PENETRATION OF ERRORS IN POWER TRANSFORMER LOSSES MEASUREMENTS IN REAL WORKING CONDITIONS

Aleksandar Damjanovic, Arif Sarwat Midzor Power Systems and Magnetics Florida International University (FIU), PO Box 2483 Oldsmar, FL, 34677 10555 West Flagler St., Miami, FL 33174 adamjan@midzor-psm.com asarwat@fiu.edu ABSTRACT Power transformer is one of the most efficient devices used in power systems. Modern electrical distribution systems typically supply a very high percentage of non-linear electronic loads. As a result, transformer losses increase, and energy efficiency decreases. The level of this deterioration is a function of harmonic voltage magnitudes at a transformers primary terminals, and load-generated harmonic currents. There are not yet recognized standards or methods for determining of transformers losses or efficiency under these non-linear load conditions. Conventional methods for measurement of transformer losses, open and short circuit tests, under nonlinear load do not apply, because measurements are possible only when the load is present, i.e, in real working conditions. The subject of this research is to determine the penetration of errors in measurements of power transformer losses and efficiency under nonlinear load conditions. We are not focused on digital process, A/D conversions, etc., we are focused on algorithms and methods of measurement and their response to the instrumentation errors. Significance of assessment of measuring errors is due to the fact that the errors of data propagate through calculations and produce errors in measuring results. Analyses are done with algebra of penetration of errors in calculations and measuring algorithms. KEY WORDS Accuracy, Penetration of Error, Power In, Power out, Voltage and Current Difference, Power Transformer, Transformer Losses, No Load Losses, Load Losses 1. Introduction Efficiency of the transformer is in the range of 95-99%, which means that up to 5% of the energy is wasted in the losses of the power transformer. The efficiency of power transformers is defined as the ratio between output and input power. Input power is the sum of the output power and transformer losses. Modern power and distribution systems have a high percentage of non-linear loads. As a result, losses in the power transformers will increase, and the efficiency decreases. The measurement of transformer losses and efficiency is very well understood and applied in the power transformer industry [1,2]. Losses measurements

are only made under linear load conditions. No-Load Losses are measured using an Open-Circuit Test, and measurement of Load Losses is done with a Short-Circuit Test. Measurement of power transformer losses under nonlinear loads can be done only when the transformer is connected to the actual loads, under real working conditions, with access to the input and output terminals of the power transformer. It is irrelevant which type of transformer is under test: power transformer, distribution transformer, control power transformers, high-voltage, medium-voltage, or low voltage. 2. Instrumentations and Digital Circuit Board Figure 1 shows a single-phase power transformer with signals from the power transformer, under the test, brought to the circuit board over the current and voltage transformers.

CT 2

VT 2

INPUT OUTPUT

V2

VT 1

I1s

V1

CT 1

I2s

LOAD

I 1 I 2TR

IdVd

V1s

DIGITAL CIRCUIT BOARD

TRV1s

I1sI2s’

V1s’

I2s

V2s

V2s

Figure1. Instrumentations connection to digital circuit board and algorithms for power transformer losses measurement in real working conditions The figure also shows measuring principle. The measuring principle and algorithms are fully described in [5-14].

Proceedings of the IASTED International Symposium

October 26 - 27, 2015 Marina del Rey, USAAdvances in Power and Energy Systems (APES 2015)

DOI: 10.2316/P.2015.831-007 126

3. Sources of Errors 3.1 Ratio Error of Power Transformer Rated voltage ratio of power transformer is not equal to the actual voltage ratio of the power transformer, because number of turns of the power transformer windings can be only integer number. Actual voltages corresponding to the number of turns because of Faraday’s Law of electromagnetic induction, are different from the rated voltages. Tolerances of the turn ratios are specified in [3,4], where “With rated voltage impressed on one winding of a transformer, all other rated voltages at no load shall be correct within 0.5% of the nameplate markings”.

