zero sequence impedance of transformer

[kostek184-coder]

How is a transformer modeled for zero sequence? There's no way to specify R0 and X0 as for a line.

[petersalemink95]

Hi @kostek184-coder,

Short answer: zero sequence parameters of a transformer are dependent on the winding types at both sides. For example:

• If both sides are floating, e.g. DD connection, the zero sequence current has no where to go, and the zero sequence parameters are thus modelled as 0: y_ff = 0, y_ft = 0, y_tf = 0, y_tt = 0.
• For DYn you can calculate the zero sequence impedance as z0_series = z1_series + 3.0 * z_grounding_to. Then you can calculate the only needed zero sequence component here: y_tt = y0_series + y_shunt. For this case I made a drawing of the model:

• For other winding types you could check the code if you're interested: https://github.com/PowerGridModel/power-grid-model/blob/d7b9e7fd7498f467f6a4be86ec252a241c5d8552/power_grid_model_c/power_grid_model/include/power_grid_model/component/transformer.hpp#L216

Hope this helps! If you;d like our documentation to be more detailed, feel free to make an issue ;)

[nitbharambe]

To start with positive and negative sequence admittance is modelled from the transformer rating in a pi model with each leg equal.

Simply put, zero sequence parameters are modelled equally as positive sequence parameters in transformers in pi model with extra conditions as follows:

• The grounding admittance is added to zero sequence based on whether ground is available or not.
• The admittances are added at respective ends only if the winding permits so. ie. there is a Yn or Zn connection. For rest the admittance is zero.
  Detailed explanation of a any branch's modelling is below. (Skip to zero sequence for your question.)

Link to Code: https://github.com/PowerGridModel/power-grid-model/blob/d7b9e7fd7498f467f6a4be86ec252a241c5d8552/power_grid_model_c/power_grid_model/include/power_grid_model/component/transformer.hpp#L216-L272

Nomenclature

• $Y_{se} =$ series admittance

• $Y_{sh} =$ shunt admittance

• $k =$ branch transformation ratio (Turns ratio for transformer)

• $\phi =$ branch phase shift (Winding clock configuration for transformer)

• $Z_{g,from}=$ grounding impedance at from side

• $Z_{g,to}=$ grounding impedance at to side

• $Y_{1,se} =$ positive sequence series admittance

• $Y_{1,sh} =$ positive sequence shunt admittance

• $Y_{0,se} =$ zero sequence series admittance

• $Y_{0,sh} =$ zero sequence shunt admittance

Symmetric modelling for any branch in PGM

Connected branches

pi-model for all branches in PGM

$$Y_{ff} = \frac{(Y_{se} + 0.5 * Y_{sh})}{k^2}$$ $$Y_{ft} = -\frac{Y_{se}}{k} \cdot e^{-j\phi}$$ $$Y_{tf} = -\frac{Y_{se}}{k} \cdot e^{j\phi}$$ $$Y_{tt} = Y_{se} + 0.5 * Y_{sh}$$

From side disconnected

$$Y_{tt} = 0.5 * Y_{sh} + \frac{1}{(\frac{1}{Y_{se}} + \frac{1}{0.5 * Y_{sh}})}$$ $$Y_{ff} = Y_{ft} = Y_{tf} = 0$$

To side disconnected

$$Y_{ff} = \frac{0.5 * Y_{sh}}{k^2} + \frac{1}{(\frac{1}{Y_{se}} + \frac{1}{0.5 * Y_{sh}})}$$ $$Y_{tt} = Y_{ft} = Y_{tf} = 0$$

Asymmetric Modelling for Transformers

Positive sequence

Same as symmetrical modelling with following:
$Y_{se} = Y_{1,se} $
$Y_{sh} = Y_{1,sh} $
$\phi = \phi_1$

Negative sequence

Same as symmetrical modelling with following:
$Y_{se} = Y_{1,se} $
$Y_{sh} = Y_{1,sh} $
$\phi = - \phi_1$

Zero Sequence

Zero sequence admittances for a transformer depends on the winding configuration.

YNyn

Same as symmetrical modelling with following:
$Y_{se} = \frac{1}{\frac{1}{Y_{1,se}} + 3 \cdot ( Z_{g,to} + \frac{Z_{g,from}}{k^2})}$
$Y_{sh} = Y_{1,sh} $
$\phi = 180^\circ$ if winding connections are configured with clocks 2, 6, 10 otherwise $\phi = 0$

YNd

$Y_{ff} = \frac{1}{k^2} \cdot (\frac{1}{\frac{1}{Y_{1,se}} + 3 \cdot \frac{Z_{g,from}}{k^2}} + Y_{1,sh})$
(If from side is connected otherwise $Y_{ff}=0$)
$$Y_{tt} = Y_{ft} = Y_{tf} = 0$$

Dyn

$Y_{tt} = \frac{1}{\frac{1}{Y_{1,se}} + 3 \cdot Z_{g,to}} + Y_{1,sh}$
(If to side is connected otherwise $Y_{tt}=0$)
$$Y_{ff} = Y_{ft} = Y_{tf} = 0$$

ZN*

$Y_{ff} = \frac{1}{k^2} \cdot (\frac{1}{\frac{1}{Y_{1,se}} * 0.1 + 3 \cdot \frac{Z_{g,from}}{k^2}})$
(If from side is connected otherwise $Y_{ff}=0$)
$$Y_{tt} = Y_{ft} = Y_{tf} = 0$$
(Note: Zero sequence impedance of zigzag winding is approximately 10% of positive sequence impedance)

*zn

$Y_{tt} = \frac{1}{\frac{1}{Y_{1,se}} * 0.1 + 3 \cdot Z_{g,to}}$
(If to side is connected otherwise $Y_{tt}=0$)
$$Y_{ff} = Y_{ft} = Y_{tf} = 0$$

All other winding configurations

$$Y_{ff} =Y_{tt} = Y_{ft} = Y_{tf} = 0$$

Phase wise admittances

Eventually these sequences admittances are converted into three phase values (012 -> abc) as PGM's solver operates on the phase domain.