# Application of the lumped parameter line model to simulate electromagnetic transients in three-phase transmission lines with vertical symmetry

## Keywords:

Electromagnetic transients, transmission lines, time domain, modal decomposition, lumped parameters

## Abstract

In this paper, we propose a transmission line model to represent three-phase untransposed transmission lines with vertical symmetry. Here, we decouple a transmission line into its propagation modes using Concordia's matrix, which is the proper orthogonal form of Clarke's matrix that has its columns normalised. Due to the usage of Concordia's matrix, the transmission line is decoupled into two coupled modes and one uncoupled mode. We represent the uncoupled mode by a single-phase transmission line, and the coupled modes by a two-phase transmission line without vertical symmetry. We compute voltages and currents for the single-phase transmission line using the classic Lumped Parameter Model. Voltages and currents in the two-phase transmission line are calculated using a cascade of coupled L circuits pairs. Modal voltages and currents are then transformed back to the phase domain. We compare the results obtained with the proposed model to those obtained with two well-known time domain models, the classic modal lumped parameter model and the lumped parameter model available in the Alternate Transients Program ATP. Results show that the proposed model agrees with time-domain models and has the advantage that is developed in the time domain without needing any inverse transformations.

## Author Biographies

### Tainá Fernanda Garbelim Pascoalato, São Paulo State University (UNESP)

Received the B.Sc. degree (with distinction) in Electrical Engineering from the Votuporanga University Center, in 2017. She received the M. Sc. degree in Electrical Engineering from the São Paulo State University, in 2020. Currently, she is a doctoral student in Electrical Engineering at the São Paulo State University, in Ilha Solteira, Brazil. Her research areas include transmission line modeling for electromagnetic transient simulations in power systems and soil models with frequency-dependent parameters.

### Pablo Torrez Caballero, São Paulo State University (UNESP)

Received the B.Sc. degree in Electromechanical Engineering from the Bolivian Private University in 2011, M.Sc. and Ph.D. in Electrical Engineering from the São Paulo State University in 2014 and 2018, respectively. Currently, he is a postdoctoral researcher at the São Paulo State University, in Ilha Solteira, Brazil. His research areas include power systems modeling and time-domain representation of frequency dependent functions.

### Sérgio Kurokawa, São Paulo State University (UNESP)

Received B. Sc degree in Electrical Engineering from S\~ao Paulo State University (UNESP) in 1990, M. Sc from Federal University of Uberlandia (UFU) in 1994 and Ph.D from University of Campinas (Unicamp) in 2003. Since 1994 he has been working as a professor and researcher at UNESP, Campus of Ilha Solteira. His current interests are electromagnetic transients in power systems and transmission line modeling.

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2021-07-17

## How to Cite

Garbelim Pascoalato, T. F., Torrez Caballero, P., & Kurokawa, S. (2021). Application of the lumped parameter line model to simulate electromagnetic transients in three-phase transmission lines with vertical symmetry. IEEE Latin America Transactions, 20(3), 379–385. Retrieved from https://latamt.ieeer9.org/index.php/transactions/article/view/5346

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