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Electric Power Systems Research 110 (2014) 9–24

Contents lists available at ScienceDirect

Electric Power Systems Research

j o ur na l ho mepage: www.elsev ier .com/ locate /epsr

A new algorithm based on Clarke’s Transform and Discrete WaveletTransform for the differential protection of three-phase powertransformers considering the ultra-saturation phenomenon

Bahram Noshad ∗, Morteza Razaz, Seyed Ghodratollah SeifossadatShahid Chamran University, Ahvaz, Iran

a r t i c l e i n f o

Article history:Received 27 July 2013Received in revised form 4 December 2013Accepted 2 January 2014Available online 29 January 2014

Keywords:Power transformerDifferential protectionsUltra-saturation phenomenonInrush currentInternal and external faultsIEEE 14-bus test system

a b s t r a c t

This paper presents, at first, a novel model for investigating the ultra-saturation phenomenon duringenergization of a loaded three-phase power transformer. Then a new approach is presented to controlthe unusual false trip of a three-phase power transformer differential protection due to ultra-saturationphenomenon based on Clarke’s Transform and Discrete Wavelet Transform (DWT). In this method, thetransient phenomena of a power transformer including the magnetic inrush current, the ultra-saturationphenomenon, the external faults, and the internal faults of the power transformer are simulated. Todistinguish between these phenomena, appropriate criteria based on Clarke’s Transform and DWT arepresented using the standard deviation of coefficients and the energy coefficients. The results of thisstudy may be used as notifications by the personnel of substation and relay manufacturers.

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

One of the most important parts in power systems is the powertransformer whose differential protection is a major concern [1].The effect of the magnetizing inrush currents has to be consideredin the power transformer protective design. The inrush currentoccurs when a power transformer is switched on, and it can bemuch higher than its nominal value, hence causing the false tripof the differential protections [2]. Power transformers must actproperly with regard to their differential protection when they facedifferent transient signals which are the results of the magnetizinginrush currents, the internal faults, and the external faults. So, thedifferential protection should operate rapidly during the internalfaults of the power transformer and should not have any actionswhile facing external faults and magnetizing inrush currents. Thecommon technique utilized to prevent the false trip of the differ-ential protection during the energization of the power transformeris the level of harmonic components. The inrush current includesa high level of second harmonic component which is used as adiscriminative feature in conventional protective schemes [1,2]. Todistinguish between the internal faults, the external faults, and theinrush current in a standard differential protection, an algorithmis used in which the differential protection operates when the

∗ Corresponding author. Tel.: +98 9163530536.E-mail address: [email protected] (B. Noshad).

amplitude of the basic component of the differential current, whichis computed by discrete Fourier transform (DFT), fixes at more than0.25 p.u. and the level of the second harmonic to the basic harmonicof the differential current, which is computed by DFT, fixes at lowerthan 15% [3–7]. In the standard differential protection, however,the mal-operation of differential protection in certain conditionsunder magnetizing inrush current leads to a tripping of healthytransformers [3–5]. Wavelet transform is a powerful and effectivetool for processing transient signals [8–16]. The wavelet transformtechnique is widely utilized in power systems for protection [8],assessment of power quality [9], fault detection [10], solving thepower quality problem [11,12], and data compression [13,14].Wavelet transform, due to its natural ability in adjusting the widthof the mother wavelet frequencies [15,16], is more appropriatethan other methods of frequency domain such as Fourier windowfor analyzing transient states. The DWT can process a signalby decomposing it into an approximate and a detail by crossingthrough low-pass and high-pass filters. The approximate is decom-posed to obtain the information of the next level and the processcontinues. Moreover, the practical implementation of DWT is verysimple. Therefore, the DWT is widely used to distinguish betweenthe transient phenomena in the power transformer differentialprotection. In addition, one of the transient phenomena that leadsto the false trip of the power transformer differential protectionduring the energization of a loaded power transformer is the ultra-saturation phenomenon. The mechanism of unusual false tripof the differential protection resulting from the ultra-saturation

0378-7796/$ – see front matter © 2014 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.epsr.2014.01.001


10 B. Noshad et al. / Electric Power Systems Research 110 (2014) 9–24

upa, upb, upc, usab, usbc, and usca the voltages of primary andsecondary windings of the power transformer

ipa, ipb, ipc, ia, ib, and ic the currents of primary and secondarywindings of the power transformer

Rp, Rs, Ldp, and Lds the resistances and the inductances ofthe primary and secondary windings of the powertransformer

ϕa, ϕb, ϕc, and ϕd the flux of the core magnetic through thewindings legs and air branch of the power trans-former

NP the number of the primary turns of the power trans-former

NS the number of the secondary turns of the powertransformer

epa, epb, epc, esa, esb, and esc the inducted primary andsecondary voltages of winding legs of the powertransformer

fa, fb, fc, and fd the magnetic potential through the three-legged and air branch of the power transformer

pa, pb, pc, k1a, k1b, k1c, k2a, k2b, k2c, f0a, f0b, and f0c related topower transformer saturation curve

Rb and Lb the resistive and the inductive load of the powertransformer

�� the flux linkages of the current transformers� s the flux linkages at the saturation knee point of the

magnetization curve of the current transformersi�0 the magnetization current at the saturation knee

point of the magnetization curve of the currenttransformers

Ls the slope of the saturation zone of the current trans-formers

ips the primary current referring to the secondary sideof the current transformers

i� the magnetizing current of the current transformersis the secondary current of the current transformerses the induced voltage in the secondary winding of the

current transformersxi, x̄, n and � coefficients, the mean of coefficients, the

number of coefficients, and the standard deviation,respectively

Id set point the activation current of the differential protectionk the percentage value of the restrain current of the

differential protectionir the value of the restraint current of the differential

protectioni2p and i2s the secondary currents of the current transform-

ers on the primary and secondary side of the powertransformer

I0 the ground mode current componentI1 and I2 the aerial mode current componentsD1I0 the detail coefficients vector at the first level of the

ground mode�Dinter, �Dinter-min and �Dinter-max the standard deviation of

D vector at the first level in various test signals ofdifferent transient phenomena, the minimum, andthe maximum of the standard deviation of D vectorat the first level related to the internal faults.

