesic ldquoElimination of overshooting effects andsuppression of numerical oscillations in transformer transientcalculationsrdquo IEEE Transactions on Power Delivery vol 23 no1 pp 243ndash251 2008

[28] A Tokic V Madzarevic and I Uglesic ldquoHysteresis modelin transient simulation algorithm based on BDF numericalmethodrdquo in Proceedings of the IEEE Power Tech Conference pp1ndash6 St Petersburg Russia 2005

[29] E Hairer and G Wanner Solving Ordinary Differential Equa-tions II Stiff and Differential-Algebraic Problems vol 14 ofSpringer Series in Computational Mathematics Springer BerlinGermany 1991

[30] C W Gear Numerical Initial Value Problems in OrdinaryDifferential Equations Prentice-Hall Englewood Cliffs NJUSA 1971

[31] RW Klopfenstein ldquoNumerical differentiation formulas for stiffsystems of ordinary differential equationsrdquoRCAReview vol 32no 3 pp 447ndash462 1971

[32] L F Shampine and M W Reichelt ldquoThe MATLAB ODE suiterdquoSIAM Journal on Scientific Computing vol 18 no 1 pp 1ndash221997

[33] J R Cash ldquoOn the integration of stiff systems of ODEsusing extended backward differentiation formulaerdquoNumerischeMathematik vol 34 no 3 pp 235ndash246 1980

[34] C Fredebeul ldquoA-BDF a generalization of the backward differ-entiation formulaerdquo SIAM Journal on Numerical Analysis vol35 no 5 pp 1917ndash1938 1998

[35] J E Frank and P J van der Houwen ldquoDiagonalizable extendedbackward differentiation formulasrdquo BIT Numerical Mathemat-ics vol 40 no 3 pp 497ndash512 2000

[36] E Alberdi and J J Anza ldquoA predictor modification to theEBDF method for Stiff systemsrdquo Journal of ComputationalMathematics vol 29 no 2 pp 199ndash214 2011

[37] S AM Yatim Z B IbrahimK I Othman andM B SuleimanldquoA quantitative comparison of numerical method for solvingstiff ordinary differential equationsrdquoMathematical Problems inEngineering vol 2011 Article ID 193691 12 pages 2011

[38] M Tadeusiewicz and S Hałgas ldquoTransient analysis of nonlineardynamic circuits using a numerical-integrationmethodrdquoCOM-PELThe International Journal for Computation and Mathemat-ics in Electrical andElectronic Engineering vol 24 no 2 pp 707ndash719 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


t = T0 RpLp

e(t) Rm Lm

Ls RsNp Ns

Figure 1 Standard 119879 equivalent transformer circuit model

120601

120601sN

120601s2

120601s1is1 is2 isN

im

Figure 2 Nonlinear curve of core inductor

[2ndash10] use the (a) based model of transformer [11ndash14] use the(b) but [15ndash18] use the (c) based model of the transformer

Typical procedures for obtaining an instantaneous non-linear magnetizing curve and nonlinear curve of core lossesby the separation technique of losses from the magnetizingcurve are described in [19 20]

The modeling of nonlinear or hysteresis inductor andnonlinear core loss resistor can be done by using the curvefitting procedure for the approximation of nonlinearity or byusing piecewise linear representation of the nonlinear curve

The well-known curve fitting approximations areobtained by using the polynomial exponential arctg andhyperbolic functions The polynomial approximation wassuggested by Mayergoyz in [21] Exponential approximationwas introduced by Trutt et al in [22] Arctg approximationwas started by Karlqvist in [23] and the hyperbolic functionswere introduced by Takacs in [24] and refined by Takacs in[25]

The piecewise linearization of a nonlinear curve forthe modeling of nonlinear or hysteretic inductor on theother hand is used in [7ndash10] Linearization of a nonlinearcurve has certain advantages and disadvantages Successfulcontrol of the numerical stability properties of the appliednumerical methods is the main advantage [26ndash28] Alsoanother advantage of this linearization is the improved speedcompared to those classical methods used in calculationswithin nonlinear algebraic-differential equation systems Onthe other hand piecewise linearization has disadvantagesrelated to overshooting effects [27]

