Electric Power Systems Research 79 (2009) 1155–1163 Contents lists available at ScienceDirect Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr Inclusion of the frequency effect in the lumped parameters transmission line model: State space formulation Sérgio Kurokawa a,∗ , Fábio N.R. Yamanaka a , Afonso J. Prado a , José Pissolato b a Faculdade de Engenharia de Ilha Solteira, Avenida Brasil Centro 56, Caixa Postal 31, Ilha Solteira, SP, CEP 15385-000, Brazil b Universidade Estadual de Campinas, Caixa Postal 6101, CEP 13081-970, Campinas, SP, Brazil article info Article history: Received 29 August 2007 Received in revised form 4 November 2008 Accepted 14 February 2009 Keywords: Electromagnetic transients Frequency dependence Time domain Transmission line Transmission line parameters State space methods abstract The objective of this paper is to include the frequency dependence of the longitudinal parameters in the lumped parameters line model. The distributed nature of the transmission line was approximated by a cascade of  circuits and the frequency effect was approximated by a rational function which was synthesized by an equivalent circuit. Then, the equivalent circuit was inserted in each  circuit of the cascade. After that, the currents and voltages along the line were described in the form of state equations. This way, it was possible to obtain a formation rule of the state matrices lumped parameters model taking into account the frequency dependence. To confirm the validation of the state matrices obtained, the lumped parameters representation of frequency-dependent lines was used to represent a single- phase line and a three-phase line. The simulations were carried out using state space techniques and an electromagnetic transient program (EMTP) (in this case, the cascade was inserted in the EMTP). It is observed that the simulation results obtained with state space representation are in agreement with those results obtained with EMTP. © 2009 Elsevier B.V. All rights reserved. 1. Introduction Transmission overhead line models used for transient simula- tions are frequently classified as being a lumped and distributed parameters model. To obtain a lumped parameters line model, a given line is rep- resented by connecting n short nominal  circuits in cascade. Considering the maximum transient frequency and the line length if a sufficiently high number of sections is chosen the distributed nature of the transmission line can be simulated adequately. When a lumped parameters line model is adopted, it is very common to use state space techniques to evaluate the currents and voltages along the line. In this way, it is possible for the model to carry out simulations directly in time domain without the explicit use of inverse transforms and it can be easily implemented. These characteristics of the lumped parameters model are the same as those used to simulate electromagnetic transients on lines with nonlinear components, such as corona effects and fault arcs [1–3], or when a detailed voltage and current profile is needed [4]. However, several papers show that the lumped parameters model is developed without taking into account the frequency dependence of the longitudinal parameters of the line [1–6] and this ∗ Corresponding author. Tel.: +55 18 3743 1129; fax: +55 18 3743 1163. E-mail addresses: kurokawa@dee.feis.unesp.br (S. Kurokawa), fabioyamanaka@gmail.com (F.N.R. Yamanaka), afonsojp@uol.com.br (A.J. Prado), pisso@dsce.fee.unicamp.br (J. Pissolato). fact limits the accuracy of the lumped parameters model because it is known that models which assume constant parameters can- not adequately simulate the response of the line over the wide range of frequencies that are present in the signals during transient conditions. Because the frequency dependence is one of the most important aspects in the modeling of transmission lines for electromagnetic transient studies and the lumped parameters line model has several applications [1–6] we decided to improve this model. To improve it, we developed the state matrices of the lumped parameters line representation taking into account the frequency dependence of the longitudinal parameters. To improve the lumped parameters model, the longitudinal parameters of the line was approximated by a rational function and, after that, this rational function was associated with an equivalent circuit. Then, the longitudinal parameters fitted by the equivalent circuit were inserted in each  circuit. Finally, we wrote the state equations for the currents and voltages along the cascade and we also found a formation rule to evaluate the matrices of the state space representation. The proposed frequency-dependent model was used to repre- sent a single-phase line and a non-transposed 440 kV three-phase line. 2. Frequency dependence of the longitudinal parameters It has long been recognized that one of the most important aspects in the modeling of transmission lines for electromagnetic 0378-7796/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2009.02.007