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2020-11-22T07:36:53Z

Created page with "== Introduction == Image:DSCI0402.JPG|right|thumb|250px|Figure 1. Three-phase, single-circuit tower line (image courtesy of [http://en.wikipedia.org/wiki/File:DSCI0402.JPG..."

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== Introduction ==

[[Image:DSCI0402.JPG|right|thumb|250px|Figure 1. Three-phase, single-circuit tower line (image courtesy of [http://en.wikipedia.org/wiki/File:DSCI0402.JPG Wikipedia])]]

[[Single-Phase Line Models|Single-phase equivalent line models]] (or positive-sequence line models) can be used quite accurately in three-phase systems when the system is balanced and the lines are perfectly transposed. However, when unbalanced systems or untransposed lines are being studied, these models break down and a full three-phase multi-conductor model is necessary.

Consider the three-phase, single circuit tower line in Figure 1, which shows three phase conductors and an earth wire. A segmental length of this multi-conductor system can be represented by the equivalent circuit in Figure 2 below:

[[Image:Multiconductor segment.png|left|thumb|700px|Figure 2. Circuit representation of a multi-conductor line segment]]
<br clear=all>

We can see that in a multi-conductor system, there is mutual coupling between the phase conductors (a, b and c), represented by the shunt inductances and capacitances. In the [[Single-Phase Line Models|single-phase equivalent line model]], these mutual couplings are ignored. Note that there could also be resistive coupling between phases, but is not shown in Figure 2 since resistive coupling is normally assumed to be negligible in overhead lines (i.e. shunt conductances = 0).

Recall that in the [[Single-Phase Line Models|single-phase equivalent line model]], we have single [[Complex_Impedance|complex quantities]] for the series impedance <math>\boldsymbol{Z} = R + j X_{L} \, </math> and shunt admittance <math>\boldsymbol{Y} = G + jB \, </math> of the line. But in the multi-conductor line, these single complex quantities are replaced by <math>n \times n</math> matrices, where n is the number of conductors in the system.

For example, the four conductor system in Figure 2 has the <math>4 \times 4 </math> impedance matrix:

: <math> [Z] = \left[ \begin{matrix}
Z_{aa} & Z_{ab} & Z_{ac} & | & Z_{ae}\\
Z_{ba} & Z_{bb} & Z_{bc} & | & Z_{be} \\
Z_{ca} & Z_{cb} & Z_{cc} & | & Z_{ce} \\
-- & -- & -- & | & -- \\
Z_{ea} & Z_{eb} & Z_{ec} & | & Z_{ee} \end{matrix} \right] \, </math>

And the <math>4 \times 4 </math> admittance matrix:

: <math> [Y] = \left[ \begin{matrix}
Y_{aa} & Y_{ab} & Y_{ac} & | & Y_{ae}\\
Y_{ba} & Y_{bb} & Y_{bc} & | & Y_{be} \\
Y_{ca} & Y_{cb} & Y_{cc} & | & Y_{ce} \\
-- & -- & -- & | & -- \\
Y_{ea} & Y_{eb} & Y_{ec} & | & Y_{ee} \end{matrix} \right] \, </math>

Since we assume that under normal conditions, the earth wire is at zero potential (i.e. no voltage between the earth wire and neutral), we can use the [[Kron Reduction|Kron reduction]] to reduce the impedance and admittance matrices from <math>n \times n</math> to <math>3 \times 3 </math> matrices, i.e.

: <math> [Z'] = \left[ \begin{matrix}
Z_{aa}' & Z_{ab}' & Z_{ac}' \\
Z_{ba}' & Z_{bb}' & Z_{bc}' \\
Z_{ca}' & Z_{cb}' & Z_{cc}' \end{matrix} \right] \, </math>

: <math> [Y'] = \left[ \begin{matrix}
Y_{aa}' & Y_{ab}' & Y_{ac}' \\
Y_{ba}' & Y_{bb}' & Y_{bc}' \\
Y_{ca}' & Y_{cb}' & Y_{cc}' \end{matrix} \right] \, </math>

We call these matrices the '''Kron reduced''' impedance and admittance matrices.