2 Transformer Iron Core Modeling

Theoriginal way ofmodeling nonlinear transformer iron corein low-frequency transient analysis using three mentionedmodels is derived in this paper

imk

Lm

(a)

imk

S120601k

Lmk i0mk

(b)

Figure 3 (a) Nonlinear inductor and (b) equivalent model

urM

uR

ur2ur1

ir1 ir2 irM

iR

Figure 4 Nonlinear curve of core resistor

21 Modeling of Nonlinear Core Inductor The nonlinear coreinductor can be described using a magnetizing curve mag-netizing current versus magnetic flux as shown in Figure 2

When this curve is piecewise linear the results are theinput vectors magnetizing currents 119868119871

119898

= [1198941199041

1198941199042

119894119904119873

]119879

and magnetic fluxes Φ119871119898

= [1206011199041

1206011199042

120601119904119873

]119879 whereas 119873

is the total number of magnetizing curve regions The mag-netizing current for the 119896th linear region of the magnetizingcurve 119896 = 1 2 (119873 minus 1) is calculated using the equation

119894119898119896

=1

119871119898119896

120601 + sign (120601) 119868119904119896

=1

119871119898119896

120601 + 119878120601119896

(1)

where 119871119898119896

= (120601119904119896+1

minus120601119904119896

)(119894119904119896+1

minus119894119904119896

) and 119868119904119896

= 119894119904119896

minus(1119871119898119896

)120601119904119896

Based on (1) it is possible to create a nonlinear inductorequivalent model using a linear inductor and a correspond-ing current source Figure 3 This model is functionallydependent on the position of the operating point 119896 At thesame time contrary to EMTP-based equivalent models theinductor model is not dependent on the integration step

22 Modeling of Nonlinear Core Resistor Analogous to thenonlinear core inductor the nonlinear core resistor can bedescribed using the curve resistor current versus corre-sponding resistor voltage as shown in Figure 4

In this case the input vectors are the resistor currents119868119877119898

= [1198941199031

1198941199032

119894119903119872

]119879 and resistor voltages 119880119877

119898

= [1199061199031

1199061199032

119906119903119872

]119879 where119872 is the total number of resistor curve


iRk Rm

(a)

iRk

Srk

Rmk i0Rk

(b)



119894119877119896

=1

119877119898119896

119906119877 + sign (119906119877) 119868119903119896

=1

119877119898119896

119906119877 + 119878119903119896

(2)

where119877119898119896

= (119906119903119896+1

minus119906119903119896

)(119894119903119896+1

minus119894119903119896

) and 119868119903119896

= 119894119903119896

minus(1119877119898119896

)119906119903119896





ℎ

= [119894ℎ1

119894ℎ2

119894ℎ119875


ℎ

=

[120601ℎ1

120601ℎ2

120601ℎ119875




119894ℎ119896

=1

119871119898119896

120601119898 + sign (Δ120601119898) 119868ℎ119896

(3)


=

(120601ℎ119896+1

minus 120601ℎ119896

)(119894ℎ119896+1

minus 119894ℎ119896

) and 119868ℎ119896

= 119894ℎ119896

minus (1119871119898119896

)120601ℎ119896



120601

1

2

3



ih



119889 = 120583119896120601 + 120578119896 = sign (Δ120601) (120601 minus 120601119898) (4)



119894ℎ119896

=1

119871119898119896

minus Δ119871119898119896

120601 + sign (Δ120601) sdot (119868ℎ119896

minus Δ119868119896) (5)

noting that Δ119871119898119896


andΔ119868119896 = (1119871119898

119896




119896


119871ℎ119896

= 119871119898119896

minus Δ119871119898119896

119878ℎ119896

= sign (Δ120601) sdot (119868ℎ119896

minus Δ119868119896)

(6)





ihkLh

(a)

ihk

Shk

Lhk i0hk

(b)





119889119883 (119905)

119889119905= 119860119895119896119883 (119905) + 119861119895119896119880 (119905) = 119865 (119883 119905) (7)


119883 (119905) = [119894119871119901

120601]119879

119880 (119905) = [119890 (119905) 119878120601119896

119878119903119895]119879

(8)


