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Table of Contents
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The distributed parameters line (DPL) theory is used to represent a half of the classic constant parameter (CP) model [1]. Overall, the CP model assumes that the line parameters R, L and C are independent of the frequency effects caused by the skin effect on phase conductors and on the ground. The model considers L and C to be distributed (ideal line) and R to be lumped at three places (R/4 on both ends and R/2 in the middle). The shunt conductance G is taken as zero.

Two half CP components must be connected through transceiver elements (or real-time simulator I/Os) to represent a transmission line. Line parameters must be the same in both components. The implementation of the half CP follows the same formulation from the standard CP line [1]. However, note that with the half CP component, the propagation delay of the line is the sum of what is accounted in a buffer inside the component and what is caused by the actual delay outside the components. The latter is referred as "extra delay" 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 I/O drivers.

For the three-phase version of this half CP model, please see half CP, 3-ph.


Mask and Parameters

Parameters

NameDescriptionUnitVariable = {Possible Values}
DescriptionUse this field to add information about the component
Description = {'string'}
LengthThe length of the linemlength = {'1e-12, 1e12'}
R'Per unit length resistanceΩ/kmResistance = {'0, 1e12'}
L'Per unit length inductanceH/km

Inductance = {'1e-12 1e12'}

C'Per unit length capacitanceF/kmCapacitance = {'1e-12, 1e12'}
Extra delay

The extra delay variable is an integer that refers to the number of delays produced outside the half CP components. For the digital simulation of the model on one simulator, the extra delay is caused by the transceiver. For the simulation on two simulators, the extra delay includes the delay in IO drive.


extradelaynb = { [0, 200] }


Ports, Inputs, Outputs and Signals Available for Monitoring

Ports

This component supports a single-phase transmission line 

Name

Description

netPower network connection of one side of the line


Inputs


Name

Description

hi

Historic current FROM the other side of the line


Outputs


Name

Description

ho

Historic current TO the other side of the line

Vt

Terminal voltage in V


Sensors

None

Theoretical Background

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:

Mathblock
alignmentcenter
 i_k (t)=\frac{v_k (t) }{Z_c } -I_{H,k} \\
 i_m (t)=\frac{v_m (t) }{Z_c } -I_{H,m} \\

where 

Mathinline
body--uriencoded--I_%7BH,k%7D
 and 
Mathinline
body--uriencoded--I_%7BH,m%7D
 are history currents defined as:

Mathblock
alignmentcenter
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)\\

where, 

Mathinline
bodyv_k
 and
Mathinline
bodyv_m
 are the nodal voltages;
Mathinline
bodyi_k
 and
Mathinline
bodyi_m
 are the injected current at both ends of the line;
Mathinline
bodyZ_c
 is the characteristic impedance and
Mathinline
body\tau
 is the propagation time delay, respectively defined as:

Mathblock
alignmentcenter
Z_c=\sqrt{\frac{L'} {C'}} \\
\tau=l \sqrt{L'C'}

where

Mathinline
bodyl
 is the length of the line, and
Mathinline
bodyL'
 and
Mathinline
bodyC'
 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

Mathinline
bodyK
 of the simulation time step
Mathinline
body\Delta t
, i.e.,
Mathinline
body\tau = K\Delta t
. Therefore, a linear interpolation is used when
Mathinline
body\tau ≠ K\Delta t
.

Inclusion of losses

To include the losses

Mathinline
bodyR'
, the line is divided into two equal lossless models of halved propagation time. Then, the total
Mathinline
bodyR=lR'
 is lumped at three places (line ends
Mathinline
bodyR/4
 and line middle
Mathinline
bodyR/2
) as shown in the following figure.

The resulted lossy line equivalent model to be implemented is given by:

where

Mathblock
alignmentcenter
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)\\

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

Mathinline
bodyh_x
 information between both ends of the line. That is, 

Mathblock
alignmentcenter
h_x =\left(\frac{1+H}{2} \right)  \left[  { \frac {v_x (t-\tau)} {Z_{imp}} +Hi_x (t-\tau)} \right] \\

Then, the output history current

Mathinline
bodyh_o
 from the end
Mathinline
bodyk
 becomes the input history current
Mathinline
bodyh_i
 of the end
Mathinline
bodym
 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 componentsNext figures show how to build a transmission line using 1 and 3 transceivers. 

References

  1. "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."