...
No | Pin Description | Pin Type | Value/Unit | Instruction |
1 | Get sending end current magnitude of wire j | O | A (RMS) | transformerID/ImagFromj where j is 1, 2 or 3 |
2 | Get receiving end current magnitude of wire j | O | A (RMS) | transformerID/ImagToj where j is 1, 2 or 3 |
3 | Get sending end current angle of wire j | O | Degree | transformerID/IangFromj where j is 1, 2 or 3 |
4 | Get receiving end current angle of wire j | O | Degree | transformerID/IangToj where j is 1, 2 or 3 |
5 | Set/Get tap position | I/O | Integer between [min_tap, max_tap] | transformerID/tap_j where j is 1, 2 or 3 |
...
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
Y_{1} = B Z_{B}^{-1} B^{T} ; Y_{w} = N Y_{1} N^{T} ; Yprim = A Y_{w} A^{T}
Y_{1} 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)*1^{2}/25 kVA = 4.4e-7 + 8e-7i. Z_{B}^{-1}= 527.831e3 - 959.692e3i
Y_{1} = B Z_{B}^{-1} B^{T} ; B is a matrix [np*m=2 x np=1]
B =
1 |
-1 |
Y_{1} =
527.831e3 - 959.692e3i | -527.831e3+959.692e3i |
-527.831e3+959.692e3i | 527.831e3 - 959.692e3i |
N is a matrix [2*np*m=4 x np*m=2]
N =
1 /7200 | 0 |
-1 /7200 | 0 |
0 | 1/120 |
0 | -1/120 |
Y_{w} = N Y_{1} N^{T} =
0.0102-0.0185i | -0.0102+0.0185i | -0.6109+1.1108i | 0.6109-1.1108i |
-0.0102+0.0185i | 0.0102-0.0185i | 0.6109-1.1108i | -0.6109+1.1108i |
-0.6109+1.1108i | 0.6109-1.1108i | 36.6549-66.6453i | -36.6549+66.6453i |
0.6109-1.1108i | -0.6109+1.1108i | -36.6549+66.6453i | 36.6549-66.6453i |
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 =
1 | 0 | 0 | 0 |
0 | 0 | 1 | 0 |
Finally the matrix Yprim is calculated
Yprim = A Y_{w} A^{T} =
0.0101-0.0185i | -0.6105+1.1100i |
-0.6105+1.1100i | 36.6007-66.5467i |
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, ZB in 1V base = (Z_{B} in pu)*1^{2}/75 kVA = 1.4667e-7 + 2.6667e-7i. Z_{B}^{-1}= 158.349e3 - 287.907e3i
Y_{1} = B Z_{B}^{-1} B^{T} ; B is a matrix [np*m=6 x np=3]
B =
1 | 0 | 0 |
-1 | 0 | 0 |
0 | 1 | 0 |
0 | -1 | 0 |
0 | 0 | 1 |
0 | 0 | -1 |
Y_{1} =
158.349e3 - 287.907e3i | -158.349e3 + 287.907e3i | 0 | 0 | 0 | 0 |
-158.349e3 + 287.907e3i | 158.349e3 - 287.907e3i | 0 | 0 | 0 | 0 |
0 | 0 | 158.349e3 - 287.907e3i | -158.349e3 + 287.907e3i | 0 | 0 |
0 | 0 | -158.349e3 + 287.907e3i | 158.349e3 - 287.907e3i | 0 | 0 |
0 | 0 | 0 | 0 | 158.349e3 - 287.907e3i | -158.349e3 + 287.907e3i |
0 | 0 | 0 | 0 | -158.349e3 + 287.907e3i | 158.349e3 - 287.907e3i |
N is a matrix [2*np*m=12 x np*m=6]
N =
1 /12470 | 0 | 0 | 0 | 0 | 0 |
-1 /12470 | 0 | 0 | 0 | 0 | 0 |
0 | 1/(208*sqrt(3)) | 0 | 0 | 0 | 0 |
0 | -1/(208*sqrt(3)) | 0 | 0 | 0 | 0 |
0 | 0 | 1 /12470 | 0 | 0 | 0 |
0 | 0 | -1 /12470 | 0 | 0 | 0 |
0 | 0 | 0 | 1/(208*sqrt(3)) | 0 | 0 |
0 | 0 | 0 | -1/(208*sqrt(3)) | 0 | 0 |
0 | 0 | 0 | 0 | 1 /12470 | 0 |
0 | 0 | 0 | 0 | -1 /12470 | 0 |
0 | 0 | 0 | 0 | 0 | 1/(208*sqrt(3)) |
0 | 0 | 0 | 0 | 0 | -1/(208*sqrt(3)) |
Y_{w} = N Y_{1} N^{T} =
0.0102-0.0185i | -0.0102+0.0185i | -0.3525+0.6409i | 0.3525-0.6409i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
-0.0102+0.0185i | 0.0102-0.0185i | 0.3525-0.6409i | -0.3525+0.6409i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
-0.3525+0.6409i | 0.3525-0.6409i | 12.2002-22.1823i | -12.2002+22.1823i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0.3525-0.6409i | -0.3525+0.6409i | -12.2002+22.1823i | 12.2002-22.1823i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0.0102-0.0185i | -0.0102+0.0185i | -0.3525+0.6409i | 0.3525-0.6409i | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | -0.0102+0.0185i | 0.0102-0.0185i | 0.3525-0.6409i | -0.3525+0.6409i | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | -0.3525+0.6409i | 0.3525-0.6409i | 12.2002-22.1823i | -12.2002+22.1823i | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0.3525-0.6409i | -0.3525+0.6409i | -12.2002+22.1823i | 12.2002-22.1823i | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.0102-0.0185i | -0.0102+0.0185i | -0.3525+0.6409i | 0.3525-0.6409i |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -0.0102+0.0185i | 0.0102-0.0185i | 0.3525-0.6409i | -0.3525+0.6409i |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -0.3525+0.6409i | 0.3525-0.6409i | 12.2002-22.1823i | -12.2002+22.1823i |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.3525-0.6409i | -0.3525+0.6409i | -12.2002+22.1823i | 12.2002-22.1823i |
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 =
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 |
Yprim = A Y_{w} A^{T} =
0.0101-0.0185i | 0 | 0 | -0.3524+0.6408i | 0.3524-0.6408i | 0 |
0 | 0.0101-0.0185i | 0 | 0 | -0.3524+0.6408i | 0.3524-0.6408i |
0 | 0 | 0.0101-0.0185i | 0.3524-0.6408i | 0 | -0.3524+0.6408i |
-0.3524+0.6408i | 0 | 0.3524-0.6408i | 24.4004-44.3645i | -12.2002+22.1822i | -12.2002+22.1822i |
0.3524-0.6408i | -0.3524+0.6408i | 0 | -12.2002+22.1822i | 24.4004-44.3645i | -12.2002+22.1822i |
0 | 0.3524-0.6408i | -0.3524+0.6408i | -12.2002+22.1822i | -12.2002+22.1822i | 24.4004-44.3645i |
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