Transformer Modeling for Simulation of Low Frequency Transients

Transcription

1 IEEE PES General Meeting July 13-17, 23, Toronto Transformer Modeling for Simulation of Low Frequency Transients J.A. MARTINEZ-VELASCO Univ. Politècnica Catalunya Barcelona, Spain B.A MORK Michigan Tech.. Univ. Houghton,, USA Introduction 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 self- and mutual inductances between coils leakage fluxes skin effect and proximity effect in coils magnetic core saturation and hysteresis eddy current losses in core capacitive effects Aim of this presentation

2 Transformer Models Matrix representation BCTRAN model Saturable Transformer Component (STC) Topology-based models Duality based models Geometric models Transformer Models Matrix representation (BCTRAN model) 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 The approach includes phase-to-phase couplings, models terminal characteristics, but does not consider differences in core or winding topology

3 Transformer Models Saturable Transformer Component (STC model) L 1 R 1 : N 2 N 1 R 2 L 2 λ L m i R m ideal.. : N N R N. L N N 1 ideal Star-circuit representation of single-phase N-N winding transformers Transformer Models Models derived using duality Core design Equivalent circuit Duality-derived model for a single-phase shell- form transformer

4 MODEL EQUATIONS CHARACTERISTICS Matrix Representation (BCTRAN model) Saturable Transformer Component (STC model) Topology-based models [R] [ωl] option [ v] = [ R ] [ i] + [ L ] [ di / dt] [A] [R] option 1 1 [ di / dt] = [ L] [ v] [ L] [ R][ i] 1 1 [ L] [ v] = [ L] [ R][ i] + [ di/ dt] Duality-based models : They are derived using a circuit-based approach without a mathematical description Geometric models [ v] = [ R][ i] + [ dλ / dt] 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 include the effects of saturation in each individual leg of the core, interphase magnetic coupling, and leakage effects. The mathematical formulation of geometric 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 Some transformer parameters are nonlinear and/or frequency-dependent due to saturation hysteresis eddy currents Saturation and hysteresis introduce distor- tion in waveforms Hysteresis and eddy currents originate losses Saturation is predominant in power transfor- mers,, but eddy current and hysteresis effects can play an important role in some transients

5 Nonlinear and Frequency-Dependent Parameters Modeling of iron cores Iron core behavior represented by a rela- tionship between the magnetic flux density B and the magnetic field intensity H Each magnetic field value is related to an infinity of possible magnetizations depen- ding 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

6 Modeling of iron cores Equivalent circuit for repre- senting a nonlinear inductor + V i R I - Hysteresis loops have a negligible influence on the magnitude of the magnetizing current 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 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 Eddy current models for transformer windings iron laminated cores

7 Eddy current effects Models for windings R 1 R 2 R N R L 1 L 2 L N Series Foster equivalent circuit Eddy current effects Models for iron laminated cores R 1 R 2 R N L 1 L 2 L N Standard Cauer equivalent L 1 L 2 L N R 1 R 2 R N Dual Cauer equivalent

8 Transformer Models Matrix representation (BCTRAN model) A B C a b c Winding Leakages BCTRAN Model Z L Z L Z L Z Y Z Y Transformer Core Equivalent BCTRAN model for a three-phase three-legged stacked core transformer Parameter Determination Data usually available for any power transformer power rating voltage rating excitation current excitation voltage excitation losses short-circuit current short-circuit voltage short-circuit losses saturation curve capacitances between terminals and between windings Excitation and short-circuit currents, voltages and losses must be provided from both direct and homopolar measurements

9 Parameter Determination An accurate representation for three-phase core transformers should be based on the core topology, include eddy current effects and saturation/hystere hystere- sis representation A very careful representation and calculation of leakage inductances is usually required Coil-capacitances have to be included for an accurate simulation of some transients Since no standard procedures have been developed, a parameter estimation seems to be required regard- less of the selected model Temperature influence should not be neglected Modeling - Case 1 Three-legged stacked-core core transformer Cross section of core and winding assembly

10 Modeling - Case 1 Rh L Rl Ll Rm Lm R Ly Rh L Ry Rl Ll Rm Lm R Ly Ry Rh L Rl Ll Rm Lm R Three-legged stacked-core core transformer Duality-based equivalent circuit Modeling - Case 1 U Three-legged stacked-core core transformer ATP implementation

11 Modeling - Case 1 1 Phase A Phase B Phase C 5 Current (A) Time (ms) Three-legged stacked-core core transformer Excitation currents Modeling - Case Phase A Phase B Phase C Time (ms) Three-legged stacked-core core transformer Inrush currents

12 Ferroresonance - Case 2 SRC U SRCX X1 Rw Leak CORE Rc 98_8Seg Magnetic Saturation - Case 2.8 Fluxlinked [Wb-T].8 Fluxlinked [Wb-T] I [A] I [A] Segment Curve vs. 2-Segment 2 Curve

13 2 [V] 15 Steady-State State Excitation - Case 2 Excitation at Rated Voltage 8-Segment Curve 2. [A] [ms] 5 (file FR_Mart.pl4; x-var t) v:xfmr c:src -XFMR Ferroresonance - Case 2A 4 [V] 3 2 Ferroresonance: 15uF, 8-Segment Magnetization Curve 12. [A] [ms] (file FR_Mart.pl4; x-var t) v:x1 v:srcx c:src -SRCX

14 35. [V] Ferroresonance - Case 2B Ferroresonance: 15uF, 2-Segment Magnetization Curve 5 [A] [ms] (file FR_Mart.pl4; x-var t) v:x1 v:srcx c:src -SRCX Effect of Mag Curve Representation 4 [V] 3 Ferroresonant Voltage: Red(2A): 8-Seg; Blue(2B): 1.A; Green: 1.5A [ms] FR_Mart.pl4: v:x1 FR_MART1p5.pl4: v:x1 FR_MART1p.pl4: v:x1

15 Effect of Mag Curve Representation 45 [A] 3 Ferroresonant Current: Red(2A): 8-Seg; Blue(2B): 1.A; Green: 1.5A [ms] 5 FR_Mart.pl4: c:src -SRCX FR_MART1p5.pl4: c:src -SRCX FR_MART1p.pl4: c:src -SRCX Conclusions 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 Several modeling levels could be conside- red since not all parameters have the same influence on all transient phenomena