Transmission Line (3ph)

Purpose

3-phase transmission line

Library

Electrical / Passive Components

Description

This component implements a three-phase transmission line. A transmission line is characterized by a uniform distribution of inductances, resistances and neutral capacitances along the line. In multi-wire lines, there are also uniformly distributed mutual inductances and coupling capacitances.

The user has the choice between two different implementations: one with series-connected pi sections of lumped elements and another one with distributed parameters based on traveling wave theory. A stiff solver is recommended for simulating models containing this component.

Pi-Section Line

In many cases, the uniformly distributed parameters of a transmission line can be approximated by a series of pi sections consisting of lumped inductors, capacitors and resistors. Fig. 196 illustrates a single pi section exemplified for a 2-phase line. Depending on the desired fidelity at higher frequencies, the number of series-connected pi sections can be configured.

Fig. 196 Schematic of a single Pi-Section for a 2-phase line

Let \(l\) be the length of the line and \(n\) the number of pi sections representing the line. The inductance \(L\), the resistance \(R\), the neutral capacitance \(C_N\) as well as the coupling capacitances \(C_{ij}\) and mutual inductances \(L_{ij}\) of the discrete elements can then be calculated from their per-unit-length counterparts \(L'\), \(R'\), \(C_{N}'\), \(C_{ij}'\) and \(L_{ij}'\) using the following equations:

\[L = \frac{l}{n} L', \qquad R = \frac{l}{n} R', \qquad C_{N} = \frac{l}{n} C_{N}'\]

\[C_{ij} = \frac{l}{n} C_{ij}', \qquad L_{ij} = \frac{l}{n} L_{ij}'\]

It is possible to specify the parameters for each phase individually in order to model asymmetric lines. In this case, the parameters must be provided in vector format. Otherwise, the parameter can be a scalar assigning the same value to all phases.

Distributed Parameter Line

The implementation of a distributed parameter line is based on the traveling wave theory, which describes the time delay phenomenon. This approach is numerically more efficient due to the absence of numerous state variables and should be used in large models.

Modeling asymmetric lines is not supported, therefore all parameters need to be scalar.

Single-Phase Lossless Line

Fig. 197 A single-phase lossless line

Consider a lossless transmission line with inductance \(L'\) and capacitance \(C'\) per unit length. At a certain point \(x\) along the total length \(d\), the relation between the line voltage and current can be described with partial differential equations:

\[-\frac{\partial e}{\partial x} = L'\cdot\frac{\partial i}{\partial t}\]

\[-\frac{\partial i}{\partial x} = C'\cdot\frac{\partial e}{\partial t}\]

Since a wave entering the sending end “s” of the line must remain unchanged when it arrives at the receiving end “r” (and vice versa), the following expression is derived:

\[i_\mathrm{s} = \frac{1}{Z}\cdot e_\mathrm{s}(t)-I_\mathrm{sh}\]

\[i_\mathrm{r} = \frac{1}{Z}\cdot e_\mathrm{r}(t)-I_\mathrm{rh}\]

where

\[I_\mathrm{sh} = \frac{1}{Z}\cdot e_\mathrm{r}(t-\tau)+i_\mathrm{r}(t-\tau)\]

\[I_\mathrm{rh} = \frac{1}{Z}\cdot e_\mathrm{s}(t-\tau)+i_\mathrm{s}(t-\tau)\]

with surge impedance \(Z = \sqrt{\frac{L'}{C'}}\) and travel time \(\tau = d\cdot\sqrt{L'C'}\). This model can be represented by a two-port equivalent circuit, where the electrical conditions at port “s” are transferred after a time delay \(\tau\) to port “r” via the controlled current source \(I_\mathrm{rh}\) as in Fig. 198.

