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Multiphase transformers can represent either a 1, 2, or 3-phase transformer. In the corresponding Excel tab, there are columns for declaration of up to 6 connection points (3 for Winding From end and 3 for Winding To end).

The sending and receiving connection points must be filled in correspondingly:

• For example, if Winding From/Bus 1 is filled, Winding To/Bus 1 must be filled as well.
• The unused connection points can be left empty. However, these empty fields must be located immediately after any completed connection point.
  For example, data in which Winding From/Bus 2 and Winding To/Bus 2 fields are filled out but Winding From/Bus 1 and Winding To/Bus 1 are empty is not valid and it causes an error.

Finally, the sending and receiving points can be connected to different phases. For example, Winding From/Bus 1 can be connected to phase A while Winding To/Bus 1 is connected to phase B.

In summary:

Multiphase 2W Transformer

Multiphase 2W Transformer with Mutual Impedance

Available I/O Pins

Model Equations 

This multiphase transformer is modeled based on the primitive nodal admittance matrix Yprim [1],[2]. 

Yprim = A N B Z_B^-1 B^T N^T A^T  matrix dimension:  np*m x np*m, np = number of phases, m= number of windings  

Y₁ = B Z_B^-1 B^T   ;   Y_w = N Y₁ N^T  ;   Yprim = A Y_w A^T

Y₁ is the ground-referenced nodal admittance matrix on a 1 volt base. Matrix dimension: np*m x np*m

N is the incidence matrix whose non-zero elements are the inverse of the numbers of turns in the windings. This matrix represents the effect of the ideal transformers shown to obtain actual windings voltages. Matrix dimension: 2*np*m x np*m

B is the incidence matrix whose elements are either 1,-1 or 0. It relates currents in the short circuit reference frame where the first winding is assumed shorted to the currents in the nodal admittance reference frame on a 1 volt base. Matrix dimension: np*m x np

A is the incidence matrix whose non-zero elements are generally either 1 and -1, that relates the winding currents to the actual terminal currents. Matrix dimension: nc x 2*np*m, nc = number of terminal currents 

Z_B is the short circuit impedance matrix. Matrix dimension: np*(m-1) x np*(m-1)

Y_w is the winding admittance matrix. Matrix dimension: 2*np*m x 2*np*m

Examples

1) A single-phase 2W transformer with the following data:  7.2/0.12 kV, 25 kVA, X = 20%, R=1.1%

In this case np = 1, m = 2.

Z_B in pu = 0.011+0.02i, ZB in 1V base = (Z_B in pu)*1²/25 kVA = 4.4e-7 + 8e-7i.  Z_B^-1= 527.831e3 - 959.692e3i 

Y₁ = B Z_B^-1 B^T ;  B is a matrix [np*m=2 x np=1] 

B =

Y₁  =  

N is a matrix  [2*np*m=4 x np*m=2] 

N = 

Y_w = N Y₁ N^T  =   

To generate matrix A is necessary to define the number of terminal currents in the model. In this case there are 2 terminal currents (see the red currents in the figure above) so nc=2 and A matrix is [nc=2 x 2*np*m=4]

A =  

Finally the matrix Yprim is calculated

Yprim = A Y_w A^T  =  

Below it can be seen how to add this single-phase transformer in the excel file. The total resistance was divided equally between the 2 windings (RW1 =  RW2 = 0.011 pu/2 = 0.0055 pu). Note that the voltages voltages must be added as phase to phase voltages even though the model is single-phase (according to the table above)

2) A three-phase 2W transformer with the following data:  12.47/0.208 kV (wye/delta), 75 kVA, X = 20%, R=1.1%

In this case np = 3, m = 2.

Z_B in pu = 0.011+0.02i, Z_B in 1V base = (Z_B in pu)*1²/75 kVA = 1.4667e-7 + 2.6667e-7i.  Z_B^-1= 158.349e3 - 287.907e3i (ohms^-1)

Y₁ = B Z_B^-1 B^T ;  B is a matrix [np*m=6 x np=3] 

B =

Y₁  =  

N is a matrix  [2*np*m=12 x np*m=6] 

N = 

Y_w = N Y₁ N^T  =   

To generate matrix A is necessary to define the number of terminal currents in the model. In this case there are 6 terminal currents (see figure above) so nc=6 and A matrix is [nc=6 x 2*np*m=12]

A =  

Yprim = A Y_w A^T  =  

The following image shows how to add this component in the excel file. 

3) Multiple transformers in the same model

See the Transformer page in phasor08_IEEE13.xls file in demo PHASOR-08.

References

[1] Roger C. Dugan, "A Perspective on Transformer Modeling for Distribution Systems Analysis". 2003 IEEE Power Engineering Society General Meeting. DOI: 10.1109/PES.2003.1267146

[2] Roger C. Dugan and Surya Santoso, "An Example of 3-phase Transformer Modeling for Distribution Systems Analysis". 2003 IEEE PES Transmission and Distribution Conference and Exposition. DOI: 10.1109/TDC.2003.1335084