Single-phase model
The half CP parameters are calculated at a given frequency; thus, it is considered as a frequency independent line model. This model is less accurate than frequency-dependent line and cable models. However, it can be successfully used to analyze cases with limited frequency dispersion.
The half CP model is based on the formulation of the classic CP line model, which neglects the frequency dependence of parameters and first assumes a lossless line. The losses are included at a later stage. The following figure shows the equivalent circuit representation of the EMT-type transmission line model.
The lossless single-phase transmission line is described by the following main equations:
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i_k (t)=\frac{v_k (t) }{Z_c } -I_{H,k} \\
i_m (t)=\frac{v_m (t) }{Z_c } -I_{H,m} \\
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where
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and
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are history currents defined as:
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I_{H,k}=\frac{v_m (t-\tau) }{Z_c } +i_m (t-\tau)\\
I_{H,m}=\frac{v_k (t-\tau) }{Z_c } +i_k (t-\tau)\\
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where,
and
are the nodal voltages;
and
are the injected current at both ends of the line;
is the characteristic impedance and
is the propagation time delay, respectively defined as:
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Z_c=\sqrt{\frac{L'} {C'}} \\
\tau=l \sqrt{L'C'} |
where
is the length of the line, and
and
are the inductance and capacitance per unit length of the line, respectively.
Note that the time-domain model described above creates a decoupling effect on the interconnected network. It is mentioned that the equation system provides an exact solution only when the propagation time is an integer multiple
of the simulation time step
, i.e.,
. Therefore, a linear interpolation is used when
.
Inclusion of losses
To include the losses
, the line is divided into two equal lossless models of halved propagation time. Then, the total
is lumped at three places (line ends
and line middle
) as shown in the following figure.
The resulted lossy line equivalent model to be implemented is given by:
where
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Z_{imp}=Z_c+\frac{R}{4} \\
I_k (t-\tau)=\left(\frac{1+H}{2} \right) \left[ { \frac {v_m (t-\tau)} {Z_{imp}} +Hi_m (t-\tau)} \right] +
\left(\frac{1-H}{2} \right) \left[ { \frac {v_k (t-\tau)} {Z_{imp}} +Hi_k (t-\tau)} \right] \\
I_m (t-\tau)=\left(\frac{1+H}{2} \right) \left[ { \frac {v_k (t-\tau)} {Z_{imp}} +Hi_k (t-\tau)} \right] +
\left(\frac{1-H}{2} \right) \left[ { \frac {v_m (t-\tau)} {Z_{imp}} +Hi_m (t-\tau)} \right] \\
H=\left( \frac{Z_c-R⁄4} {Z_c+R⁄4} \right)\\
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History current buffer
The half CP line is used to represent one Norton equivalent circuit of the lossy equivalent line. Therefore, two half CP model must be linked to each other to represent a transmission line. Both half CP components must be set with the same line parameters and connected through transceiver components. This connection is used to interchange the history current
information between both ends of the line. That is,
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h_x =\left(\frac{1+H}{2} \right) \left[ { \frac {v_x (t-\tau)} {Z_{imp}} +Hi_x (t-\tau)} \right] \\
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Then, the output history current
from the end
becomes the input history current
of the end
and vice versa. In the implementation of the model, a history buffer is kept and rotated for calculating the history current sources. The buffer length depends on the propagation delay and the simulation time step. The propagation delay must be greater to the integration time-step.
Note that the propagation delay of the line is the sum of what is accounted in the buffer and what is caused by the actual delay outside the components. The latter is referred as "extra delay", which must be indicated in the form of the model. For the digital simulation of the model on one simulator, the actual delay is caused by the transceiver. For the simulation on 2 simulators, the actual delay includes the delay in IO drive.
Example
Two half CP components must be connected through transceiver elements to represent a transmission line. Line parameters must be the same in both components. Next figures show how to build a transmission line using 1 and 3 transceivers.
References
"H. W. Dommel, "Digital computer solution of electromagnetic transients in single and multiphase networks," IEEE Trans. Power App. Syst., vol. pas-88, pp. 388-99, 04/ 1969."