3.2 Error of Current and Voltage Sensors Voltage and current sensors are used to transfer volts and currents from the power transformer to the circuit board. Transferring of signals via these sensors introduce error. Generated signal could be in analog, voltage, current or even digital form. It can be then utilized to display measured values or can be used for further analysis. A variety of sensors are used: Current and Voltage Transformers, Hall Effect Sensors, Resistive Dividers, and Rogovski Coil. 3.3 Current Transformers Mismatch Multiplier Since the measurement is done numerically by digital instrument, interposing CT is not needed, but mismatch of the primary and secondary CTs need to be considered in the calculations by introducing mismatch multiplier, equivalent to interposing CTs ratio. If the secondary current is transferred to the primary side, the mismatch multiplier is:

1

21CTCT

TRCTM ⋅= (1)

When the secondary current is transferred to the primary side, the current difference is obtained by the difference of the primary and secondary current on the primary side:

( )

⋅⋅−=−=∆

1

22121

1CTCT

TRiiiii ssss (2)

3.4 Errors in A/D Conversions and Power Measurements Periodic power signals with voltage, and current having a period of T can deliver an average power P of:

( ) ( ) dttvtiT

PT

⋅⋅= ∫01 (3)

The digital wattmeter calculates average power by numerical integration. Voltage and current waveforms are sampled simultaneously, converted to digital values, and

the sum of their product is the average power. If the voltage at time kt is denoted as ( )ktv , and the current as ( )kti , the average power can be calculated as:

( ) ( )[ ]∑=

⋅⋅=n

kkk tvti

nP

1

1 (4)

Where n is number of samples, the average power from (4) is not equal to the average power from (3). The difference is due to truncation error and sampling process. In addition to the errors due to the sampling process, quantization, and truncation, the errors comes from A/D conversion as well. 4. Propagation of Errors Through to Calculations The importance of estimating data errors is due to the fact that data errors propagate through the calculations to produce errors in results. [15-18]. Error propagation through calculations depends on the nature of the calculations, whether the function of the variables with errors are added, subtracted, multiplied, divided, integrated, or any other function or combination of functions. For example, if we have two functions with absolute errors, then the difference of these two numerically integrated functions is:

( ) ( ) ( )[ ]

( ) ( )

( ) ( )

( ) ( )

−+

+

−=

=+−+=

=+−+=

∑∑

∑∑

∑∑

∑

==

==

==

=

N

iii

N

iii

N

iii

N

iii

N

iiii

N

iiii

N

iiiiiiini

fN

fN

xfN

xfN

xfN

xfN

xfxfN

fR

022

011

02

01

022

011

02211

11

11

11

1

εε

εε

εεε

(5)

The total absolute error of the difference of two functions is the difference of the absolute errors of these two functions: ( ) ( ) ( )( ) ( )( ) ( )( )xfxfxfxff 221121 εεεε −=−= (6)

5. Methods and Algorithms in Power Transformer Losses Measurements 5.1 Difference of Power In and Power The transformers Total Losses are calculated as a difference of the input and output power. Assumption is that the load has unity power factor. The average values of the total Losses are:

( )

( ) OUTIN

T

SS

T

SSLOS

PPdtctbCTiPTvT

dtctbCTiPTvT

P

−=⋅⋅⋅⋅⋅−

−⋅⋅⋅⋅⋅=

∫

∫

02222

01111

1

1

(7)

127

Average value of the total Losses with numerical integration is:

( )

( ) NOUTNIN

N

kSkSk

N

kSkSkNUMLOS

ppCTiPTvN

CTiPTvN

p

−−=

−−

=−−−

−=⋅⋅−

−⋅⋅=

∑

∑

12212

12111

1

1

(8)

Where: TR -Ratio of the Transformer Under the Test

1CT - Primary Side Current Transformer ratio 2CT - Secondary Side Current Transformer ratio 1PT - Primary Side Voltage Transformer ratio 2PT - Secondary Side Voltage Transformer ratio