ED1, ED1min and ED1max the energy coefficients at the firstlevel in various test signals of different transientphenomena, the minimum, and maximum of energycoefficients at the first level related to the internalfaults, respectively

phenomenon is related to the saturation of the magnetic core of thepower transformer, which occurs during a rapid jump of the termi-nal voltages and is a widely occurring phenomenon. This happenswhen a power transformer is switched on or when a short circuitfault that occurred near a power transformer is removed. In thesesituations, slow and high decaying inrush currents may be createdwhich may well be much higher than the full load value. Conse-quently, even in conditions where saturation is very heavy, there isalways a considerable second harmonic in the current in the stateof having the same polarity for the residual flux in the magneticcore of the power transformer as the DC flux which results from thevoltage jump and does not reach below 15% of the basic one. Thus,the ratio of the second harmonic is an appropriate criterion forpreventing an unusual mal-operation in differential protections.If the differential protection discovers the second harmonic thatis higher than 15% of the basic component, the operation of thedifferential protection will be blocked. But the mal-operation ofthe differential protection in certain conditions under magnetizinginrush current has led to the tripping of healthy transformers[3–5]. In this case, the inrush current during the period of theultra-saturation will miss some specifications such as the highratio of the second harmonic and the dead angle. Also, the DC fluxin the magnetic core of the power transformer in the primary stageof the process tends to increase rather than decrease, resulting inthe ultra-saturation phenomenon [3–5]. Hence, the amplitude ofthe basic component of the differential current gets higher, and thelevel of the second harmonic decreases under that of relay restrain[3–5]. So, the standard differential protection is not able to identifythe ultra-saturation phenomenon and the description and controlof the ultra-saturation phenomenon is necessary for preventingthe false trip of the differential protection. In all previous studiesof ultra-saturation phenomenon, the model of a loaded powertransformer was considered as single-phase [3–5]. Also, severalstudies have been done to distinguish between the inrush current,the internal and the external faults by various algorithms [8–16],but none has taken the ultra-saturation phenomenon into account.

This paper presents a novel model for investigating theultra-saturation phenomenon during energization of a loadedthree-phase power transformer. It also presents a new approach fora three-phase power transformer differential protection, consider-ing the effect of ultra-saturation phenomenon based on Clarke’sTransform and DWT. To model the ultra-saturation phenomenon,the nonlinear specification of the transformer core and the satu-ration effect of current transformers are taken into account. It isassumed that the load of the transformer is a resistive and induc-tive load. In this algorithm, the ultra-saturation phenomenon, theexternal and internal faults of the power transformer (includingthe three-phase fault (ABC fault), the three-phase-to-ground fault(ABCG fault), the phase-to-ground fault (AG fault), the phase-to-phase fault (AB fault), the phase-to-phase-to-ground fault (ABGfault)) and the magnetic inrush current are simulated. Also, theturn-to-turn and turn-to-ground internal faults are simulated. Todistinguish between these phenomena, appropriate criteria arepresented using Clarke’s Transform and DWT by the use of energycoefficients and the standard deviation of coefficients. In this paper,the DWT is utilized since it provides enough information for theanalysis of the main signal in a considerably short computationtime. Moreover, the practical implementation of DWT is very sim-ple. To distinguish between transient phenomena in the powertransformer, energy coefficients and the standard deviation of coef-ficients at the first level in a time window involving transientstates are used. Since the criteria are defined at the first level,the proposed algorithm is totally appropriate in terms of speed,accuracy and computational cost. Practically, the algorithm is alsovery easy to implement. To verify the results of the proposedalgorithm, the ultra-saturation phenomenon in IEEE 14-bus test


B. Noshad et al. / Electric Power Systems Research 110 (2014) 9–24 11

Fig. 1. The basic structure of a power transformer.

system is also investigated and the proposed algorithm in IEEE14-bus test system is implemented. In this paper, simulation isdone by the programs PSCAD and MATLAB. The main advantagesof this paper are the followings: (1) a new model is presented forinvestigating the ultra-saturation phenomenon during energiza-ton of a loaded three-phase power transformer, (2) a new model ispresented for current transformer which is very simple and effec-tive, (3) a new algorithm is presented for the three-phase powertransformer differential protection considering the effect of ultra-saturation phenomenon based on Clarke’s Transform and DWT,and (4) in this paper, attempts have been made to select realisticparameters. Therefore, the data from power transformer and cur-rent transformers have been taken from Irantransfo Co. in Zanjan,Iran.

2. Ultra-saturation modeling

For modeling the ultra-saturation phenomenon, modeling ofthe current transformer and the power transformer are required.The modeling of the power transformer is always done by usingand combining magnetic and electrical circuits. The basic structure

Fig. 2. The magnetic equivalent circuit of power transformer.

of a three-phase power transformer is shown in Fig. 1. Fig. 2 showsthe magnetic equivalent circuit of the power transformer shownin Fig. 1 and, regarding the fact that most three-phase powertransformers are connected as YN/d connections, the electricequivalent circuit is shown in Fig. 3. According to Figs. 1–3 and byusing Eqs. (8)–(29), the power transformer is modeled. Moreover,in this paper, a new model is presented for a current transformerwhich is very simple and effective. Fig. 4 displays the currenttransformer equivalent circuit referred to the secondary side. Inthis circuit, R2 and L2 are the resistive and the inductive winding

Fig. 3. The electric equivalent circuit of an YN/d power transformer.



Lb

Rb

isL2R

iµN pi ips p N s

=

Fig. 4. The current transformer model referred to secondary side.

of the secondary side of the current transformer. Rb and Lb are theresistive and the inductive burden of the current transformer. Sincethe core loss does not affect the behavior of the current transformersaturation, it is neglected [17]. The magnetization curve illustratedin Fig. 5 is used for the current transformer core. This single-valuedmagnetization curve is used for the current transformer, sincethe hysteresis characteristic does not drastically affect the currenttransformer transient behavior [18]. According to Figs. 4 and 5 andby using Eqs. (30)–(37), the current transformer is modeled.

3. Proposed algorithm for the differential protection

In this study, Clarke’s Transform and DWT are used to distin-guish between transient phenomena in the power transformer.DWT can provide sufficient information for analyzing the main sig-nal with a considerably short computation time. Daubechies4 (Db4)is used as the mother wavelet since the successful application ofthe family “mother wavelet of Daubechie’s (Db4)” in the extrac-tion of transient states of the power systems has been reported inseveral studies [15,16,19]. Also, according to [20,21], Daubechies’smother wavelet is a proper choice for this purpose. The results ofa wavelet transformation analysis are largely dependent on theselection of the mother wavelet [20,21]. To choose the motherwavelet, then, this study has utilized different mother waveletsto evaluate the accuracy in the proposed algorithm. The motherwavelets studied in this paper are Morlet (Mo), Coiflet (Coif), Haar(Hr), Symlet (Sy) and Daubechies (Db). Table 1 displays the correctoperation number for various test signals of transient phenomenaof the power transformer. According to this table, the Db motherwavelet demonstrates the excellent performance for all transientphenomena. Therefore, Db is used as the mother wavelet. In thispaper, the standard deviation of coefficients and the energy coeffi-cients are used to distinguish between transient phenomena in thepower transformer. The energy coefficients and the standard devi-ation of coefficients in a time window which involves a transient

Fig. 5. The magnetizing characteristic of transformer core.

