This example shows the use of the Three-Phase Transformer Inductance Matrix Type block to model a three-phase core-type saturable transformer. It also shows that using three single-phase transformers to simulate a Yg/Yg core-type transformer is not acceptable.
Gilbert Sybille (Hydro-Quebec, IREQ)
The model shows two identical circuits with a three-phase transformer rated 225 kVA, 2400 V/600V, 60Hz, connected to a 1 MVA, 2400 V power network. A 45 kW resistive load (20 % of transformer nominal power) is connected on the 600 V side. Each phase of the transformer consists of two windings both connected in wye with a grounded neutral.
The transformers in circuit 1 and circuit 2 use two different models:
1) Circuit 1 uses a physical model where the core geometry and the B-H characteristic of the iron used to build the core are the basic parameters used for modelling the magnetic properties of the transformer.
2) Circuit 2 uses the Three-Phase Transformer Inductance Matrix Type (Two Windings) block for modeling the linear part of the model. Saturation is modeled in the "Saturation" subsystem by three single-phase saturable transformers connected on the primary side of the linear transformer model.
In order to minimize the quantity of iron, the transformer core uses the core-type construction. Contrary to a three-phase transformer built with three independent units, the three phases of a core-type transformer are coupled. Because of these couplings, the transformer reactances in positive- and zero-sequence are quite different. When the three voltages applied on primary side are balanced, (positive-sequence voltage) the fluxes set up in each limb are also balanced and they stay trapped inside the magnetic core. However, when the voltage source or the load is unbalanced, a zero-sequence voltage is added to the positive- and negative sequences voltages. This zero-sequence voltage produces three flux components in phase in each limb, resulting in a zero-sequence flux component which has to circulate outside the iron core, through the air and transformer tank or casing. Due to the high reluctance (low permeability) of the flux return path through the air, the zero-sequence no-load excitation current is much higher than in positive sequence. For this particular model, zero-sequence excitation current exceeds 3 times the nominal current (344 %), as compared to only 2.2 % in positive sequence. Excitation current, no load active losses and short-circuit impedance R+j*X (where R=winding resistance, X=leakage reactance) have been measured for the physical model of circuit 1. Results are shown in the table below.
positive-sequence zero-sequence ------------------ ------------------- No-load excitation current (% of nominal current) 2.28 % 353 %
No load active power losses (winding losses + iron losses) (% of nominal power) 2.0 % 14.0 %
Short-circuit impedance R+jX (pu) 0.02 + j*0.10 pu 0.0168 + j*0.0914 pu
Note : You can verify these values by using the "Sequence Measurements" block provided in the model. Specify either "Positive " or "Zero" sequence in the block menu. To perform the no-load measurements you must disconnect the load and use an infinite voltage source (source impedance bypassed), either in positive-sequence or a zero-sequence. In order to apply a zero-sequence voltage connect all three terminals of B1 measurement block to the same A terminal of the source. Also, in order to initialize fluxes in zero-sequence, specify the following vector of Voltages for flux initialization in the block menu:
[VmagA VmagB VmagC (pu) VangleA VangleB VangleC (deg)] =[ 1 1 1 0 0 0]
A schematic of the core geometry specified in the physical model is shown below.
L2 L2 =================== || || || || || || A B C L1 || || || || || || ====================
L1 = average height of the three limbs bearing the windings (2 windings per limb) = 53 inches L2 = average length of the core yokes interconnecting the limbs = 21 inches A1=A2 = cross section of the limbs and yokes = 45.48 square inches Number of turns of high-voltage windings (2400/sqrt(3)= 1386 V) = 128 Number of turns of low-voltage windings (600/sqrt(3) = 346.4 V)= 32
Look under the mask of the transformer of circuit 1 to see how the electrical and magnetic circuit models are built. The electrical part is implemented by six controlled current sources (one source per winding). These current sources are driven by the magnetomotive force developed by each winding. The "Core" subsystem uses the electric/magnetic analogy to implement the magnetic circuit which consists of 7 steel elements and 7 air elements representing flux leakages for each of the six coils and flux zero-sequence return path.
