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{"@context":"https://schema.org","@type":"QAPage","mainEntity":{"@type":"Question","name":"zero sequence impedance of transformer","text":"
How is a transformer modeled for zero sequence? There's no way to specify R0 and X0 as for a line.
","upvoteCount":1,"answerCount":2,"acceptedAnswer":{"@type":"Answer","text":"
To start with positive and negative sequence admittance is modelled from the transformer rating in a pi model with each leg equal.
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Simply put, zero sequence parameters are modelled equally as positive sequence parameters in transformers in pi model with extra conditions as follows:
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The grounding admittance is added to zero sequence based on whether ground is available or not.
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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.
\nDetailed explanation of a any branch's modelling is below. (Skip to zero sequence for your question.)
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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
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Nomenclature
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$Y_{se} =$ series admittance
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$Y_{sh} =$ shunt admittance
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$k =$ branch transformation ratio  (Turns ratio for transformer)
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$\\phi =$ branch phase shift (Winding clock configuration for transformer)
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$Z_{g,from}=$ grounding impedance at from side
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$Z_{g,to}=$ grounding impedance at to side
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$Y_{1,se} =$ positive sequence series admittance
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$Y_{1,sh} =$ positive sequence shunt admittance
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$Y_{0,se} =$ zero sequence series admittance
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$Y_{0,sh} =$ zero sequence shunt admittance
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Symmetric modelling for any branch in PGM
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Connected branches
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pi-model for all branches in PGM
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$$Y_{ff} = \\frac{(Y_{se} + 0.5 * Y_{sh})}{k^2}$$\n$$Y_{ft} = -\\frac{Y_{se}}{k} \\cdot e^{-j\\phi}$$\n$$Y_{tf} = -\\frac{Y_{se}}{k} \\cdot e^{j\\phi}$$\n$$Y_{tt} = Y_{se} + 0.5 * Y_{sh}$$
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From side disconnected
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$$Y_{tt} = 0.5 * Y_{sh} + \\frac{1}{(\\frac{1}{Y_{se}} + \\frac{1}{0.5 * Y_{sh}})}$$\n$$Y_{ff} = Y_{ft} = Y_{tf} = 0$$
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To side disconnected
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$$Y_{ff} = \\frac{0.5 * Y_{sh}}{k^2} + \\frac{1}{(\\frac{1}{Y_{se}} + \\frac{1}{0.5 * Y_{sh}})}$$\n$$Y_{tt} = Y_{ft} = Y_{tf} = 0$$
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Asymmetric Modelling for Transformers
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Positive sequence
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Same as symmetrical modelling with following:
\n$Y_{se} = Y_{1,se} $
\n$Y_{sh} = Y_{1,sh} $
\n$\\phi = \\phi_1$
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Negative sequence
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Same as symmetrical modelling with following:
\n$Y_{se} = Y_{1,se} $
\n$Y_{sh} = Y_{1,sh} $
\n$\\phi = - \\phi_1$
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Zero Sequence
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Zero sequence admittances for a transformer depends on the winding configuration.
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YNyn
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Same as symmetrical modelling with following:
\n$Y_{se} = \\frac{1}{\\frac{1}{Y_{1,se}} + 3 \\cdot ( Z_{g,to} + \\frac{Z_{g,from}}{k^2})}$
\n$Y_{sh} = Y_{1,sh} $
\n$\\phi = 180^\\circ$ if winding connections are configured with clocks 2, 6, 10 otherwise $\\phi = 0$
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YNd
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$Y_{ff} = \\frac{1}{k^2} \\cdot (\\frac{1}{\\frac{1}{Y_{1,se}} + 3 \\cdot \\frac{Z_{g,from}}{k^2}} + Y_{1,sh})$
\n(If from side is connected otherwise $Y_{ff}=0$)
\n$$Y_{tt} = Y_{ft} = Y_{tf} = 0$$
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Dyn
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$Y_{tt} = \\frac{1}{\\frac{1}{Y_{1,se}} + 3 \\cdot Z_{g,to}} + Y_{1,sh}$
\n(If to side is connected otherwise $Y_{tt}=0$)
\n$$Y_{ff} = Y_{ft} = Y_{tf} = 0$$
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ZN*
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$Y_{ff} = \\frac{1}{k^2} \\cdot (\\frac{1}{\\frac{1}{Y_{1,se}} * 0.1 + 3 \\cdot \\frac{Z_{g,from}}{k^2}})$
\n(If from side is connected otherwise $Y_{ff}=0$)
\n$$Y_{tt} = Y_{ft} = Y_{tf} = 0$$
\n(Note: Zero sequence impedance of zigzag winding is approximately 10% of positive sequence impedance)
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*zn
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$Y_{tt} = \\frac{1}{\\frac{1}{Y_{1,se}} * 0.1 + 3 \\cdot Z_{g,to}}$
\n(If to side is connected otherwise $Y_{tt}=0$)
\n$$Y_{ff} = Y_{ft} = Y_{tf} = 0$$
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All other winding configurations
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$$Y_{ff} =Y_{tt} = Y_{ft} = Y_{tf} = 0$$
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Phase wise admittances
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Eventually these sequences admittances are converted into three phase values (012 -> abc) as PGM's solver operates on the phase domain.
","upvoteCount":4,"url":"https://github.com/orgs/PowerGridModel/discussions/72#discussioncomment-14860375"}}}