Frequency-Dependent Overhead Line (Three-Phase)

Three-phase overhead line which includes effects that vary as a function of frequency

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Libraries:
Simscape / Electrical / Passive / Lines

Description

The Frequency-Dependent Overhead Line (Three-Phase) block represents a high-fidelity frequency-dependent overhead line that offers accurate transient response simulation from 0.01 Hz to 100 kHz.

The block computes frequency-dependent impedance and admittance matrices. The calculations also depend on the loss, inductance, and capacitance of the return path.

For more information on using a frequency-dependent overhead line (three-phase), see Engineering Applications.

Equations

Line Model

The electromagnetic behavior of a multiconductor transmission line is described by the telegrapher's equation.

$-\frac{dV}{dx}=ZI$

$-\frac{dI}{dx}=YV$

Where:

• V is the vector of line phase voltages.

• I is the vector of phase currents.

• Z is the series impedance matrix in per unit length.

• Y is the shunt admittance matrix in per unit length.

These are general solutions for the vector of currents and voltages.

$I\left(x\right)=e^{-\text{Ψ}x}{C}_1+e^{\text{Ψ}x}{C}_2$

(1)

$Y_{c}V\left(x\right)= e^{-\text{Ψ}x}{C}_1-e^{\text{Ψ}x}{C}_2$

(2)

Where $$\text{Ψ}=\sqrt{YZ}$$ is the propagation matrix and $Y_c={(\sqrt{YZ})}^{-1}Y$ is the characteristic admittance.

Consider now a transmission line segment of length x = l. At the beginning of one of its ends, when x = 0, the equations 1 and 2 are evaluated as follows.

$I_1=C_1+C_2$

(3)

$Y_c{V}_1= C_1-C_2$

(4)

The integration constant vectors C₁ and C₂ can be expressed in terms of I₀ and V₀.

$C_1=\frac{(I_1+Y_c{V}_1)}{2}$

(5)

$C_2=\frac{(I_1-Y_c{V}_1)}{2}$

(6)

Similarly, at x = l, equations 1 and 2 are evaluated as follows.

$I_2=e^{-\Psi \text{l}}{C}_1+e^{\Psi \text{l}}{C}_2$

(7)

$Y_c{V}_2=e^{-\Psi \text{l}}{C}_1-e^{\Psi \text{l}}{C}_2$

(8)

To obtain the two fundamental equations for the transmission line model, first you use equations 7 and 8 to perform this calculation.

$I_2-Y_c{V}_2=-2e^{-\Psi l}{C}_1$

(9)

Then you substitute equation 5 into equation 9.

$I_2-Y_c{V}_2=-H(I_1+Y_c{V}_1)$

(10)

Where $H=e^{-\text{Ψ}l}$ is the propagation factor matrix. Equation 10 establishes the relation between voltages and currents at the terminals of a multi-conductor line section.

You can obtain a companion expression providing the model for the terminal 1 at x = 0.

$I_1-Y_c{V}_1=-H(I_2+Y_c{V}_2)$

(11)

Define:

• $I_{sh,1}=Y_c{V}_1$ — Shunt current vector produced at terminal 1 by injected voltages V₁

• $I_{sh,2}=Y_c{V}_2$ — Shunt current vector produced at terminal 2 by injected voltages V₂

• $I_{rfl,1}=\frac{1}{2}(I_1+Y_c{V}_1)$ — Reflected currents of terminal 1

• $I_{rfl,2}=\frac{1}{2}(I_2+Y_c{V}_2)$ — Reflected currents of terminal 2

And finally rewrite equations 10 and 11 as follows:

$I_1=I_{sh,1}-2H{I}_{rfl,2}$

$I_2=I_{sh,2}-2H{I}_{rfl,1}$

These equations constitute a traveling wave line model for the segment of length L.

Particularly for transmission lines with ground return, the parameters are highly dependent on frequency. Model solutions are then carried out directly in the phase domain. The block computes automatically the impedance and the characteristic admittance matrices on the full frequency range, performing an approximation through rational fitting.

Rational fitting for this model is performed by using the vector fitting (VF) procedure. For more information on the phase domain line model and the state-space analysis, see Wide-Band Line Model Implementation in Matlab for EMT Analysis [1].

Variables

To set the priority and initial target values for the block variables before simulation, use the Initial Targets section in the block dialog box or Property Inspector. For more information, see Set Priority and Initial Target for Block Variables.

Nominal values provide a way to specify the expected magnitude of a variable in a model. Using system scaling based on nominal values increases the simulation robustness. You can specify nominal values using different sources, including the Nominal Values section in the block dialog box or Property Inspector. For more information, see System Scaling by Nominal Values.

Ports

Conserving

~1 — Three-phase AC voltage 1

electrical

Expandable electrical conserving three-phase port associated with AC voltage 1.

~2 — Three-phase AC voltage 2

electrical

Expandable electrical conserving three-phase port associated with AC voltage 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.

X position of the line — Horizontal line position

[-6, 0, 6] m (default) | vector of three elements

Position of the line in terms of the x-axis of the reference frame.

Y position of the line — Vertical line position

[16, 26, 16] m (default) | vector of three positive elements

Position of the line in terms of the y-axis of the reference frame. The elements of the vector must be greater than 0.

Resistivity of the conductor — Conductor resistivity

2.7e-8 m*Ohm (default) | positive scalar

Conductor resistivity. The value must be greater than 0.

Resistivity of the earth — Earth return resistivity

100 m*Ohm (default) | positive scalar

Earth return resistivity. The value must be greater than 0.

Radius of the conductor — Conductor radius

0.017 m (default) | positive scalar

Conductor radius. The value must be greater than 0.

Length of the line — Line length

150e3 m (default) | scalar in the range [1, 300000]

Length of the frequency-dependent overhead line.

Ground configuration — Ground node accessibility

Internally grounded (default) | Accessible ground nodes

Configuration of the ground nodes.

More About

Engineering Applications

Three-phase frequency-dependent overhead lines are typically used in:

• Simulations of transient phenomena in electrical power systems

• Design and operation of power systems and power apparatuses

References

[1] Ramos-Leanos, O, Iracheta R.. Wide-Band Line Model Implementation in Matlab for EMT Analysis. Arlington, TX: IEEE North American Power Symposium (NAPS), 2010.

[2] Ramos-Leanos, O. Wideband Line/Cable Models for Real-Time and Off-Line Simulations of Electromagnetic Transients. Diss. École Polytechnique de Montréal, 2013.

[3] Ramos-Leanos, O., J. L. Naredo, J. Mahseredjian, I. Kocar, C. Dufour, and J. A. Gutierrez-Robles. A wideband line/cable model for real-time simulations of power system transients. IEEE Transactions on Power Delivery, 27.4 (2012): 2211-2218.

[4] Iracheta, R., and O. Ramos-Leanos. Improving computational efficiency of FD line model for real-time simulation of EMTS. Arlington, TX: IEEE North American Power Symposium (NAPS), 2010.

[5] Dommel, H. W. Electromagnetic transients program (EMTP) theory book. Portland OR: Bonneville Power Administration, 1986

Extended Capabilities

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

Version History

Introduced in R2019a