119860119895119896 =

[[[[[[

[

minus

119877119901 + 119877119898119895

119871119901

119877119898119895

119871119901119871119898119896

119877119898119895

minus

119877119898119895

119871119898119896

]]]]]]

]

119861119895119896 =

[[[[[[[[[

[

1

119871119901

0

119877119898119895

119871119901

minus119877119898119895

119877119898119895

119871119901

minus119877119898119895

]]]]]]]]]

]

119879

(9)


120585119895119896 =

max119894100381610038161003816100381610038161003816Re (120582(119894)

119895119896)100381610038161003816100381610038161003816

min119894100381610038161003816100381610038161003816Re (120582(119894)

119895119896)100381610038161003816100381610038161003816

120577119895119896 = max119894

10038161003816100381610038161003816Re (120582(119894)

119895119896)10038161003816100381610038161003816

(10)




119883119899+1 = 119883119899 +Δ119905

2[119865 (119883119899 119905119899) + 119865 (119883119899+1 119905119899+1)]

119899 = 1 2 3

(11)



119901

sum

119898=1

1

119898nabla119898119883119899+1 = Δ119905119865 (119883119899+1 119905119899+1) (12)



e(t)

t = T0

Rp Lp

iLp

Rmj Srjum Lmk Shk

Ls RsNp Ns



minus15

minus1

minus05

0

05

1

15

2


Mag

netic

flux

(Vs)




119901

sum

119898=1

1

119898nabla119898119883119899+1 = Δ119905119865 (119883119899+1 119905119899+1)

+ 120581120574119901 (119883119899+1 minus 119883[0]

119899+1)

(13)


119898=1(1119898) the predicted

value is119883[0]119899+1

= sum119901



0 0025 005 0075 01 0125 015 0175 020

100

200

300

400

500

600

700

800


Volta

ge (V

)








0 002 004 006 008 01 012 014

0

2

4

6

8

10

Time (s)

Curr

ent (

A)


005 0055 006minus05

0

05














0 014minus16

minus155


Time (s)

Model 1

0 014minus45

minus25


Time (s)

Model 2

0 014minus49

minus47


Time (s)

Model 3

Time (s)

Model 1

Time (s)

Model 2

Time (s)

Model 3

0 014minus100

minus50

0Max eigenvalue

0 014minus100

minus50

0Max eigenvalue

0 014minus100

minus50

0Max eigenvalue



0 002 004 006 008 01 012 014

0

2

4

6

8

10

Time (s)

Curr

ent (

A)


MeasuredModel 1

Model 2Model 3




01 011 012 013 014 015 016minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time (s)

Curr

ent (

A)


MeasuredModel 1

Model 2Model 3





1 2 3 4 5 60

2

4

6

8

10

Peak number

Relat

ive e

rror

()



0 002 004 006 008 01 012 0140

05

1

15

2

25

3

35

Time (s)

1st h

arm

onic

(A)

MeasuredModel 1

Model 2Model 3

MeasuredModel 1

Model 2Model 3

0 002 004 006 008 01 012 0140

05

1

15

2

25

3

35

Time (s)

2nd

harm

onic

(A)

0 002 004 006 008 01 012 0140

05

1

15

2

25

3

35

Time (s)

3rd

harm

onic

(A)

0 002 004 006 008 01 012 0140

05

1

15

2

25

3

35

Time (s)

4th

harm

onic

(A)





6

1965 sdot 104le 120585119895119896 le 4949 sdot 10

4


6

1256 sdot 104le 120585119895119896 le 1183 sdot 10

7


6

4622 sdot 104le 120585119895119896 le 1566 sdot 10

7








119883119899+1 = (1 +120579

3)119883119899 minus

120579

3119883119899minus1 + (1 minus 120579)

Δ119905

2119865 (119883119899 119905119899)

+ (1 +120579

3)Δ119905

2119865 (119883119899+1 119905119899+1)

(14)



119883119899+1 = (1 +120579

5)119883119899 minus

3120579

10119883119899minus1 + (1 minus 120579)

Δ119905

2119865 (119883119899 119905119899)

+ (1 +120579

5)Δ119905

2119865 (119883119899+1 119905119899+1) +

120579

10119883[0]

119899+1

(15)


5 Conclusions


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










iRk Rm

(a)

iRk

Srk

Rmk i0Rk

(b)