Table 1The percentage of correct operation for various mother wavelets in transientphenomena.

Phenomenon type Mother wavelet type Correct operation (%)

Ultra-saturationphenomenon

Db 98.21Hr 65.2Sy 85.21Coif 92.1Mo 85.41

Inrush current

Db 98.7Hr 63.3Sy 86.3Coif 92.65Mo 84.32

AG internal fault

Db 98.9Hr 61.02Sy 86.78Coif 91.24Mo 83.5

AB internal fault

Db 98.63Hr 59.88Sy 84.6Coif 90.21Mo 84.78

ABG internal fault

Db 98.21Hr 59.14Sy 88.06Coif 91.36Mo 87.021

ABC internal fault

Db 98.11Hr 65.02Sy 87.5Coif 92.01Mo 83.24

ABCG internal fault

Db 98.99Hr 64.2Sy 89.5Coif 93.27Mo 84.52

Turn-to-groundinternal fault (phase A)

Db 98.45Hr 62.5Sy 83.47Coif 90.8Mo 85.12

Turn-to-turn internalfault (phase A)

Db 98.51Hr 62.4Sy 82.14Coif 91Mo 86.08

AG external fault

Db 98.17Hr 59.7Sy 86.8Coif 93.1Mo 83.56

AB external fault

Db 98.1Hr 59.6Sy 85.32Coif 92.5Mo 82.58

ABG external fault

Db 98.32Hr 59.66Sy 87.14Coif 92.77Mo 84.6

ABC external fault

Db 98.52Hr 59.44Sy 88.54Coif 92.38Mo 85.3



Table 1 (Continued)

Phenomenon type Mother wavelet type Correct operation (%)

ABCG external fault

Db 98.75Hr 59.82Sy 84.65Coif 90.86Mo 83.65

state are used as a distinguishing factor. The energy coefficients andthe standard deviation of coefficients are obtained in a time win-dow of 0.01 s. Transient state information is maintained throughthe use of a moving window in the analysis. After the initial sur-vey, the time window period is obtained as a compromise betweenaccuracy and response time. The energy of a discrete signal x(n) isobtained from the following formula:

Energy =+∞∑n=−∞

x2[n] (1)

The total energy is the sum of squares of wavelet coefficients. Also,the standard deviation of a discrete signal x(n) is obtained from thefollowing formula:

� =

√√√√ n∑i=1

(xi − x̄)2

n(2)

The differential protection should be able to distinguish theinternal faults from the external faults, the magnetizing inrush cur-rent and the ultra-saturation phenomenon. It should also operateonly under the internal faults. In this algorithm, the differential cur-rents of phases are first obtained from the subtraction of secondarycurrents from current transformers on the primary and secondarysides of the power transformer. Then, the following steps are per-formed to distinguish between different transient phenomena:

(1) In this algorithm, the activation current is first calculated. Theactivation current of the differential protection is defined asfollows:

|Id|set point = |k · ir | =∣∣∣∣k · i2p + i2s

2

∣∣∣∣ (3)

According to [3–7], the value of Id set point is taken to be 0.25 p.u.If the three-phase differential currents are lower than theactivation current, the normal condition occurs. Otherwise, atransient phenomenon may occur.

(2) In the next step, input signals are analyzed by DWT for extract-ing the information of the transient signal in the time and thefrequency domain. After this process, approximate and detailcoefficients of the differential current of the filter bank specifiedare extracted. Under normal conditions and stable operations,the changes in the values of detail coefficients are very small.If a sudden change in detail coefficients is observed, the sys-tem may be under a fault condition. In this case, if one of thedifferential currents is larger than the activation current, usingClarke’s Transform, the modal current components are first cal-culated according to Eq. (4) to generate the ground mode (I0),aerial mode 1 (I1) and aerial mode 2 (I2):⎡⎢⎣ I0I1I2

⎤⎥⎦ =(

1√3

)⎡⎢⎢⎢⎣1 1 1

√2

−1√2

−1√2

0

√3

2−

√3

2

⎤⎥⎥⎥⎦⎡⎢⎣ IaIbIc

⎤⎥⎦ (4)

It should be emphasized that Clarke’s Transform could beimplemented for instantaneous values as well as the phasors

[22]. In this study, sampling frequency is 10 kHz. Since the timewindow width is considered half of a cycle, there are 100 sam-ples in each window in which detail coefficients are separated.Detail coefficients are 50 samples. In the proposed algorithm,detail coefficients are used. To distinguish between the tran-sient phenomena in the power transformer, D vector is firstdefined as follows:

D = D1I0 (5)

In this step, the standard deviation of D vector in various testsignals of different transient phenomena is calculated. If thestandard deviation of D vector related to the internal faults,according to Eq. (6), is considered as the threshold value, theexternal faults are distinguished from the internal faults:

�Dinter−min ≤ �Dinter ≤ �Dinter−max (6)

In the next step, to distinguish between the internal faults,the inrush current and the ultra-saturation phenomenon, theenergy coefficients of the aerial mode current component (I1)at the first level in various test signals of different transientphenomena will be calculated. The energy coefficients of theaerial mode (I1) at the first level related to the internal faults areconsidered as the threshold value and are defined as follows:

ED1 min ≤ ED1 ≤ ED1 max (7)

If the energy coefficients are not between the minimum andthe maximum range in Eq. (7), the magnetizing inrush cur-rent or the ultra-saturation phenomenon may occur. In thispaper, and at various transient phenomena in the power trans-former, 1115 test signals involving the internal and the externalfaults, the inrush current and the ultra-saturation phenomenonare simulated and the standard deviation of D vector (relatedto the ground mode) and the energy coefficients of the aerialmode (I1) in different test signals are calculated. In the proposedalgorithm, the ground mode and the aerial mode current com-ponents (I0 and I1) are used instead of the three-phase currentsof power transformer by using Clerk’s transform, which led todecreased computations. The proposed algorithm’s flowchartis displayed in Fig. 6.