== Nominal <math>\pi </math> Line ==

[[Image:Multi_pi_line.png|right|thumb|400px|Figure 3. Multi-conductor nominal <math>\pi</math> line model]]

The multi-conductor nominal <math>\pi</math> line model shown in Figure 3 is a direct substitution of the <math> 1 \times 1 </math> complex parameters in the [[Single-Phase_Line_Models#Nominal_.CF.80_Line|single-phase nominal <math>\pi</math> Line]] with <math> n \times n </math> matrices.

If we consider the [[Kron_Reduction|Kron reduced case]], [Z] and [Y] are <math> 3 \times 3 </math> matrices as described in the previous section. The voltages and currents are <math> 3 \times 1 </math> complex vectors of the form:

: <math> \boldsymbol{V_{s}} = \left[ \begin{matrix} \boldsymbol{V_{s,a}} \\ \boldsymbol{V_{s,b}} \\ \boldsymbol{V_{s,c}} \end{matrix} \right] \, </math>, <math> \boldsymbol{I_{s}} = \left[ \begin{matrix} \boldsymbol{I_{s,a}} \\ \boldsymbol{I_{s,b}} \\ \boldsymbol{I_{s,c}} \end{matrix} \right] \, </math>, <math> \boldsymbol{V_{r}} = \left[ \begin{matrix} \boldsymbol{V_{r,a}} \\ \boldsymbol{V_{r,b}} \\ \boldsymbol{V_{r,c}} \end{matrix} \right] \, </math> and <math> \boldsymbol{I_{r}} = \left[ \begin{matrix} \boldsymbol{I_{r,a}} \\ \boldsymbol{I_{r,b}} \\ \boldsymbol{I_{r,c}} \end{matrix} \right] \, </math>

The [[Overhead_Line_Models#ABCD_Parameters_.28Generalised_Line_Constants.29|ABCD parameters]] of the multi-conductor nominal <math>\pi</math> line are:

: <math> \left[ \begin{matrix} \boldsymbol{V_{s}} \\ \\ \boldsymbol{I_{s}} \end{matrix} \right] = \left[ \begin{matrix}
\left( I + \frac{[Z] [Y]}{2} \right) & [Z] \\ \\
Y \left( I + \frac{[Z] [Y]}{4} \right) & \left( I + \frac{[Z] [Y]}{2} \right) \end{matrix} \right]
\left[ \begin{matrix} \boldsymbol{V_{r}} \\ \\ \boldsymbol{I_{r}} \end{matrix} \right] \, </math>

<br clear=all>

== Distributed Parameter Line ==
''Refer to the [[Distributed_Parameter_Line_Model|distributed parameter model article]] for the detailed derivation of the model.''

In the [[Single-Phase Line Models|single-phase line model]], we saw that it was possible to represent a [[Single-Phase_Line_Models#Equivalent_.CF.80_Line|distributed parameter line model]] using the same equivalent circuit as the nominal <math>\pi</math> model, but with adjusted line parameters '''Z''' and '''Y'''. This was called the [[Single-Phase_Line_Models#Equivalent_.CF.80_Line|equivalent <math>\pi</math> model]] and the adjusted parameters could be calculated as follows:

: <math>\boldsymbol{Z'} = \boldsymbol{Z}_{c} \sinh(\boldsymbol{\gamma} l) </math>

: <math>\frac{\boldsymbol{Y'}}{2} = \frac{1}{\boldsymbol{Z}_{c}} \tanh \left( \frac{\boldsymbol{\gamma} l}{2} \right) </math>

Where <math>\gamma = \sqrt{\boldsymbol{zy}} </math> is the propagation constant <math>m^{-1}</math>

However in the multi-conductor line, the propagation constant would be a matrix of the form:

: <math>[\gamma] = \left( [Z][Y] \right)^{\frac{1}{2}} </math>

There is no straightforward method for calculating the hyperbolic sine and tangent functions of a matrix (the hyperbolic functions can be expanded as a [https://en.wikipedia.org/wiki/Hyperbolic_function#Taylor_series_expressions Taylor series], but this is still not particularly easy to compute). This led to the development of the modal transformation, which is a method for decoupling the phases of the impedance and admittance matrices.