119894119877119896

=1

119877119898119896

119906119877 + sign (119906119877) 119868119903119896

=1

119877119898119896

119906119877 + 119878119903119896

(2)

where119877119898119896

= (119906119903119896+1

minus119906119903119896

)(119894119903119896+1

minus119894119903119896

) and 119868119903119896

= 119894119903119896

minus(1119877119898119896

)119906119903119896





ℎ

= [119894ℎ1

119894ℎ2

119894ℎ119875


ℎ

=

[120601ℎ1

120601ℎ2

120601ℎ119875




119894ℎ119896

=1

119871119898119896

120601119898 + sign (Δ120601119898) 119868ℎ119896

(3)


=

(120601ℎ119896+1

minus 120601ℎ119896

)(119894ℎ119896+1

minus 119894ℎ119896

) and 119868ℎ119896

= 119894ℎ119896

minus (1119871119898119896

)120601ℎ119896



120601

1

2

3



ih



119889 = 120583119896120601 + 120578119896 = sign (Δ120601) (120601 minus 120601119898) (4)



119894ℎ119896

=1

119871119898119896

minus Δ119871119898119896

120601 + sign (Δ120601) sdot (119868ℎ119896

minus Δ119868119896) (5)

noting that Δ119871119898119896


andΔ119868119896 = (1119871119898

119896




119896


119871ℎ119896

= 119871119898119896

minus Δ119871119898119896

119878ℎ119896

= sign (Δ120601) sdot (119868ℎ119896

minus Δ119868119896)

(6)





ihkLh

(a)

ihk

Shk

Lhk i0hk

(b)





119889119883 (119905)

119889119905= 119860119895119896119883 (119905) + 119861119895119896119880 (119905) = 119865 (119883 119905) (7)


119883 (119905) = [119894119871119901

120601]119879

119880 (119905) = [119890 (119905) 119878120601119896

119878119903119895]119879

(8)


119860119895119896 =

[[[[[[

[

minus

119877119901 + 119877119898119895

119871119901

119877119898119895

119871119901119871119898119896

119877119898119895

minus

119877119898119895

119871119898119896

]]]]]]

]

119861119895119896 =

[[[[[[[[[

[

1

119871119901

0

119877119898119895

119871119901

minus119877119898119895

119877119898119895

119871119901

minus119877119898119895

]]]]]]]]]

]

119879

(9)


120585119895119896 =

max119894100381610038161003816100381610038161003816Re (120582(119894)

119895119896)100381610038161003816100381610038161003816

min119894100381610038161003816100381610038161003816Re (120582(119894)

119895119896)100381610038161003816100381610038161003816

120577119895119896 = max119894

10038161003816100381610038161003816Re (120582(119894)

119895119896)10038161003816100381610038161003816

(10)




119883119899+1 = 119883119899 +Δ119905

2[119865 (119883119899 119905119899) + 119865 (119883119899+1 119905119899+1)]

119899 = 1 2 3

(11)



119901

sum

119898=1

1

119898nabla119898119883119899+1 = Δ119905119865 (119883119899+1 119905119899+1) (12)



e(t)

t = T0

Rp Lp

iLp

Rmj Srjum Lmk Shk

Ls RsNp Ns



minus15

minus1

minus05

0

05

1

15

2


Mag

netic

flux

(Vs)




119901

sum

119898=1

1

119898nabla119898119883119899+1 = Δ119905119865 (119883119899+1 119905119899+1)

+ 120581120574119901 (119883119899+1 minus 119883[0]

119899+1)

(13)


119898=1(1119898) the predicted

value is119883[0]119899+1

= sum119901



0 0025 005 0075 01 0125 015 0175 020

100

200

300

400

500

600

700

800


Volta

ge (V

)








0 002 004 006 008 01 012 014

0

2

4

6

8

10

Time (s)

Curr

ent (

A)


005 0055 006minus05

0

05














0 014minus16

minus155


Time (s)

Model 1

0 014minus45

minus25


Time (s)

Model 2

0 014minus49

minus47


Time (s)

Model 3

Time (s)

Model 1

Time (s)

Model 2

Time (s)

