Three-Winding Autotransformer

Model Two-Winding Transformer Inductance Matrix Type It is a three-phase transformer with two windings per phase. Unlike other models of three-phase transformers, which are represented by three separate single-phase transformers, this model takes into account the connections between windings of different phases. The transformer core and windings are shown in the following figure.

The phase windings of the transformer are numbered as follows:

This type of core implies that phase winding 1 is connected to all other phase windings (from 2 to 6), whereas in the block Three-Phase Transformer (Two Windings) (three-phase transformer with three independent cores) winding 1 is connected only to winding 4.

Model Two-Winding Transformer Inductance Matrix Type implements the following matrix relation:

,

where

Own and mutual The terms of the inductance matrix are calculated based on the ratio of voltages, the inductive component of the no-load currents, and the short-circuit reactants at the rated frequency. Two sets of values in the forward and zero sequence allow us to calculate 6 diagonal terms and 15 non-diagonal terms of the symmetric inductance matrix.

The terms of the inductance matrix (6×6) are calculated from the values of the no-load currents (one three-phase winding is excited, and the other three-phase winding is open) and the short-circuit reactants of the direct and zero sequence and measured when the three-phase primary winding is excited and the three-phase secondary winding is short-circuited.

Let’s take the following parameters of the direct sequence:

The intrinsic and mutual reactants of a direct sequence are determined by the expressions:

,

,

.

Intrinsic reactive resistances , and mutual reactive resistances zero sequences are defined using similar expressions.

Expansion of the following two (2x2) reactance matrices of the forward and zero sequences

up to the matrix (6x6) it is carried out by replacing each of the four pairs ] a submatrix (3x3) of the form:

,

where proper and reciprocal terms are defined as

,

.

To simulate core losses (active power and direct and zero sequences), additional shunt active resistances are also connected to the terminals of one of the three-phase windings.

If the primary winding is selected, the resistances are calculated as:

,

.

The block takes into account the selected connection type, and the block icon is automatically updated. The input port n1 is added to the block if the connection Y to the available neutral of the primary winding is selected (Wye with floating neutral). If you select the available neutral of the secondary winding, an additional port n2 will be created.

If the parameter value is Core type installed Three single-phase cores then the model uses three independent circuits with and matrices (3x3). In this case, the parameters of the forward and zero sequences are identical and only the values of the forward sequence are specified.

,

,

.

For the model Two-Winding Autotransformer The same connections must be specified for the three-phase primary and secondary windings. If you choose a connection Wye with floating neutral for primary and secondary windings, common neutral port n1 displayed on the left. The figure shows the connection of the windings of one phase of the autotransformer, when both three-phase windings are connected to a star (with a grounded neutral).

If

If

The following numbers of phase windings correspond to windings W1, W2:

Model Three-Winding Transformer Inductance Matrix Type It is a three-phase transformer with three windings per phase. Unlike other models of three-phase transformers, which are represented by three separate single-phase transformers, this model takes into account the connections between windings of different phases. The transformer core and windings are shown in the following figure.

The phase windings of the transformer are numbered as follows:

This type of core implies that phase winding 1 is connected to all other phase windings (from 2 to 9).

Model Three-Winding Transformer Inductance Matrix Type implements the following matrix relation:

,

where

Own and mutual The terms of the inductance matrix are calculated based on the ratio of voltages, the inductive component of the no-load currents, and the short-circuit reactants at the rated frequency. Two sets of values of the forward and zero sequences allow us to calculate 9 diagonal terms and 36 non-diagonal terms of the symmetric inductance matrix.

Terms of the inductance matrix (9x9) They are calculated from the values of the no-load currents (one three-phase winding is energized, and the other two three-phase windings are open) and short-circuit reactants.

Short-circuit reactants:

and — reactants of the forward and zero sequences when the three-phase primary winding is excited and the three-phase secondary winding is short-circuited.

and — reactive resistances of the forward and zero sequences when the three-phase primary winding is excited and the three-phase tertiary winding is short-circuited.

and — reactants of the forward and zero sequences when the three-phase secondary winding is excited and the three-phase tertiary winding is short-circuited.

Let’s take the following parameters of the direct sequence for three-phase windings and ( and have values 1, 2, or 3):

The intrinsic and mutual reactants of a direct sequence are determined by the expressions:

,

,

.

Intrinsic reactive resistances , and mutual reactive resistances zero sequences are defined using similar expressions.

Expansion of the following two (3x3) reactance matrices of the forward and zero sequences:

,

up to the matrix (9x9) it is carried out by replacing each of the four pairs ] a submatrix (3x3) of the form:

,

where proper and reciprocal terms are defined as

,

.

