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A STATE SPACE SINGLE-PHASE TRANSFORM ER MODEL INCORPO RATING NONLINEARPHENOMENA OF MAGNETIC SATURATION AND HYSTERESIS FOR TRANSIENT AND PERIODICSTEADY-STATE ANALYSISSigridt Garcia Aurelio Medina Carlos Perez

Facultad de Ingenieria ElCctrica, U.M.S.N.H.Edificio A, 2 . Piso, Ciudad UniversitariaMorelia, Michoacin, MEXI Osigridt@jupiter.ccu.umich.mx amedina6iJzeus.ccu.umich.m cprojas@zeus.ccu.umich.mx

Abstract: An efficient state space single-phase transformer model ispresented in this contribution, which incorporates the nonlinearphenomena of magnetic saturation and hysteresis. It is analyzedunder transient and periodic steady-state operation conditions. Thenonlinearity of saturation and hysteresis is efficiently representedwith a novel and simple formulation. Fast periodic steady-statesolutions are achieved with the application of a Newton techniquefor the acceleration of time domain computations. Comparisons aremade between conventional methods used for the numericalsolution of the transformer represented by a set of OrdinaryDifferential Equations ODES) and the Newton method in terms offull cycles and CPU times needed to obtain the periodic steady-statesolutionof the network.Keywords: State Space, Single-phase, Magnetic Saturation,Hysteresis, Periodic Steady-State,Newton Method..1 INTRODU TIONThe simplest transformer model is used in single-phasestudies at fundamental fiequency, such as short-circuit andload flow analysis. Here the transformers is represented bythe leakage reactance in series with an deal autotransformers.In one of the fEst models propose d [l], the transformer isrepresented with a x equivalent circu it incorporating the effectof the turns ratio in each branch. However, to better describethe transformer behaviour, the electric and magnetic circuitsshould be incorporated in the equivalent model. The magneticcircuit has a higher degree of complexity for its appropriaterepresentation since it is dependent on the magnetic andphysical characteristics.Digital models have been developed, more complex andrigorous, that combine the electric and magnetic circuits.

However, the main interest has been around the analysis ofelectromagnetic transients [2], since for this type of analysis aswell as for the steady-state analysis of power systems underunbalanced conditions detailed transformer representations areneeded [3]. Transform er models have been described in theform of shunt admittances [4] or admittance matrices 15-71,which can be obtained form excitation and short-circuit testdata. The period ic behaviour of power transformers requires ofa more com plete and rigorous model which could appropriatelypredict the harmonic distortion in the power system.The transformer behaviour during transients and interruptionsare more or less difficult to simulate, depending on the natureof the phenomenon involved. There are transformer models,where the simulation results are identical with those obtainedwith the equivalents circuits of positive and zero sequence[8]. Other mathematical model [9] for the digital simulationof multiple windings transformers is based on thedecomposition and step by step linearization and is mainlyapplied for the simulation of transients in circuits withnonlinear elements.A basic electro-magnetic model for the magnetizing branchof a transformer was developed using a methodology based onits representation by means of a Norton harmonic equivalent[4]. The generalization in the Harmonic Domain, representingthree-phase netw orks containing multiple transformer units waspresented in other contribution [6]. Here each electromagneticharmonic equivalent is described by a network circuit, in theform of a Norton equivalen t which interfac es with the rest ofthe network according to the transformer electricalconnections.A procedure for the determination of the saturation andhysteresis in the transformer has been proposed in acontribution [lo] where test field data are not required. In [1I ]the saturation and hysteresis effects are incorporated. Thenumerical analysis depends closely on the material propertiesunder study. In this contribution the harmonic analysis for asingle-phase transformers is carried-out using a flux linkagehysteresis effects is made with a simple an efficient arctangentformulation [121 is used.formulation; for the incorpor tion of magnetic saturation and

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2. STATE SPACE FORMULATIONn e quiva lent c ircu it of h e transformer is illusbated in theFig.1. state variables.

It can be observed from the(7), (8) and (10) that thecurrents are calculated using the initial conditions for the

0 4 5

0 4

0 35s2E= 0.3B 0 2s.-.

0 2= 0 1s

RI Lll R z LIZ 3. NONLINEAR PHENOMENAThe incorporation of nonlinear effects such as magneticsaturation and hysteresis is achieved in the transformer modelwith the appropriate modification of the differential equations11)-(13) and the algebraic equations (7)-(10).3.1 Magnetic SaturationIn Fig. 2 the saturation curve is illustrated. This curve isapproximated with the equation,

.