The three figures below show respectively:
1) Iron B-H characteristic
ans = Figure (1) with properties: Number: 1 Name: '' Color: [0.9400 0.9400 0.9400] Position: [1 1 786 551] Units: 'pixels' Use GET to show all properties
2) Saturation characteristics for the three phases (flux in pu as function of peak magnetizing current in pu) when the transformer is excited in positive sequence (3 balanced voltages). These saturation characteristics obtained in positive sequence are used in the three single-phase saturable transformers to model saturation of the core-type transformer.
ans = Figure (2) with properties: Number: 2 Name: '' Color: [0.9400 0.9400 0.9400] Position: [105 495 560 420] Units: 'pixels' Use GET to show all properties
Using the positive-sequence saturation characteristics to model core saturation gives acceptable results even in presence of zero-sequence voltages. This is because the magnetic circuit used for conducting zero-sequence flux is mainly linear due to its large air gap. The large zero-sequence currents required to magnetize the high reluctance air path are taken into account in the linear model. Therefore, connecting a saturable transformer outside the three-limb linear model with a flux-current characteristic obtained in positive sequence will produce currents required for magnetization of the iron core.
3) Waveforms of excitation currents when a 1.5 pu voltage is applied at the 2400 V terminals.
ans = Figure (3) with properties: Number: 3 Name: '' Color: [0.9400 0.9400 0.9400] Position: [674 501 560 414] Units: 'pixels' Use GET to show all properties
Notice on Figure 2 that, because of the core asymmetry, the magnetizing current of phase B is lower than the current obtained for phase A and phase C. See for example on Figure 3 the excitation currents obtained with 1.5 pu voltage.
In order to emphasis the importance of a correct representation of transformer zero-sequence parameters, the transient performance of the Inductance Matrix Type transformer of circuit 2 is compared to the physical model of circuit 1 when a single-phase to ground fault is applied on phase A. A six-cycle fault is applied at 2400 V terminals at t=0.05 sec and cleared at t=0.15 sec.
Before starting simulation, open the Three-Phase Transformer Inductance Matrix Type block menu. Check that the "Core type" parameter is set to "Three-limb or five-limb core". Now, select the "Parameters" tab and check that the positive- and zero-sequence parameters are set according to the table given in the Circuit Description section.
1. Comparison of transient performance of transformer operating in linear region
Start the simulation. Observe on Scope1 and Scope2 respectively for circuit 1 and circuit 2 the following waveforms at 2400 V terminals: three-phase voltages, three-phase currents, three-phase fluxes.
When the fault is applied, the three currents flowing in the 2400 V windings increase from their steady state value (0.20 pu) to 1 pu and contain mainly a zero-sequence component (3 components in phase). During the fault, a DC flux is trapped in phase A, close to its value at fault application (~ -1 pu), whereas the sinusoidal fluxes in phases B and C do not exceed 1.3 pu, Therefore, transformer is operating mainly in the linear region (see Figure 2). Voltage and current waveforms of both models compare well, indicating that the Inductance Matrix Type transformer accurately represents the linear part of the core-type transformer.
2. Comparison of transient performance with saturated transformer
At fault clearing, a flux offset is produced on phase A, driving transformer into saturation. Flux in phase A reaches 1.5 pu, resulting in a strongly non linear current in phase A. Comparison of phase A currents is still acceptable although larger peak values are observed with the transformer of circuit 2. The reason is that the three saturable transformers modelling saturation in positive sequence are connected at winding terminals rather than being connected close to the core, behind the winding resistance and leakage reactances.
3. Simulating the core-type transformer with three single-phase transformers
You will now observe the impact of simulating the core-type transformer by using three single-phase transformers. Open the Three-Phase Transformer Inductance Matrix Type block menu and change the "Core type" parameter to "Three single-phase cores".
Restart simulation and compare waveforms of the two circuits. Notice that during fault the transformer currents of circuit 2 stay unchanged for phases B and C whereas current in phase A falls to zero. This test clearly shows that simulating a Yg/Yg core-type transformer with three single-phase units is unacceptable. The reason is that, in case of three single-phase units, positive-sequence parameters are assumed to be equal to zero-sequence parameters, and the low zero-sequence shunt reactance seen from the transformer input terminals does not exist anymore.
However, if the 600 V winding would be connected in Delta, simulation results would still be acceptable with three single-phase units because the delta connection would now allow circulation of zero-sequence current. To check the effect of using a Delta connection for the 600 V winding, change the Winding 2 connection of the Three-Phase Transformer Inductance Matrix Type to "Deta D1". In the transformer of circuit 1 you have to manually reconnect the secondary in Delta and, in its block menu, change the phase voltage of Winding 2 from 600/sqrt(3) to 600 V. Restart simulation and check that waveforms compare well for both circuits.
In summary, using three single-phase transformers with positive-sequence parameters to simulate a core-type transformer is acceptable only if one of the windings uses a Delta connection.