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Frequency-Dependent Transmission Line

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This example shows a custom frequency-dependent transmission line model. The characteristic admittance and propagation function are first derived from the frequency-dependent resistance, reactance, and susceptance. The derived values are fitted using RF Toolbox. The Universal Line Model (ULM) [1] is then implemented in Simscape based on the fitted parameters. The results from the frequency-dependent transmission line model and the classic pi-section transmission line model are compared.

Model

Specification of Parameters

Import the frequency-dependent parameters of the transmission line. These parameters are computed for an overhead line that is 20 m above the ground [2]. The ground resistivity and the skin effect of the conductor are considered negligible. The following parameters are pre-computed for the simulation:

  • Frequency-dependent series resistance per unit length, $R$

  • Frequency-dependent series reactance per unit length, $X$

  • Frequency-dependent shunt susceptance per unit length, $B$

  • Corresponding frequency, $freq$

  • Length of the transmission line, $len$

  Name         Size            Bytes  Class     Attributes

  B         1000x1              8000  double              
  R         1000x1              8000  double              
  X         1000x1              8000  double              
  freq      1000x1              8000  double              
  len          1x1                 8  double

The frequency-dependent R, L and C are shown in these figures:

Characteristic Admittance and Propagation Function

The characteristic admittance is expressed as $Y_c = \sqrt{Y/Z}$, where $Z$ and $Y$ are the frequency-dependent series impedance and shunt admittance per unit length.

The propagation velocity is expressed as: $V = \omega / Imag(\Gamma)$, where $\Gamma = \sqrt{YZ}$ is the propagation constant, and $\omega$ is the corresponding angular velocity.

The propagation function, $H$, is then expressed as $H = e^{-\Gamma len}$.

Rational Fitting for Characteristic Admittance

To convert the characteristic admittance to the rational form, use the rational fit function rationalfit (RF Toolbox) from RF Toolbox.

$Y_c = \displaystyle\sum_{i=1}^{n} \frac{YcResidues_i}{s-YcPoles_i}$

where:

  • $n$ is the number of poles (order of the fit).

  • $YcPoles_i$ is the $i_{th}$ pole of $Y_c$.

  • $YcResidues_i$ is the $i_{th}$ residue of $Y_c$.

In this case, an eighth order fit is performed.

These figures show the comparison between the characteristic admittance before and after rational fitting.

Rational Fitting for Propagation Function

The time delay of the propagation function is first removed to help reduce the order of rational fitting, where $\tau$ is the propagation time delay and $H_k$ is the propagation function without time delay. The time delay is represented by a delay unit in the model.

To convert the propagation function without time delay to the rational form, use the rationalfit (RF Toolbox) function from RF Toolbox.

$H_k = \displaystyle\sum_{i=1}^{n} \frac{HkResidues_i}{s-HkPoles_i}$

where:

  • $n$ is the number of poles (order of the fit).

  • $HkPoles_i$ is the $i_{th}$ pole of $H_k$.

  • $HkResidues_i$ is the $i_{th}$ residue of $H_k$.

In this case, an eighth order fit is performed.

These figures show that the propagation function H (with time delay) before and after rational fitting agree.

Universal Line Model Implemented in Simscape

In this example, only a single conductor and the ground return is considered. The equivalent circuit of the line in Laplace domain can be deduced from the Universal Line Model (ULM) [1]. Key variables are:

  • $V_j$ is the voltage at terminal $j$.

  • $I_j$ is the current at terminal $j$.

  • $I_{sh,j}$ is the shunt current at terminal $j$.

  • $I_{rfl,j}$ is the reflected current from terminal $j$.

  • $I_{aux,j}$ is the auxiliary current from terminal $j$.

  • $H$ is the propagation function.

From this equivalent circuit, the system of equations can be written as:

  • $I_1 = I_{sh,1} - I_{aux,2}$

  • $I_2 = I_{sh,2} - I_{aux,1}$

where:

  • $I_{aux,1} = 2HI_{rfl,1}$

  • $I_{aux,2} = 2HI_{rfl,2}$

  • $I_{rfl,1} = (I_1 + I_{sh,1})/2$

  • $I_{rfl,2} = (I_2 + I_{sh,2})/2$

  • $I_{sh,1} = Y_c V_1$

  • $I_{sh,2} = Y_c V_2$

Considering the rational form of the characteristic admittance, the shunt current on terminal one is:

  • $I_{sh,1} = Y_c V_1 = \displaystyle\sum_{i=1}^{n} W_i$

  • $W_i = \frac{YcResidues_i V_i}{s-YcPoles_i}$

To transform these equations from Laplace domain to time domain, an inverse Laplace transform is performed. This transformation leads to:

  • $i_{sh,1} = Y_c v_1 = \displaystyle\sum_{i=1}^{n} w_i$

  • $\frac{d w_i}{d t} = YcPoles_i w_i + YcResidues_i v_1$

where $w_i$, $v_1$ and $i_{sh,1}$ are the time domain representation of $W_i$, $V_1$ and $I_{sh,1}$.

Similarly, considering the rational form of the propagation function, the auxiliary current on terminal one is:

  • $I_{aux,1} = 2HI_{rfl,1} = 2 \displaystyle\sum_{i=1}^{n} X_i$

  • $X_i = \frac{HkResidues_i}{s-HkPoles_i} e^{-s \tau} I_{rfl,1}$

To transform these equations from Laplace domain to time domain, an inverse Laplace transform is performed. This transformation leads to:

  • $i_{aux,1} = 2 \displaystyle\sum_{i=1}^{n} x_i$

  • $\frac{d x_i}{d t} = HkPoles_i x_i + HkResidues_i i_{rfl,1}(t-\tau)$

Currents on terminal two can be deduced using the same procedure. Time domain equations are then implemented in Simscape using Simscape language.

Simulation Results from Simscape Logging

For the first simulation case, the voltage source is generating a 60 Hz sine wave. The Pi-Section Transmission Line uses an RLC parameterized assuming a 60 Hz input, which matches the frequency of the voltage source. The plot shows the input and output terminal voltages of the transmission line. The two models show good agreement at steady state.

For the second simulation case, the voltage source is generating a 60 Hz sine wave with a 10 kHz modulation. The Pi-Section Transmission Line still uses an RLC parameterized assuming a 60 Hz input. It is clear that the custom frequency-dependent transmission line model is suitable for broader band signals whereas the pi-section model is only applicable for extremely narrow band signals.

References

[1] Morched, Atef, Bjorn Gustavsen, and Manoocher Tartibi. "A universal model for accurate calculation of electromagnetic transients on overhead lines and underground cables." IEEE Transactions on Power Delivery 14.3 (1999): 1032-1038.

[2] Dommel, Herman W. "Overhead line parameters from handbook formulas and computer programs." IEEE Transactions on Power Apparatus and Systems 2 (1985): 366-372.