# Transmission Line (Three-Phase)

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

• 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:

${V}_{1}^{\prime }-{V}_{2}^{\prime }=\left[\begin{array}{ccc}R+2{R}_{m}& & \\ & R-{R}_{m}& \\ & & R-{R}_{m}\end{array}\right]{I}_{1}^{\prime }+\left[\begin{array}{ccc}L+2M& & \\ & L-M& \\ & & L-M\end{array}\right]\frac{d{I}_{1}^{\prime }}{dt}$

${I}_{1}^{\prime }+{I}_{2}^{\prime }=\left[\begin{array}{ccc}{C}_{g}& & \\ & {C}_{g}+3{C}_{l}& \\ & & {C}_{g}+3{C}_{l}\end{array}\right]\frac{d{V}_{2}^{\prime }}{dt}$

${I}_{1}^{\prime }=T\prime {I}_{1}$

${I}_{2}^{\prime }=T\prime {I}_{2}$

${V}_{1}^{\prime }=T\prime {V}_{1}$

${V}_{2}^{\prime }=T\prime {V}_{2}$

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

where:

• R is the line resistance for the segment.

• Rm is the mutual resistance for the segment.

• L is the line inductance for the segment.

• M is the value of the Mutual inductance parameter.

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

• Cl 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:

${R}_{0}=R+2{R}_{m}$

${R}_{1}=R-{R}_{m}$

${L}_{0}=L+2M$

${L}_{1}=L-M$

${C}_{0}={C}_{g}$

${C}_{1}={C}_{g}+3{C}_{l}$

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

$R=\frac{2{R}_{1}+{R}_{0}}{3}$

${R}_{m}=\frac{{R}_{0}-{R}_{1}}{3}$

$L=\frac{2{L}_{1}+{L}_{0}}{3}$

$M=\frac{{L}_{0}-{L}_{1}}{3}$

${C}_{l}=\frac{{C}_{1}-{C}_{0}}{3}$

${C}_{g}={C}_{0}$

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

To model a fault in the Transmission Line (Three-Phase) block, in the Faults section, click the Add fault hyperlink in the parameter that corresponds to the specific fault that you want to model. When the Create Fault window opens, you use it to specify the fault properties. For more information about fault modeling, see Fault Behavior Modeling and Fault Triggering.

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)

## Ports

### Conserving

expand all

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

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

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

#### Dependencies

To enable this port, set Accessible ground connections at section interface to `Yes`

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

#### Dependencies

To enable this port, set Accessible ground connections at section interface to `Yes`

## Parameters

expand all

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

Length of the transmission line.

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 of the transmission line per phase per-unit length.

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

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

Line-line capacitance per-unit length.

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

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

Option to model the transmission line as two sections. You must enable this parameter to add a non-intrusive fault in the Transmission Line (Three-Phase) block.

Number of segments in the pi-line model.

#### Dependencies

To enable this parameter, set Model two line sections to `No`.

Position of the interface in the pi-line model. If you add a non-intrusive fault, this parameter represents the position of the fault in the pi-line model.

#### Dependencies

To enable this parameter, set Model two line sections to `Yes`.

Number of segments in the first section of the pi-line model. If you add a non-intrusive fault, this parameter represents the number of segments before the fault occurs.

#### Dependencies

To enable this parameter, set Model two line sections to `Yes`.

Number of segments in the second section of the pi-line model. If you add a non-intrusive fault, this parameter represents the number of segments after the fault occurs.

#### Dependencies

To enable this parameter, set Model two line sections to `Yes`.

Option to access the ground connections at the fault position. If you set this parameter to `No`, the block internally connects the ground connections at the fault position to the electrical reference.

#### Dependencies

To enable this parameter set Model two line sections to `Yes`.

### Parasitics

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

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

### Faults

Option to add a fault to the section interface of the pi-line model.

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)```

#### Dependencies

To enable this parameter, add a fault to the Transmission Line (Three-Phase) block by clicking the Add fault hyperlink in the Section interface fault parameter.

After you create the fault, you can change the properties in the Fault Inspector window. When you open a block that has a fault, the Open Fault Inspector hyperlink appears instead of the Add fault hyperlink. For an example that shows how to include faults, see Analyze a DC Armature Winding Fault.

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

#### Dependencies

To enable this parameter, add a fault to the Transmission Line (Three-Phase) block by clicking the Add fault hyperlink in the Section interface fault parameter and 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)```

After you create the fault, you can change the properties in the Fault Inspector window. When you open a block that has a fault, the Open Fault Inspector hyperlink appears instead of the Add fault hyperlink. For an example that shows how to include faults, see Analyze a DC Armature Winding Fault.

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

#### Dependencies

To enable this parameter, add a fault to the Transmission Line (Three-Phase) block by clicking the Add fault hyperlink in the Section interface fault parameter and 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)```

After you create the fault, you can change the properties in the Fault Inspector window. When you open a block that has a fault, the Open Fault Inspector hyperlink appears instead of the Add fault hyperlink. For an example that shows how to include faults, see Analyze a DC Armature Winding Fault.

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

#### Dependencies

To enable this parameter, add a fault to the Transmission Line (Three-Phase) block by clicking the Add fault hyperlink in the Section interface fault parameter and 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)```

After you create the fault, you can change the properties in the Fault Inspector window. When you open a block that has a fault, the Open Fault Inspector hyperlink appears instead of the Add fault hyperlink. For an example that shows how to include faults, see Analyze a DC Armature Winding Fault.

## Version History

Introduced in R2013b