Frequency Domain Transient Analysis Applied to Transmission System Restoration Studies Pablo Gómez, Pilar Arellano, Ricardo O. Mota Abstract--In this work, a frequency domain method to evaluate transient overvoltages produced in the restoration process of transmission systems is described. During this process, long simulation times are necessary given that several switching operations are performed to interconnect different parts of the network. The method is applied to analyze a particular transmission system, for which maximum overvoltages derived from the sequential energization of transmission lines at different restoration stages are evaluated. For the frequency domain analysis, the Numerical Laplace Transform (NLT) is applied, comparing its results with those obtained directly in time domain using the ATP-EMTP. Keywords: Frequency domain analysis, restoration, switching transients. I. INTRODUCTION EVERAL types of disturbances can produce power system complete blackout or partial outage. Restoration process must be performed at the minimum possible time and with the minimum number of operations. Results from several types of analysis of the system, e.g., power flows, small disturbance stability, transient stability and electromagnetic transients are fundamental in the restoration process. Many analytical tools are available to perform these studies; however, electromagnetic transient analysis is usually neglected or greatly simplified. Therefore, large transient overvoltages due to inadequate switching operations are some of the main causes of restoration delay and equipment damage, being of particular concern the transmission line energization [1]. Over the last decades, switching overvoltages related to line energization have been studied with different methods. At the present time, time domain methods are preferred for transient analysis, given their simplicity to simulate changes in network topology and the inclusion of non-linear elements. Among these methods, the Electromagnetic Transient Program (EMTP), initially introduced by Dommel [2], is nowadays the most widely known and applied tool for the analysis of electromagnetic transients in power systems. This work was supported by the National Polytechnic Institute under project CGPI 20070211. P. Gómez, P. Arellano and R. O. Mota are with the Grad. Program in Electrical Eng., SEPI-ESIME-Zacatenco, National Polytechnic Institute, México D. F. MÉXICO (e-mail: pgomezz@ipn.mx). Presented at the International Conference on Power Systems Transients (IPST’07) in Lyon, France on June 4-7, 2007 The inclusion of frequency dependent elements, such as transmission lines, has always been an inherent difficulty of time domain methods. Several approaches have been applied to overcome this problem since early 70s [3]-[8]. However, in a recent paper [12], it has been shown that two of the most advanced time domain line models used nowadays, namely the J. Marti model [7] and the Phase Domain model [8], can still present errors when simulating systems with strong frequency dependence. On the other hand, when using frequency domain methods for electromagnetic transient studies [9]-[12], frequency dependence of the line parameters can be included in a straightforward manner. An important shortcoming of these methods is its difficulty to deal with changes in the network topology and with non-linear elements. This has been dealt with in previous works through the application of the superposition principle with good results [11], [12]. In this work a frequency domain method, based on the Numerical Laplace Transform (NLT) [13], [14], is applied to evaluate switching transient overvoltages produced in the restoration process of a particular transmission system, for which maximum overvoltages derived from the sequential energization of transmission lines at each restoration stage are evaluated and comparisons with ATP-EMTP are provided. II. GENERAL METHODOLOGY This section reviews the methodology applied to analyze switching transients in the frequency domain, previously described in [12]. A. Transmission Line Model A multiconductor transmission line is considered as a distributed parameter model having series impedance matrix Z=R+sL and shunt admittance matrix Y=sC per unit length, being s the Laplace variable. Taking into account skin and ground return effects, both resistance and inductance are considered as frequency dependent and computed from Gary's formulae [15]. Applying nodal analysis, a multiconductor transmission line can be represented in frequency domain as ( ) ( ) ( ) ( ) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - - = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ L L l l l l V V Ψ Y Ψ Y Ψ Y Ψ Y I I 0 0 0 0 0 0 coth h csc h csc coth (1) where V 0 and I 0 are voltage and current vectors at the sending end, V L and I L are the respective values at the receiving end, l is the line length, Y 0 and Ψ are the characteristic admittance and voltage propagation matrices given by S