Model 3

0 014minus100

minus50

0Max eigenvalue

0 014minus100

minus50

0Max eigenvalue

0 014minus100

minus50

0Max eigenvalue



0 002 004 006 008 01 012 014

0

2

4

6

8

10

Time (s)

Curr

ent (

A)


MeasuredModel 1

Model 2Model 3




01 011 012 013 014 015 016minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time (s)

Curr

ent (

A)


MeasuredModel 1

Model 2Model 3





1 2 3 4 5 60

2

4

6

8

10

Peak number

Relat

ive e

rror

()



0 002 004 006 008 01 012 0140

05

1

15

2

25

3

35

Time (s)

1st h

arm

onic

(A)

MeasuredModel 1

Model 2Model 3

MeasuredModel 1

Model 2Model 3

0 002 004 006 008 01 012 0140

05

1

15

2

25

3

35

Time (s)

2nd

harm

onic

(A)

0 002 004 006 008 01 012 0140

05

1

15

2

25

3

35

Time (s)

3rd

harm

onic

(A)

0 002 004 006 008 01 012 0140

05

1

15

2

25

3

35

Time (s)

4th

harm

onic

(A)





6

1965 sdot 104le 120585119895119896 le 4949 sdot 10

4


6

1256 sdot 104le 120585119895119896 le 1183 sdot 10

7


6

4622 sdot 104le 120585119895119896 le 1566 sdot 10

7








119883119899+1 = (1 +120579

3)119883119899 minus

120579

3119883119899minus1 + (1 minus 120579)

Δ119905

2119865 (119883119899 119905119899)

+ (1 +120579

3)Δ119905

2119865 (119883119899+1 119905119899+1)

(14)



119883119899+1 = (1 +120579

5)119883119899 minus

3120579

10119883119899minus1 + (1 minus 120579)

Δ119905

2119865 (119883119899 119905119899)

+ (1 +120579

5)Δ119905

2119865 (119883119899+1 119905119899+1) +

120579

10119883[0]

119899+1

(15)


5 Conclusions


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










ihkLh

(a)

ihk

Shk

Lhk i0hk

(b)





119889119883 (119905)

119889119905= 119860119895119896119883 (119905) + 119861119895119896119880 (119905) = 119865 (119883 119905) (7)


119883 (119905) = [119894119871119901

120601]119879

119880 (119905) = [119890 (119905) 119878120601119896

119878119903119895]119879

(8)


119860119895119896 =

[[[[[[

[

minus

119877119901 + 119877119898119895

119871119901

119877119898119895

119871119901119871119898119896

119877119898119895

minus

119877119898119895

119871119898119896

]]]]]]

]

119861119895119896 =

[[[[[[[[[

[

1

119871119901

0

119877119898119895

119871119901

minus119877119898119895

119877119898119895

119871119901

minus119877119898119895

]]]]]]]]]

]

119879

(9)


120585119895119896 =

max119894100381610038161003816100381610038161003816Re (120582(119894)

119895119896)100381610038161003816100381610038161003816

min119894100381610038161003816100381610038161003816Re (120582(119894)

119895119896)100381610038161003816100381610038161003816

120577119895119896 = max119894

10038161003816100381610038161003816Re (120582(119894)

119895119896)10038161003816100381610038161003816

(10)




119883119899+1 = 119883119899 +Δ119905

2[119865 (119883119899 119905119899) + 119865 (119883119899+1 119905119899+1)]

119899 = 1 2 3

(11)



119901

sum

119898=1

1

119898nabla119898119883119899+1 = Δ119905119865 (119883119899+1 119905119899+1) (12)



e(t)

t = T0

Rp Lp

iLp

Rmj Srjum Lmk Shk

Ls RsNp Ns



minus15

minus1

minus05

0

05

1

15

2


Mag

netic

flux

(Vs)




119901

sum

119898=1

1

119898nabla119898119883119899+1 = Δ119905119865 (119883119899+1 119905119899+1)

+ 120581120574119901 (119883119899+1 minus 119883[0]

119899+1)

(13)


119898=1(1119898) the predicted

value is119883[0]119899+1

= sum119901



0 0025 005 0075 01 0125 015 0175 020

100

200

300

400

500

600

700

800


Volta

ge (V

)








