Modeling and Simulation the Current Transformer for ...

International Review on Modelling and Simulations (I.RE.MO.S.), Vol. 4, N. 4 August 2011

Modeling and Simulation the Current Transformer for Measuring Current Harmonics Using MATLAB-SIMULINK B. Vahidi1, V. Farahani2

Abstract – The present describe a method to teach power system engineers how to represent the hysteresis in current transformer modeling. This algorithm allows multi-valued hysteresis behavior to be represented. This method has been implemented into MATLAB-SIMULINK to analyzed the current transformer when excited with nonsinusoidal currents, a current transformer is simulated and the results are shown. Evaluation of the simulation with 40 students is very positive in terms of their developing confidence in and understanding of this simulation. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: MATLAB-SIMULINK, Current Transformer, Hysteresis

Nomenclature H B Hm Bm Hc NS Lc Ac g

λm

Dout Din, Height Drev

Magnetic Field (A/M) Magnetic Induction (T) Peak Value Of H Peak Value Of B Coercive Field Secondary Turn Magnetic Core Mean Path Length Core Sectional Area Air Gap Peak Value Of λ External Diameter Of CT Internal Diameter Of CT Effective Height Of CT Dmax1

I.

Introduction

Power engineering education is receiving a lot of attention lately due to restructuring and deregulation of utility industry. This has resulted in a significant turnover of the work force where quite a few professionals have either changed their jobs or retired as a sequence. Therefore new university graduates should be equipped with required knowledge needed in a more competitive industry. The modeling and simulation in power engineering education is not a new concept. A number of excellent educational papers on this subject have been published. A variety of software tools are available to simulate electrical circuits; one is OrCAD. Many simulations of different aspects of a power system and engineering application using PSpice, MATLAB and EMTP have been presented by different researchers [1]-[14].

Manuscript received and revised July 2011, accepted August 2011

All current transformers (CTs) used in measurement are basically similar in construction, standard transformers in that they consist of magnetically coupled primary and secondary windings, wound on a common iron core, the primary winding being connected in series with the network unlike voltage transformers. They must therefore withstand the networks short-circuit current [15]. There are two types of current transformers: 1- Wound primary type, 2- Bar primary type. The wound primary type is used for smaller current. Instruments and meters are required to work accurately up to full-load current, but above this, it is advantageous to saturate and protect the instruments under fault conditions. Hence, it is common to have metering CTs with a very sharp knee-point voltage [15]. Today’s power systems planning and maintenance requires monitoring and recording of harmonic currents and harmonics powers [16]. The goals of these measurements are: 1) Frequency spectrum of current flowing in network [17]. 2) To detect resonance or electromagnetic interference conditions [18]. 3) Harmonic pollution sources location [19], [20]. The measure of CT performance is the ability to reproduce accurately the primary current on the secondary side of the CT in terms of its magnitude and shape [21]. CT saturation leads to inaccurate current measurement [21].

II.

CT Equivalent Circuit

The general equivalent circuit of a CT should contain the primary section and the secondary section combined with the burden [22]. Fig. 1(a) shows an equivalent

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circuit of a CT and its excitation circuit. Fig. 1(b) shows the phasor diagram. It consist of an inductor Lp and Rp which represent the primary system inductance and resistance respectively, Ls and Rs represent the equivalent secondary inductance and resistance respectively. RB and LB represent the burden equivalent resistance and inductance respectively. Lm and Rm represent the magnetizing inductance and core loss resistance respectively.

imag =

(

Lc H = aλ + bv 1 − cλ 2 Ns

)

(2 )

where: λ = N s φ = N s Ac B = ∫ Vs dt

is the linkage flux, where a =

(3 )

Lc H m LH , b= c c , N s λm Ns

Vs 1 ξ →0. , λm = N s Ac Bm and v = 2 ξ +V λm s NS, Lc and Ac are the secondary turn, magnetic core mean path length and cross sectional area, respectively. g is air gap and λm is peak value of λ . Following equations are the best fitting empirical expressions that describe the coercive and the maximum magnetic fields as function of the peak magnetic induction [22], [27]: c=

(a)

H c = 2 Bm − 0.032 Bm12.42

(b) Figs. 1. Equivalent circuit of CT

⎛A⎞ H m = 3.0797 Bm 0.6989 + 7.9577 × 10-5 gBm ⎜ ⎟ ⎝m⎠

III. Hysteresis Model Various approaches for modeling the hysteresis behavior have been described in the literature. The steady state and transient behavior are defined by exponential flux paths expressed in terms of three nonlinear parameters [23]. The model selected in the present paper is based on the approaches described in references [22], [24], [25], [26]. As described in reference [27], the hysteresis element can be defined of major and minor loops. The minor loop trajectories of a hysteresis element can be defined in terms of its major loop. All points of operation are assumed to be within the major loop. The points, beyond which the characteristic becomes single-valued are defined as the positive and negative saturation points [25]. During normal operation (low flux density in core), the H-B relation for ferromagnetic core is [22]: H=

