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Basic transformer model

L11 and L22 are the self-inductance of winding 1 and 2 respectively, and L12 L11 and L22 are the self-inductance of winding 1 and 2 respectively, and L12 and and L21 are the mutual inductance between the windings.L21 are the mutual inductance between the windings.

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Basic transformer modelBasic transformer model

Example: Consider a transformer with a 10% leakage reactance equally divided between the two windings and a magnetising current of 0.01 p.u.

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Basic transformer modelBasic transformer model

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Numerical implementationNumerical implementation

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Numerical implementationNumerical implementation

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Numerical implementationNumerical implementation

Transformer equivalent after discretisationTransformer equivalent after discretisation

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Modelling of non-linearitiesModelling of non-linearities

Typical studies requiring the modelling of saturation are: Inrush Typical studies requiring the modelling of saturation are: Inrush current on energising a transformer, steady-state overvoltage current on energising a transformer, steady-state overvoltage studies, core-saturation instabilities and ferro-resonance.studies, core-saturation instabilities and ferro-resonance.

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Modelling of non-linearitiesModelling of non-linearities

to impose a decay time on the inrush currents, as would occur on to impose a decay time on the inrush currents, as would occur on energisation or fault recovery:energisation or fault recovery:

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Three-Phase Transformer Inductance Matrix Type (Two Windings)

This core geometry implies that phase winding 1 is coupled to all This core geometry implies that phase winding 1 is coupled to all ((other phase other phase windings (2 to 6)windings (2 to 6)

The phase windings of the transformer are numbered as The phase windings of the transformer are numbered as followsfollows::

11 and 4 on phase Aand 4 on phase A 22 and 5 on phase Band 5 on phase B 33 and 6 on phase Cand 6 on phase C

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Three-Phase Transformer Inductance Matrix Type (Two Windings)

Transformer ModelTransformer ModelThe Three-Phase Transformer Inductance Matrix Type: The Three-Phase Transformer Inductance Matrix Type:

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Three-Phase Transformer Inductance Matrix Type (Two Windings)

R1 to R6 represent the winding resistances.R1 to R6 represent the winding resistances. The self inductance terms The self inductance terms LiiLii and the mutual inductance terms Lij and the mutual inductance terms Lij are computed from the voltage ratios, the inductive component of the are computed from the voltage ratios, the inductive component of the no load excitation currents and the short-circuit reactances at nominal no load excitation currents and the short-circuit reactances at nominal frequency. frequency. Two sets of values in positive-sequence and in zero-sequence allow Two sets of values in positive-sequence and in zero-sequence allow calculation of the 6 diagonal terms and 15 off-diagonal terms of the calculation of the 6 diagonal terms and 15 off-diagonal terms of the symmetrical inductance matrix.symmetrical inductance matrix.

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Three-Phase Transformer Inductance Matrix Type (Two Windings)

The self and mutual terms of the (6x6) L matrix are obtained from The self and mutual terms of the (6x6) L matrix are obtained from excitation currents (one three-phase winding is excited and the other excitation currents (one three-phase winding is excited and the other three-phase winding is left open) and from positive- and zero-three-phase winding is left open) and from positive- and zero-sequence short-circuit reactances X1sequence short-circuit reactances X11212 and X0 and X01212 measured with three- measured with three-phase winding 1 excited and three-phase winding 2 short-circuited.phase winding 1 excited and three-phase winding 2 short-circuited.

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Three-Phase Transformer Inductance Matrix Type (Two Windings)

Q1Q111= Three-phase reactive power absorbed by winding 1 at no load when = Three-phase reactive power absorbed by winding 1 at no load when winding 1 is excited by a positive-sequence voltage Vnom1 with winding 2 winding 1 is excited by a positive-sequence voltage Vnom1 with winding 2 openopenQ1Q122= Three-phase reactive power absorbed by winding 2 at no load when = Three-phase reactive power absorbed by winding 2 at no load when winding 2 is excited by a positive-sequence voltage Vnom2 with winding 1 winding 2 is excited by a positive-sequence voltage Vnom2 with winding 1 openopenX1X11212= Positive-sequence short-circuit reactance seen from winding 1 = Positive-sequence short-circuit reactance seen from winding 1 when winding 2 is short-circuitedwhen winding 2 is short-circuitedVnom1, Vnom2= Nominal line-line voltages of windings 1 and 2Vnom1, Vnom2= Nominal line-line voltages of windings 1 and 2

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Three-Phase Transformer Inductance Matrix Type (Two Windings)

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Three-Phase Transformer Inductance Matrix Type (Two Windings)

Extension from the following two (2x2) reactance matrices in positive-sequence Extension from the following two (2x2) reactance matrices in positive-sequence and in zero-sequenceand in zero-sequence

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Three-Phase Transformer Inductance Matrix Type (Two Windings)

In order to model the core losses (active power P1 and P0 in positive- and zero-In order to model the core losses (active power P1 and P0 in positive- and zero-sequences), additional shunt resistances are also connected to terminals of one of sequences), additional shunt resistances are also connected to terminals of one of the three-phase windings. If winding 1 is selected, the resistances are computed the three-phase windings. If winding 1 is selected, the resistances are computed as: as:

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Three-Phase Transformer Inductance Matrix Type (Two Windings)

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Three-Phase Transformer Inductance Matrix Type (Two Windings)

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Three-Phase Transformer Inductance Matrix Type (Two Windings)

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UMEC (Unified Magnetic Equivalent Circuit) model

Single-phase UMEC model

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UMEC (Unified Magnetic Equivalent Circuit) model

Single-phase UMEC model

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UMEC (Unified Magnetic Equivalent Circuit) model

Three-limb three-phase UMEC

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UMEC (Unified Magnetic Equivalent Circuit) model

Three-limb three-phase UMEC