0 002 004 006 008 01 012 014

0

2

4

6

8

10

Time (s)

Curr

ent (

A)


005 0055 006minus05

0

05














0 014minus16

minus155


Time (s)

Model 1

0 014minus45

minus25


Time (s)

Model 2

0 014minus49

minus47


Time (s)

Model 3

Time (s)

Model 1

Time (s)

Model 2

Time (s)

Model 3

0 014minus100

minus50

0Max eigenvalue

0 014minus100

minus50

0Max eigenvalue

0 014minus100

minus50

0Max eigenvalue



0 002 004 006 008 01 012 014

0

2

4

6

8

10

Time (s)

Curr

ent (

A)


MeasuredModel 1

Model 2Model 3




01 011 012 013 014 015 016minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time (s)

Curr

ent (

A)


MeasuredModel 1

Model 2Model 3





1 2 3 4 5 60

2

4

6

8

10

Peak number

Relat

ive e

rror

()



0 002 004 006 008 01 012 0140

05

1

15

2

25

3

35

Time (s)

1st h

arm

onic

(A)

MeasuredModel 1

Model 2Model 3

MeasuredModel 1

Model 2Model 3

0 002 004 006 008 01 012 0140

05

1

15

2

25

3

35

Time (s)

2nd

harm

onic

(A)

0 002 004 006 008 01 012 0140

05

1

15

2

25

3

35

Time (s)

3rd

harm

onic

(A)

0 002 004 006 008 01 012 0140

05

1

15

2

25

3

35

Time (s)

4th

harm

onic

(A)





6

1965 sdot 104le 120585119895119896 le 4949 sdot 10

4


6

1256 sdot 104le 120585119895119896 le 1183 sdot 10

7


6

4622 sdot 104le 120585119895119896 le 1566 sdot 10

7








119883119899+1 = (1 +120579

3)119883119899 minus

120579

3119883119899minus1 + (1 minus 120579)

Δ119905

2119865 (119883119899 119905119899)

+ (1 +120579

3)Δ119905

2119865 (119883119899+1 119905119899+1)

(14)



119883119899+1 = (1 +120579

5)119883119899 minus

3120579

10119883119899minus1 + (1 minus 120579)

Δ119905

2119865 (119883119899 119905119899)

+ (1 +120579

5)Δ119905

2119865 (119883119899+1 119905119899+1) +

120579

10119883[0]

119899+1

(15)


5 Conclusions


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










e(t)

t = T0

Rp Lp

iLp

Rmj Srjum Lmk Shk

Ls RsNp Ns



minus15

minus1

minus05

0

05

1

15

2


Mag

netic

flux

(Vs)




119901

sum

119898=1

1

119898nabla119898119883119899+1 = Δ119905119865 (119883119899+1 119905119899+1)

+ 120581120574119901 (119883119899+1 minus 119883[0]

119899+1)

(13)


119898=1(1119898) the predicted

value is119883[0]119899+1

= sum119901



0 0025 005 0075 01 0125 015 0175 020

100

200

300

400

500

600

700

800


Volta

ge (V

)








0 002 004 006 008 01 012 014

0

2

4

6

8

10

Time (s)

Curr

ent (

A)


005 0055 006minus05

0

05














0 014minus16

minus155


Time (s)

Model 1

0 014minus45

minus25


Time (s)

Model 2

0 014minus49

minus47


Time (s)

Model 3

Time (s)

Model 1

Time (s)

Model 2

Time (s)

Model 3

0 014minus100

minus50

0Max eigenvalue

0 014minus100

minus50

0Max eigenvalue

0 014minus100

minus50

0Max eigenvalue



0 002 004 006 008 01 012 014

0

2

4

6

8

10

Time (s)

Curr

ent (

A)


MeasuredModel 1

Model 2Model 3




01 011 012 013 014 015 016minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time (s)

Curr

ent (

A)


MeasuredModel 1

Model 2Model 3





1 2 3 4 5 60

2

4

6

8

10

Peak number

Relat

ive e

rror

()



0 002 004 006 008 01 012 0140

05

1

15

2

25

3

35

Time (s)

1st h

arm

onic

(A)

