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# Half CP, 3-ph

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 components, 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 IO drive.

The single-phase version half CP, 1-ph is extended to the three-phase line by using a modal transformation to decouple the equations from phase domain to modal domain. The decoupled circuits are solved separately and transformed back to the phase domain. For a continuously transposed line, only sequence 0 and 1 are considered and a built-in Clarke transformation matrix is used. For an untransposed line (3 distinct modes) an input matrix is used.

# Mask and Parameters

## Parameters

Name | Description | Unit | Variable = {Possible Values} | |||
---|---|---|---|---|---|---|

Length | The length of the line | km | Length = {'1e-12, 1e12'} | |||

Continuously transposed | Transposition (Untransposed/Transposed) | transp = { 0, 1} | ||||

No {0} | Untransposed line | |||||

Yes {1} | Transposed line | |||||

R | Per unit length resistance for each phase (mode) | Ω/km | Resistance = {'0 1e12'} | |||

L | Per unit length inductance for each phase (mode) | H/km | Inductance = {'1e-12, 1e12'} | |||

C | Per unit length capacitor for each phase (mode) | F/km | Capacitance = {'1e-12, 1e12'} | |||

Transformation matrix (Ti) | Transformation matrix between mode current and phase current ([Iphase] = [Ti] x [Imode]); not used in the case of transposed line. For a continuously transposed line, only sequence 0 and 1 are considered and a built-in Clarke transformation matrix is used. | Ti = { [-1e64, 1e64] } | ||||

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 |
---|---|

net(a,b,c) | Power network connection of phases (a,b,c) of one side of the line |

## Inputs

Name | Description |
---|---|

hi_(a,b,c) | Historic current of phases (a,b,c) |

## Outputs

Name | Description |
---|---|

ho_(a,b,c) | Historic current of phases (a,b,c) |

Vt_(a,b,c) | Terminal voltage of phases (a,b,c) in V |

ols_(a,b,c) | Terminal current of phases (a,b,c) in A |

## 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:

where and are history currents defined as:

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:

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

### 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,

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.

## Three-phase model

The single-phase model is extended to the three-phase line by using a modal transformation to decouple the equations from phase domain to modal domain. The hatted vector are modal quantities:

where and are the series impedance and shunt admittance of the line, respectively. The transformation matrices are given from the following eigenvalue problem:

where is a diagonal matrix of the eigenvalues of the product . Also, the transformation matrices follow the relation:

For an untransposed line there are as many distinct modes as phases. In the half CP model, a constant and real transformation is used, which is calculated at a given model frequency. The LineData model can be used to obtain this matrix. For a continuously transposed line, only sequence 0 and 1 are considered and a built-in Clarke transformation matrix is used.

To find a solution in the three-phase system, the decoupled circuits are solved separately and transformed back to the phase domain. See the following figure:

## 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."

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 components, 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 IO drive.

The single-phase version half CP, 1-ph is extended to the three-phase line by using a modal transformation to decouple the equations from phase domain to modal domain. The decoupled circuits are solved separately and transformed back to the phase domain. For a continuously transposed line, only sequence 0 and 1 are considered and a built-in Clarke transformation matrix is used. For an untransposed line (3 distinct modes) an input matrix is used.

# Mask and Parameters

## Parameters

Name | Description | Unit | Variable = {Possible Values} | |||
---|---|---|---|---|---|---|

Length | The length of the line | km | Length = {'1e-12, 1e12'} | |||

Continuously transposed | Transposition (Untransposed/Transposed) | transp = { 0, 1} | ||||

No {0} | Untransposed line | |||||

Yes {1} | Transposed line | |||||

R | Per unit length resistance for each phase (mode) | Ω/km | Resistance = {'0 1e12'} | |||

L | Per unit length inductance for each phase (mode) | H/km | Inductance = {'1e-12, 1e12'} | |||

C | Per unit length capacitor for each phase (mode) | F/km | Capacitance = {'1e-12, 1e12'} | |||

Transformation matrix (Ti) | Transformation matrix between mode current and phase current ([Iphase] = [Ti] x [Imode]); not used in the case of transposed line. For a continuously transposed line, only sequence 0 and 1 are considered and a built-in Clarke transformation matrix is used. | Ti = { [-1e64, 1e64] } | ||||

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 |
---|---|

net(a,b,c) | Power network connection of phases (a,b,c) of one side of the line |

## Inputs

Name | Description |
---|---|

hi_(a,b,c) | Historic current of phases (a,b,c) |

## Outputs

Name | Description |
---|---|

ho_(a,b,c) | Historic current of phases (a,b,c) |

Vt_(a,b,c) | Terminal voltage of phases (a,b,c) in V |

ols_(a,b,c) | Terminal current of phases (a,b,c) in A |

## 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:

where and are history currents defined as:

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:

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

### 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,

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.

## Three-phase model

The single-phase model is extended to the three-phase line by using a modal transformation to decouple the equations from phase domain to modal domain. The hatted vector are modal quantities:

where and are the series impedance and shunt admittance of the line, respectively. The transformation matrices are given from the following eigenvalue problem:

where is a diagonal matrix of the eigenvalues of the product . Also, the transformation matrices follow the relation:

For an untransposed line there are as many distinct modes as phases. In the half CP model, a constant and real transformation is used, which is calculated at a given model frequency. The LineData model can be used to obtain this matrix. For a continuously transposed line, only sequence 0 and 1 are considered and a built-in Clarke transformation matrix is used.

To find a solution in the three-phase system, the decoupled circuits are solved separately and transformed back to the phase domain. See the following figure:

## 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."

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