Fig. 198 Two-port equivalent circuit for model

Approximation of Series Resistance

Fig. 199 Approximation of Series Resistance

Since the shunt conductance is usually negligible, the series resistance is responsible for the major part of the power losses. Such series resistance can be approximated by three lumped resistors, two of which with the value \(\frac{R}{4}\) are placed at both ends of the line while one with the value \(\frac{R}{2}\) is placed in the middle (Fig. 199). \(R = R'\cdot d\) is the total series resistance of the line. After aggregation and substitution, the original expression of the two equivalent current source becomes:

\[I_\mathrm{sh} = \frac{1+h}{2}\cdot\left(\frac{1}{Z_\mathrm{R}}\cdot e_\mathrm{r}(t-\tau)+h\cdot i_\mathrm{r}(t-\tau)\right)+\frac{1-h}{2}\left(\frac{1}{Z_\mathrm{R}}\cdot e_\mathrm{s}(t-\tau)+h\cdot i_\mathrm{s}(t-\tau)\right)\]

\[I_\mathrm{rh} = \frac{1+h}{2}\cdot\left(\frac{1}{Z_\mathrm{R}}\cdot e_\mathrm{s}(t-\tau)+h\cdot i_\mathrm{s}(t-\tau)\right)+\frac{1-h}{2}\left(\frac{1}{Z_\mathrm{R}}\cdot e_\mathrm{r}(t-\tau)+h\cdot i_\mathrm{r}(t-\tau)\right)\]

with \(Z_\mathrm{R} = Z+\frac{R}{4}\) and \(h = \frac{Z-\frac{R}{4}}{Z+\frac{R}{4}}\).

Three-Phase Line

Fig. 200 Three-phased line

The differential equations of a 3-phase system with vector variables \(\vec{e}=[e_a,e_b,e_c]^T\), \(\vec{i}=[i_a,i_b,i_c]^T\) can be expressed as:

\[-\frac{\partial \vec{e}}{\partial x} = \mathbf{L'}\cdot\frac{\partial \vec{i}}{\partial t} + \mathbf{R'}\cdot\vec{i}\]

\[-\frac{\partial \vec{i}}{\partial x} = \mathbf{C'}\cdot\frac{\partial \vec{e}}{\partial t}\]

Under the assumption of symmetrical phase parameters, the per unit length inductance, capacitance and resistance can be written in matrix form:

\[\begin{split}\mathbf{L'} = \left[ \begin{array}{ccc} L_S' & L_M' & L_M' \\ L_M' & L_S' & L_M' \\ L_M' & L_M' & L_S' \\ \end{array} \right]\end{split}\]

\[\begin{split}\mathbf{C'} = \left[ \begin{array}{ccc} C_E'+2\cdot C_K' & -C_K' & -C_K' \\ -C_K' & C_E'+2\cdot C_K' & -C_K' \\ -C_K' & -C_K' & C_E'+2\cdot C_K' \\ \end{array} \right]\end{split}\]

\[\begin{split}\mathbf{R'} = \left[ \begin{array}{ccc} R' & 0 & 0 \\ 0 & R' & 0 \\ 0 & 0 & R' \\ \end{array} \right]\end{split}\]

The presence of off-diagonal elements (mutual inductance and coupling capacitance) in the matrix make it difficult to solve the equation system. However, this can be overcome with the help of modal transformations. If the differential equations are multiplied by a transformation matrix \(\mathbf{T}\) on the left side

\[-(\mathbf{T} \cdot \frac{\partial \vec{e}}{\partial x}) = (\mathbf{T} \cdot \mathbf{L'} \cdot \mathbf{T}^\mathrm{-1})\cdot (\mathbf{T} \cdot\frac{\partial \vec{i}}{\partial t}) + (\mathbf{T}\cdot \mathbf{R'} \cdot \mathbf{T}^\mathrm{-1})\cdot (\mathbf{T} \cdot \vec{i})\]

\[-(\mathbf{T} \cdot\frac{\partial \vec{i}}{\partial x}) = (\mathbf{T} \cdot \mathbf{L'} \cdot \mathbf{T}^\mathrm{-1})\cdot (\mathbf{T}\cdot\frac{\partial \vec{e}}{\partial t})\]

with

\[\begin{split}\mathbf{T} = \left[ \begin{array}{ccc} 1 & 1 & 1 \\ 1 & -2 & 1 \\ 1 & 1 & -2 \\ \end{array} \right]\end{split}\]

the off-diagonal elements of the inductance, capacitance and resistance matrix can be eliminated:

\[\begin{split}\mathbf{L'} _\mathrm{mod} = \mathbf{T} \cdot \mathbf{L'} \cdot \mathbf{T}^\mathrm{-1} = \left[ \begin{array}{ccc} L_u' & 0 & 0 \\ 0 & L_v' & 0 \\ 0 & 0 & L_w' \\ \end{array} \right]\end{split}\]

