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IEEE PES General Meeting July 13 - 17, 2003, Toronto Transformer Modeling for Simulation of Low Frequency Transients J.A. MARTINEZMARTINEZ-VELASCO Univ. Politècnica Catalunya Barcelona, Spain B.A MORK Michigan Tech. Tech. Univ. Houghton, Houghton, USA Introduction z z z Large number of core designs Some of transformer parameters are both nonlinear and frequency dependent Physical attributes whose behavior may need to be correctly represented core and coil configurations selfself- and mutual inductances leakage fluxes skin effect and between coils proximity effect in coils hysteresis magnetic core saturation and eddy current losses in core capacitive z effects Aim of this presentation 1 Transformer Models z Matrix representation BCTRAN model z Saturable Transformer Component (STC) z Topology-based models Duality based models Geometric models Transformer Models Matrix representation (BCTRAN model) z Branch impedance matrix of a multi-phase multiwinding transformer Steady state equations Transient equations [V ] = [ Z ] [ I ] [ v] = [ R] [i ] + [ L] [ di / dt ] [R] and jω[L] are the real and the imaginary part of [Z], whose elements can be derived from excitation tests z The approach includes phase-to-phase couplings, models terminal characteristics, but does not consider differences in core or winding topology 2 Transformer Models Saturable Transformer Component (STC model) L1 R1 Lm N1 : N2 λ i R2 L2 Rm ideal N1 : NN RN . . . LN ideal StarStar-circuit representation of singlesingle-phase NNwinding transformers Transformer Models Models derived using duality Core design Equivalent circuit DualityDuality-derived model for a singlesingle-phase shellshellform transformer 3 MODEL EQUATIONS • [R] − [ωL] option Matrix Representation (BCTRAN model) [ v ] = [ R] [ i ] + [ L] [ di / dt ] • [A] − [R] option [di / dt ] = [ L]−1[v ] − [ L]−1[ R][i ] −1 Saturable Transformer Component (STC model) Topology-based models −1 [ L] [v ] = [ L] [ R][i ] + [di / dt ] CHARACTERISTICS • These models include all phase-to-phase coupling and terminal characteristics. • Only linear models can be represented. • Excitation may be attached externally at the terminals in the form of non-linear elements. • They are reasonable accurate for frequencies below 1 kHz. • It cannot be used for more than 3 windings. • The magnetising inductance is connected to the star point. • Numerical instability can be produced with 3-winding models. • Duality-based models : They are de- • Duality-based models include the effects of saturation in each individual leg of the rived using a circuit-based approach core, interphase magnetic coupling, and without a mathematical description leakage effects. • Geometric models • The mathematical formulation of geome[v ] = [ R][i ] + [dλ / dt ] tric models is based on the magnetic equations and their coupling to the electrical equations, which is made taking into account the core topology. Models differ from each other in the way in which the magnetic equations are derived. Nonlinear and Frequency-Dependent Parameters z Some transformer parameters are nonlinear and/or frequency-dependent due to saturation hysteresis eddy z z z currents Saturation and hysteresis introduce distortion in waveforms Hysteresis and eddy currents originate losses Saturation is predominant in power transformers, but eddy current and hysteresis effects can play an important role in some transients 4 Nonlinear and Frequency-Dependent Parameters z Modeling of iron cores Iron core behavior represented by a relationship between the magnetic flux density B and the magnetic field intensity H Each magnetic field value is related to an infinity of possible magnetizations depending on the history of the sample To characterize the material behavior fully, a model has to be able to plot • major and minor hysteresis loops (minor loops can be symmetric or asymmetric) Nonlinear and Frequency-Dependent Parameters B Initial curve Anhysteretic curve Major loop Symmetric minor loop H Asymmetric minor loop Magnetization curves and hysteresis loops 5 Modeling of iron cores + z i Equivalent circuit for reprerepresenting a nonlinear inductor V R I z Hysteresis loops have a negligible influence on the magnitude of the magnetizing current z Hysteresis losses can have some influence on some transients; the residual flux has a major influence on the magnitude of inrush currents The