Transmission Line (Three-Phase)

Three-phase transmission line using lumped-parameter pi-section line model

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Libraries:
Simscape / Electrical / Passive / Lines

Description

The Transmission Line (Three-Phase) block models a three-phase transmission line using the lumped-parameter pi-line model. This model takes into account phase resistance, phase self-inductance, line-line mutual inductance and resistance, line-line capacitance, and line-ground capacitance.

To simplify the block-defining equations, Clarke’s transformation is used. The resulting equations are:

$V_1^{\prime }-V_2^{\prime }=\left[\begin{array}{ccc}R+2R_m& & \\ & R-R_m& \\ & & R-R_m\end{array}\right]I_1^{\prime }+\left[\begin{array}{ccc}L+2M& & \\ & L-M& \\ & & L-M\end{array}\right]\frac{d{I}_1^{\prime }}{dt}$

$I_1^{\prime }+I_2^{\prime }=\left[\begin{array}{ccc}{C}_g& & \\ & C_g+3C_l& \\ & & C_g+3C_l\end{array}\right]\frac{d{V}_2^{\prime }}{dt}$

$$I_1^{\prime }=T\prime I_1$$

$I_2^{\prime }=T\prime I_2$

$V_1^{\prime }=T\prime V_1$

$V_2^{\prime }=T\prime V_2$

$T=\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}1& \sqrt{2}& 0\\ 1& \raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}$}\right.& \sqrt{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\\ 1& \raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}$}\right.& -\sqrt{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\end{array}\right]$

where:

• R is the line resistance for the segment.

• R_m is the mutual resistance for the segment.

• L is the line inductance for the segment.

• M is the value of the Mutual inductance parameter.

• C_g is the line-ground capacitance for the segment.

• C_l is the line-line capacitance for the segment.

• T is the Clarke’s transformation matrix.

• I1 is the three-phase current flowing into the ~1 port.

• I2 is the three-phase current flowing into the ~2 port.

• V1 is the three-phase voltage at the ~1 port.

• V2 is the three-phase voltage at the ~2 port.

The positive and zero-sequence parameters are defined by the diagonal terms in the transformed equations:

$R_0=R+2R_m$

$R_1=R-R_m$

$$L_0=L+2M$$

$L_1=L-M$

$C_0=C_g$

$C_1=C_g+3C_l$

Rearranging these equations gives the physical line quantities in terms of positive and zero-sequence parameters:

$R=\frac{2R_1+R_0}{3}$

$$R_m=\frac{R_0-R_1}{3}$$

$L=\frac{2L_1+L_0}{3}$

$$M=\frac{L_0-L_1}{3}$$

$$C_l=\frac{C_1-C_0}{3}$$

$C_g=C_0$

The figure shows the equivalent electrical circuit for a single-segment pi-line model using Clarke’s transformation.

To increase fidelity, you can use the Number of segments parameter to repeat the pi-section N times, resulting in an N-segment transmission line model. More segments significantly slows down your simulation.

To improve numerical performance, you can add parasitic resistance and conductance components. Choosing large values for these components improves simulation speed but decreases simulation accuracy.

Propagation Speeds

The Transmission Line (Three-Phase) calculates the positive-sequence propagation speeds by using these equations:

• $$Z_1=R_1+j\omega L_1$$ is the series impedance, in Ω/Km.

• $$Y_1=j\omega C_1$$ is the shunt admittance, in s/Km.

• $${\gamma }_1=\sqrt{Z_1{Y}_1}$$ is the propagation constant.

• $${\nu }_1=\frac{\omega }{Imag\left({\gamma }_1\right)}$$ is the propagation speed, in Km/s, where $$\omega =2\pi f$$.

For zero-sequence propagation speed, the equations are the same but R₁, L₁, C₁, Z₁, Y₁, γ₁, and ν₁ are R₀, L₀, C₀, Z₀, Y₀, γ₀, and ν₀.

Faults

The Transmission Line (Three-Phase) block allows you to model these types of faults at specific position along the transmission line:

• Single-phase-to-ground fault (a-g, b-g, or c-g)

• Two-phase fault (a-b, b-c, or c-a)

• Two-phase-to-ground fault (a-b-g, b-c-g, or c-a-g)

• Three-phase fault (a-b-c)

• Three-phase-to-ground fault (a-b-c-g)

An input physical signal, F, triggers the fault. If the input physical signal is greater than or equal to 0.5, the fault is active. If the input physical signal is less than 0.5, the fault is inactive.

Examples

Marine Full Electric Propulsion Power System

A representative marine half-ship electrical power system with base load, hotel load, bow thrusters and electric propulsion.

Open Script

Ports

Input

F — Fault triggering signal

physical

Physical signal port associated with the fault control signal.

Dependencies

To enable this port, set Enable fault to Yes

Conserving

~1 — Three-phase port

electrical

Expandable three-phase port.

~2 — Three-phase port

electrical

Expandable three-phase port.

g1 — Ground connection at end ~1

electrical

Electrical conserving port corresponding to ground connection at ~1 end of the transmission line.

g2 — Ground connection at end ~2

electrical

Electrical conserving port corresponding to ground connection at ~2 end of the transmission line.

g1' — Ground connection at fault position 1

electrical

Electrical conserving port corresponding to ground connection at fault position 1.

Dependencies

To enable this port, set Accessible ground connections at fault position to Yes

g2' — Ground connection at fault position 2

electrical

Electrical conserving port corresponding to ground connection at fault position 2.