4. Simulation results

Several studies have distinguished between the inrush current,the internal and the external faults via various algorithms, butnone has taken into account the ultra-saturation phenomenon. Inthis paper, a new model is first presented for investigating theultra-saturation phenomenon during the energizaton of a loadedthree-phase power transformer. In order to distinguish betweenthe external faults, the internal faults, and the inrush current in thestandard differential protection, an algorithm is used in which thedifferential protection operates when the amplitude of the basiccomponent of the differential current fixes at more than 0.25 p.u.and the level of the second harmonic to basic harmonic of the dif-ferential current fixes at lower than 15%. But it has been describedthat, in certain conditions, the false trip of the differential protec-tion under ultra-saturation phenomenon has led to the trippingof healthy transformers [3–5]. So, in the standard differential pro-tection, the differential protection operates under ultra-saturationphenomenon. This study, however, shows that the proposed algo-rithm can help distinguish the internal faults from the inrushcurrent, the external faults and the ultra-saturation phenomenon.To prove the proposed algorithm, some cases of the external andinternal faults, the inrush current and the ultra-saturation phe-nomenon are presented in the next parts.



Fig. 6. The flowchart of the proposed algorithm.

4.1. Simulation of the ultra-saturation phenomenon

It is supposed that the loaded three-phase power transformer isswitched on from the high-voltage side at t = 0. The source and thethree-phase power transformer parameters are:

ϕa(0) = 1.5878e−3 Wb, ϕb(0) = 4.7619e−3 Wb, ϕc(0) = −6.3518e−3

Wb, upa = Um sin(wt + �), upb = Um sin(wt + � − 120), upc = Um sin(wt + � − 240), Um = 400 kV, w = 100� rad, � = 80, k = 400/230 kV,Rp = 0.9 �, Ldp = 0.1019 H, Rs = 0.5 �, Lds = 0.0337 H, Np = 610,Ns = 352, Rb = 5 �, Lb = 0.3029 H, S = 500 MVA, Rd = 2500 A.t/Wb,f0a = 0.0 A t, f0b = 0.03 A t, f0c = 0.05 A t, pa = 2, pb = 2,pc = 2, k1a = 28 Wb/A t, k2a = 0.995 Wb/A t, k1b = 45 Wb/A t,k2b = 0.9158 Wb/A t, k1c = 28 Wb/A t, k2c = 0.9134 Wb/A t

Parameters for the current transformers on the low-voltage andthe high-voltage sides of the power transformer are:

Low-voltage side: k = 1000/5, Bs = 1.9 T, Ls = 0.7 mH, A = 3.53e−3 m2,� s = 1.34 (Wb turns), i�0 = 0.03 mA, R = 0.15 �

High-voltage side: k = 600/5, Bs = 1.8 T, Ls = 0.7 mH, A = 3.472e−3 m2,� s = 0.75 (Wb turns), i�0 = 0.05 mA, R = 0.05 �

For modeling the ultra-saturation phenomenon, ipa, ipb, ipc, ia, ib,ic, ϕa, ϕb and ϕc can be solved from Eqs. (8)–(29). ipa, ipb, ipc, ia, iband ic represent the primary and secondary currents of the three-phase power transformer. ipa, ipb and ipc are the primary currents ofcurrent transformers on the primary side of the power transformerand ia, ib and ic are the primary currents of current transformers onthe secondary side of the power transformer. The secondary cur-rents of current transformers on the primary and secondary sidesof the power transformer should then be calculated. �� is relatedto current transformers on the primary and secondary sides of thepower transformer and can be solved from Eqs. (33), (35) and (37)using the forth-order Runge–Kutta method with a 10 �s time step.The magnetic current i�, according to Eq. (30), and the secondarycurrent of current transformers, according to what is given in Eqs.(32), (34) and (36), have all been calculated using the computed��. In these relations, ips is the primary current of the power trans-former. Therefore, by using Eqs. (8)–(37) for the power transformerand the current transformers, the ultra-saturation phenomenon ismodeled and the unusually false trip of the differential protectiondue to the ultra-saturation phenomenon is presented, using DFTalgorithm of the differential currents. The wave shapes of the mag-netic linkage of transformer core, the primary currents referred tothe secondary side on the primary side and the secondary currentsreferred to the secondary side on the secondary side of the three-phase power transformer due to the ultra-saturation phenomenonare all displayed in Figs. 7 and 8, respectively. The differential cur-rents (id) due to the ultra-saturation phenomenon, calculated fromthe difference between secondary currents of current transformerson the primary and secondary sides of the power transformer, arealso displayed in Fig. 8. As illustrated in Fig. 8, the primary currentsreferred to the secondary side on the primary side of the powertransformer involve a much higher aperiodic component becauseof the nonlinearity of the transformer core, but the aperiodic com-ponent on the secondary currents referred to the secondary sideon the secondary side of the power transformer is too much low. Inthis case, the current transformers of primary and secondary sidesof the power transformer differ in their transforming behavior somuch that the false differential currents (Fig. 8) with considerablemagnitudes and low harmonic contents will probably be created.Fig. 9 displays the changes of the amplitudes of the basic com-ponent and the ratio change of the second harmonic to the basicharmonic of id in Fig. 8, which are obtained with the DFT algorithm.In this Fig., the amplitudes of the basic component of the differ-ential currents are normalized according to the secondary currentof current transformers. As illustrated in Fig. 9, the basic compo-nent of the differential currents is more than 0.25 p.u. from thebeginning of the energization, and approximately after 6 cycles, itis stabilized at above 0.25 p.u. Also, according to Fig. 9, as the ener-gization time exceeds 0.1364 s, 0.2149 s and 0.2321 s for A, B, and Cphases respectively, the ratio of the second harmonic to the basicharmonic stabilizes at lower than 15%. So, according to Fig. 9, if thedifferential protection uses 0.25 p.u. as the operating threshold forthe amplitude of the basic component of differential current and15% as the second harmonic restraint ratio, the false trip occurs at0.1364 s. To prove the results, in the ultra-saturation phenomenon,different signals are generated by changing the residual flux, theload level, the inception angle and the different states of currenttransformer saturation. Due to the limitation of the paper length,only one case of the ultra-saturation phenomenon resulting fromchanging the load level is shown in Fig. 10. According to this figure,in a standard differential protection, the amplitudes of the basiccomponent and the ratio change of the second harmonic to the



0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-1000

-500

0

500

1000

t(s)

flux

of p

hase

A (W

b*tu

rns)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-500

0

500

1000

1500

t(s)

flux

of p

hase

B (W

b*tu

rns)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-1500

-1000

-500

0

500

flux

of p

hase

C (W

b*tu

rns)

t(s)

Fig. 7. The magnetic linkage of transformer core due to the ultra-saturation phenomenon.

basic harmonic of the differential currents due to ultra-saturationphenomenon has exceeded the threshold values, resulting in theoperation of the differential protection. So, in a standard differen-tial protection, the mal-operation of differential protection underultra-saturation phenomenon may occur.

