Multi-conductor Line Models
Introduction
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:
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 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 equivalent line model, we have single complex quantities for the series impedance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{Z} = R + j X_{L} \, } and shunt admittance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{Y} = G + jB \, } of the line. But in the multi-conductor line, these single complex quantities are replaced by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \times n} matrices, where n is the number of conductors in the system.
For example, the four conductor system in Figure 2 has the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4 \times 4 } impedance matrix:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [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] \, }
And the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4 \times 4 } admittance matrix:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [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] \, }
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 to reduce the impedance and admittance matrices from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \times n} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3 \times 3 } matrices, i.e.
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [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] \, }
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [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] \, }
We call these matrices the Kron reduced impedance and admittance matrices.
Nominal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi } Line
The multi-conductor nominal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi} line model shown in Figure 3 is a direct substitution of the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 \times 1 } complex parameters in the single-phase nominal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi} Line with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \times n } matrices.
If we consider the Kron reduced case, [Z] and [Y] are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3 \times 3 } matrices as described in the previous section. The voltages and currents are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3 \times 1 } complex vectors of the form:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{V_{s}} = \left[ \begin{matrix} \boldsymbol{V_{s,a}} \\ \boldsymbol{V_{s,b}} \\ \boldsymbol{V_{s,c}} \end{matrix} \right] \, } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{I_{s}} = \left[ \begin{matrix} \boldsymbol{I_{s,a}} \\ \boldsymbol{I_{s,b}} \\ \boldsymbol{I_{s,c}} \end{matrix} \right] \, } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{V_{r}} = \left[ \begin{matrix} \boldsymbol{V_{r,a}} \\ \boldsymbol{V_{r,b}} \\ \boldsymbol{V_{r,c}} \end{matrix} \right] \, } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{I_{r}} = \left[ \begin{matrix} \boldsymbol{I_{r,a}} \\ \boldsymbol{I_{r,b}} \\ \boldsymbol{I_{r,c}} \end{matrix} \right] \, }
The ABCD parameters of the multi-conductor nominal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi}
line are:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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] \, }
Distributed Parameter Line
Refer to the distributed parameter model article for the detailed derivation of the model.
In the single-phase line model, we saw that it was possible to represent a distributed parameter line model using the same equivalent circuit as the nominal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi} model, but with adjusted line parameters Z and Y. This was called the equivalent Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi} model and the adjusted parameters could be calculated as follows:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{Z'} = \boldsymbol{Z}_{c} \sinh(\boldsymbol{\gamma} l) }
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\boldsymbol{Y'}}{2} = \frac{1}{\boldsymbol{Z}_{c}} \tanh \left( \frac{\boldsymbol{\gamma} l}{2} \right) }
Where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma = \sqrt{\boldsymbol{zy}} } is the propagation constant Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m^{-1}}
However in the multi-conductor line, the propagation constant would be a matrix of the form:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\gamma] = \left( [Z][Y] \right)^{\frac{1}{2}} }
There is no straightforward method for calculating the hyperbolic sine and tangent functions of a matrix (the hyperbolic functions can be expanded as a 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 ABCD parameters of the multi-conductor distributed parameter line are (in modal form):
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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]\, }
Where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{V_{s}'} } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{I_{s}'} } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{V_{r}'} } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{I_{r}'} } are modal sending end and receiving end voltages and currents respectively:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{V_{s}'} = \left[ \begin{matrix} \boldsymbol{V_{s0}} \\ \boldsymbol{V_{s1}} \\ \boldsymbol{V_{s2}} \end{matrix} \right] \, } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{I_{s}'} = \left[ \begin{matrix} \boldsymbol{I_{s0}} \\ \boldsymbol{I_{s1}} \\ \boldsymbol{I_{s2}} \end{matrix} \right] \, } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{V_{r}'} = \left[ \begin{matrix} \boldsymbol{V_{r0}} \\ \boldsymbol{V_{r1}} \\ \boldsymbol{V_{r2}} \end{matrix} \right] \, } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{I_{r}'} = \left[ \begin{matrix} \boldsymbol{I_{r0}} \\ \boldsymbol{I_{r1}} \\ \boldsymbol{I_{r2}} \end{matrix} \right] \, }
The ABCD parameters are diagonal sub-matrices of the form:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[ \cosh{(\gamma l)} \right] = \left[ \begin{matrix} \cosh{(\gamma_0 x)} & & \\ & \cosh{(\gamma_1 x)} & \\ & & \cosh{(\gamma_2 x)}\end{matrix} \right] } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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] }
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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] }
For details on the derivation of the modal equations, see the main distributed parameter model article.
References
- [1] Wedepohl, L. M., "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., "Propagation on Overhead Transmission Lines I-Theory of Modal Analysis ", IEEE Transactions on Power Apparatus and Systems, Vol 84(3), 1965