⎡ ⎛ B ⎞2 ⎤ B H m ± ⎢1 − ⎜ ⎟ ⎥ Hc Bm ⎢⎣ ⎝ Bm ⎠ ⎥⎦

⎛A⎞ ⎜ ⎟ ⎝m⎠

(4 )

(5 )

Fig. 2 shows the linkage flux and Fig. 3 shows hysteresis loop.

Fig. 2. Linkage flux variation

(1 )

where H and B are the magnetic field (A/m) and magnetic induction (T), respectively, Hm and Bm are peak value and Hc is the coercive field. From equation (1) the following equation can be obtained [22]: Fig. 3. Hysteresis loop

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To generate the closed minor loops the following fundamental can be applied [24]: inside a minor loop the magnetizing current depends only on the last two reversal points, and each curve tends to return to previous-to-last reversal point. Effectively, after detecting a reversal at point 2, it tends to return to previous-to last reversal point (point 1), a new trajectory 2-1 is calculated. First, Dmax, which is the distance between the reversal point 2 and the appropriate half of the major loop is calculated (Fig. 3). In addition, the vertical distance Dmin between 1 and the major loop is calculated (0 in this example). By assuming, that the vertical distance decrease linearly with φ , all the points

As can be seen in Fig. 5 the measured voltage is sent to integral block in order to compute the λ . Hysteresis equation block is made the λ -Imag relationship. Following blocks are in hysteresis equation block (All of parts of these blocks are from Matlab-Simulink).

of trajectory are then defined. A linear relation D( φ ) of D as function of the flux is found at each reversal point and is stored in memory along with the coordinates of this point. Memorization is important in order to represent properly the behavior of embedded loops as the evolution on Fig. 3. From Fig. 3:

Fig. 6. Hm product block

D ( φ ) = Dslope × φ+ Dincpt

Hm is output of Fig. 6 where: Dout = External diameter of CT, Din = Internal diameter of CT, height = Effective height of CT. The effective cross section of CT (A) is: A = ( Dout - Din )× height

(6 )

where:

(9 )

The Hc is obtained from Fig. 7. Dslope =

Dmax1 − 0 ϕmax − ϕrev

Dincpt = Dmax1 − Dslope × ϕrev

(7 )

(8 )

Minor loop trajectories for decreasing values of flux are created in a similar manner [25].

Fig. 7. Hc product block

The a, b, c coefficients are obtained from Fig. 8. In this calculation K =

III.1. Major Loop Modeling The simulated circuit is shown in Fig. 4.

path =

( Dout − Din ) ×

π 2

π

2

and Average length of flux

.

Fig. 4. Simulated circuit

Fig. 8. a, b, c coefficients product blocks

In Fig. 4 the current transformer is an ideal transformer. The hysteresis modeling block is a current control source that control the magnetize current by voltage which is across the source. Fig. 5 shows this block circuit.

The magnetize current of major loop are obtained from Fig. 9 where: slipe = +1 if slope of λ >0 slipe = -1 if slope of λ 0) & (U2 = U3)

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There is the following block (Fig. 17) in the If action Subsytem block of Fig. 16.

Fig. 17. If Action Subsystem block of Fig. 16

Therefore rec = -2 (Fig. 16) for reversal trajectory of minor loops and is zero for other trajectory. Therefore: Count + rec 0 for major loop The following circuit (Fig. 18) which is placed in hysteresis equation block (Fig. 5), is selected the proper magnetized current at output. Production of Imag of major loop is shown in Fig. 9.

Fig. 18. Block for producing proper magnetized current Fig. 15. Flux variation + count variation

The Imag of minor loops are produced as follows. In order to construct the minor loops, we need to have λ ptl , λrev , Dmax1 and Dmax2. For computing Dmax1 and Dmax2 we need to have iptl and irev. The values of

λ ptl

,

λrev , iptl and irev are outputs of block of Fig. 13, block of Fig. 19 stores these values.