4.2. Simulation of the inrush current, the external and internalfaults

For simulation of the inrush current, the external and internalfaults, the power system in Fig. 11 is considered. The external faultsoccurred outside the protective zone of differential relay on the load

side and the internal faults occurred inside the protective zone ofdifferential relay on the secondary side of the power transformer.The simulated external and internal fault types involve the ABCfault, the ABCG fault, the AG fault, the AB fault and the ABG fault.Also, according to [23], the turn-to-ground and turn-to-turn inter-nal faults are simulated. In the turn-to-ground internal fault, thefault has occurred on the secondary winding of phase A at 25% posi-tion of winding and in the turn-to-turn internal fault, the fault hasoccurred on the secondary winding of phase A in which 10% of turnsis a short circuit. For simulation of the external and internal faults,the time for fault occurrence is considered 0.2 s. Also, for simulationof the inrush current, the power system in Fig. 11 is considered in

0 0.1 0.2 0.3 0.4-5

0

5

t(s)

ipa

(A)

0 0.1 0.2 0.3 0.4-5

0

5

10

t(s)

ipb

(A)

0 0.1 0.2 0.3 0.4-10

-5

0

5

t(s)

ipc

(A)

0 0.1 0.2 0.3 0.4-10

-5

0

5

10

t(s)

ia (A

)

0 0.1 0.2 0.3 0.4-10

-5

0

5

10

t(s)

ib (A

)

0 0.1 0.2 0.3 0.4-20

-10

0

10

t(s)

ic (A

)

0 0.1 0.2 0.3 0.4-10

-5

0

5

t(s)

id o

f pha

se A

(A)

0 0.1 0.2 0.3 0.4-10

-5

0

5

10

t(s)

id o

f pha

se B

(A)

0 0.1 0.2 0.3 0.4-5

0

5

10

15

t(s)

id o

f pha

se C

(A)

Fig. 8. The primary and secondary currents refer to secondary sides and the differential currents due to the ultra-saturation phenomenon.



0 0.1 0.2 0.3 0.4

0.4

0.5

0.6

0.7

t(s)

mag

nitu

deof

pha

se A

(p.u

.)

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

50

100

t(s)

ratio

of p

hase

A (%

)

0 0.1 0.2 0.3 0.4

0.4

0.5

0.6

0.7

t(s)

mag

nitu

deof

pha

se B

(p.u

.)

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

50

100

t(s)

ratio

of p

hase

B (%

)

0 0.1 0.2 0.3 0.40.2

0.4

0.6

0.8

1

t(s)

mag

nitu

deof

pha

se C

(p.u

.)

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

50

100

t(s)

ratio

of p

hase

C (%

)

Fig. 9. The changes of the amplitudes of the basic component and the ratio change of the second harmonic to the basic harmonic of the differential currents with DFTalgorithm.

which the external and internal faults are first removed and thenthe unloaded transformer is energized at 0.2 s. In the inrush current,different signals are generated by changing the residual flux, theswitching time, and the inception angle. In the internal and externalfaults, different signals are generated by changing the residual flux,the fault resistance, the load level, the inception angle and the fault

occurrence time. Also, in the external faults, current transformersaturation is considered. Due to the limitation of the paper length,one case of the inrush current, the external faults with a consider-ation of current transformer saturation and the internal faults bychanging the residual flux, the fault occurrence time and the incep-tion angle is shown in Figs. 12–14. According to Fig. 12, the ratio

0 0.1 0.2 0.3 0.4-15

-10

-5

0

5

t(s)

id o

f pha

se A

(A)

0 0.1 0.2 0.3 0.4-20

-10

0

10

t(s)

id o

f pha

se B

(A)

0 0.1 0.2 0.3 0.4-10

0

10

20

t(s)

id o

f pha

se C

(A)

0 0.1 0.2 0.3 0.40.25

0.3

0.35

0.4

0.45

t(s)

mag

nitu

de o

fph

ase

A (p

.u.)

0 0.1 0.2 0.3 0.40.26

0.28

0.3

0.32

t(s)

mag

nitu

de o

fph

ase

B (p

.u.)

0 0.1 0.2 0.3 0.40.27

0.28

0.29

0.3

0.31

t(s)

mag

nitu

de o

fph

ase

C (p

.u.)

0.1 0.2 0.3 0.40

50

100

t(s)

Rat

io o

fph

ase

A (%

)

0.1 0.2 0.3 0.40

50

100

t(s)

Rat

io o

fph

ase

B (%

)

0.1 0.2 0.3 0.40

50

100

t(s)

Rat

io o

fph

ase

C (%

)

Fig. 10. The differential currents, the changes of the amplitudes of the basic component and the ratio change of the second harmonic to the basic harmonic of the differentialcurrents due to the ultra-saturation phenomenon by the change of the load level.



Fig. 11. Simulated power system for analysis of the inrush current, the external andinternal faults.

change of the second harmonic to the basic harmonic of the differ-ential currents due to the inrush current always stabilizes at morethan 15% and thus the differential protection under the inrush cur-rent condition will not operate. According to Fig. 13, the amplitudesof the differential currents due to the external faults stabilize at lessthan 0.25 and, consequently, the differential protection under theexternal faults condition will not operate. According to Fig. 14, theamplitudes of the basic component and the ratio change of the sec-ond harmonic to the basic harmonic of the differential currents

0 0.1 0.2 0.3 0.4-40

-20

0

20

t(s)

id o

f pha

se A

(A)

0 0.1 0.2 0.3 0.4-20

0

20

40

t(s)

id o

f pha

se B

(A)

0 0.1 0.2 0.3 0.4-20

-10

0

10

20

t(s)

id o

f pha

se C

(A)

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

50

100

t(s)

Rat

io o

fph

ase

A (%

)

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

20

40

6080

t(s)

Rat

io o

fph

ase

B (%

)

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

50

100

t(s)

Rat

io o

fph

ase

C (%

)

Fig. 12. The differential currents and the ratio of the second harmonic to basic harmonic of the differential currents due to inrush current by the change of the residual flux.

0 0.1 0.2 0.3 0.4-4

-2

0

2

4

t(s)

id o

f pha

se A

(A)

0 0.1 0.2 0.3 0.4-2

-1

0

1

2

t(s)

id o

f pha

se B

(A)

0 0.1 0.2 0.3 0.4-0.1

0

0.1

0.2

0.3

t(s)

id o

f pha

se C

(A)

0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

t(s)

mag

nitu

de o

fph

a se

A (p

.u.)