Fig. 1 Transformer equivalent circuit

.

..

Applying the Kirchhoff s Law for voltages to the circuit of

1)

(2)

A, i,)= 0.32atan(i, , ,)the Fig. 1, the following equation s are obtained, where i, in terms of A is expressed as,v = R,i,+ dt

d 4v = R,i,t

/ z:

Experimental...... ....(-pproximated

.where v2 is also obtained as,v = -RLi2

0 1 .

The flux inkages A nd d2 an be written as follows,(4)

4.= 4 2 5 )A = L i (6)

The general flux linkage law is written as,

where the currents can be calculated as,i =-4 - 4

L,l

Fig.Fig. 2Saturation Curve

(7) The Fig. 3 shows the comparison between theexperimental saturation characteristics and the approximatedcurve using (14) and (1 5) . T h e experimental saturation curvehas been reproduced with a very good precision.8 )

Applying the Kirchhoffs Law for currents to the node 1results in,i i2 = i i

i = il + 2-9 )

10)where

From the inner loop of Fig. 1the following equation isobtained,

dn = i,R,dt .where a state varialde is obtained; th e other state variables areobtained from (1) and (2) as,dn, v -R,i,dt

= - i2 RL R, 13)dt

0.5 . , . . , . ,

:I,. . . , , I00 2 a 4 5 e 7 B 10Magnet iz ing currentFig. 3 Comparison of saturation characteristicsWhen magnetic saturation is to be represented then (10)is modified, the branch loss is not included and 1 1 is notconsidered.

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3.2 HysteresisThe Fig. 4 illustrates a simple and efficient model for therepresentation of the hysteresis phenomenon. Here R,represents the hysteresis losses and L, the nonlinearinductance where saturation characteristics should beincluded. The magnetizing curren t i can be obtained as,

The Table 1shows themaximum error obtained d uring theconvergence of the transformer state variables includingsaturation effects. The B F method is considerably slower thatthe ND, hereas the BF method requires of a number of fullcycles (NFC)much larger than 208 to ob tain the steady-statesolution, the NDmethod requires of only 11 periods of time.Similarly in Table 2, summarizes the resulting errorsduring the convergence of the transformer state variableswhen hysteresis effects are taken in to account.

Fig. 4 Inductor model for hysteresis representation -Ai

mm -deriving (15) gives,

Substituting (17) in 16) results in,

Then (18) remains only a function of and can becalculated from the initial condition.4. CASESTUDYThe Newton methodology described in [13-151 to acceleratethe convergence of the state variables in the time domain tothe Limit Cycle [161 is now applied to obtain the steady-statesolution of the transformer o f Fig. 1, where nonlinear effectsof magnetic saturation and hysteresis are incorporated. Thetest system data are given in A ppend ix A.The required number of state variables depends on theoperation of the transformer, i.e., if the transform er isoperated with or without saturation. For the representation ofsaturation two state variables are required whereas for themodeling of hysteresis effects three sate v ariables are needed.The state variables are solved with the Brute Force (BF) andNumerical Differentiation DN) methods, respectively. Bothapproaches used the Fourth Order Runge-Kutta integrationmethod. The Limit Cycle and thus the periodic steady-statesolution is obtained once the maximum error in the statevariables is within IOe-1Op.u.Table 1 Errors during convergence of the transformer with

NFC I BF ND5 I 6.8811e-4 6.8811e-48 1.5563e-5 I 3.1429e-81 1 I l.SS84e-S 15.1847~-14

I 208 I 1.6959e-5 I

Table 2 Errors during convergence of the transformer withhysteresisBF I

When hysteresis is included in the transformer model, thenumber of periods required to reach the distorted final steady-state solution is noticeably less because the system has ahigher damping comp ared with the case of saturation.a Harmonic AnalysisThe Fig. 5 illustrates the transformer steady-state operationincorporating magnetic saturation and hysteresis effects fordifferent loads: R,RL nd RC. Only primary side transformercurrent is plotted.

