AC Cable (Three-Phase)

Three-phase AC power cable

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Libraries:
Simscape / Electrical / Passive / Lines

Description

The AC Cable (Three-Phase) block represents a three-phase AC power cable with a conducting sheath surrounding each phase. The figure shows a single-phase conductor inside a conducting sheath. The inner cylinder represents the main conductor for the phase, and the outer cylinder represents the conducting sheath.

You can model composite or expanded three-phase ports by setting the Modeling option parameter to either:

• Composite three-phase ports — Contains three-phase connection ports for the sheaths and phases and a single-phase connection port for each electrical reference node.

• Expanded three-phase ports — Contains single-phase connection ports for each sheath, phase, and electrical reference node.

The AC Cable (Three-Phase) block includes inductances and mutual inductances between each phase, sheath, and return path. Therefore, you can connect an ideal electrical reference block to both return ports, g1 and g2, while maintaining loss modeling in the Earth- or neutral-return line.

To facilitate simulation convergence when you connect the AC Cable (Three-Phase) block to a source block, include source impedance using one of these methods:

• Configure the source block to include impedance.

• Insert a block that models impedance between the source block and the AC Cable (Three-Phase) block.

To model unbonded sheaths, connect the unbonded sheaths to an Open Circuit (Three-Phase) block. The figure shows a model of single-point bonding when you set Modeling option to Composite three-phase ports.

For high performance modeling, in terms of simulation speed, use a single AC Cable (Three-Phase) block. To improve model fidelity in terms of frequency behavior, connect several AC Cable (Three-Phase) blocks in series. For series-connected blocks, the sheaths and main conductors act as coupled transmission lines with perfect transposition of the phases. The number of AC Cable (Three-Phase) blocks that you use to model a particular physical length of cable must be less than the number of transpositions in the physical system that you are modeling. Types of continuous multi-segment cables that you can model include:

• Unbonded continuous cables

  

• Single-point bonded continuous cables

  

• Double-point bonded continuous cables

  

You can also model cross-bonded cables using the AC Cable (Three Phase) block.

This three pi-segment cable model implements cross-bonding using expanded three-phase ports and single-phase connection lines. The sheath in the model is two-point bonded.

This model of blocks with composite three-phase-ports uses Phase Permute blocks to implement cross bonding. The sheath in the model is unbonded.

For an example that allows you to choose the number of segments and type of bonding, see AC Cable with Bonded Sheaths.

Three-Phase AC Cable Model

The AC Cable (Three-Phase) block uses the concept of partial inductances to calculate the inductance values. These values include the partial self-inductance of each phase, sheath, and return path and the partial mutual inductances between each:

• Phase and each other phase

• Phase and the sheath of that phase

• Phase and the sheath of neighboring phases

• Phase and the return

• Sheath and each neighboring sheath

• Sheath and the return

For three equivalent phases, the matrix that defines the resistance relationships for the vector [phase A; sheath A; phase B; sheath B; phase C; sheath C] is

$R=\left[\begin{array}{cccccc}{R}_a+R_g& R_g& R_g& R_g& R_g& R_g\\ R_g& R_s+R_g& R_g& R_g& R_g& R_g\\ R_g& R_g& R_a+R_g& R_g& R_g& R_g\\ R_g& R_g& R_g& R_s+R_g& R_g& R_g\\ R_g& R_g& R_g& R_g& R_a+R_g& R_g\\ R_g& R_g& R_g& R_g& R_g& R_s+R_g\end{array}\right]$

$R_a=R_a^{\text{'}}l$

$R_g=R_{\text{return}}^{\text{'}}l,$

for which R'_return depends on the return parameterization method such that:

• For a return parameterization based on distance and resistance $R_{\text{return}}^{\text{'}}=R_g^{\text{'}}.$

• For a return parameterization based on frequency and Earth resistivity $R_{\text{return}}^{\text{'}}={\pi }^2{10}^{-7}f$

and

$R_s=R_s^{\text{'}}l,$

where:

• R is the resistance matrix.

• R_a is the resistance of a particular phase.

• R_s is the resistance of a particular sheath.

• R_g is the resistance of the Earth- or neutral-return.

• R'_a is the resistance per unit length for the phase.

• l is the cable length.

• R'_s is the resistance per unit length for the sheath.

