Implement threephase threewinding transformer with configurable winding connections and core geometry
Fundamental Blocks/Elements
The ThreePhase Transformer Inductance Matrix Type (Three Windings) block is a threephase transformer with a threelimb core and three windings per phase. Unlike the ThreePhase Transformer (Three Windings) block, which is modeled by three separate singlephase transformers, this block takes into account the couplings between windings of different phases. The transformer core and windings are shown in the following illustration.
The phase windings of the transformer are numbered as follows:
1, 4, 7 on phase A
2, 5, 8 on phase B
3, 6, 9 on phase C
This core geometry implies that phase winding 1 is coupled to all other phase windings (2 to 9), whereas in ThreePhase Transformer (Three Windings) block (a threephase transformer using three independent cores) winding 1 is coupled only with windings 4 and 7.
Note Do not confuse the phase winding numbers (1, 2, and 3) with the numbers used to identify the threephase windings of the transformer. Threephase winding 1 consists of phase windings 1, 2, 3, threephase winding 2 consists of phase windings 4, 5, 6, and threephase winding 3 consists of phase windings 7, 8, 9. 
The ThreePhase Transformer Inductance Matrix Type (ThreeWindings) block implements the following matrix relationship:
$$\left[\begin{array}{c}{V}_{1}\\ {V}_{2}\\ \vdots \\ {V}_{9}\end{array}\right]=\left[\begin{array}{cccc}{R}_{1}& 0& \dots & 0\\ 0& {R}_{2}& \dots & 0\\ \vdots & \vdots & \ddots & \vdots \\ 0& 0& \dots & {R}_{9}\end{array}\right]\cdot \left[\begin{array}{c}{I}_{1}\\ {I}_{2}\\ \vdots \\ {I}_{9}\end{array}\right]+\left[\begin{array}{cccc}{L}_{11}& {L}_{12}& \dots & {L}_{19}\\ {L}_{21}& {L}_{22}& \dots & {L}_{29}\\ \vdots & \vdots & \ddots & \vdots \\ {L}_{91}& {L}_{92}& \dots & {L}_{99}\end{array}\right]\cdot \frac{d}{dt}\left[\begin{array}{c}{I}_{1}\\ {I}_{2}\\ \vdots \\ {I}_{9}\end{array}\right].$$
R_{1} to R_{9} represent the winding resistances. The selfinductance terms L_{ii } and the mutual inductance terms L_{ij} are computed from the voltage ratios, the inductive component of the no load excitation currents and the shortcircuit reactances at nominal frequency. Two sets of values in positivesequence and in zerosequence allow calculation of the 9 diagonal terms and 36 offdiagonal terms of the symmetrical inductance matrix.
When the parameter Core type is
set to Three singlephase cores
, the model
uses three independent circuits with (3x3) R and L matrices. In this
condition, the positivesequence and zerosequence parameters are
identical and you need only specifying positivesequence values.
The self and mutual terms of the (9x9) L matrix are obtained from excitation currents (one threephase winding is excited and the other two threephase windings are left open) and from shortcircuit reactances.
The following shortcircuit reactances are specified in the mask parameters:
X_{112}, X_{012} — positive and zerosequence reactances measured with threephase winding 1 excited and threephase winding 2 shortcircuited
X_{113}, X_{013} — positive and zerosequence reactances measured with threephase winding 1 excited and threephase winding 3 shortcircuited
X_{123}, X_{023} — positive and zerosequence reactances measured with threephase winding 2 excited and threephase winding 3 shortcircuited
Assuming the following positivesequence parameters for threephase windings i and j (where i=1, 2, or 3 and j=1, 2, or 3):
Q_{1i}= Threephase
reactive power absorbed by winding i at no load when winding i is
excited by a positivesequence voltage Vnom_{i} with
winding j open
Q_{1j}= Threephase
reactive power absorbed by winding j at no load when winding j is
excited by a positivesequence voltage Vnom_{j} with
winding i open
X_{1ij }= positivesequence
shortcircuit reactance seen from winding i
when winding j is shortcircuited
Vnom_{i}, Vnom_{j }= nominal lineline voltages of windings i and j_{. }
The positivesequence self and mutual reactances are given by:
$$\begin{array}{c}{X}_{1}(i,i)=\frac{{V}_{{\text{nom}}_{i}}^{2}}{Q{1}_{i}}\\ {X}_{1}(j,j)=\frac{{V}_{{\text{nom}}_{j}}^{2}}{Q{1}_{j}}\\ {X}_{1}(i,j)={X}_{1}(j,i)=\sqrt{{X}_{1}(j,j)\cdot ({X}_{1}(i,i)X{1}_{ij})}.\end{array}$$
The zerosequence selfreactances X_{0}(i,i), X_{0}(j,j), and mutual reactance X_{0}(i,j) = X_{0}(j,i) are also computed using similar equations.