5.2 Measurement of Losses with Voltage and Current Difference 5.2.1 No load losses The no load losses are calculated as a product of the primary and secondary current difference transferred on the primary side and multiplied by the primary voltages. The secondary current is transferred on the primary side multiplied by the current mismatch multiplier. The average value is:

dtPTvCTTR

iCTiT

PT

SSSNLOS ⋅

⋅⋅

⋅⋅−⋅= ∫

0112211

11 (9)

and the average value by numerical integration is:

∑=

−−−−

⋅⋅

⋅⋅−⋅=

N

kSkSkSkNNLOS PTvCT

TRiCTi

Np

111211

11

(10) 5.2.2 Load losses The load losses are calculated by primary and secondary voltage difference, transferred on the secondary side, and multiplied by the secondary current. The average value is:

dtiCTPTvTRPTv

TP

T

SSSLLOS ⋅

⋅⋅

⋅−⋅= ∫0

22221

1

1 (11)

and numerically integrated average value is:

∑=

−−−−

⋅

⋅−⋅=N

kSkSkSkNLLOS iCTPTv

TRPTv

Np

12222

11

1

(12) 5.2.3 Total losses The total losses calculated by the voltage and current difference are equivalent to the total losses calculated by the difference of the power in and power out. The Total losses are a sum of the no load and load

Losses:

( ) ( )

NOUTNIN

N

kSkSk

N

kSkSk

N

kSkSkSk

N

kSkSkSkNNLOS

pp

CTiPTvN

CTiPTvN

iCTPTvTRPTv

N

PTvCTTR

iCTiN

p

−−

=−−

=−−

=−−−

=−−−−

−=

⋅⋅−−⋅⋅=

=

⋅

⋅−⋅+

⋅⋅

⋅⋅−⋅=

∑∑

∑

∑

12212

12111

12222

11

111211

11

1

11

(13) 6. Penetration of Errors in Power Transformer Losses Measurements Through to Calculations 6.1 Errors in Power Transformer Losses Measurements by Difference of Power In and Power Out The error in the power transformer losses measurements by the difference of power in and power out is:

( )( )

( )( )22222222

11111111

2221111

22222222

11111111

11

AVCTPTSS

AVCTPTSS

SSSS

AVCTPTSS

AVCTPTSSTLOS

CTiPTvCTiPTv

CTiPTvCTiPTvCTiPTv

CTiPTvp

εεεεεεεε

εεεεεεεεε

+++⋅⋅⋅⋅−−+++⋅⋅⋅⋅+

+⋅⋅⋅−⋅⋅⋅==++++⋅⋅⋅⋅−

−++++⋅⋅⋅⋅=∆

(14)

.ε - are the errors of the instrumentations CT’s , VT’s and instruments, or numerical integration 6.2 Errors in Power Transformer Losses Measurements by Voltage and Current Difference 6.2.1 Errors in no load losses measurements

( )

( )

( )

( )

( )

( )[ ]111111

2222

11112211

1111

2222

1111

1

1

1

111

VPTSS

TRACTS

ACTSSS

VPTS

TRACTS

ACTS

rNLOS

PTvPTv

CTTR

i

CTiCTTR

iCTi

PTv

CTTR

i

CTip

εε

εεε

εε

εε

εεε

εεε

+⋅⋅+⋅⋅

−+⋅⋅⋅−

−+⋅⋅+

⋅⋅−⋅

=

=++⋅⋅⋅

⋅

−++⋅⋅⋅−

−++⋅⋅=

(15) If we rearrange equation (15) the No load losses are:

128

( )

( )TRACTVPTSS

VPTACTSS

SSSNLS

PTvCTTR

i

CTiPTv

PTvCTTR

iCTip

εεεεε

εεεε

ε

−+++⋅⋅⋅⋅⋅−

−+++⋅⋅⋅⋅+

+⋅⋅

⋅⋅−⋅=

22111122

11111111

112211

1

1

(16)

6.2.2 Errors in load losses measurements

( )( )

( )

( )