saturation characteristic can be modeled by a piecewise linear inductance with two slopes, except in some cases, e.g. ferroresonance Eddy current effects z Excitation losses are mostly iron-core losses hysteresis and eddy current losses they cannot be separated hysteresis losses are much smaller than eddy current losses z Eddy current models for transformer windings iron laminated cores 6 Eddy current effects Models for windings R1 R2 RN L1 L2 LN R0 Series Foster equivalent circuit Eddy current effects Models for iron laminated cores R1 L1 R2 L2 L1 R1 RN L2 R2 Standard Cauer equivalent LN LN RN Dual Cauer equivalent 7 Transformer Models Matrix representation (BCTRAN model) A B C a b c Winding Leakages BCTRAN Model ZL ZY ZL ZL ZY Transformer Core Equivalent BCTRAN model for a threethree-phase threethree-legged stacked core transformer Parameter Determination z Data usually available for any power transformer z power rating voltage rating excitation current excitation voltage excitation losses shortshort-circuit current shortshort-circuit voltage shortshort-circuit losses saturation curve capacitances between terminals and between windings Excitation and shortshort-circuit currents, voltages and losses must be provided from both direct and homopolar measurements 8 Parameter Determination z An accurate representation for threethree-phase core transformers should be based on the core topology, include eddy current effects and saturation/hystere saturation/hystere-sis representation z A very careful representation and calculation of leakage inductances is usually required z CoilCoil-capacitances have to be included for an accurate simulation of some transients z Since no standard procedures have been developed, a parameter estimation seems to be required regardregardless of the selected model z Temperature influence should not be neglected Modeling - Case 1 Three-legged stacked-core transformer Cross section of core and winding assembly 9 Modeling - Case 1 L0 Rh Rl Ll R0 L0 Rh Lm Rm Lm Rm Lm Ly Ry Ll R0 Rm Rl Ly Ry L0 Rh Rl Ll R0 Three-legged stacked-core transformer Duality-based equivalent circuit Modeling - Case 1 U Three-legged stacked-core transformer ATP implementation 10 Modeling - Case 1 10 Phase A Phase B Phase C Current (A) 5 0 -5 -10 100 120 140 160 Time (ms) 180 200 Three-legged stacked-core transformer Excitation currents Modeling - Case 1 Phase A Phase B Phase C 180 120 60 0 -60 -120 180 120 60 0 -60 -120 180 120 60 0 -60 -120 0 100 200 Time (ms) 300 400 500 Three-legged stacked-core transformer Inrush currents 11 Ferroresonance - Case 2 CORE U SRC SRCX X1 Rw Leak Rc 98_8Seg Magnetic Saturation - Case 2 0.8 0.8 Fluxlinked [Wb-T] Fluxlinked [Wb-T] 0.6 0.6 0.4 0.4 0.2 0.2 I [A] 0.0 I [A] 0.0 0.0 62.7 125.3 188.0 250.7 0.0 62.7 125.3 188.0 250.7 8-Segment Curve vs. 2-Segment Curve 12 Steady-State Excitation - Case 2 Excitation at Rated Voltage – 8-Segment Curve 200 2.0 [V] [A] 150 1.5 100 1.0 50 0.5 0 0.0 -50 -0.5 -100 -1.0 -150 -1.5 -200 0 10 20 (file FR_Mart.pl4; x-var t) v:XFMR 30 40 [ms] -2.0 50 c:SRC -XFMR Ferroresonance - Case 2A Ferroresonance: 15uF, 8-Segment Magnetization Curve 400 12.0 [V] [A] 300 6.8 200 100 1.6 0 -3.6 -100 -200 -8.8 -300 -400 0.00 11.11 (file FR_Mart.pl4; x-var t) v:X1 22.22 v:SRCX 33.33 44.44 55.56 [ms] -14.0 66.67 c:SRC -SRCX 13 Ferroresonance - Case 2B Ferroresonance: 15uF, 2-Segment Magnetization Curve 350.0 50 [V] [A] 262.5 35 175.0 20 87.5 0.0 5 -87.5 -10 -175.0 -25 -262.5 -350.0 0.00 11.11 (file FR_Mart.pl4; x-var t) v:X1 22.22 v:SRCX 33.33 44.44 55.56 [ms] -40 66.67 c:SRC -SRCX Effect of Mag Curve Representation 400 [V] 300 Ferroresonant Voltage: Red(2A): 8-Seg; Blue(2B): 2-Seg @ 1.0A; Green: 2-Seg @ 1.5A 200 100 0 -100 -200 -300 -400 0.00 11.11 22.22 33.33 44.44 55.56 [ms] 66.67 FR_Mart.pl4:v:X1 FR_MART1p5.pl4:v:X1 FR_MART1p0.pl4:v:X1 14 Effect of Mag Curve Representation Ferroresonant Current: Red(2A): 8-Seg; Blue(2B): 2-Seg @ 1.0A; Green: 2-Seg @ 1.5A 45 [A] 30 15 0 -15 -30 -45 0 10 20 30 40 [ms] 50 FR_Mart.pl4:c:SRC -SRCX FR_MART1p5.pl4:c:SRC -SRCX FR_MART1p0.pl4:c:SRC -SRCX Conclusions z z There is no agreement on the most adequate model Modeling difficulties great variety of core designs nonlinear and frequency dependent parameters inadequacy for acquisition and determination of some transformer parameters z Several modeling levels could be considered since not all parameters have the same influence on all transient phenomena 15