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Transmission Lines and Cables
Guidelines for transmission line/cable models
Line models
Transmission lines are characterized by two electrical parameters, i.e., the series impedance Z (longitudinal field effects), and the shunt admittance Y (transversal field effects). Considering the nature of these parameters, time-domain line models can be divided in two groups: lumped- and distributed-parameters models. In the lumped-parameter models, both Z and Y are calculated at a single frequency. These models are adequate for steady-state studies, where there is only one frequency of interest. For transient studies, the most appropriate models are those that consider the parameters distributed along the distance. This group of models may consider the parameters Z and Y as constant or frequency-dependent.
The following figure depicts a general classification of the line models available in HYPERSIM.
RL coupled
In the case of very short lines in distribution systems, the capacitance of the line is negligible. Therefore, these distribution lines are typically modeled based on conductor series resistance and inductance only, i.e., as RL coupled branches.
Model advantages: Simple model.
Model disadvantages: Cannot be used as a decoupled element in the network. Only recommended to represent very short transmission lines in distribution systems.
PI model
The PI-line model is an RLC approximation of the transmission line that is exact only at the specified frequency, typically the power frequency. A line can be modeled by several PI sections in series, which helps to emulate traveling wave effects; however, this is not quite accurate. The application of the PI model is only adequate for steady-state purposes.
Model advantages: Simple model.
Model disadvantages: Only adequate for steady state analysis. Cannot be used as a decoupled element in the network.
Constant parameter (CP) model
The constant parameter (CP) model, also known as Bergeron model [1], uses the traveling wave method to model a transmission line. This model is characterized by an ideal transfer function of traveling wave to which resistive losses are added, [1]. The CP model requires a small computational burden. However, since the propagation (traveling wave) modes cannot be properly represented at high frequencies, the CP model is only recommended for modeling lines in analysis of problems with limited frequency dispersion.
Model advantages: It can emulate the traveling wave effects in transients, for example during faults. Can be used as a decoupling element in parallel simulations.
Model disadvantages: Surge impedance and traveling wave (propagation) function are only valid at one frequency. Typically, high‐frequency damping is neglected in this model.
Frequency-dependent (FD) line model (Modal-type)
The FD-line model is based on modal decomposition techniques [2]. This model improves the frequency response of CP line by correctly fitting the surge impedance and propagation functions for each mode. This model provides accurate representations for a large class of lines. However, its accuracy is restricted to aerial lines with symmetric or nearly symmetric configurations. The reason for this is that the FD-line model uses a constant modal transformation matrix, evaluated at a fixed frequency, which is an inaccurate assumption for unbalanced systems.
Please note that cables are always unbalanced since the capacitance to ground from inner wire and outer wire cannot be equal. Therefore, this FD model-domain approach should not be used to simulate cables. The frequency dependent option for cables in HYPERSIM is the wideband model, which is developed in phase domain.
Model advantages: Accurate representation of frequency-dependent parameters for balanced lines. Relatively fast model. Can be used as a decoupling element in parallel simulations.
Model disadvantages: Restricted to aerial lines with symmetric or nearly symmetric configurations. Not applicable for cables.
Wideband (WB) line/cable model (Phase-type)
The wideband line/cable model, also called the universal line model (ULM) [3], is a fully accurate line and cable model in which both surge impedance and propagation functions are calculated into the phase domain directly. Thus, it provides highly accurate results for the simulation of coupling effects between parallel lines, compared to the FD model. It is highly recommended for transient studies involving cables.
Model advantages: Most accurate model for lines and cables. Can be used as a decoupling element in parallel simulations.
Model disadvantages: The model is slower than all other model due to its complexity. Surge impedance and propagation function must be approximated by rational functions with an external “fitter”.
Basic modeling guidelines
The following table summarizes the line models and their main characteristics, [4].
Line model | Accuracy | Computational burden | Recommended applications |
RL coupled | Low | Low-medium | General 50 Hz – 60 Hz steady state studies for very short lines in distribution systems |
PI | Medium | Low-medium | General 50 Hz – 60 Hz, steady state studies |
CP | Good | Low | Transient simulations, fault studies |
FD | Very good | Medium | High‐accuracy transients simulation involving overhead lines, balanced and nearly unbalanced |
WB | Excellent | High | High‐accuracy transient simulations especially with cables and unbalanced lines |
It is noted that for the distributed parameter (traveling wave) models, the simulation time step must be based on the shortest response of the transmission line. Thus, the propagation delay must be greater than the simulation time step. The travel time or propagation delay of the line τ is defined as:
where is the length of the line and is the propagation velocity defined as:
where and are respectively the inductance and capacitance of the line per unit length. The propagation delay can be compared to the time step ∆t to evaluate if a lumped-parameter (for instance PI) or a traveling wave model (for instance CP) is appropriate. The following figure shows a decision tree for the selection of the appropriate transmission line model according to the propagation delay. Note that to achieve the best accuracy for the high-frequency studies, the user can select the FD-line models instead of the CP. However, the FD models are slower than CP models; thus, they are only recommended to study reduced networks.
Overhead lines may be classified according to length. Considering typical values for R, L and C per unit length (see the note in Typical Electrical Parameters), they can be classified as [5]:
- Short lines: lines shorter than about 80 km. They have negligible capacitance and may be represented by their series impedance (RL coupled model).