Dependencies

To enable this port, set Accessible ground connections at fault position to Yes

Parameters

Modeling option — Whether to model composite or expanded three-phase ports

Composite three-phase ports (default) | Expanded three-phase ports

Whether to model composite or expanded three-phase ports.

Composite three-phase ports represent three individual electrical conserving ports with a single block port. You can use composite three-phase ports to build models that correspond to single-line diagrams of three-phase electrical systems.

Expanded three-phase ports represent the individual phases of a three-phase system using three separate electrical conserving ports.

Main

Line length — Length of the transmission line

1 km (default)

Length of the transmission line.

Frequency used for rlcg specification — Frequency used for rlcg specification

60 Hz (default) | positive scalar

Frequency used for the R, L, C, G specification, where:

• R is line resistance per unit length.

• L is the line inductance per unit length.

• C is the line capacitance per unit length.

• G is the line conductance per unit length.

Resistance — Resistance of the transmission line

0.02 Ohm/km (default)

Resistance of the transmission line per phase per-unit length.

Inductance — Self-inductance of the transmission line

0.5 mH/km (default)

Self-inductance of the transmission line per phase per-unit length.

Mutual inductance — Line-line mutual inductance

0.1 mH/km (default)

Line-line mutual inductance per-unit length. Set this to 0 to remove mutual inductance.

Line-line capacitance — Line-line capacitance

0.3 μF/km (default)

Line-line capacitance per-unit length.

Line-ground capacitance — Line-ground capacitance

0 μF/km (default)

Line-ground capacitance per-unit length. The default value is 0μF/km (no line-ground capacitance).

Mutual resistance — Line-line mutual resistance

0 Ohm/km (default)

Line-line mutual resistance per unit length. The default value is 0Ohm/km (no line-line mutual resistance).

Number of segments — Number of segments

1 (default)

Number of segments in the pi-line model.

Parasitics

Parasitic series resistance — Parasitic series resistance

1e-6 Ohm (default)

Resistance value, divided by the number of segments, that is added in series with every capacitor in the model.

Parasitic parallel conductance — Parasitic parallel conductance

1e-6 1/Ohm (default)

Conductance value, divided by the number of segments, that is added in parallel with every series resistor and inductor in the model.

Faults

Enable fault — Fault enabling

No (default) | Yes

Whether to enable faults.

Fault type — Type of fault

Single-phase to ground (a-g) (default) | Single-phase to ground (b-g) | Single-phase to ground (c-g) | Two-phase (a-b) | Two-phase (b-c) | Two-phase (c-a) | Two-phase to ground (a-b-g) | Two-phase to ground (b-c-g) | Two-phase to ground (c-a-g) | Three-phase (a-b-c) | Three-phase to ground (a-b-c-g)

Type of fault to model. The visibility of related parameters depends on the fault model. Options are:

• Single-phase to ground (a-g)

• Single-phase to ground (b-g)

• Single-phase to ground (c-g)

• Two-phase (a-b)

• Two-phase (b-c)

• Two-phase (c-a)

• Two-phase to ground (a-b-g)

• Two-phase to ground (b-c-g)

• Two-phase to ground (c-a-g)

• Three-phase (a-b-c)

• Three-phase to ground (a-b-c-g)

Faulted phase-neutral resistance — Faulted phase-neutral resistance

1e-3 Ohm (default)

Resistance between the phase connection and the neutral point when the fault is active.

Dependencies

To enable this parameter set the Fault type parameter to:

• Single-phase to ground (a-g)

• Single-phase to ground (b-g)

• Single-phase to ground (c-g)

• Two-phase (a-b)

• Two-phase (b-c)

• Two-phase (c-a)

• Two-phase to ground (a-b-g)

• Two-phase to ground (b-c-g)

• Two-phase to ground (c-a-g)

• Three-phase (a-b-c)

• Three-phase to ground (a-b-c-g)

Faulted neutral-ground resistance — Faulted neutral-ground resistance

1e-3 Ohm (default)

Resistance between the neutral point and the electrical reference when fault is active.

Dependencies

To enable this parameter set the Fault type parameter to:

• Single-phase to ground (a-g)

• Single-phase to ground (b-g)

• Single-phase to ground (c-g)

• Two-phase to ground (a-b-g)

• Two-phase to ground (b-c-g)

• Two-phase to ground (c-a-g)

• Three-phase to ground (a-b-c-g)

Unfaulted phase-neutral conductance — Unfaulted phase-neutral conductance

1e-6 1/Ohm (default)

Conductance between the phase connection and the neutral point when the fault is not active.

Dependencies

To enable this parameter set the Fault type parameter to:

• Single-phase to ground (a-g)

• Single-phase to ground (b-g)

• Single-phase to ground (c-g)

• Two-phase (a-b)

• Two-phase (b-c)

• Two-phase (c-a)

• Two-phase to ground (a-b-g)

• Two-phase to ground (b-c-g)

• Two-phase to ground (c-a-g)

• Three-phase (a-b-c)

• Three-phase to ground (a-b-c-g)

Fault location — Location of fault

1 Km (default)

Location where the fault occurs

Dependencies

To enable this parameter set Enable fault to Yes.

Number of segments before fault — Number of segments before fault

1 (default)

Number of segments in the pi-line model before the fault occurs

Dependencies

To enable this parameter set Enable fault to Yes.

Accessible ground connections at fault position — Accessible ground connections at fault position

No (default) | Yes

Select Yes to access the ground connections at the fault position, otherwise the ground connections at the fault position are connected to the electrical reference internally

Dependencies

To enable this parameter set Enable fault to Yes.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2013b