( )( )[ ]222222

2222

111

1221

1

2222

2222

111

1

11

1

ACTSS

VPTS

TRVPTSSS

ACTS

VPTS

TRVPTSLLS

CTiCTiPTv

TRPTvPTv

TRPTv

CTiPTv

TRPTv

p

εε

εε

εεε

εε

εε

εεεε

+⋅⋅⋅+⋅⋅

+⋅⋅−

−−+⋅⋅+

⋅−⋅=

=++⋅⋅⋅

⋅

++⋅⋅−

−−++⋅⋅=

(17) If we rearrange equation (17) the load losses are:

( )( ) =+++⋅⋅⋅⋅−

−++−+⋅⋅⋅⋅+

+⋅⋅

⋅−⋅=

2222222

2211221

1

22221

1

ACTVPTSS

ACTTRVPTSS

SSSLLS

CTiPTv

CTiTRPTv

CTiPTvTRPTvp

εεεε

εεεεε

ε

(18)

6.2.3 Error in total losses measurements Error in the total losses measured by the difference of the voltages and currents is a sum of the no load losses and load losses

( )

( )

( )( )2222222

2211221

1

22221

1

22111122

11111111

112211

1

1

ACTVPTSS

ACTTRVPTSS

SSS

TRACTVPTSS

VPTACTSS

SSSTLS

CTiPTv

CTiTRPTv

CTiPTvTRPTv

PTvCTTR

i

CTiPTv

PTvCTTR

iCTip

εεεε

εεεεε

εεεεε

εεεε

ε

+++⋅⋅⋅⋅−

−++−+⋅⋅⋅⋅+

+⋅⋅

⋅−⋅+

−+++⋅⋅⋅⋅⋅−

−+++⋅⋅⋅⋅+

+⋅⋅

⋅⋅−⋅=∆

(19) Equation (19) can be written as:

( )( )2222222

11111111

2221111

ACTVPTSS

VPTACTSS

SSSSTLS

CTiPTvCTiPTv

CTiPTvPTvCTip

εεεεεεεε

ε

+++⋅⋅⋅⋅−−+++⋅⋅⋅⋅++⋅⋅⋅−⋅⋅⋅=∆

(20)

The error in total losses measured by the voltage and current difference, equation (20), is equivalent to the error

of total losses measured by the difference of power in and power out, equation (14). 7. Example Calculations of Penetration of Errors Through to Calculations in Power Transformer Losses Measurements To present the process of penetration of error through to calculations, we calculated – simulated, measurement of power transformer losses, with developed software, on 50 kVA, 480/120V, single phase transformer. The transformer is loaded with a linear resistive load of 17.236 kW. Power transformer ratio error is assumed to be 0.5%. Calculated values without implementation of errors of instrumentations are used as a reference. Table 1. Open and Short Circuit Transformer Tests

Open and Short Circuit Test

Open Circuit Test [W]

Short Circuit Test[W]

Total Losses[W]

230.380 377.744 608.135 Table 2. Transformer losses from difference of power in and power out, and voltage and current difference without errors of instrumentations

Power In-Power Out

Power In [W]

Power Out [W]

Total Losses[W]

17843.667 17236.231 607.431

VA Difference

No Load Losses[W]

Load Losses[W]

Total Losses[W]

229.686 377.744 607.431 Current Difference Primary Side [A]

Voltage Difference Secondary Side [V]

0.4785 2.5735 Table 3. Transformer primary and secondary volts and currents

Primary Volts [V]

Primary Current [A]

Secondary Volts [V]

Secondary Current[A]

480.0000 37.1742 117.4265 146.7831 Table 4. Transformer primary and secondary volts and currents without errors on the secondary sides on the VT’s and CT’s

Primary Volts [V]

Primary Current [A]

Secondary Volts [V]

Secondary Current[A]

120.0000 3.7174 117.4265 4.8928 Table 5: Instrumentation Data of CT’s and VT’s

1CT (A) 1PT (V) 2CT (A) 2PT (V)

Ratio 50:5 480:120 150:5 120:120

129

Accuracy Class [%]