Extension from the following two (3x3) reactance matrices in positivesequence and in zerosequence
$$\begin{array}{l}\left[\begin{array}{ccc}{X}_{1}(1,1)& {X}_{1}(1,2)& {X}_{1}(1,3)\\ {X}_{1}(2,1)& {X}_{1}(2,2)& {X}_{1}(2,3)\\ {X}_{1}(3,1)& {X}_{1}(3,2)& {X}_{1}(3,3)\end{array}\right]\\ \left[\begin{array}{ccc}{X}_{0}(1,1)& {X}_{0}(1,2)& {X}_{0}(1,3)\\ {X}_{0}(2,1)& {X}_{0}(2,2)& {X}_{0}(2,3)\\ {X}_{0}(3,1)& {X}_{0}(3,2)& {X}_{0}(3,3)\end{array}\right]\end{array}$$
to a (9x9) matrix, is performed by replacing each of the nine [X_{1} X_{0}] pairs by a (3x3) submatrix of the form:
$$\left[\begin{array}{ccc}{X}_{s}& {X}_{m}& {X}_{m}\\ {X}_{m}& {X}_{s}& {X}_{m}\\ {X}_{m}& {X}_{m}& {X}_{s}\end{array}\right]$$
where the self and mutual terms are given by:
X_{s} = (X_{0} +
2X_{1})/3
X_{m} =
(X_{0} – X_{1})/3
In order to model the core losses (active power P1 and P0 in positive and zerosequences), additional shunt resistances are also connected to terminals of one of the threephase windings. If winding i is selected, the resistances are computed as:
$$R{1}_{i}=\frac{{V}_{{\text{nom}}_{i}}^{2}}{P{1}_{i}}\text{\hspace{1em}}R{0}_{i}=\frac{{V}_{{\text{nom}}_{i}}^{2}}{P{0}_{i}}.$$
The block takes into account the connection type you select,
and the icon of the block is automatically updated. An input port
labeled N
is added to the block if you select the
Y connection with accessible neutral for winding 1. If you ask for
an accessible neutral on threephase winding 2 or 3, an extra outport
port labeled n2
or n3
is generated.
Often, the zerosequence excitation current of a transformer with a 3limb core is not provided by the manufacturer. In such a case, a reasonable value can be guessed as explained below.
The following figure shows a threelimb core with a single threephase winding. Only phase B is excited and voltage is measured on phase A and phase C. The flux Φ produced by phase B shares equally between phase A and phase C so that Φ/2 is flowing in limb A and in limb C. Therefore, in this particular case, if leakage inductance of winding B would be zero, voltage induced on phases A an C would be k.V_{B=}V_{B}/2. In fact, because of the leakage inductance of the three windings, the average value of induced voltage ratio k when windings A, B, and C are successively excited must be slightly lower than 0.5
Assume:
Z_{s} = average value
of the three selfimpedances
Z_{m} =average
value of mutual impedance between phases
Z_{1} =
positivesequence impedance of threephase winding
Z_{0} =
zerosequence impedance of threephase winding
I_{1} =
positivesequence excitation current
I_{0} =
zerosequence excitation current
$$\begin{array}{c}{V}_{B}={Z}_{s}{I}_{B}\\ {V}_{A}={Z}_{m}{I}_{B}={V}_{B}/2\\ {V}_{C}={Z}_{m}{I}_{B}={V}_{B}/2\\ {Z}_{s}=\frac{2{Z}_{1}+{Z}_{0}}{3}\\ {Z}_{m}=\frac{{Z}_{0}{Z}_{1}}{3}\\ {V}_{A}={V}_{C}=\frac{{Z}_{m}}{{Z}_{s}}{V}_{B}=\frac{\frac{{Z}_{1}}{{Z}_{0}}1}{2\frac{{Z}_{1}}{{Z}_{0}}+1}{V}_{B}=\frac{\frac{{I}_{0}}{{I}_{1}}1}{2\frac{{I}_{0}}{{I}_{1}}+1}{V}_{B}=k{V}_{B},\end{array}$$
where k= ratio of induced voltage (with k slightly lower than 0.5)
Therefore, the I_{0}/I_{1 }ratio can be deduced from k:
$$\frac{{I}_{0}}{{I}_{1}}=\frac{1+k}{12k}.$$
Obviously k cannot be exactly 0.5 because this would lead to an infinite zerosequence current. Also, when the three windings are excited with a zerosequence voltage the flux path returns through the air and tank surrounding the iron core. The high reluctance of the zerosequence flux path results in a high zerosequence current.