0 Ir . g r r r k W -. I (b)Fig. 5 Primary current waveform i . (a) With Saturation,(b)With HysteresisWith the satura tion characteristic represented the harmonicconten t varies accord ing to the lo ad type. In all cases there isa large third harmonic content, see Fig 6(a) where for the Rload it is approximately 25% of the f i en ta 1. The Fig.6(b) illustrates the case when the hysteresis effect isincorporated, the harmonic content increases notoriously forthe R load. For the others types of loads the harmonics do notchange noticeably.

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2520151050

3 5 7 9 1 1

primary circuit going to zero, as observed from the thirdsegment of Fig. 7.At t=0.0588 secs. the fault is clearedproducing the transient shown in the fourth section. Thesolution process is again accelerated until reaching the newperiodic steady-state of the system; identical to the obtainedpreviously and illustrated by the seco nd segm ent of Fig. 7.With this technique of acceleration of time domaincomputations it is possible to ob tain a detailed solution of thefew transient cycles of interest and once the fault is cleared,to reach the periodic steady-state solution considerably fasterthan digital procedures based on conventional methods forthe numerical solution of ODES. For example, it is observedfrom Fig. 7that despite of the large current peaks takingplace in the first and second transients the periodic steady-state is obtained in a reduced number of cycles.

C4RRLRC

EiRW RL 5. CONCLUSIONSORC A model for the single phase transformer incorporating themagnetic saturation and hysteresis effects has been p resented.A simple an efic ien t arctangent formulation is used for therepresentation of the nonlinear saturation and hysteresis

characteristics. Ihe transform er behav iour was analyzed withFig. 6 Harmonic content of the Fig. 5 , respectively(a) With saturation, (b) With hysteresis

6) Transient AnalysisA transient study was carried-out applying an open sourceperturbation. This fault consists in opening the vo ltage sourceat t~0.0539 ecs.; this source supplies the transformerprimary circuit. The application of this fault condition willmake evident the advantage and efficiency of obtaining anaccelerated time d omain periodic steady-state solution oncethe fault is cleared.

Ik

Fig. 7Transient and steady-sate solutionThe Fig. 7illustrates @eevolution in time of the current atthe primary side of the transformer. In the f m t section thereis a marked transient peak as a result of the transformerenergization, this solution is accelerated until the periodicsteady-state is achieved. The second section illustrates thefact that being in steady-state a fault was applied at 0 005secs. by opening the source; this results in the current at the

three different loads: R, RL and RC, respectively.Comparisons were made b etween a conventional method andthe Numerical Differentiation technique for obtaining theperiodic steady-state solution of the system. The potential ofthis technique for obtaining a fast periodic steady-statesolution of the syste m after any pe rturbatio n has beendem onstrate d. Th is new alternative for transient and steady-state analysis of circuits and electric networks has theadvantages of follow ing closely the few transient cycles andaccelerating the time dom ain com putations immediately afterclearing the fault to obtain a fast steady-state solution.

6 ACKNOWLEDGMENTSThe authors gratefully acknowledge financial support fromUniversidad Michoacana de San NicolL de Hidalgo to carry-outthis investigation.7. R E F E R E N C E S[I J Stagg. G.W. nd El-Ab iad, A.H.,Comp uter Me/lrods in Power Sys/emsAnnlysis, McGraw-H l l , London.[2] L c h , F., Sem lyen, A., Complete Transformer Model forElectromagnetic Transients, EEE Trnnsnc/ionson Powe r Delivery. Vol. 9,No. I January 199 4, pp. 23 1-23 9.[3] Medina, A., hrrillaga. J., Simulation of Multilimb Power Transfomersin the Harmonic Domain, IEE Proceedings, Vol. 139, No. 3, May 1992,[4] Semlyen. A., hcha, E. y Arrillaga, J. , Harmon ic Norton Equivalent forthe Magnetisin Branch of a Transformer, IEE Proceedings, Part. C, Vol.134, No. 2. March, pp. 162-169.[ 5 ] Brandwajn, V., Dommel, H.W., Dommel, I. ., Matrix representation ofThree-phase N-Winding Transformers for Steady-State and TransientStudies, EEE Trnnsncfions on Pow er Appnrntrrs and Sysfems Vol. PAS-101 No. 6, June 19 82. pp. 1369 -1378.[6] Acha, E., Arrillaga. J. , Medina, A.. Semlyen A.. General Frame ofReference for Analysis of Harmonic Distortion in Systems with MultipleTransformer Nonlinearities, IEE Proceedings, Vol. 136. Part. C. No. 5,September 1989. pp. 27 1-278.

