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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:
To represent a ... | Fill these columns to add connection points... |
|---|---|
single-phase transformer | Only Winding From/Bus 1 and Winding To/Bus 1 columns. |
two-phase transformer | The Winding From/Bus1 and Winding From/Bus 2 as well as Winding To/Bus 1 and Winding To/Bus 2. |
three-phase transformer | The Winding From/Bus1, Winding From/Bus 2, Winding From/Bus 3 as well as Winding To/Bus 1, Winding To/Bus 2 and Winding To/Bus 3. |
Multiphase 2W Transformer
Symbol | Description | Unit | |
|---|---|---|---|
ID | Transformer name | unique name | |
Status | Connect/Disconnect status | Initial value 1 (0 for disconnected) | |
Number of Phases | Phase count in use | 1, 2, or 3 | |
Winding From | Bus1 | Primary side: Bus 1 | a unique name |
Bus2 | Primary side: Bus 2 | a unique name | |
Bus3 | Primary side: Bus 3 | a unique name | |
V (kV) | Primary winding nominal voltage (phase-to-phase) | kV | |
S_base (kVA) | Nominal power in primary side | kVA | |
Conn. type (*) | Primary winding connection type | ‘wye’ and ‘delta’ | |
Winding To | Bus1 | Secondary side: Bus 1 | a unique name |
Bus2 | Secondary side: Bus 2 | a unique name | |
Bus3 | Secondary side: Bus 3 | a unique name | |
V (kV) | Secondary winding nominal voltage (phase-to-phase) | kV | |
S_base (kVA) | Nominal power in secondary side | NOT APPLICABLE | |
Conn. type (*) | Secondary winding connection type | ‘wye’ and ‘delta’ | |
Tap 1 | Initial tap position: winding 1 | Integer between Lowest and Highest Tap | |
Tap 2 | Initial tap position: winding 2 | ||
Tap 3 | Initial tap position: winding 3 | ||
Lowest Tap | The lowest tap position | Integer value | |
Highest Tap | The highest tap position | Integer value | |
Min Range (%) | Max voltage buck | 0 < value < 100 | |
Max Range (%) | Max voltage boost | value > 0 | |
X (pu) | Total reactance | p.u. | |
Rw1 (p.u.) | Primary winding resistance | p.u. | |
Rw2 (p.u.) | Secondary winding resistance | p.u. | |
Multiphase 2W Transformer with Mutual Impedance
Symbol | Description | Unit | |
|---|---|---|---|
ID | Transformer name | a unique name | |
Status | Connect/Disconnect status | Initial value 1 (0 for disconnected) | |
Number of Phases | Phase count in use | 1, 2, or 3 | |
Winding From | Bus1 | Primary side: Bus 1 | a unique name |
Bus2 | Primary side: Bus 2 | a unique name | |
Bus3 | Primary side: Bus 3 | a unique name | |
V (kV) | Primary winding nominal voltage (phase-to-phase) | kV | |
S_base (kVA) | Nominal power in primary side | kVA | |
Conn. type (*) | Primary winding connection type | ‘wye’ and ‘delta’ | |
Winding To | Bus1 | Secondary side: Bus 1 | a unique name |
Bus2 | Secondary side: Bus 2 | a unique name | |
Bus3 | Secondary side: Bus 3 | a unique name | |
V (kV) | Secondary winding nominal voltage (phase-to-phase) | kV | |
S_base (kVA) | Nominal power in secondary side | NOT APPLICABLE | |
Conn. type (*) | Secondary winding connection type | ‘wye’ and ‘delta’ | |
Tap 1 | Initial tap position: winding 1 | Integer between Lowest and Highest Tap | |
Tap 2 | Initial tap position: winding 2 | ||
Tap 3 | Initial tap position: winding 3 | ||
Lowest Tap | The lowest tap position | Integer value | |
Highest Tap | The highest tap position | Integer value | |
Min Range (%) | Max voltage buck | 0 < value < 100 | |
Max Range (%) | Max voltage boost | value > 0 | |
Z0 leakage (pu) | Zero-sequence impedance | transformer p.u. | |
Z1 leakage (pu) | Positive-sequence impedance | transformer p.u. | |
X0/R0 | Zero-sequence reactance to resistance ratio | ratio | |
X1/R1 | Positive-sequence reactance to resistance ratio | ratio | |
No Load Loss (kW) | No-load power loss | NOT APPLICABLE | |
Note: (*) Four types of winding configurations are supported:
DD0
YgYg0
DYg1
YgD1
Available I/O Pins
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 ZB-1 BT NT AT matrix dimension: np*m x np*m, np = number of phases, m= number of windings
Y1 = B ZB-1 BT ; Yw = N Y1 NT ; Yprim = A Yw AT
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
ZB is the short circuit impedance matrix. Matrix dimension: np*(m-1) x np*(m-1)
Yw 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.
ZB in pu = 0.011+0.02i, ZB in 1V base = (ZB in pu)*12/25 kVA = 4.4e-7 + 8e-7i. ZB-1= 527.831e3 - 959.692e3i
Y1 = B ZB-1 BT ; B is a matrix [np*m=2 x np=1]
B =
1 |
-1 |
Y1 =
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 |
Yw = N Y1 NT =
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 Yw AT =
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.
ZB in pu = 0.011+0.02i, ZB in 1V base = (ZB in pu)*12/75 kVA = 1.4667e-7 + 2.6667e-7i. ZB-1= 158.349e3 - 287.907e3i (ohms-1)
Y1 = B ZB-1 BT ; 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 |
Y1 =
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)) |
Yw = N Y1 NT =
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 Yw AT =
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.xlsx file in ePHASORSIM example phasor08.
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