Let us assume I_{1}= 0.5%. A reasonable value for I_{0} could be 100%. Therefore I_{0}/I_{1}=200. According to the equation for I_{0}/I_{1} given above, one can deduce the value of k. k=(200−1)/(2*200+1)= 199/401= 0.496.
Zerosequence losses are also higher than the positivesequence losses because of the additional eddy current losses in the tank.
Finally, the value of the zerosequence excitation current and the value of the zerosequence losses are not critical if the transformer has a winding connected in Delta because this winding acts as a short circuit for zerosequence.
The threephase windings can be configured in the following manner:
Y
Y with accessible neutral
Grounded Y
Delta (D1), delta lagging Y by 30 degrees
Delta (D11), delta leading Y by 30 degrees
Note The D1 and D11 notations refer to the following clock convention. It assumes that the reference Y voltage phasor is at noon (12) on a clock display. D1 and D11 refer respectively to 1 PM (delta voltages lagging Y voltages by 30 degrees) and 11 AM (delta voltages leading Y voltages by 30 degrees). 
Select the core geometry: Three singlephase cores
or Threelimb
or fivelimb core
(default). If you select the first
option only the positivesequence parameters are used to compute the
inductance matrix. If you select the second option, both the positive
and zerosequence parameters are used.
The winding connection for threephase winding 1. Choices are Y
, Yn
, Yg
(default), Delta
(D1)
, and Delta (D11)
.
The winding connection for threephase winding 2. Choices are Y
, Yn
, Yg
(default), Delta
(D1)
, and Delta (D11)
.
The winding connection for threephase winding 3. Choices are Y
, Yn
, Yg
(default), Delta
(D1)
, and Delta (D11)
.
Select to connect the threephase windings 1 and 2 in autotransformer (threephase windings 1 and 2 in series with additive voltage). Default is cleared.
If the first voltage specified in the Nominal lineline voltages parameter is higher than the second voltage, the low voltage tap is connected on the right side (a2,b2,c2 terminals). Otherwise, the low voltage tap is connected on the left side (A,B,C terminals).
In autotransformer mode you must specify the same winding connections
for the threephase windings 1 and 2. If you select Yn
connection
for both winding 1 and winding 2, the common neutral N connector is
displayed on the left side.
The following figure illustrates winding connections for one phase of an autotransformer when the threephase windings are connected respectively in Yg,Yg, and Delta.
If V1 > V2:
If V2 > V1:
Windings W1,W2,W3 correspond to the following phase winding numbers:
Phase A: W1=1, W2=4, W3=7
Phase B: W1=2, W2=5, W3=8
Phase C: W1=3, W2=6, W3=9
Select Winding voltages
to measure
the voltage across the winding terminals of the ThreePhase Transformer
block.
Select Winding currents
to measure
the current flowing through the windings of the ThreePhase Transformer
block.
Select All measurements
to measure
the winding voltages and currents.
Default is None
.
Place a Multimeter block in your model to display the selected measurements during the simulation. In the Available Measurements list box of the Multimeter block, the measurements are identified by a label followed by the block name.
If the Winding 1 connection parameter
is set to Y
, Yn
,
or Yg
, the labels are as follows.
Measurement  Label 

Winding 1 voltages 
or

Winding 1 currents 
or

If the Winding 1 connection parameter
is set to Delta (D11)
or Delta (D1)
, the
labels are as follows.
Measurement  Label 

Winding 1 voltages 

Winding 1 currents 

The same labels apply for threephase windings 2 and 3, except
that 1
is replaced by 2
or 3
in
the labels.
The nominal power rating, in voltamperes (VA), and nominal
frequency, in hertz (Hz), of the transformer. Default is [260e6,
60]
.
The phasetophase nominal voltages of windings 1, 2, 3 in volts
RMS. Default is [315e3, 120e3, 43e3]
.