\[\begin{split}\mathbf{C'} _\mathrm{mod} = \mathbf{T} \cdot \mathbf{C'} \cdot \mathbf{T}^\mathrm{-1} = \left[ \begin{array}{ccc} C_u' & 0 & 0 \\ 0 & C_v' & 0 \\ 0 & 0 & C_w' \\ \end{array} \right]\end{split}\]

\[\begin{split}\mathbf{R'} _\mathrm{mod} = \mathbf{T} \cdot \mathbf{R'} \cdot \mathbf{T}^\mathrm{-1} = \left[ \begin{array}{ccc} R' & 0 & 0 \\ 0 & R' & 0 \\ 0 & 0 & R' \\ \end{array} \right] = \mathbf{R'}\end{split}\]

Thus the original system, in which the three phases are coupled, has been converted to three decoupled systems in the modal domain (denoted as \(u\), \(v\), \(w\); see Fig. 201). They can be treated separately in the same way as the single-phase system.

Fig. 201 Three decoupled systems in the modal domain

The simulation output in the modal domain should be eventually transformed back into the phase domain via the inverse of the matrix \(\mathbf{T'}\).

Parameters

Self inductance per unit length

The series self inductance \(L_\mathrm{S}'\) per unit length. If the length \(l\) is specified in meters \((\mathrm{m})\), the unit of \(L_\mathrm{S}'\) is henries per meter \((\mathrm{H}/\mathrm{m})\).

For a pi-section line, the self inductance can be specified individually per phase by providing a 3-element vector of the form \(L' =\left[ L_1' \quad L_2' \quad L_3' \right]\).

Mutual inductance per unit length

The series mutual inductance \(L_\mathrm{M}'\) per unit length. If the length \(l\) is specified in meters \((\mathrm{m})\), the unit of \(L_\mathrm{M}'\) is henries per meter \((\mathrm{H}/\mathrm{m})\).

In a pi-section line, the mutual inductances \(M_{ij}'\) between the i-th and j-th phase can be specified individually by providing a 3-element vector \(\mathrm{M'} = \left[ M_{12}' \quad M_{13}' \quad M_{23}' \right]\) containing the upper triangular coupling matrix.

Resistance per unit length

The series resistance \(R'\) per unit length. If the length \(l\) is specified in meters \((\mathrm{m})\), the unit of \(R'\) is ohms per meter \((\Omega / \mathrm{m})\).

For a pi-section line, the parameter can be a vector of the form \(R' =\left[ R_1' \quad R_2' \quad R_3' \right]\).

Neutral capacitance per unit length

The line-to-neutral capacitance \(C_\mathrm{N}'\) per unit length. If the length \(l\) is specified in meters \((\mathrm{m})\), the unit of \(C_\mathrm{N}'\) is farads per meter \((\mathrm{F}/\mathrm{m})\).

For a pi-section line, this parameter can be a vector of the form \(C_\mathrm{N}' = \left[ C_\mathrm{N1}' \quad C_\mathrm{N2}' \quad C_\mathrm{N1}' \right]\).

Coupling capacitance per unit length

The line-to-line capacitance \(C_\mathrm{C}'\) per unit length. If the length \(l\) is specified in meters \((\mathrm{m})\), the unit of \(C_\mathrm{C}'\) is farads per meter \((\mathrm{F}/\mathrm{m})\).

In a pi-section line, the coupling capacitance \(C_{ij}'\) between the i-th and j-th phase can be specified individually by providing a 3-element vector \(\mathrm{C'} = \left[ C_{12}' \quad C_{13}' \quad C_{23}' \right]\) containing the upper triangular coupling matrix.

Length

The length \(l\) of the line. The unit of \(l\) must match the units \(L_\mathrm{S}'\), \(L_\mathrm{M}'\), \(R'\), \(C_\mathrm{N}'\) and \(C_\mathrm{C}'\) are based on.

Number of pi sections

Number of sections used to model the transmission line. The default is 3. This parameter only affects the pi-section implementation.

References

• H. Dommel: “Digital Computer Solution of Electromagnetic Transients in Single and Multiple Networks”, IEEE Transactions on Power Apparatus and Systems, Vol. PAS88, No. 4, April 1969