• R'_return is the resistance per unit length of the return. The value of R'_return varies depending on the return parameterization method.

• R'_g is the resistance per unit length for the Earth- or neutral return.

• f is the frequency that the block uses to calculate Earth-return parameters if you parameterize the block using the frequency and Earth resistivity method.

The block uses standard expressions to calculate the capacitances between:

• Concentric or adjacent cylinders

• Each phase and its own sheath

• Each sheath and the return

The matrix that defines these capacitance relationships is

$C=\left[\begin{array}{cccccc}{C}_{a{s}_a}& -C_{a{s}_a}& 0& 0& 0& 0\\ -C_{a{s}_a}& C_{a{s}_a}+C_{s_{a}g}& 0& 0& 0& 0\\ 0& 0& C_{a{s}_a}& -C_{a{s}_a}& 0& 0\\ 0& 0& -C_{a{s}_a}& C_{a{s}_a}+C_{s_{a}g}& 0& 0\\ 0& 0& 0& 0& C_{a{s}_a}& -C_{a{s}_a}\\ 0& 0& 0& 0& -C_{a{s}_a}& C_{a{s}_a}+C_{s_{a}g}\end{array}\right]$

$C_{a{s}_a}=\frac{2\pi {\epsilon }_{\text{r}}{\epsilon }_{0}l}{\ln \left(\frac{r_s}{r_a}\right)}$

$$r_a=GMR\cdot e^{\frac{1}{4}}$$

$C_{s_{a}g}=\frac{2\pi {\epsilon }_{\text{env}}{\epsilon }_{0}l}{\ln \left(\frac{r_{cable}}{r_{s,outer}}\right)},$

where:

• C is the capacitance matrix.

• C_asa is the capacitance between each phase and the sheath of that phase.

• C_sag is the capacitance between each sheath and return.

• ϵ_r is the permittivity of the dielectric.

• ϵ₀ is the permittivity of free space.

• r_s is the radius of the sheath.

• r_a is the effective radius of the conductor. For a single-strand conductor, r_a is the radius of the strand.

• r_cable is the cable radius and r_cable is greater than r_s,outer.

• GMR is the geometric mean radius of the conductor. For a single-strand conductor, $$GMR=r_{\text{strand}}{e}^{-\frac{1}{4}}$$, where r_strand is the radius of the strand.

• ϵ_env is the permittivity of the material between the sheathed lines and the return path.

The block uses the concept of partial inductances to calculate inductance values. These values include the partial self-inductance of each phase, sheath, and return path and the partial mutual inductances between each:

• Phase and each other phase

• Phase and the sheath of that phase

• Phase and the sheath of neighboring phases

• Phase and the return

• Sheath and each neighboring sheath

• Sheath and the return

The equations that define these inductance relationships are:

$L=\left[\begin{array}{cccccc}{D}_a& \delta & A& \alpha & A& \alpha \\ \delta & D_s& \alpha & S& \alpha & S\\ A& \alpha & D_a& \delta & A& \alpha \\ \alpha & S& \delta & D_s& \alpha & S\\ A& \alpha & A& \alpha & D_a& \delta \\ \alpha & S& \alpha & S& \delta & D_s\end{array}\right]$

$D_a=L_a-M_{ag}$

$L_a=2\times {10}^{-7}l\left[\ln \left(\frac{2l}{r_a}\right)-\frac{3}{4}\right]$

$M_{ag}=M_{sg}=2\times {10}^{-7}l\left[\ln \left(\frac{2l}{D_{\text{return}}}\right)-1\right]$

for which D_return depends on the return parameterization method such that:

• For a return parameterization based on distance and resistance $D_{\text{return}}=D_{\text{e}}.$

• For a return parameterization based on frequency and Earth resistivity $D_{\text{return}}=1650\sqrt{\frac{\rho }{2\pi f}}.$

$D_s=L_s-M_{sg}$

$L_s=M_{a{s}_a}=2\times {10}^{-7}l\left[\ln \left(\frac{2l}{r_s}\right)-\frac{3}{4}\right]$

$\delta =M_{a{s}_a}-M_{ag}$

$\alpha =M_{a{s}_b}-M_{ag}$

$M_{a{s}_b}=M_{s_a{s}_b}=M_{ab}=2\times {10}^{-7}l\left[\ln \left(\frac{2l}{d_{ab}}\right)-1\right],$

for which d_ab depends on the line formation parameterization method, such that:

• For a trefoil line formation parameterization $d_{ab}=D_{ab}.$

• For a flat line formation parameterization $d_{ab}=D_{ab}\sqrt[3]{2}.$

$A=M_{ab}-M_{ag}$

$S=M_{s_a{s}_b}-M_{sg},$

where:

• L is the inductance matrix.

