Three-Phase Autotransformer with Tertiary Winding

Three-phase autotransformer with tertiary winding

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• 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.

Ports

Conserving

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Electrical conserving port associated with the phase A terminal on the high-voltage side of the transformer.

Electrical conserving port associated with the phase B terminal on the high-voltage side of the transformer.

Electrical conserving port associated with the phase C terminal on the high-voltage side of the transformer.

Electrical conserving port associated with the neutral terminal of the transformer.

Electrical conserving port associated with the phase A terminal on the low-voltage side of the transformer.

Electrical conserving port associated with the phase B terminal on the low-voltage side of the transformer.

Electrical conserving port associated with the phase C terminal on the low-voltage side of the transformer.

Electrical conserving port associated with the phase A terminal of the tertiary winding.

Electrical conserving port associated with the phase B terminal of the tertiary winding.

Electrical conserving port associated with the phase C terminal of the tertiary winding.

Parameters

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Nominal power rating of the transformer, in volt-amperes (VA).

Nominal frequency of the transformer, in hertz.

Line-to-line nominal voltage, in volts RMS, on the high-voltage side of the transformer.

Line-to-line nominal voltage, in volts RMS, on the low-voltage side of the transformer.

Line-to-line nominal voltage, in volts RMS, of the tertiary winding.

Short-circuit resistance, in pu, from the high-voltage terminals when the low-voltage terminals are short-circuited.

Short-circuit resistance, in pu, from the high-voltage terminals when the tertiary terminals are short-circuited.

Short-circuit resistance, in pu, from the low-voltage terminals when the tertiary terminals are short-circuited.

Select evaluated from RHL, RHT, RLT to use the standard model to determine the winding resistances.

Select Evaluated from RHL, RHT ; same losses in R1 & R2 to use the alternate model to determine the winding resistances.

Short-circuit inductance, in pu, from the high-voltage terminals when the low-voltage terminals are short-circuited.

Short-circuit inductance, in pu, from the high-voltage terminals when the tertiary terminals are short-circuited.

Short-circuit inductance, in pu, from the low-voltage terminals when the tertiary terminals are short-circuited.

Select Three single-phase transformers to implement a three-phase transformer using three single-phase transformer models. This core type is used in very large power transformers (hundreds of MW) found in utility grids.

Select Three-limb core (core-type) to implement a three-limb core three-phase transformer. In most applications, three-phase transformers use a three-limb core (core-type transformer). This option produces accurate simulation results during an asymmetrical fault for both linear and nonlinear models, including saturation. During asymmetrical voltage conditions, the zero-sequence flux of a core-type transformer returns outside the core, through an air gap, structural steel, and a tank. Thus, the natural zero-sequence inductance L0 (without delta winding) of such a core-type transformer is usually very low (typically 0.3 pu < L0 < 2 pu) compared with a three-phase transformer using three single-phase units (L0 > 100 pu). This low L0 value affects voltages, currents, and flux unbalances during linear and saturated operation.

Inductance L0, in pu, of the three-limb core transformer.

Dependencies

To enable this parameter, set Core Type to Three single-phase transformers.

Magnetization resistance Rm, in pu.

Magnetization inductance Lm, in pu, for a nonsaturable core.

Dependencies

To enable this parameter, clear Simulate saturation.

Select to implement a saturable core.

Saturation characteristic for the saturable core. Specify a series of current and flux pairs, in pu, starting with the pair (0, 0).

Dependencies

To enable this parameter, select Simulate saturation.

Select to define the initial fluxes with the Initial fluxes [phi0A, phi0B, phi0C] parameter.

When the Specify initial fluxes parameter is not selected upon simulation, the Simscape™ Electrical™ Specialized Power Systems software automatically computes the initial fluxes to start the simulation in steady state. The computed values are saved in the Initial Fluxes parameter and overwrite any previous values.

Dependencies

To enable this parameter, select Simulate saturation.

Specifies initial fluxes for each phase of the transformer.

When the Specify initial fluxes parameter is not selected upon simulation, the Simscape Electrical Specialized Power Systems software automatically computes the initial fluxes to start the simulation in steady state. The computed values are saved in the Initial Fluxes parameter and overwrite any previous values.

Dependencies

To enable this parameter, select Simulate saturation and Specify initial fluxes.

Select Winding voltages to measure the voltage across the winding terminals.

Select Fluxes and excitation currents ( Imag + IRm ) to measure the flux linkage, in volt seconds (V.s), and the total excitation current including iron losses modeled by Rm.

Select Fluxes and magnetization currents ( Imag ) to measure the flux linkage, in volt seconds (V.s), and the magnetization current, in amperes (A), not including iron losses modeled by Rm.

Select All measurements (V I Flux) to measure the winding voltages, magnetization currents, and the flux linkages.

Place a Multimeter block in your model to display the selected measurements during the simulation. In the Available Measurements parameter of the Multimeter block, the measurements are identified by a label followed by the block name.

To enable these settings, in the powergui block, set Simulation type to Discrete.

Select to insert a delay at the output of the saturation model that computes the magnetization current as a function of flux linkage (the integral of input voltage computed by a trapezoidal method). This delay eliminates the algebraic loop that results from trapezoidal discretization methods and speeds up the simulation of the model. However, the delay introduces one time step time delay in the model and can cause numerical oscillations if the sample time is too large. The algebraic loop is required in most cases to get an accurate solution.

When this parameter is cleared, the discretization method of the saturation model is specified by the Discrete solver model parameter.

Select one of these methods to resolve the algebraic loop.

• Trapezoidal iterative — Although this method produces correct results, it is not recommended because Simulink® tends to slow down and may fail to converge and stop simulation, especially if the model has a large number of transformers Also, because of the Simulink algebraic loop constraint, this method cannot be used in real time. In R2018b and previous releases, this is setting is used when Break Algebraic loop in discrete saturation model parameter is cleared.

• Trapezoidal robust — This method is slightly more accurate than the Backward Euler robust method. However, it may produce slightly damped numerical oscillations on transformer voltages when the transformer is at no load.

• Backward Euler robust — This method provides good accuracy and prevents oscillations when the transformer is at no load.

The maximum number of iterations for the robust methods is specified in the Preferences tab of the powergui block, in the Solver details for nonlinear elements section. For real-time applications, you may need to limit the number of iterations. The two robust solvers are the recommended methods for discretizing the saturation model of the transformer.

Dependencies

To enable this parameter, clear Break Algebraic loop in discrete saturation model.