MeasuredModel 1

Model 2Model 3

MeasuredModel 1

Model 2Model 3

0 002 004 006 008 01 012 0140

05

1

15

2

25

3

35

Time (s)

2nd

harm

onic

(A)

0 002 004 006 008 01 012 0140

05

1

15

2

25

3

35

Time (s)

3rd

harm

onic

(A)

0 002 004 006 008 01 012 0140

05

1

15

2

25

3

35

Time (s)

4th

harm

onic

(A)





6

1965 sdot 104le 120585119895119896 le 4949 sdot 10

4


6

1256 sdot 104le 120585119895119896 le 1183 sdot 10

7


6

4622 sdot 104le 120585119895119896 le 1566 sdot 10

7








119883119899+1 = (1 +120579

3)119883119899 minus

120579

3119883119899minus1 + (1 minus 120579)

Δ119905

2119865 (119883119899 119905119899)

+ (1 +120579

3)Δ119905

2119865 (119883119899+1 119905119899+1)

(14)



119883119899+1 = (1 +120579

5)119883119899 minus

3120579

10119883119899minus1 + (1 minus 120579)

Δ119905

2119865 (119883119899 119905119899)

+ (1 +120579

5)Δ119905

2119865 (119883119899+1 119905119899+1) +

120579

10119883[0]

119899+1

(15)


5 Conclusions


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












0 002 004 006 008 01 012 014

0

2

4

6

8

10

Time (s)

Curr

ent (

A)


005 0055 006minus05

0

05














0 014minus16

minus155


Time (s)

Model 1

0 014minus45

minus25


Time (s)

Model 2

0 014minus49

minus47


Time (s)

Model 3

Time (s)

Model 1

Time (s)

Model 2

Time (s)

Model 3

0 014minus100

minus50

0Max eigenvalue

0 014minus100

minus50

0Max eigenvalue

0 014minus100

minus50

0Max eigenvalue



0 002 004 006 008 01 012 014

0

2

4

6

8

10

Time (s)

Curr

ent (

A)


MeasuredModel 1

Model 2Model 3




01 011 012 013 014 015 016minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time (s)

Curr

ent (

A)


MeasuredModel 1

Model 2Model 3





1 2 3 4 5 60

2

4

6

8

10

Peak number

Relat

ive e

rror

()



0 002 004 006 008 01 012 0140

05

1

15

2

25

3

35

Time (s)

1st h

arm

onic

(A)

MeasuredModel 1

Model 2Model 3

MeasuredModel 1

Model 2Model 3

0 002 004 006 008 01 012 0140

05

1

15

2

25

3

35

Time (s)

2nd

harm

onic

(A)

0 002 004 006 008 01 012 0140

05

1

15

2

25

3

35

Time (s)

3rd

harm

onic

(A)

0 002 004 006 008 01 012 0140

05

1

15

2

25

3

35

Time (s)

4th

harm

onic

(A)





6

1965 sdot 104le 120585119895119896 le 4949 sdot 10

4


6

1256 sdot 104le 120585119895119896 le 1183 sdot 10

7


6

4622 sdot 104le 120585119895119896 le 1566 sdot 10

7








119883119899+1 = (1 +120579

3)119883119899 minus

120579

3119883119899minus1 + (1 minus 120579)

Δ119905

2119865 (119883119899 119905119899)

+ (1 +120579

3)Δ119905

2119865 (119883119899+1 119905119899+1)

(14)



119883119899+1 = (1 +120579

5)119883119899 minus

3120579

10119883119899minus1 + (1 minus 120579)

Δ119905

2119865 (119883119899 119905119899)

+ (1 +120579

5)Δ119905

2119865 (119883119899+1 119905119899+1) +

120579

10119883[0]

119899+1

(15)


5 Conclusions


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 014minus16

minus155


Time (s)

Model 1

0 014minus45

minus25


Time (s)

Model 2

0 014minus49

minus47


Time (s)

Model 3

Time (s)

Model 1

Time (s)

Model 2

Time (s)

Model 3

0 014minus100

minus50

0Max eigenvalue

0 014minus100

minus50

0Max eigenvalue

0 014minus100

minus50

0Max eigenvalue



0 002 004 006 008 01 012 014

0

2

4

6

8

10

Time (s)

