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:

$V_{1}^{'} - V_{2}^{'} = [\begin{matrix} R + 2 R_{m} \\ R - R_{m} \\ R - R_{m} \end{matrix}] I_{1}^{'} + [\begin{matrix} L + 2 M \\ L - M \\ L - M \end{matrix}] \frac{d I_{1}^{'}}{d t}$

$I_{1}^{'} + I_{2}^{'} = [\begin{matrix} C_{g} \\ C_{g} + 3 C_{l} \\ C_{g} + 3 C_{l} \end{matrix}] \frac{d V_{2}^{'}}{d t}$

$I_{1}^{'} = T' I_{1}$

$I_{2}^{'} = T' I_{2}$

$V_{1}^{'} = T' V_{1}$

$V_{2}^{'} = T' V_{2}$

$T = \frac{1}{\sqrt{3}} [\begin{matrix} 1 & \sqrt{2} & 0 \\ 1 & \frac{- 1}{\sqrt{2}} & \sqrt{\frac{3}{2}} \\ 1 & \frac{- 1}{\sqrt{2}} & - \sqrt{\frac{3}{2}} \end{matrix}]$

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:

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

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

$L_{0} = L + 2 M$

$L_{1} = L - M$

$C_{0} = C_{g}$

$C_{1} = C_{g} + 3 C_{l}$

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

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

$R_{m} = \frac{R_{0} - R_{1}}{3}$

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

$M = \frac{L_{0} - L_{1}}{3}$

$C_{l} = \frac{C_{1} - C_{0}}{3}$

$C_{g} = C_{0}$

Alternatively, you can specify the positive-sequence and zero-sequence parameters directly by setting the Parameterization parameter to By positive and zero-sequence parameters. (since R2026a)

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:

$Z_{1} = R_{1} + j ω L_{1}$ is the series impedance, in Ω/Km.
$Y_{1} = j ω C_{1}$ is the shunt admittance, in s/Km.
$γ_{1} = \sqrt{Z_{1} Y_{1}}$ is the propagation constant.
$ν_{1} = \frac{ω}{I m a g (γ_{1})}$ is the propagation speed, in Km/s, where $ω = 2 π f$ .

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 Add fault next to the fault that you want to model. 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:

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)

You can model a fault at one of the two three-phase connection ports, at both connection ports at the same time, or at a specific position along the transmission line.

Fault Triggers

You specify how and when faults occur by using the Trigger type parameter.

If you set the Trigger type to Timed, the Transmission Line (Three-Phase) block triggers faults when the simulation time reaches the value you specify for the Trigger fault at time parameter.

If you set the Trigger type to Conditional, you can choose whether faults in the Transmission Line (Three-Phase) block are reversible. To model irreversible faults, click Open fault properties to open the Property Inspector and select the Trigger stays on once activated parameter. The block enters the faulted state when the trigger condition becomes true for the first time and remains in the faulted state for the rest of the simulation. To model reversible faults, clear the Trigger stays on once activated parameter. The block enters the faulted state when the trigger condition is true and enters the unfaulted state when the trigger condition is false.

For more information about adding faults to blocks and specifying fault triggers, see Introduction to Simscape Faults.

Examples

IEEE 9-Bus System

Model a 9-bus three-phase power system network. This example is based on the IEEE® benchmark test case. For more information, see "Power System Control and Stability" by P. M. Anderson and A. A. Fouad (IEEE Press, 2003).

Open Model

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 Model

Mixed AC/DC System Loadflow

Use the Load Flow Analyzer to review the load flow results of a mixed AC/DC system. The model to which this analysis is applied includes an AC load flow source, a three-phase rectifier, and three loads. One of the loads is AC, one is permanently connected on the DC side, and one is switched on the DC side.

Open Model

Ports

Conserving

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~1 — Three-phase port 1
electrical

Expandable three-phase electrical conserving port associated with the first port of the transmission line.

Dependencies

To enable this port, set Electrical connection to Composite three-phase ports.

~2 — Three-phase port 2
electrical

Expandable three-phase electrical conserving port associated with the second port of the transmission line.

Dependencies

To enable this port, set Electrical connection to Composite three-phase ports.

a1 — Phase a of port 1
electrical

Electrical conserving port associated with the phase a of port 1.

Dependencies

To enable this port, set Electrical connection to Expanded three-phase ports.

a2 — Phase a of port 2
electrical

Electrical conserving port associated with the phase a of port 2.

Dependencies

To enable this port, set Electrical connection to Expanded three-phase ports.

b1 — Phase b of port 1
electrical

Electrical conserving port associated with the phase b of port 1.

Dependencies

To enable this port, set Electrical connection to Expanded three-phase ports.

b2 — Phase b of port 2
electrical

Electrical conserving port associated with the phase b of port 2.

Dependencies

To enable this port, set Electrical connection to Expanded three-phase ports.

c1 — Phase c of port 1
electrical

Electrical conserving port associated with the phase c of port 1.

Dependencies

To enable this port, set Electrical connection to Expanded three-phase ports.

c2 — Phase c of port 2
electrical

Electrical conserving port associated with the phase c of port 2.

Dependencies

To enable this port, set Electrical connection to Expanded three-phase ports.

g1 — Ground connection at end ~1
electrical

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

g2 — Ground connection at end ~2
electrical

Electrical conserving port corresponding to ground connection at the ~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 section interface 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 section interface to Yes.

Parameters

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Main

Electrical connection — Whether to model composite or expanded three-phase ports
`Composite three-phase ports` (default) | `Expanded three-phase ports`

Since R2026a

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.

Programmatic Use

To set the block parameter value programmatically, use the set_param function.

Parameter:	`port_option`
Values:	`"ee.enum.threePhasePort.composite"` (default) \| `"ee.enum.threePhasePort.expanded"`

Line length — Length of transmission line
`1` `km` (default) | positive scalar

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.

Parameterization — Option to parameterize block
`By physical line parameters` (default) | `By positive and zero-sequence parameters`

Since R2026a

Option to parameterize the block by using physical line parameters or positive and zero-sequence parameters.

Resistance — Resistance of transmission line
`0.02` `Ohm/km` (default) | positive scalar

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

Dependencies

To enable this parameter, set Parameterization to By physical line parameters. (since R2026a)

Mutual resistance — Line-line mutual resistance
`0` `Ohm/km` (default) | nonnegative scalar

Line-line mutual resistance per unit length. The value of this parameter must be less than the value of the Resistance parameter.

Dependencies

To enable this parameter, set Parameterization to By physical line parameters. (since R2026a)

Inductance — Self-inductance of transmission line
`0.5` `mH/km` (default) | positive scalar

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

Dependencies

To enable this parameter, set Parameterization to By physical line parameters. (since R2026a)

Mutual inductance — Line-line mutual inductance
`0.1` `mH/km` (default) | nonnegative scalar

Line-line mutual inductance per-unit length. Set this to 0 to remove mutual inductance. The value of this parameter must be less than the value of the Inductance parameter.

Dependencies

To enable this parameter, set Parameterization to By physical line parameters. (since R2026a)

Line-line capacitance — Line-line capacitance
`0.3` `μF/km` (default) | positive scalar

Line-line capacitance length, in per-unit.

Dependencies

To enable this parameter, set Parameterization to By physical line parameters. (since R2026a)

Line-ground capacitance — Line-ground capacitance
`0.1` `μF/km` (default) | positive scalar

Line-ground capacitance length, in per-unit.

Dependencies

To enable this parameter, set Parameterization to By physical line parameters. (since R2026a)