0 0.1 0.2 0.3 0.40

0.05

0.1

0.15

0.2

t(s)

mag

nitu

de o

fph

ase

B (p

.u.)

0 0.1 0.2 0.3 0.40

0.00 5

0.01

0.01 5

t(s)

mag

nitu

de o

fph

ase

C (p

.u.)

Fig. 13. The differential currents and the amplitudes of the basic component of the differential currents due to the AB external fault by the change of the occurrence time.



0 0.1 0.2 0.3 0.4-40

-20

0

20

40

t(s)

id o

f pha

se A

(A)

0 0.1 0.2 0.3 0.4-1

-0.5

0

0.5

t(s)

id o

f pha

se B

(A)

0 0.1 0.2 0.3 0.4-1

-0.5

0

0.5

1

t(s)

id o

f pha

se C

(A)

0 0.1 0.2 0.3 0.40

1

2

3

t(s)

mag

nitu

de o

fph

ase

A (p

.u.)

0 0.1 0.2 0.3 0.40

0.02

0.04

0.06

t(s)

mag

nitu

de o

fph

ase

B (p

.u.)

0 0.1 0.2 0.3 0.40

0.02

0.04

0.06

0.08

t(s)

mag

nitu

de o

fph

ase

C (p

.u.)

0.1 0.2 0.3 0.40

50

100

t(s)

Rat

io o

fph

ase

A (%

)

0.1 0.2 0.3 0.40

50

100

t(s)

Rat

io o

fph

ase

B (%

0.1 0.2 0.3 0.40

50

100

t(s)

Rat

io o

fph

ase

C (%

Fig. 14. The differential currents, the amplitudes of the basic component of the differential currents and the ratio of the second harmonic to basic harmonic of the differentialcurrents due to the AG internal fault by the change of the inception angle.

due to internal faults exceed the threshold values and differen-tial protection will operate. Therefore, according to these figures,the standard differential protection has a proper action when itfaces different transient signals which are the result of an inrushcurrent, internal faults and external faults. But in a standard differ-ential protection, the mal-operation of differential protection underultra-saturation phenomenon may occur.

4.3. The results of the proposed algorithm

According to the proposed algorithm, D vector is first calculatedbased on Eq. (5), and then the standard deviation of D vector in vari-ous test signals is calculated. The minimum and maximum standarddeviations of D vector related to the internal faults are considered asthe threshold values. The minimum and maximum standard devi-ations of D vector in various test signals are presented in Table 2.Accordingly, if the minimum and maximum standard deviations ofD vector related to the internal faults are considered as the thresh-old values, the external faults will not be between the minimumand the maximum according to Eq. (6), and therefore, the exter-nal faults are distinguished from the internal faults. The minimumand the maximum energy coefficients of the aerial mode (I1) at thefirst level are also presented in Table 2. If the minimum and themaximum energy coefficients of the aerial mode (I1) related to theinternal faults are considered as the threshold values, the inrushcurrent and the ultra-saturation phenomenon, according to Eq. (7),will not be between the minimum and the maximum range andhence these phenomena are distinguished from the internal faults.Therefore, according to Table 2, the minimum and the maximumthreshold values for energy coefficients and standard deviation ofD vector due to the internal faults are 0.00973, 1.196038, 11.0175,and 115.455. For example, the minimum and maximum values forenergy coefficients and standard deviation of D vector of a signaltest due to the ABC internal fault by changing the fault resistanceare 0.183, 0.391, 72.41, and 105.71. In the same line, the minimum

and maximum values for energy coefficients and standard devia-tion of D vector of a signal test due to the AB external fault by achange in the residual flux are 0.00173, 0.1091, 0.5112, and 2.711.The minimum and maximum values for energy coefficients andstandard deviation of D vector of a signal test due to the inrush cur-rent upon changing the inception angle are 9.211E−05, 4.124E−04,0.4839, and 20.131. The minimum and maximum values for energycoefficients and standard deviation of D vector of a signal testdue to the ultra-saturation phenomenon under very heavy cur-rent transformer saturation are 8.091E−05, 9.07E−05, 4.013, and13.767. Therefore, these results show the accuracy of the proposedalgorithm.

All the previous simulations have been done for a symmetri-cal conductor configuration. In this paper, a dedicated simulationis done for an unbalanced fault occurring in a line (connected tothe targeted transformer) that does not have a symmetrical con-ductor configuration. The simulated external faults involve the AGfault, the AB fault, and the ABG fault. For this study, 270 test sig-nals are considered. The results are shown in Table 3. Accordingto this Table, the proposed algorithm is over performing comparedto standard protections. The results also show the accuracy of theproposed algorithm. Also, one of the most important criteria forthe digital relays is the time for making a decision. In the proposedalgorithm, the main program has been simulated by the MATLABprogram. However, to calculate the operating time for relay, thisprogram is coded into PIC 80 MHz 16 BIT microcontroller. So, thisprogram has been simulated by using PIC 80 MHz 16 BIT microcon-troller in which the operating time was 11.8ms. Note that we haveto consider the time which is consumed Analog/Digital (A/D) for itsconversion. The conversion time of A/D is very low (according to80 MHz). We can, therefore, ignore this time compared to the totaloperating time of relay. According to [24], the operating time of astandard differential relay is within 30–40 ms. Thus, a comparisonof the results with [24] shows the response time of the proposedrelay is very high.



Table 2The minimum and maximum standard deviation of D vector and energy coefficients of the aerial mode at the first level in various test signals of different transient phenomena.

Phenomenon type Type event The minimumstandarddeviation of Dvector

The Maximumstandarddeviation of Dvector

The minimumenergycoefficients ofthe aerial mode

The maximumenergycoefficients ofthe aerial mode


Changing the residual flux 1.7094 10.153 7.11E−05 8.12E−05Changing the load level 2.631 12.708 7.83E−05 9.69E−05Changing the inception angle 2.112 11.441 6.28E−05 9.11E−05The different states of current transformer saturation 4.013 14.2657 8.09E−05 9.07E−05

Inrush currentChanging the residual flux 0.409 22.8614 1.83E−06 7.12E−05Changing the switching time 0.418 18.013 5.12E−05 9.07E−05Changing the inception angle 0.3839 20.131 9.11E−05 6.24E−04

AG internal fault

Changing the residual flux 13.011 14.71 0.42100 1.19604Changing the fault resistance 14.8 15.031 0.01238 0.79000Changing the load level 11.0175 13.21 0.09300 0.91100Changing the inception angle 12.023 12.72 0.51100 1.00100Changing the fault occurrence time 12.98 15.597 0.11700 0.83000