The resistances in pu for windings 1, 2, and 3. Default is [0.005,
0.005, 0.005]
.
The noload excitation current in percent of the nominal current
when positivesequence nominal voltage is applied at any threephase
winding terminals (ABC, abc2, or abc3). Default is 0.06
.
The core losses plus winding losses at noload, in watts (W),
when positivesequence nominal voltage is applied at any threephase
winding terminals (ABC, abc2, or abc3). Default is 260e6*0.04/100
.
The positivesequence shortcircuit reactances X12, X23, and
X13 in pu. Xij is the reactance measured from winding i when winding
j is shortcircuited. Default is [0.087, 0.166, 0.067]
.
When the Connect windings 1 and 2 in autotransformer parameter is selected, the shortcircuit reactances are labeled XHL, XHT, and XLT. H, L, and T indicate the following terminals: H=high voltage winding (either winding 1 or winding 2), L=low voltage winding (either winding 1 or winding 2), and T=tertiary (winding 3).
The noload excitation current in percent of the nominal current
when zerosequence nominal voltage is applied at any threephase winding
terminals (ABC, abc2, or abc3) connected in Yg or Yn. Default is 100
.
Note: If your transformer contains deltaconnected windings (D1 or D11), the zerosequence current flowing into the Yg or Yn winding connected to the zerosequence voltage source does not represent the net excitation current because zerosequence currents are also flowing in the delta winding. Therefore, you must specify the noload zerosequence circulation current obtained with the delta windings open. 
If you want to measure this excitation current, you must temporarily change the delta windings connections from D1 or D11 to Y, Yg, or Yn, and connect the excited winding in Yg or Yn to provide a return path for the source zerosequence currents.
The core losses plus winding losses at noload, in watts (W),
when zerosequence nominal voltage is applied at any threephase winding
terminals (ABC, abc2, or abc3) connected in Yg or Yn. The Delta windings
must be temporarily open to measure these losses. Default is 260e6*1/100
.
Note: Note: If your transformer contains deltaconnected windings (D1 or D11), the zerosequence current flowing into the Yg or Yn winding connected to the zerosequence voltage source does not represent the net excitation current because zerosequence currents are also flowing in the delta winding. Therefore, you must specify the noload zerosequence circulation current obtained with the delta windings open. 
The zerosequence shortcircuit reactances X12, X23, and X13
in pu. Xij is the reactance measured from winding i when winding j
is shortcircuited. If the Zerosequence X12
measured with winding 3 Delta connected check box is not
selected, X12 represents the shortcircuit reactance when winding
3 is not connected in Delta. Default is [0.1, 0.2, 0.3 ]
.
When the Connect windings 1 and 2 in autotransformer check box is selected, the shortcircuit reactances are labeled XHL, XHT, and XLT. H, L, and T indicate the following terminals: H=high voltage winding (either winding 1 or winding 2), L=low voltage winding (either winding 1 or winding 2), and T=tertiary (winding 3).
Select this check box if the available zerosequence short circuit tests are obtained with tertiary winding (winding 3) connected in Delta. Default is cleared.
This transformer model does not include saturation. If you need modeling saturation, connect the primary winding of a saturable ThreePhase Transformer (Two Windings) in parallel with the primary winding of your model. Use the same connection (Yg, D1 or D11) and same winding resistance for the two windings connected in parallel. Specify the Y or Yg connection for the secondary winding and leave it open. Specify appropriate voltage, power ratings, and the saturation characteristics that you want. The saturation characteristic is obtained when the transformer is excited by a positivesequence voltage.
If you are modeling a transformer with three singlephase cores or a fivelimb core, this model produces acceptable saturation currents because flux stays trapped inside the iron core.
For a threelimb core, the saturation model also gives acceptable results even if zerosequence flux circulates outside of the core and returns through the air and the transformer tank surrounding the iron core. As the zerosequence flux circulates in the air, the magnetic circuit is mainly linear and its reluctance is high (high magnetizing currents). These high zerosequence currents (100% and more of nominal current) required to magnetize the air path are already taken into account in the linear model. Connecting a saturable transformer outside the threelimb linear model with a fluxcurrent characteristic obtained in positive sequence produces currents required for magnetization of the iron core. This model gives acceptable results whether the threelimb transformer has a delta or not.
Linear Transformer, Multimeter, ThreePhase Transformer (Two Windings), ThreePhase Transformer (Three Windings), ThreePhase Transformer Inductance Matrix Type (Two Windings)