The [[Overhead_Line_Models#ABCD_Parameters_.28Generalised_Line_Constants.29|ABCD parameters]] of the multi-conductor distributed parameter line are (in modal form):

: <math> \left[ \begin{matrix} \boldsymbol{V_{s}'} \\ \\ \boldsymbol{I_{s}'} \end{matrix} \right] = \left[ \begin{matrix}
\left[ \cosh{(\gamma l)} \right] & \left[ \boldsymbol{Z_{c}} \sinh{(\gamma l)} \right] \\ \\
\left[ \frac{1}{\boldsymbol{Z_{c}}} \sinh{(\gamma l)} \right] & \left[ \cosh{(\gamma l)}\right] \end{matrix} \right]
\left[ \begin{matrix} \boldsymbol{V_{r}'} \\ \\ \boldsymbol{I_{r}'} \end{matrix} \right]\, </math>

Where <math> \boldsymbol{V_{s}'} </math>, <math> \boldsymbol{I_{s}'} </math>, <math> \boldsymbol{V_{r}'} </math> and <math> \boldsymbol{I_{r}'} </math> are modal sending end and receiving end voltages and currents respectively:

: <math> \boldsymbol{V_{s}'} = \left[ \begin{matrix} \boldsymbol{V_{s0}} \\ \boldsymbol{V_{s1}} \\ \boldsymbol{V_{s2}} \end{matrix} \right] \, </math>, <math> \boldsymbol{I_{s}'} = \left[ \begin{matrix} \boldsymbol{I_{s0}} \\ \boldsymbol{I_{s1}} \\ \boldsymbol{I_{s2}} \end{matrix} \right] \, </math>, <math> \boldsymbol{V_{r}'} = \left[ \begin{matrix} \boldsymbol{V_{r0}} \\ \boldsymbol{V_{r1}} \\ \boldsymbol{V_{r2}} \end{matrix} \right] \, </math> and <math> \boldsymbol{I_{r}'} = \left[ \begin{matrix} \boldsymbol{I_{r0}} \\ \boldsymbol{I_{r1}} \\ \boldsymbol{I_{r2}} \end{matrix} \right] \, </math>

The ABCD parameters are diagonal sub-matrices of the form:

: <math> \left[ \cosh{(\gamma l)} \right] =
\left[ \begin{matrix}
\cosh{(\gamma_0 x)} & & \\
& \cosh{(\gamma_1 x)} & \\
& & \cosh{(\gamma_2 x)}\end{matrix} \right] </math>, <math> \left[ \boldsymbol{Z_{c}} \sinh{(\gamma l)} \right] = \left[ \begin{matrix}
\boldsymbol{Z_0} \sinh{(\gamma_0 x)} & & \\
& \boldsymbol{Z_1} \sinh{(\gamma_1 x)} & \\
& & \boldsymbol{Z_2} \sinh{(\gamma_2 x)} \end{matrix} \right] </math>

: <math> \left[ \frac{1}{\boldsymbol{Z_{c}}} \sinh{(\gamma l)} \right] =
\left[ \begin{matrix}
\frac{1}{\boldsymbol{Z_0}} \sinh{(\gamma_0 x)} & & \\
& \frac{1}{\boldsymbol{Z_1}} \sinh{(\gamma_1 x)} & \\
& & \frac{1}{\boldsymbol{Z_2}} \sinh{(\gamma_2 x)}\end{matrix} \right] </math>

For details on the derivation of the modal equations, see the main [[Distributed_Parameter_Line_Model#Multi-conductor_Distributed_Parameter_Model|distributed parameter model article]].

== References ==

:* [1] Wedepohl, L. M., "[http://dx.doi.org/10.1049/piee.1963.0314 Application of matrix methods to the solution of travelling-wave phenomena in polyphase systems]", Proceedings of the IEE, Vol. 110(12), 1963

:* [2] Hedman, D. E., "[http://dx.doi.org/10.1109/TPAS.1965.4766176 Propagation on Overhead Transmission Lines I-Theory of Modal Analysis ]", IEEE Transactions on Power Apparatus and Systems, Vol 84(3), 1965

== Related Topics ==

:* [[Overhead Line Models]]
:* [[Single-Phase Line Models]]
:* [[Distributed Parameter Line Model]]

[[Category:Modelling / Analysis]]

Multi-conductor Line Models - Revision history

Jules: Created page with "== Introduction == Image:DSCI0402.JPG|right|thumb|250px|Figure 1. Three-phase, single-circuit tower line (image courtesy of [http://en.wikipedia.org/wiki/File:DSCI0402.JPG..."