AB internal fault


ABG internal fault


ABC internal fault


ABCG internal fault




Turn-to-Turn internalfault (phase A)


AG external fault

Changing the residual flux 0.312 0.511 2.63E−06 4.41E−06Changing the fault resistance 0.27146 0.409 4.21E−06 5.12E−06Changing the load level 0.292 0.6977 3.10E−06 4.68E−06Changing the inception angle 0.281 0.611 2.98E−06 4.35E−06Changing the fault occurrence time 0.301 0.603 3.41E−06 5.95E−06Current transformer saturation 0.311 0.657 3.5E−06 5.89E−06

AB external fault

Changing the residual flux 0.512 2.11 0.00123 0.11000Changing the fault resistance 0.702 3.4167 0.00714 0.09810Changing the load level 0.491 2.71 3.96E−05 0.09210Changing the inception angle 0.36217 2.91 0.00290 0.12904Changing the fault occurrence time 0.3901 3.01 0.00900 0.08110Current transformer saturation 0.814 3.161 0.009875 0.11324

ABG external fault

Changing the residual flux 2.81 3.211 0.00821 0.05100Changing the fault resistance 2.061 4.2141 0.00634 0.03110Changing the load level 1.98 3.05 0.00690 0.07587Changing the inception angle 1.78 2.99 0.00780 0.04200Changing the fault occurrence time 1.6412 2.56 0.00810 0.06100Current transformer saturation 1.9875 3.864 0.00954 0.06345

ABC external fault




Table 2 (Continued)





ABCG external fault


Fig. 15. The IEEE 14-bus test system.

5. The study of ultra-saturation phenomenon in IEEE14-bus test system

Fig. 15 displays the single line diagram of the IEEE 14-bus testsystem. The IEEE 14-bus test system has two generators at buses1 and 2 and three synchronous condensers at buses 3, 6 and 8.To study the ultra-saturation phenomenon at the IEEE 14-bus test

system, a two-winding transformer between buses 5 and 6 is con-sidered with the following characteristics:

S = 100 MVA, k = 69 kV13.8 kV

, f = 60 Hz, Yd connection

To simulate the ultra-saturation phenomenon, the power trans-former between buses 5 and 6 is energized by loads at t = 0. Thecurrent transformers on the primary and secondary sides are con-sidered 600/5 and 3000/5, respectively. The differential currentsthat are calculated from the subtraction between secondary cur-rents of current transformers on the primary and secondary sides ofthe power transformer due to the ultra-saturation phenomenon areshown in Fig. 16. Fig. 16 also displays the changes in the amplitudesof the basic component and the ratio change of the second har-monic to the basic harmonic of differential currents. As illustratedin Fig. 16, the amplitudes of the basic component of differentialcurrents are more than 0.25 p.u. from the beginning of energiza-tion, and, approximately after 5 cycles (f = 60 Hz), they are stabilizedat more than 0.25 p.u. Also, according to Fig. 16, if the energiza-tion time exceeds 0.1538 s, 0.1507 s and 0.1742 s for A, B, and Cphases respectively, the ratio of the second harmonic to the basicharmonic of the differential currents stabilizes at lower than 15%.So, according to Fig. 13, the false trip occurs at 0.1507 s.

6. Investigation of the proposed algorithm in IEEE 14-bustest system

To study the proposed algorithm in IEEE 14-bus test system, thepower transformer between bus 5 and 6 is considered. Accordingto the proposed algorithm, the minimum and maximum stan-dard deviations of D vector in various test signals are presented

Table 3The minimum and maximum standard deviation of D vector and energy coefficients of the aerial mode at the first level in various test signals of different external faults.





AG external fault


AB external fault

Changing the residual flux 1.754 3.864 0.0547 1.047Changing the fault resistance 1.9845 4.653 0.0967 0.953Changing the load level 1.524 4.097 8.714E−03 0.867Changing the inception angle 1.367 3.215 0.0905 1.914Changing the fault occurrence time 1.0741 4.91 0.0875 0.7801Current transformer saturation 2.685 4.763 0.0857 1.6201

ABG external fault




Table 4The minimum and maximum standard deviation of D vector and energy coefficients of the aerial mode at the first level in various test signals of different transient phenomenain IEEE 14-bus test system.






Changing the residual flux 2.81 3.011 6.27E−05 7.21E−05Changing the load level 2.601 3.022 6.58E−05 8.05E−05Changing the inception angle 2.3666 2.981 6.34E−05 7.11E−05The different states of current transformer saturation 2.452 3.13626 7.01E−05 8.03E−05

Inrush currentChanging the residual flux 7.012 17.5753 7.86E−05 8.12E−05Changing the switching time 5.211 15.241 6.08E−05 8.41E−05Changing the inception angle 4.9419 16.105 7.12E−05 9.97E−05

AG internal fault


AB internal fault


ABG internal fault


ABC internal fault


ABCG internal fault




Turn-to-turn internalfault (phase A)


AG external fault


AB external fault


ABG external fault




Table 4 (Continued)





ABC external fault


ABCG external fault


0 0.1 0.2 0.3 0.4-5

0

5

10

t(s)

id o

f pha

se A

(A)

0 0.1 0.2 0.3 0.4-10

-5

0

5

t(s)

id o

f pha

se B

(A)

0 0.1 0.2 0.3 0.4-5

0

5

10

t(s)

id o

f pha

se C

(A)

0 0.1 0.2 0.3 0.40.2

0.3

0.4

0.5

t(s)

mag

nitu

deof

pha

se A

(p.u

.)

0 0.1 0.2 0.3 0.40.2

0.3

0.4

0.5

t(s)

mag

nitu

deof

pha

se B

(p.u

.)

0 0.1 0.2 0.3 0.40.25

0.3

0.35

0.4

0.45

t(s)

mag

nitu

deof

pha

se C

(p.u

.)

0.1 0.2 0.3 0.40

50

100

t(s)

Rat

io o

f pha

se B

(%)

0.1 0.2 0.3 0.40

50

100

t(s)

Rat

io o

f pha

se C

(%)

0.1 0.2 0.3 0.40

50

100

t(s)

Rat

io o

f pha

se A

(%)

Fig. 16. The differential currents, the amplitude of the basic component and the ratio of the second harmonic to basic harmonic of the differential currents.

in Table 4. According to Table 4, if the minimum and maximumstandard deviations of D vector related to the internal faults areconsidered as the threshold values, the external faults will not bebetween the minimum and maximum amounts according to Eq.(6), and therefore, the external faults are distinguished from theinternal faults. The minimum and maximum energy coefficients ofthe aerial mode (I1) at the first level are also presented in Table 4. Ifthe minimum and maximum energy coefficients of the aerial mode(I1) related to the internal faults are considered as the thresholdvalues, the inrush current and the ultra-saturation phenomenonaccording to Eq. (7) will not be between the minimum and max-imum range, and hence these phenomena are distinguished fromthe internal faults.