0.3 0.3 -0.3 -0.3

Table 6: Data of Digital Measurements of Volts and Currents

1A (A) 1V (V) 2A (A) 2V (V)

Rated Values 50 480 150 120

Accuracy Class [%] 0.1 0.1 -0.1 -0.1

7.1 Errors in Power Transformer Losses Measurements by Difference of Power In and Power Out

The Total losses are:

( )( )

W

CTiPTvCTiPTvp

AVCTPTSS

AVCTPTSSTLOS

0703.8883417.170984121.17986100

1.0100

1.0100

3.0100

3.0100.308928.400.1000.120

1001.0

1001.0

1003.0

1003.0100.107174.300.4000.120

11

22222222

11111111

=−=

−−−−⋅⋅⋅⋅−

++++⋅⋅⋅⋅=

=++++⋅⋅⋅⋅−++++⋅⋅⋅⋅=∆

εεεεεεεεε

Power In without errors is: 66275.178431111 =⋅⋅⋅= CTiPTvp SSIN [W]

Power In with errors is: ( )

][4121.179861 11111111

WCTiPTvp AVCTPTSSIN

==++++⋅⋅⋅⋅= εεεε

Absolute Error of Power In is: ( ) [ ]WCTiPTvp AVCTPTSSIN 7493.14211111111 =+++⋅⋅⋅⋅=∆ εεεεε

Relative Error of the Power In is:

[%]800.01006627.17843

6627.178434121.17986

100[%}

=⋅−

=

=⋅∆

∆−∆=∆

IN

INININ p

ppp ε

Power Out without errors is: 2316.172362222 =⋅⋅⋅= CTiPTvp SSOUT [W ]

Power Out with errors is:

( )][3417.17098

1 22222222

WCTiPTvp AVCTPTSSOUT

=++++⋅⋅⋅⋅= εεεεε

Absolute Error of Power Out is: ( )

][8899.13722222222

WCTiPTvp AVCTPTSSOUT

=+++⋅⋅⋅⋅=∆ εεεεε

Relative Error of the Power Out is:

[%]800.01002316.17236

3417.170982316.17236

100[%}

=⋅−

=

=⋅∆

∆−∆=∆

OUT

OUTOUTOUT p

ppp ε

Transformer Total Power Losses without errors are: ( ) 4312.607=−=∆ OUTINTLS ppp [W]

Transformer Total Power Losses with errors are: ( ) 0703.888=−=∆ εεε OUTINTLS ppp [W]

Transformer Total Losses absolute error is: ( ) 6391.280=−∆=∆ TLSTLSTLS ppp εε [W]

Relative error of Transformer Power Losses with Power In-Out Method is:

[%]2010.46

1004312.607

4312.6070703.888100[%]

=

⋅−

=⋅∆

∆−∆=∆

ε

ε

LS

LSLSTLS p

ppP

7.2 Errors in Power Transformer Losses Measurements by Voltage and Current Difference 7.2.1 Errors in no load loss measurements Current difference without errors is:

4785.012211 =

⋅⋅−⋅= CT

TRiCTii SSdiff [A]

Current difference with errors is: ( )

( ) ][9575.011

1

2222

1111

ACT

TRi

CTii

TRACTS

ACTS

rdiff =

−++⋅⋅⋅

−++⋅⋅=

εεε

εεε

Primary voltage without errors is:

48011 =⋅= PTvv SP [V] Primary voltage with errors is:

( ) 9200.4811 1111 =++⋅⋅= VPTSrP PTvv εεε [V] No Load Losses without errors are:

6863.2291112211 =⋅⋅

⋅⋅−⋅= PTvCT

TRiCTip SSSNLS [W]

No Load Losses with errors are: ( )

( )

( ) ][4250.4611

111

1111

2222

1111

WPTv

CTTR

i

CTip

VPTS

TRACTS

ACTS

rNLOS

==++⋅⋅⋅

⋅

−++⋅⋅⋅

−++⋅⋅=

εε

εεε

εεε

Absolute error of No Load Losses measurement is: 7387.231][ =−=∆ NLSrNLSWNLS ppp ε [W]