• D_a is the self-inductance of a single phase through its entire path and return.

• L_a is the partial self-inductance of each phase.

• M_ag is the partial mutual inductance between each phase and the Earth- or neutral-return.

• M_sg is the partial mutual inductance between each sheath and the Earth- or neutral-return.

• The factor, $$2\times {10}^{-7}$$ is equal to $${\mu }_0/2\pi$$, because permeability of free space, μ₀, is equal to $$1.257\times {10}^{-6}$$ or $$4\pi \times {10}^{-7}$$ H/m.

• D_s is the self-inductance of a single sheath through its entire path and return.

• L_s is the partial self-inductance of each sheath.

• M_asa is the partial mutual inductance between each phase and the sheath of that phase.

• δ is the effective mutual inductance between a phase and the sheath of that phase.

• α is the effective mutual inductance between a phase and a neighboring sheath.

• M_asb is the partial mutual inductance between each phase and the sheath of each neighboring phase.

• M_sasb is the partial mutual inductance between sheaths of different phases.

• M_ab is the partial mutual inductance between each phase and each other phase.

• D_return is the effective distance to the return. The value of D_return varies if you use the distance/return parameterization method.

• D_e is the effective distance to the Earth- or neutral-return.

• ρ is the effective Earth resistivity for an Earth-return.

• f is the frequency that is used to determine the return path properties.

• d_ab is the effective distance between adjacent phases. The value of d_ab varies depending on the line parameterization method.

• D_ab is the center-to-center distance between adjacent phases.

• A is the effective mutual inductance between phases.

• S is the effective mutual inductance between sheaths.

A modal transformation that is related to the Clarke transform simplifies the equivalent circuit. The six-by-six transformation, T, is

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

As $T^{†}=T^{-1}$, applying the T transform yields the modal resistance matrix, R_m, the modal capacitance matrix, C_m, and the modal inductance matrix, L_m.

The transformed matrices are:

$R_m=T^{†}RT=\left[\begin{array}{cccccc}{R}_a+3R_g& 3R_g& 0& 0& 0& 0\\ 3R_g& R_s+3R_g& 0& 0& 0& 0\\ 0& 0& R_a& 0& 0& 0\\ 0& 0& 0& R_s& 0& 0\\ 0& 0& 0& 0& R_a& 0\\ 0& 0& 0& 0& 0& R_s\end{array}\right]$

$C_m=T^{†}CT=\left[\begin{array}{cccccc}{C}_{a{s}_a}& -C_{a{s}_a}& 0& 0& 0& 0\\ -C_{a{s}_a}& C_{a{s}_a}+C_{s_{a}g}& 0& 0& 0& 0\\ 0& 0& C_{a{s}_a}& -C_{a{s}_a}& 0& 0\\ 0& 0& -C_{a{s}_a}& C_{a{s}_a}+C_{s_{a}g}& 0& 0\\ 0& 0& 0& 0& C_{a{s}_a}& -C_{a{s}_a}\\ 0& 0& 0& 0& -C_{a{s}_a}& C_{a{s}_a}+C_{s_{a}g}\end{array}\right]=C$

$L_m=T^{†}LT=\left[\begin{array}{cccccc}{D}_a+2A& \delta +2\alpha & 0& 0& 0& 0\\ \delta +2\alpha & D_s+2A& 0& 0& 0& 0\\ 0& 0& D_a-A& \delta -\alpha & 0& 0\\ 0& 0& \delta -\alpha & D_s-S& 0& 0\\ 0& 0& 0& 0& D_a-A& \delta -\alpha \\ 0& 0& 0& 0& \delta -\alpha & D_s-S\end{array}\right].$

The transformation changes each six-by-six matrix into three uncoupled two-by-two matrices. The capacitance matrix is invariant under this transformation. The power is invariant in the transformed and untransformed domains because T is unitary.