7. Conclusion

In this paper, a novel approach was presented to control theunusual mal-operation of a three-phase power transformer dif-ferential protection due to ultra-saturation phenomenon based onClarke’s Transform and DWT. In this algorithm, the ultra-saturationphenomenon, the external and the internal faults of the power

transformer and the magnetic inrush current were simulated. Todistinguish between these phenomena, appropriate criteria usingClarke’s Transform and DWT using the energy coefficients and thestandard deviation of coefficients were presented. The results ofthis study may serve as notifications for the personnel of substationand relay manufacturers.

Appendix A. Modeling of a loaded three-phase transformer

According to Fig. 2:

−Npipea + fa − �d�d = 0 (8)

−Npipeb + fb − �d�d = 0 (9)

−Npipec + fc − �d�d = 0 (10)

�a + �b + �c + �d = 0 (11)



By considering the saturation curve, a, b, and c reluctances inFig. 2 are nonlinear and defined as follows [25]:

�a(fa)−1 = k1a

(1 + (|fa|/f0a)pa )1/pa+ k2a (12)

�b(fb)−1 = k1b

(1 + (|fb|/f0b)pb )1/pb+ k2b (13)

�c(fc)−1 = k1c

(1 + (|fc |/f0c)pc )1/pc+ k2c (14)


fa = �a(fa) · �a (15)

fb = �b(fb) · �b (16)

fc = �c(fc) · �c (17)

Due to Eqs. (12)–(17), ϕa, ϕb and ϕc can be calculated. Regardingthe fact that most three-phase power transformers are connectedas YN/d connections, the electric equivalent circuit can be shown inFig. 3.

upa = Rpipa + Ldpdipadt

+ Npd�adt

(18)

upb = Rpipb + Ldpdipbdt

+ Npd�bdt

(19)

upc = Rpipc + Ldpdipcdt

+ Npd�cdt

(20)

usab = −Rsisa − Ldsdisadt

+ Nsd�adt

(21)

usbc = −Rsisb − Ldsdisbdt

+ Nsd�bdt

(22)

usca = −Rsisc − Ldsdiscdt

+ Nsd�cdt

(23)

For simulating the ultra-saturation phenomenon, the three-phase power transformer is loaded, and it is supposed that thebalanced three-phase load of the power transformer is a resis-tive and inductive load. By connecting complex load (Rb − Lb) inWye form to the equivalent circuit of the three-phase power trans-former in Fig. 3, and based on Kirchhoff voltage and current laws insecondary side, it can be written:

ia = isa − isc (24)

ib = isb − isa (25)

ic = isc − isb (26)

Due to Eqs. (21)–(26);

−Rsisa − Ldsdisadt

+ Nsd�adt

= Rb(isa − isc) + Lbd(isa − isc)

dt

− Rb(isb − isa) − Lbd(isb − isa)

dt(27)

−Rsisb − Ldsdisbdt

+ Nsd�bdt

= Rb(isb − isa) + Lbd(isb − isa)

dt

− Rb(isc − isb) − Lbd(isc − isb)

dt(28)

−Rsisc − Ldsdiscdt

+ Nsd�cdt

= Rb(isc − isb) + Lbd(isc − isb)

dt

− Rb(isa − isc) − Lbd(isa − isc)

dt(29)

Appendix B. Current transformer modeling


i� =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

� − sLs

+ i�0 , � > s

� + sLs

− i�0 , � < − s

i�0

� s, | �| ≤ s

(30)

In Fig. 4, we defined:

R = R2 + Rb, L = L2 + Lb (31)

According to Fig. 5, the magnetization curve of the current trans-former has three regions. For region 1, �� > � s:

According to Fig. 4 and Eq. (30):

is = ips − 1Ls

( � − s) − i�0 (32)

d �dt

= RLsLs + L

(ips − ( � − s)

Ls− i�0

)+ LLsLs + L

(dipsdt

)(33)

For region 2, �� < −� s:

is = ips − 1Ls

( � + s) + i�0 (34)

d �dt

= RLsLs + L

(ips − ( � + s)

Ls+ i�0

)+ LLsLs + L

(dipsdt

)(35)

For region 3, |��| ≤ � s:

is = ips − i�0

� s

(36)

d �dt

= R s s + Li�0

(ips − i�0

� s

)+ L s s + Li�0

(dipsdt

)(37)

References

[1] C.A. Mathews, An improved transformer differential relay, AIEE Transactions73 (Pt. III) (1954) 645–650.

[2] T.R. Specht, Transformer magnetizing inrush current, AIEE Transactions 70 (Pt.I) (1951) 323–328.

[3] X. Lin, P. Liu, The ultra-saturation phenomenon of loaded transformer ener-gization and its impacts on differential protection, IEEE Transactions on PowerDelivery 20 (2) (2005) 1265–1272.

[4] H. Weng, X. Lin, P. Liu, Studies on the operation behavior of differential pro-tection during a loaded transformer energization, IEEE Transactions on PowerDelivery 22 (3) (2007) 1386–1391.

[5] A. Wiszniewski, W. Rebizant, D. Bejmert, L. Schiel, Ultra saturation phe-nomenon in power transformers—myths and reality, IEEE Transactions onPower Delivery 23 (3) (2008) 1327–1334.

[6] A. Wiszniewski, H. Ungrad, W. Winkler, Protection Techniques in ElectricalEnergy Systems, Marcel Dekker, New York, 1995.

[7] Numerical Differential Protection Relay for Transformers Generators MotorsMini Bus bars SIEMENS. A.G. 7UT613/63x V.4.06 Instruction Manual Order.C53000-G1176-2006 C160-2.

[8] O. Chaari, M. Meunier, F. Brouaye, Wavelets: a new tool for the resonantgrounded power distribution systems relaying, IEEE Transactions on PowerDelivery 11 (3) (1996) 1301–1308.

[9] S. Santoso, E.J. Powers, W.M. Grady, P. Hofmann, Power quality assessment viawavelet transform analysis, IEEE Transactions on Power Delivery 11 (2) (1996)924–930.



[10] F. Jiang, Z.Q. Bo, M.A. Redfern, A new generator fault detection scheme usingwavelet transform, in: Proc. 33rd Universities Power Engineering Conference(UPEC’98), Edinburgh, UK, September, 1998, pp. 360–363.

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