Relative error of No Load Losses measurement is:

894.100100[%] =⋅−

=∆NLS

NLSrNLSNLS p

ppp ε [%]

7.2.2 Errors in Load Losses measurements Voltage difference without error is:

5735.2221

1 =

⋅−⋅= PTvTRPTvv SSS [V]

Voltage difference with errors is:

( )( )

][9232.21

1

2222

111

1 VPTv

TRPTv

vVPTS

TRVPTSS =

++⋅⋅−

−−++⋅⋅=

εε

εεεε

Secondary current without error is: 7831.14622 =⋅= CTii SS [A]

Secondary Current with error is: ( ) 190.1461 2222 =++⋅⋅⋅= ACTSS CTii εεε [A]

130

Load Losses without error are:

749.37722221

1 =⋅⋅

⋅−⋅=∆ CTiPTvTRPTvp SSSLLOS [W]

Load Losses with errors are:

( )( )

( ) ][3596.42711

1

2222

2222

111

1

WCTiPTv

TRPTv

p

ACTS

VPTS

TRVPTSLLS

=++⋅⋅⋅

⋅

++⋅⋅−

−−++⋅⋅=∆

εε

εε

εεεε

Absolute error of Load Losses measurement is: 6146.49][ =−−= NLSrLLSLLSWLLS pppp ε [W]

Relative error of No Load Losses measurement is:

1344.13100[%] =⋅−

=∆LLS

rLLSLLSLLS p

ppp ε [%]

7.2.3 Errors in Total Losses measurements Transformer Total Losses without errors are:

4312.607=+=− LLSNLSVATLS ppp [W] Transformer Total Losses with errors are:

7846.888=+=− εε LLSNLSVATLS ppp [W] Absolute error in Transformer Total Losses measurements is:

( ) 6391.280=−=∆ −−− VATLSVATLSVATLS ppp εε [W] Relative error in the Power Transformer Losses measurements with VA Difference Method is:

318.46100[%] =⋅−

=∆ −

TLS

TLSTLSVATLS p

ppP ε [%]

8. Power Transformers Losses Measurements with Digital Instrument The transformer losses are measured with the instrumentation and equipment shown in Figure 2, and the results are shown in Figure 3.

Figure 2. Measurement of transformer losses with digital Instrument Transformer Data: 3 kVA, 480/208, DY Nonlinear Load: Three single phase nonlinear load banks of 1 kW resistive load banks with bridge rectifiers to

simulate IEEE 519 Standard single phase nonlinear load profile. Table 7: Data of used CT’s and VT’s

1CT (A) 1PT (V) 2CT (A) 2PT (V) Ratio 5:5 480:120 5:2.88 120:120 Accuracy Class [%]

0.3 0.3 0.3 0.3

Figure 3. Measurement of the transformer losses under nonlinear load From Figure 3, the Total Losses with the VA Difference method are 96.616 W, and the Total Losses with Power In- Out method are 1429.426-1332.810=96.616 W, which are exactly the same number. Figure 4. and Figure 5, show the voltage and current differences.