Assumptions and Limitations

• For resistance calculations, the phases are equivalent.

• Relative to the phase-to-sheath capacitance and the sheath-return capacitances all other capacitances, are negligible due to the shielding provided by the conducting sheaths.

Examples

AC Cable with Bonded Sheaths

A three-phase cable model comprised of multiple pi-sections. Each phase is enclosed in a conductive sheath. The conductive sheath is connected to ground at either end of the cable through a simple resistance. A high-voltage source provides power to an unbalanced resistive load through the power cable. You can configure the sheath to be either series-bonded or cross-bonded. You can also configure the number of pi-sections. Increasing the number of pi-sections improves the accuracy but slows down the simulation. To facilitate convergence, the voltage source includes an internal impedance.

• Open Model

Comparison of Three-Phase Port Types

A comparison of Composite three-phase ports versus Expanded three-phase ports. The first circuit shows a Voltage Source configured with a Composite three-phase port. A Phase Splitter block provides the interface to the Simscape™ foundation library electrical elements. The second circuit shows a Voltage Source configured with an Expanded three-phase port which can connect directly to the Simscape foundation library electrical elements. The Voltage Source can be changed from Composite to Expanded three-phase ports by use of the Modeling option mask parameter.

• Open Model

Ports

Conserving

~S1 — Sheath

electrical

Expandable three-phase electrical conserving port associated with sheath 1.

~N1 — Phase

electrical

Expandable three-phase electrical conserving port associated with a, b, and c phases 1.

g1 — Ground

electrical

Electrical conserving port associated with ground 1.

~S2 — Sheath

electrical

Expandable three-phase electrical conserving port associated with sheath 2.

~N2 — Phase

electrical

Expandable three-phase electrical conserving port associated with a, b, and c phases 2.

g2 — Ground

electrical

Electrical conserving port associated with ground 2.

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.

Cable length — Length

120 km (default)

Length of the cable.

Geometric mean radius of conductor — Radius

5 mm (default)

Geometric mean radius of the conductor, which is a function of the number and type of individual strands in the conductor of the AC cable.

Sheath radius — Radius

10 mm (default)

Average radius of the sheath. To ensure that the sheath radius is greater than the physical radius of a single-stranded conductor with a particular GMR, the sheath radius must be greater than $$GMR*e^{\frac{1}{4}}$$.

Outer cable radius — Outer radius

20 mm (default)

Outer radius of the cable, in mm. The cable radius must be greater than the Sheath radius parameter. This ensures that the sheath and conductor are both enclosed inside an insulating outer cable layer.

Line-line spacing (center-to-center) — Distance

25 mm (default)

Distance between the line centers.

Line formation — Line configuration

Trefoil (default) | Flat

Cable line formation.

Conductor resistance per length — Resistance

1 Ohm/km (default)

Resistance per length of a conductor.

Sheath resistance per length — Resistance

10 Ohm/km (default)

Resistance per length of a sheath.

Insulation relative permittivity — Permittivity

2.4 (default)

Relative permittivity of the insulation.

Relative permittivity between lines and return path — Relative permittivity

1 (default)

Relative permittivity of the circuit.

Return parameterization — Model

Use frequency and Earth resistivity (default) | Use distance and resistance

Parameterization method.

Dependencies

Enabling either option enables other parameters.

Frequency for Earth-return impedance — Frequency

60 Hz (default)

Frequency at which the Earth-return impedance is calculated.

Dependencies

Selecting Use frequency and Earth resistivity for the Return parameterization parameter enables this parameter.

Earth resistivity — Resistance

100 m*Ohm (default)

Earth-return resistivity.

Dependencies

Selecting Use frequency and Earth resistivity for the Return parameterization parameter enables this parameter.

Effective distance to return path — Return path distance

1 km (default)

Effective distance between the phases and the return path.

Dependencies

Selecting Use distance and resistance for the Return parameterization parameter enables this parameter.

Return path resistance per length — Return path unit resistance

0.1 Ohm/km (default)

Resistance per length of the return path.

Dependencies

Selecting Use distance and resistance for the Return parameterization parameter enables this parameter.

Extended Capabilities

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

Version History

Introduced in R2017b