Curr

ent (

A)


MeasuredModel 1

Model 2Model 3




01 011 012 013 014 015 016minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time (s)

Curr

ent (

A)


MeasuredModel 1

Model 2Model 3





1 2 3 4 5 60

2

4

6

8

10

Peak number

Relat

ive e

rror

()



0 002 004 006 008 01 012 0140

05

1

15

2

25

3

35

Time (s)

1st h

arm

onic

(A)

MeasuredModel 1

Model 2Model 3

MeasuredModel 1

Model 2Model 3

0 002 004 006 008 01 012 0140

05

1

15

2

25

3

35

Time (s)

2nd

harm

onic

(A)

0 002 004 006 008 01 012 0140

05

1

15

2

25

3

35

Time (s)

3rd

harm

onic

(A)

0 002 004 006 008 01 012 0140

05

1

15

2

25

3

35

Time (s)

4th

harm

onic

(A)





6

1965 sdot 104le 120585119895119896 le 4949 sdot 10

4


6

1256 sdot 104le 120585119895119896 le 1183 sdot 10

7


6

4622 sdot 104le 120585119895119896 le 1566 sdot 10

7








119883119899+1 = (1 +120579

3)119883119899 minus

120579

3119883119899minus1 + (1 minus 120579)

Δ119905

2119865 (119883119899 119905119899)

+ (1 +120579

3)Δ119905

2119865 (119883119899+1 119905119899+1)

(14)



119883119899+1 = (1 +120579

5)119883119899 minus

3120579

10119883119899minus1 + (1 minus 120579)

Δ119905

2119865 (119883119899 119905119899)

+ (1 +120579

5)Δ119905

2119865 (119883119899+1 119905119899+1) +

120579

10119883[0]

119899+1

(15)


5 Conclusions


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 2 3 4 5 60

2

4

6

8

10

Peak number

Relat

ive e

rror

()



0 002 004 006 008 01 012 0140

05

1

15

2

25

3

35

Time (s)

1st h

arm

onic

(A)

MeasuredModel 1

Model 2Model 3

MeasuredModel 1

Model 2Model 3

0 002 004 006 008 01 012 0140

05

1

15

2

25

3

35

Time (s)

2nd

harm

onic

(A)

0 002 004 006 008 01 012 0140

05

1

15

2

25

3

35

Time (s)

3rd

harm

onic

(A)

0 002 004 006 008 01 012 0140

05

1

15

2

25

3

35

Time (s)

4th

harm

onic

(A)





6

1965 sdot 104le 120585119895119896 le 4949 sdot 10

4


6

1256 sdot 104le 120585119895119896 le 1183 sdot 10

7


6

4622 sdot 104le 120585119895119896 le 1566 sdot 10

7








119883119899+1 = (1 +120579

3)119883119899 minus

120579

3119883119899minus1 + (1 minus 120579)

Δ119905

2119865 (119883119899 119905119899)

+ (1 +120579

3)Δ119905

2119865 (119883119899+1 119905119899+1)

(14)



119883119899+1 = (1 +120579

5)119883119899 minus

3120579

10119883119899minus1 + (1 minus 120579)

Δ119905

2119865 (119883119899 119905119899)

+ (1 +120579

5)Δ119905

2119865 (119883119899+1 119905119899+1) +

120579

10119883[0]

119899+1

(15)


5 Conclusions


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












6

1965 sdot 104le 120585119895119896 le 4949 sdot 10

4


6

1256 sdot 104le 120585119895119896 le 1183 sdot 10

7


6

4622 sdot 104le 120585119895119896 le 1566 sdot 10

7








119883119899+1 = (1 +120579

3)119883119899 minus

120579

3119883119899minus1 + (1 minus 120579)

Δ119905

2119865 (119883119899 119905119899)

+ (1 +120579

3)Δ119905

2119865 (119883119899+1 119905119899+1)

(14)



119883119899+1 = (1 +120579

5)119883119899 minus

3120579

10119883119899minus1 + (1 minus 120579)

Δ119905

2119865 (119883119899 119905119899)

+ (1 +120579

5)Δ119905

2119865 (119883119899+1 119905119899+1) +

120579

10119883[0]

119899+1

(15)


5 Conclusions


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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