Figure 4. Currents Differences

Figure 5. Voltage Differences

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9. Conclusion The study was focused on investigation of penetration of errors of power transformer losses measurements in real working conditions. The research and development was carried out for more than ten years. Over this period of time we did many site and laboratory experiments, and measurements of different sizes and types of power transformers. Importance of estimating of errors in measurements is due to the fact that data errors propagate through to calculations and produce errors in the final results. Error propagation through calculations depend on the nature of the calculations. The errors penetrate through calculations and algorithms of measurements. Although distribution of errors in the instrumentations, includes CTs, VTs, and, A/D conversion, V and A-meters, etc. are of stochastic nature, the penetration of error through to calculations is of deterministic nature. When the measured results are obtained by complex calculations and algorithms, the total errors can be calculated only by algebra of the penetration of errors. Neglecting penetration of errors through to calculations can lead to erroneous conclusion about the results of measurements. If the errors of the devices are known, the measurement can be calibrated, but the calibration will be applied to both methods. The total losses measured by both methods are the same. References [1] IEEE Std 57.12 2001 – “ Test Code for Dry-Type Distribution and Power Transformers”. [2] NEMA TP 2, “ Standard Test Method for measuring the Energy Consumption of Distribution Transformers”. [3] IEEE Std. C57.12.00, “ IEEE Standard General Requirements for Liquid-Immersed Distribution, Power, and Regulating Transformers”. [4] IEEE Std. C57.12.01, “IEEE Standard General Requirements for Dry-Type Distribution and Power Transformers Including Those with Solid- Cast and/or Resin-Encapsulated Windings”. [5] E.Arri and N. Locci, “ Uncertainty analysis in a novel method for transformer power losses measurement”, Elsevier Science,Measurement, 12(4), 1994, 315–320. [6] D. Lin, E.F. Fuchs, M. Doyle ‘Computer-Aided Testing of Electrical Apparatus Supplying Non-Linear Loads’, IEEE Transactions on Power Systems, Vol.12, No.1, February 1997. [7] Ernesto Arri, Nicola Locci, Francesco Mocci, ‘Measurement of Transformer Power Losses and Efficiency in Real Working Conditions’, IEEE Transactions on Instrumentation and Measurement. Vol.40 No.2 April 1991.

[8] E.F.Fuchs, R.Fei” A new Computer-Aided Method for Efficiency Measurements of Low-Loss Transformers and Inductors Under Nonsinusoidal Operations”, IEEE Transactions on Power Delivery , Vol.11. No.1 January 1996. [9] E.F. Fuchs, D, Yildrim, T. Batan “ Innovative procedure for measurements oflosses of transformers supplying nonsinusoidal loads”, IEE Proceedings, Generation, Transmission and Distribution, Vo. 46, No.6, November 1999. [10] Ewald F. Fuchs, Fellow, IEEE, Dingsheng Lin, and Jonas Martynaitis “ Measurement of Three-Phase Transformer Derating and Reactive Power Demand under Nonlinear Loading Conditions”, IEEE Transactions on Power Delivery , Vol. 21 No.2 April 2006. [11] Dingsheng Lin and Ewald F. Fuchs, “Real-Time Monitoring of Iron-Core and Copper Losses of Transformers Under (Non)Sinusoidal Operation”, IEEE Transactions on Power Delivery , Vol. 21. No.3 July 2006. [12] Ewald F.Fuchs and Mohammed A.S. Masoum” Power Quality in Power Systems and Electrical Machines” , Elsevier Academic Press, san Diego, USA, 2008. [13] A.Damnjanovic and G.Ferguson, “Distribution Transformer Losses Under Nonlinear Load Measurement and Evaluation”, IEEE Power Engineering Society General Meeting 2004 , Denver Colorado . [14] A.Damnjanovic and G.Ferguson,“ The accuracte Measurement of Transformer Losses Under Non-Linear Load Conditions” DISTRIBUTECH Conference and Exhibition , January 2004, Orlando Florida. [15] Arghya Sarkar,and S. Sengupta, “Design and Implementation of a High Accuracy Sampling Wattmeter under Non-sinusoidal and Time Varying Environments”, 2008 IEEE. [16] Lu-Zu-Liang, “An Error Estimate for Quasi-Integer-Period Sampling and an Approach for Improving its Accuracy”, IEEE Transactions on Instrumentation and Measurements, Vol. 37. No. 2 June 1988. [17] G. N. Stenbakken, O. B. Laug, T. H. Kibalo, B. A. Bell, and A. G. Perrey, “ NBS Wideband Sampling Wattmeter” , NBS Technical Note 1221, Electro Systems Division, Center for Electronics and Electrical Engineering, National Engineering Laboratory, National Bureau of Standards, Gaithersburg, MD 20899. [18] J. Topping , F. Inst.P. “ Errors Observation and Their Treatment”, The Institute of Physics and Physical Theory, Chapmen and Hill Limited, London, 19.

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