To simulate core losses (active power and direct and zero sequences), additional shunt active resistances are also connected to the terminals of one of the three-phase windings.

If the winding is selected , resistances are calculated as:

,

.

The block takes into account the selected connection type, and the block icon is automatically updated. Port n1 is added to the block if connection Y is selected to the available neutral of the primary winding (Wye with floating neutral). If you select the available neutral of the secondary or tertiary winding, an additional port n2 or n3 will be created.

If the parameter value is Core type installed Three single-phase cores then the model uses three independent circuits with and matrices (3x3). In this case, the parameters of the forward and zero sequences are identical and only the values of the forward sequence are specified.

,

,

.

For the model Three-Winding Autotransformer The same connections must be specified for the three-phase primary and secondary windings. If you choose a connection Wye with floating neutral for primary and secondary windings, common neutral port n1 displayed on the left. The figure shows the connection of the windings of one phase of the autotransformer, when the primary and secondary three-phase windings are connected in a star (with a grounded neutral), and the tertiary three-phase winding in a triangle.

If

If

The following numbers of phase windings correspond to windings W1, W2, W3:

Often, the value of the no-load current of the zero sequence of a transformer with a three-rod core is not provided by the manufacturer. In this case, its value can be approximately calculated as described below.

The following figure shows a three-core core with one three-phase winding. Only phase B is excited, the voltage is measured in phase A and phase C. Flow , created by phase B, is equally distributed between phase A and phase C, so that the flow /2 flows into rod A and into rod C. Therefore, in this particular case, if the scattering inductance of winding B is zero, the voltage induced at phases A and C will be equal to . In fact, due to the scattering inductance of the three windings, the average value of the induced voltage coefficient is when the windings A, B and C are excited sequentially, it should be slightly below 0.5.

Let’s accept:

— the average value of the three intrinsic resistances.

— the average value of the mutual resistance between the phases.

— the resistance of the direct sequence of the three-phase winding.

— resistance of the zero sequence of the three-phase winding.

— the idling current of the direct sequence.

— zero-sequence idle current.

,

,

,

,

,

,

where — the coefficient of induced voltage (slightly below 0.5).

Hence, the ratio can be obtained relatively :

.

It is obvious that it cannot be exactly equal to 0.5, as this would lead to an infinite current of the zero sequence. In addition, when the three windings are energized by a zero-sequence voltage, the magnetic flux must be closed through the air and the tank surrounding the iron core. The high magnetic resistance of the zero sequence flow path leads to a high zero sequence current.

Suppose that . Reasonable value it can be 100%. Thus . According to the above equation for you can define the value :

.

The zero sequence losses should also be higher than the direct sequence losses due to additional eddy current losses in the tank.

Finally, the magnitude of the zero-sequence excitation current and the magnitude of the zero-sequence losses are not significant if the transformer has a winding connected according to a triangle circuit, since this winding acts as a short circuit for the zero sequence.

The three-phase transformer windings can be connected as follows:

For more information about the operating modes of a three-phase system connected according to the Triangle scheme, see Recommendations for delta connection of transformer windings.

These models do not provide for saturation. To simulate saturation, connect the primary winding of the saturating transformer Two-Winding Transformer (Three-Phase) parallel to the primary winding. Use the same connection (Yg, D1 or D11) and the same resistance for two windings connected in parallel. Specify the connection Y or Yg for the secondary winding and leave it open. Specify the required voltage, rated power, and saturation characteristic. The saturation characteristic is obtained when the transformer is energized by a direct sequence voltage. If a transformer with three single-phase cores or a five-core core is modeled, then such a model creates acceptable saturation currents, since the flow remains inside the steel core. For a three-core core, this saturation model still gives acceptable results, even if the zero-sequence flow circulates outside the core and returns through the air and the transformer tank surrounding the steel core. Since the zero-sequence flow circulates in the air, the magnetic circuit is basically linear and its magnetic resistance is high (high magnetization currents). These high zero-sequence currents (100% or more of the rated current) required for the magnetization of the airway have already been taken into account in the linear model. Connecting a saturating transformer outside a three-rod linear model with a magnetic current characteristic obtained for a direct sequence creates the currents necessary for magnetization of the steel core. This model gives acceptable results regardless of whether a three-rod transformer has a winding connected in a triangle or not.

Use the parameter group Initial Targets to set the priority and initial target values for the block parameter variables before modeling. For more information, see Configuring physical blocks using target values.