Saturable Transformer with Hysteresis
This example shows the simulation of hysteresis in a saturable transformer.
S. Casoria (Hydro-Quebec) and P. Brunelle (TransEnergie)
One phase of a three-phase transformer is connected on a 500 kV, 5000 MVA network. The transformer is rated 500 kV/230 kV, 450 MVA (150 MVA per phase). The flux-current saturation characteristic of the transformer is modeled with the hysteresis or with a simple piecewise nonlinear characteristic.
A Three-Phase Programmable Voltage Source is used to vary the internal voltage of the equivalent 500 kV network. During the first 3 cycles source voltage is programmed at 0.8 pu. Then, at t = 3 cycles (0.05 s) voltage is increased by 37.5% (up to 1.10 pu). In order to illustrate remnant flux and inrush current at transformer energization, the circuit breaker which is initially closed is first opened at t = 6 cycles (0.1 s), then it is reclosed at t = 9 cycles (0.15 s).
The Initial flux phi0 in the transformer is set at zero and source phase angle is adjusted at 90 degrees so that flux remains symmetrical around zero when simulation is started.
A Multimeter block and a Scope block are used to monitor waveforms of flux, magnetization current (not including the eddy currents which are modeled by the Rm resistance), excitation current (including eddy current modeled by Rm), voltages and current flowing into the primary winding. A X-Y Graph block is used to monitor the transformer operating point moving on the flux-current characteristic.
1) Simulation of saturation with hysteresis
Open the transformer menu and select the 'Simulate hysteresis' check box. Using the 'Hysteresis design' tool of the Powergui, load the hysteresis characteristic (load file 'hysteresis.mat'). Notice that flux and excitation current are displayed in pu. Using the 'Parameter units' popup menu, you can convert the pu units to SI units. The saturation characteristics consists of two regions: 1) The main hysteresis loop: In this region there are 2 different flux values for a single current value. 2) The saturated region defined by a simple line segment starting from the maximum point (Is,Fs) of the main loop. The hysteresis loop is defined by the following 3 points marked by red crosses on the main loop: [I=0; Remnant flux(Fr = 0.85 pu)], [Coercive Current(Ic = 0.004 pu); F = 0], [Saturation current(Is = 0.015 pu); Saturation flux(Fs = 1.2 pu)] plus the slope dF/dI at coercive current(F = 0). Using the 'Zoom around hysteresis' checkbox and 'Display' button you can view the whole characteristic or zoom on the hysteresis.
Start the simulation and observe the following phenomena on the two scope blocks:
From 0 to 0.05 sec: Voltage and flux peak values are at 0.8 pu. Notice typical square wave of magnetization current. As no remnant flux was specified, magnetization current and flux are symmetrical. Flux travels on inner loops (inside the main loop).
From 0.05 to 0.1 sec: Voltage is 1.1 pu. Flux now reaches approximately +1.1pu. A slight flux asymmetry is produced at voltage change. and the flux which varies between +1.14 pu and -1.05 pu now travels on the main loop. Current pulses appear on the magnetization current (yellow trace, Imag), indicating beginning of saturation.
From 0.1 to 0.15 sec: At first zero crossing after the breaker opening order, the current is interrupted, and a flux of 0.84 pu stays trapped in the transformer core.
From 0.15 to 0.2 sec: The breaker is reclosed at t = 9 cycles, at a zero crossing of source voltage, producing an additional flux offset of approximately 1 pu. The peak flux now reaches 1.85 pu, driving the transformer into the saturated region. Peak excitation current now reaches 0.81 pu.
2) Simulation of saturation with a piecewise nonlinear characteristic
Open the transformer menu and deselect 'Simulate hysteresis'. The saturation will now be simulated by a piecewise nonlinear single-valued characteristic defined by 7 points. Current/Flux pairs (in pu) are: [0 0; 0.0 0.85; 0.015 1.2; 0.03 1.35; 0.06 1.5; 0.09 1.56; 0.12 1.572]. The saturated region is the same, but the hysteresis loop is not simulated. Note that this single valued characteristics still allows specification of a remnant flux (keep phi0=0 flux as with hysteresis saturation model). Start simulation and compare waveforms.