- Medium-length lines: lines with lengths in the range of 80 km to about 200 km. They may be represented by PI or CP models.
- Long lines: longer than about 200 km. For such lines the distributed effects of the parameters are significant. They need to be represented by CP line models.
Note that for very short lines in distribution systems, the capacitance of the line is negligible. Therefore, these distribution lines are typically modeled based on conductor series resistance and inductance only, i.e., as RL coupled branches.
CIGRE guidelines
The study of transients in power systems involves frequency range from dc to about 50 MHz and in specific cases even more. Usually, transients above power frequency involve electromagnetic phenomena. Below power frequency, electromechanical transients in rotating machines are involved [6]. The following table shows the type of line models with respect to frequency and transmission line parameters based on CIGRE guidelines.
Topic | Low Frequency Transients (0.1 Hz–3 kHz) | Slow Front Transients (50/60 Hz–20 kHz) | Fast Front Transients (10 kHz–3 MHz) | Very Fast Front Transients (100 kHz–50 MHz) |
Representation of transposed lines | Lumped parameters, PI models | Distributed parameter, multiphase model, PI model possible for 50/60 Hz | Distributed parameter, multiphase model | Distributed parameter, single-phase model |
Line asymmetry | Important | Capacitive and inductive asymmetry important, for statistical study inductive asymmetry not important | Negligible for single-phase simulations, otherwise important | Negligible |
Frequency-dependent parameters | Important | Important | Important | Important |
For all except very short transmission lines, distributed parameter line models are preferable. If the frequency dependence is important then a frequency dependent model should be used. Details of the transmission line geometry and conductor data are then required to calculate accurately the frequency-dependent electrical parameters of the line. More information is available in Line Data.
It is noted that for very fast-front transients, a single-phase model with distributed parameters should be used [6]. This means that each phase is represented individually (phase and ground wire conductors). The calculation of lightning-caused studies requires a more detailed model including towers, footing impedances, etc. Usually a few line spans will suffice. Frequency-dependence of parameters is also important for the ground propagation mode.
Additional Notes
PI versus CP line models
The following circuit shows a simple network to observe the differences between the PI and CP models. On the figure below, one can observe the traveling wave edges caused during the energization in the CP model response (red line). The PI line is a lumped parameter model that cannot represent the distributed nature of the line parameters (blue line). The two PI‐line model in cascade emulates the traveling wave effect effect but with much less accuracy (green line). It is noted that an infinite number of PI models in cascade would be necessary to represent traveling wave phenomenon in the line model.
The user can manually fill the data form for the CP and PI models (for instance PI Section, 3-ph and Constant Param, 3-ph) by providing the positive and zero sequence data for the R, L and C of the transmission line. Typical line data is provided in Typical Electrical Parameters. The calculation of the electrical parameters for a CP line in HYPERSIM can also be done from the line geometry data with the Line Data module. The *.pun file generated with this module must be loaded in the CP component form.
Frequency-dependent effect
FD‐line models (modal and phase type) include wire skin effect and ground return frequency dependence [4].
- CP line model: Propagation function considers a single frequency; thus, high-frequency components cannot be adequately damped.
- FD‐line model: Characteristic and propagation functions are computed in the modal domain.
- Wideband model: Characteristic and propagation functions are computed in the phase domain directly.
The following figure shows the typical differences between FD and non‐FD line models in terms of their surge impedance Z_{c} and their propagation function H.
FD line fitter
The calculation of the electrical parameters for the FD line in HYPERSIM can be done with the Line Data module. Note that this module uses an internal fitter to compute rational functions in the modal domain. The resulting output file (*.pun) contains the electrical line parameters required by the FD model in HYPERSIM. The user provides the parameters of the FD line by loading the *.pun file in the data form. An example is provided in Frequency Dependent, 3-ph.
Wideband line/cable fitter
To use wideband models into a simulation, one must first use a fitter tool to build the tabulated values of the line or cable impedance and admittance (RLCG matrices) into Laplace domain rational functions with similar responses. These Laplace domain approximations are then discretized by the circuit solver to compute the line responses in the time domain. The fitter available in HYPERSIM is described in Wideband Line/Cable Fitter. This device is used for generating data (*.dat) for the Wideband Line/Cable device. Note that the WB fitter device accepts model data file from the Line Data (*.lyz) and Cable Data (*.cyz) modules.
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.
- J. R. Marti, "Accurate modelling of frequency-dependent transmission lines in electromagnetic transient simulations," IEEE Trans. Power App. Syst., vol. PAS-101, pp. 147-55, 01/1982.
- Morched A., Gustavsen B. and Tartibi M.: “A universal model for accurate calculation of electromagnetic transients on overhead lines and underground cables”. Power Delivery, IEEE Transactions, Volume: 14, Issue: 3, July 1999, pp. 1032 -1038
- Application note, Power line models for dummies, Christian Dufour OPAL-RT, 2019
- Kundur, P., Neal J. Balu, and Mark G. Lauby. Power System Stability and Control. New York: McGraw-Hill, 1994.
- Das, J.C.. Transients in Electrical Systems: Analysis, Recognition, and Mitigation. US: McGraw-Hill Professional, 2010.
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