Fig. 16. Reversal trajectory detection block

The output of this block is: -2 where λ ptl ≤ λ ≤ λrev ; -0 for other values of λ . Fig. 19. Block for storing λ ptl , λrev , iptl and irev

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For the moment that condition of Fig. 13 satisfied, the linkage flux and magnetized current will be stored (Fig. 19). If count < 0 (forward trajectory of minor loops), the values of λ ptl and iptl will be equal to values of λ and i respectively (Fig. 18). If rec > 0 (it contain the reversal trajectory of minor loops as well), the values of λrev and irev will be equal to values of λ and i respectively (Fig. 18). The linkage flux and magnetized current for other times are the same as this moment. Now we have to calculate the Drev: for known Irev the value of λ1 (linkage flux at point 1) is computed from [25]:

(

)

I rev = aλ + b 1 - cλ 2 ⇒ bcλ 2 - aλ + I rev - b = 0 ⇒λ =

a ± a 2 - 4bc ( I rev - b )

(10)

Then:

D ptl = λ ptl - λ2

(15)

Dλ = Dslope × λ + Dincpt

(16)

Dslope =

D ptl - 0

λ ptl - λrev

(17)

Dincpt = D ptl - Dslope × λ ptl

(18)

2 I mag = a ( λ - Dλ ) - b ⎡1 - c ( λ - Dλ ) ⎤ ⎣ ⎦

(19)

Fig. 20 shows circuit for computing the Drev , this circuit is placed in If Action Subsystem block (Fig. 13).

2bc

The following value (greater) of linkage flux is acceptable:

λ1 =

a + a 2 - 4bc ( I rev - b ) 2bc

Fig. 20. Block for computing the Drev

Then: Dmax1 = λ1 - λrev

(11)

Dλ = Dslope × λ + Dincpt

(12)

Circuit of Fig. 21 is used to compute the magnetize current of minor loops. This circuit is placed in hysteresis equation block (Fig. 5).

and:

where: Dslope =

Dmax1 - 0

λmax - λrev

Dincpt = Dmax1 - Dslope × λrev

The value of Imag is computed due to the assumption of linear dependence of between linkage flux, and distance between major and minor loops. The Imag is computed from: 2 I mag = a ( λ + D ( λ ) ) + b ⎡1 - c ( λ + D ( λ ) ) ⎤ ⎢⎣ ⎥⎦

(13)

For reverse trajectory at minor hysteresis loop, the value of λ2 will be computed from following formula:

λ2 =

-a - a 2 + 4bc ( I rev + b ) 2bc

(14)

Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved

Fig. 21. Block to compute minor loop hysteresis

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Fig. 22 shows the circuit which is placed in Fig. 21 block.

Loop of Hysteresis of Fig. 22 to compute the magnetized current.

Fig. 25. Circuit for computing the magnetized current

IV. Fig. 22. Block for computing the Dλ

In Fig. 22, the Dλ is computed by Dλ block. Minor Loop of Hysteresis block is used to computed the magnetize current of minor loops. Following circuit (Fig. 23) is placed in the Dλ block of Fig. 22 to compute the Dslope and Dincpt.

Simulation Results

A CT with Following specifications are simulated [16]: Current ratio: 100/5 A f = 50 Hz NP = 1 NS = 20 Zs = 0.5 25.84 Ω S = 12.5 VA Din = 0.13 m Dout = 0.2 m Rec = 500 Ω w (rectangular cross section effective height) = 0.06 m Fig. 26 shows harmonic phase angle error versus phase difference for different harmonics when the CT is feed by following current:

i p = 2 ⎡⎣ I p1 sin ( wt + α1 ) + I ph sin ( hwt + α h ) ⎤⎦

(20)

where: g = air-gap α h = the phase angle of h order of current harmonics The effect of

Ih on phase angle error is shown in I1

Fig. 27.

Fig. 23. Circuit for computing Dslope and Dincpt

Circuit of Fig. 24 is placed in block Dλ (Fig. 22) to compute the Dλ .

Fig. 24. Circuit for computing the Dλ

The following circuit (Fig. 25) is placed in Minor Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved

Fig. 26. Harmonic phase angle error versus phase difference for different harmonics order (g = 10-5 m)

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Table II shows the average scores for each question out of students’ feedback. TABLE II AVERAGE SCORE OBTAINED FROM STUDENTS’ ANSWERS Average score Question 1 3.42 Question 2 3.57 Question 3 3.37 Question 4 3.52 Total 3.47

VI.

Fig. 27. Phase Angle error versus

Ih

for different harmonics

I1 ( α h - α1 = 0 , g = 10 -5 m )

V.

