# Three-Phase Autotransformer with Tertiary Winding

**Libraries:**

Simscape /
Electrical /
Specialized Power Systems /
Power Grid Elements

## Description

The Three-Phase Autotransformer with Tertiary Winding block represents a
three-phase autotransformer. The high-voltage side is identified by the **A**, **B**, and **C** ports, the low-voltage side by the **a**, **b**, and **c** ports, and the tertiary-winding by the **a3**, **b3**, and **c3** ports.

### Equivalent Circuit

The equivalent circuit of one phase is shown in the diagram. Each phase consists of three coupled windings: a series winding between the high-voltage terminals and low-voltage terminals, a common winding between the low-voltage terminals and the neutral terminal, and a tertiary winding connected in Delta D1.

**Standard Model for Winding Resistances and Inductances**

The resistances and leakage inductances of the three windings are determined from short-circuit test parameters using the following standard equations:

*R1* =
(*R12**(1+*k*)/(1-*k*)
+ *R13* -
*R23*_pu)/(1-*k*)/2
*R2* = (*R12* - *R13*
+ *R23*_pu)/(1-*k*)/2 *
R3* = (-*R12* + *R13* +
*R23**(1-2**k*))/(1-*k*)/2

*L1* =
(*L12**(1+*k*)/(1-*k*)
+ *L13* -
*L23*)/(1*-k*)/2
*L2* = (*L12* - *L13*
+ *L23*)/(1-*k*)/2
*L3* = (-*L12* +
*L13* +
*L23**(1-2**k*))/(1-*k*)/2

where:

*R12*,*R13*, and*R23*are the RHL, RHT, and RLT short-circuit test resistances.*L12*,*L13*, and*L23*are the LHL, LHT, and LLT short-circuit test inductances.*k*is the voltage ratio between the high-voltage side and low-voltage side nominal voltages.

All parameters are in pu based on the nominal power and nominal voltage of the windings.

The standard equations listed above may produce negative winding resistances and
inductances. Although negative values are permitted in phasor models (at 50 Hz
or 60 Hz) using algebraic equations, these negative parameters may result in
numerical instability in EMT models that use differential equations. In this
case, a warning message suggests you modify the *R23* or the *L23* parameter and
proposes a range of values that produce positive resistance and inductance
values.

**Alternate Model for Winding Resistances**

To avoid the limitations of the standard model that may result in negative resistances or
very uneven sharing of losses between winding 1 and winding 2, you may choose an
alternate model for computing winding resistances. This alternate model assumes
that the Joules losses corresponding to *R12* are
equally shared between winding w1 and winding w2 (*R2* = *R1*), which is close to real
life. *R1* and *R2* are in pu and are computed as follows:

*R1* =
*R12*/2/(1-*k*)^2
*R2* = *R1 *

*R3* is adjusted to obtain the specified *R13* value, as given by the following
equation:

*R3* = *R13* -
*R1**(1-*k*)^2 -
*R2***k*^2.

Although this model returns an error on *R23*, this has a
limited impact because it affects only the tertiary winding, which is frequently
unloaded or feeds a maximum of 10% of autotransformer nominal power. This model
more accurately represents the sharing of currents between winding 1 and winding
2 for DC or very low-frequency phenomena. For example, during a geomagnetic
disturbance that produces very low-frequency earth electrical fields, the
resulting geomagnetically induced currents (GICs) throughout the network are
dependent on the DC autotransformer model.

## Examples

## Ports

### Conserving

## Parameters

## Extended Capabilities

## Version History

**Introduced in R2021b**