pp.269-276

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171Medina, A., Amlla ga J., Analysis of Transformer-Generator Interactionin the Harmonic Domain , IEE Proc-Gener. Transm. Distrib., Vol. 141,No. 1, January 1994, pp. 38 46 .[SI Dommel, H.W., Transformer Mo dels in the Simu lation ofElectrom agnetic Transients , 5th Power Systems Comp utation Conference,Cambridge, England, Paper 3.1/4, pp.1-16.[9] Ivanoff, D., Simulation of Multiwinding Nonlinear Tmsformers ,Escola Politkcnica,Silo Paulo-Brasil. 1977.[IO]Chin E. Lin, Jong-Bi Wei, Ching-Lien Huang, Chi-Jen Huang, A NewModel for Transformer Saturation Ch aracteristic by Including HysteresisLoops , IEEE Transactions on Magnetics, Vol. 25, No. 3, May 1 989 , pp.[I 1 Hassani M.M., Lachiver G. and Jasmin G., 'Wumerical simulation ofthe magnetic core of a transformer in transient operations , CanadianConference in Eleceical and Computer Engineering, September 1989, pp.[12] Pkrez-Rojas C. Represe ntation of the saturation and hysteresis,approa ching through arctan gent function , (in Spanish). RVPAA-99, IEEE,Acapulco, Gro., Mtxico, July, 1999. pp. 118-121.[U ] Garcia, S. Soluci6n de redes elictr icas en el dominio del tiempo conaceleracih de la cowergencia al ciclo Iimite , BEng Thesis (in Spanish),Facultad de Ingenim'a Elbctrica, UMSNH, March 1998, Morelia, Mich.,Mbxico.[14] A. Medina, S.Garcia, Time Domain Accelerated Solu tion of Systemswith Non-Linear and Time-Varying , Proceedings of the IntemacionalConference 19 98 NSF Design and Manufacturing Grantees Conference,Monterrey, N.L., Mexico, January 5-8,1998, pp. 797-798.[I51 Medina, A., Garcia, S., Determinaci6n de la Solucih Peribdica enEstado Estacionario de Redes Elbcb-icas Mediante T h i c a Newton en elDomini0 del Tiempo , Proceeding of the Intemacional Conference IEEERVP98 (in Spanish), Acapu lco, G o . , Mexico, July, 19 98,pp. 223-227.[I61Parker, T.S., Chua, L.O., ' ractical Numerical Algorithms for ChaoticSystems , Springer-Verlag,New York, 989.

2706-2712.

289-291.

[17] Krause , P. C. Analysisof Electric Machinery McGraw Hill, 1986.BIOGRAPHIESSigridt Garcia. Obtained her BEng Degree from the UnivesidadMichoacana de San Nicolis d e Hidalgo UMSNH, Morelia, Mich., Mtxico1998. At present she is an Academ ic Associa te at the Facultad d e IngenieriaElectrica, UMSNH and is wo rking towards her Masters degree.Aurelio Medina Obtained his Ph.D. from the University of Canterbury,Christch urch, New Zealan d in 1992 . He has worked as a Post-DoctoralFellow at the Universities of Canterbury, New Zealand 1 year) andToronto, Canada (2 years). At present he holds a permanent academicposition at the Facultad de Ingenieria Eltctrica. UMSNH, M orelia, Mexico,where he is the Head of the Division for Graduate Studies. His researchinterests are in the dynam ic and steady-sta te analysis of power systems.Carlos Perez-Rojas. Obtain ed the B.S.E.E. from the UMSNH UniversidadMichoacana de San Nic olL de H idalgo, Mexico) in 1990. He obtained in1993 the M. S. n Power Systems from the UANL (Universidad Au th om ade Nuevo h 6 n , Mexico). At present he is a Staff member at the Faculty ofElectrical Engineering of the UMSNH. He is working towards his Ph. D.Degree at the DIE-FIME-UAN L, Mexico. He is an IEEE and SIAM studentmember.APPEND IX A. TEST SYSTEMDATAR,=10,&=1.25, R,,=lOOO %=200 P ILf,=Lf2=0.01 05 y L,,,=0.27952 [HIV,=156 en(a ) [vl

The parameters of the transformer are the following, [17]:

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