Student Feedback

The methodology described in the present paper has explained for 40 senior undergraduate students in power system. They used the methodology and filled a questionnaire form. The questionnaire consisted of four questions shown in Table I. TABLE I QUESTIONNAIRE ANSWERED BY THE STUDENTS Questions 1- I had previous knowledge of MATLAB-SIMULINK 2- The content of this practical is valuable for an engineer 3- Are you understanding the concept of current harmonics better after using this utility 4- Are you more familiar with the subject after using this utility

In the present paper MATLAB-SIMULINK is used to simulate the CT for measuring current harmonics. By using blocks of MATLAB saturation effect is modeled and in this modeling major and minor hysteresis loops are considered. By considering the minor hysteresis loops the error will be reduced. Effects of different parameters on phase angle error are shown, which shows the ability of method for this job. Evaluation of the project involving 40 students indicates benefits of this project for learning and mastering the subject. This method cannot be used for modeling the transient state of the current transformer and can just perform the steady state analysis of the current transformer.

References [1]

Score [2]

[3]

The students graded them as 1 (poor), 2 (not much), 3 (good) and 4 (very good). Fig. 28 shows the global results obtained from the students’ questionnaire.

[4]

35

25

Question 1 Question 2 Question 3

20

Question 4

Number of students

30

[5]

15

[6]

10 5 0 Poor

Not Much

Good

Very Good

[7] Fig. 28. Answers of students to the questionnaire

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Conclusion

B. Vahidi, A. Agheli, S. Jazebii, “Teaching short-circuit withstand test on power transformers to M.Sc. students And junior engineers using Matlab-Simulink”, Computer Applications in Engineering Education, http://www3.interscience.wiley.com/journal/38664/home, Accessed 22 Feb. 2010, DOI: 10.1002/cae.20416. B. Vahidi, M. M. Arabshahi, Teaching current tests on surge arresters to undergraduate students using Malab-Simulink”, Computer Applications in Engineering Education, http://www3.interscience.wiley.com/journal/38664/home, Accessed 6 Jan. 2010, DOI: 10.1002/cae.20405. B. Vahidi, A. Khorsandi, “Simulation of effect of voltage sag on inrush current using Matlab-Simulink”, Computer Applications in Engineering Education, http://www3.interscience.wiley.com/journal/38664/home, Accessed 29 March 2010, DOI: 10.1002/cae.20431. B. Vahidi, M. Taherkhani, “Teaching short circuit breaking test on high voltage circuit breakers to undergraduate students by using Matlab-Simulink”, Computer Applications in Engineering Education, http://www3.interscience.wiley.com/journal/38664/home, Accessed 11 Oct. 2010, DOI: 10.1002/cae.20491. B. Vahidi, A. Haghani Abandansari, “Teaching ferroresonance in power system to undergraduate students by using MatlabSimulink”, Computer Applications in Engineering Education, http://www3.interscience.wiley.com/journal/38664/home, Accessed 24 June 2009, DOI: 10.1002/cae.20316. B. Vahidi, M. R. Bank Tavakoli, “Simulation of Ward Leonard test set and induction motor on Matlab for teaching performance characteristics of induction motor to undergraduate students”, Computer Applications in Engineering Education, http://www3.interscience.wiley.com/journal/38664/home, Accessed 13 Apr. 2009, DOI: 10.1002/cae.20256. B. Vahidi, J. Beiza, Using Pspice in teaching impulse voltage testing of power transformers to senior undergraduate students, IEEE Trans. Educ., vol. 48, no. 2, May 2005, pp 307-312.

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[8]

[9]

[10]

[11]

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[13]

[14]

[15] [16]

[17]

[18]

[19]

[20]

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B. Vahidi, S. M. Tabatabaei, M. M. Baharlou, Power System Lightning Transient Simulation on OrCAD with Corona Effect Consideration for Educational Purposes, Computer Applications in Engineering Education, http://www3.interscience.wiley.com/journal/38664/home, Accessed 16 May. 2011, DOI: 10.1002/cae.20552. B. Vahidi, M. R. Bank Tavakoli, Standstill Frequency Response Test Analyzer by Using Excel Macro for Educational Purpose, Computer Applications in Engineering Education, http://www3.interscience.wiley.com/journal/38664/home, Accessed 22 Feb., 2010, DOI: 10.1002/cae.20415. B. Vahidi, E. Esmaeeli, MATLAB-SIMULINK Based Simulation for Digital Differential Relay Protection of Power Transformer for Educational Purpose”, Computer Applications in Engineering Education, http://www3.interscience.wiley.com/journal/38664/home, Accessed 22 Feb., 2010, DOI: 10.1002/cae.20493. B. Vahidi, A. A. Damaki Aliabad, A Software Based on MATLAB for Teaching Substation Lightning Protection Design to Undergraduate Students with Emphasize on Different Striking Distance Models”, Computer Applications in Engineering Education, http://www3.interscience.wiley.com/journal/38664/home, Accessed 23 Jun. 2009, DOI: 10.1002/cae.20307 B. Vahidi, M. R. Bank Tavakoli, W. Gawlik, Determining Parameters of Turbine’s Model Using Heat Balance Data of Steam Power Unit for Educational Purposes, IEEE Trans. Power Systems, Vol. 22, No. 4, pp. 1547-1553, 2007. M. R. Bank Tavakoli, B. Vahidi, W. Gawlik, An Educational Guide to Extract the Parameters of Heavy Duty Gas Turbines Model in Dynamic Studies Based on Operational Data, IEEE Trans. on Power Systems, Vol. 24, No. 3, pp. 1366-1374, 2009. R. A. Naghizadeh, B. Vahidi, M. R. Bank Tavakoli, Estimating the Parameters of Dynamic Model of Drum Type Boilers Using Heat Balance Data as an Educational Procedure, IEEE Trans. on Power Systems, DOI: 10.1109TPWRS.2010.2061879. L. G. Hewitson, M. Brown and R. Balakrishnan, Practical power systems protection, 1st Edition, Oxford, Elsevier, 2005. A. E. Emanuel, J. A. Orr, “Current harmonics measurement by means of current transformers, IEEE Transactions on Power Delivery, vol.22, July 2007, pp. 1318-1325. R. C. Dugan, M. F. McGranaghan, S. Santoso, H. B. Beaty, Electrical power system quality, New York, McGraw Hill, 2003, Ch. 6. R. L. Almonte, A. W. Ashley, Harmonics at the utility industrisal interface: A real world example, IEEE Transactions on Industrial Application, vol. 31, Nov./Dec. 1995, pp. 1419-26. A. E. Emanuel, Powers in nonsinusoidal situations: A review of definitions and physical meaning, IEEE Transactions on Power Delivery, vol. 5, July 1990, pp. 1377-1389. L. Cristaldi, A. Ferrero, A method and related digital instrument for the measurement of electric power quality, IEEE Transactions on Power Delivery, vol. 10, July 1995, pp. 1183-1189. J. Pan, K. Vu, Hu Yi, An efficient compensation algorithm for current transformer saturation effects, IEEE Transactions on Power Delivery, vol.19, Oct. 2004, pp. 1623-1627. M. Tumay, R. R. S. Simpson, H. El. Khatroush, Dynamic model of a current transformer, International Journal of Electrical Engineering Education, vol. 37, July 2000, pp. 268-276. D. O’Kelly, Simulation of transient and steady state magnetization characteristics with hysteresis, Proc. IEE, vol. 124, June 1977, pp. 578-582. S. Casoria, P. Brunelle, G. Sybille, Hysteresis modelingin the Matlab/Power system blockset, Mathematics and Computers in Simulation, vol. 63, Nov. 2003, pp. 237-248. J. G. Frame, N. Mohan, T. H. Liu, Hysteresis Modeling in an electro-magnetic transients program, IEEE Transactions on Power Apparatus and systems, vol. PAS-101, Sept. 1982, pp. 3403-3412. S. N. Talukdar, J. R. Bailey, Hysteresis model for system studies, IEEE Transactions on Power Apparatus and systems, vol. PAS95, July/Aug. 1976, pp. 1429-1434. J. Meisel, Current instrument transformer error calculation, AIEE, Dec. 1963, pp. 1082-1085.

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Authors’ information 1

Department of Electrical Engineering, Amirkabir University of Technology, Tehran, Iran. Behrooz Vahidi was born in Abadan, Iran in 1953. He received the B.S. in electrical engineering from Sharif University of Technology, Tehran, Iran in 1980 and M.S. degree in electrical engineering from Amirkabir University of Technology, Tehran, Iran in 1989. He also received his Ph.D. in electrical engineering from UMIST, Manchester, UK in 1997. From 1980 to 1986 he worked in the field of high voltage in industry as chief engineer. From 1989 to present he has been with the department of electrical engineering of Amirkabir University of Technology where he is now a professor. His main fields of research are high voltage, electrical insulation, power system transient, lightning protection and pulse power technology. He has authored and coauthored five books and more than 220 papers on high voltage engineering and power system. Prof. Vahidi is senior member of IEEE. Vahid Farahani was born in Arak, Iran in 1986. He received the B.S. in electrical engineering from Amirkabir University of Technology, Tehran, Iran in 2008. Currently he is M.Sc. student at Amirkabir University of Technology, Tehran, Iran. His main fields of research are harmonics in in power system and reconfiguration of power distribution systems.

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