Using Evolutionary Algorithms to Determine Frequency-Dependent Network Equivalents
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Using Evolutionary Algorithms to Determine Frequency-Dependent Network Equivalents Carlos F. M. Almeida · Nelson Kagan
Received: 3 August 2012 / Revised: 8 January 2013 / Accepted: 16 September 2013 / Published online: 16 October 2013 © Brazilian Society for Automatics–SBA 2013
Abstract The present paper presents an alternative methodology to determine frequency-dependent network equivalents from the frequency response of electric power networks. According to this methodology, one can substitute part of the power network through a simplified electric circuit, reducing the number of elements to be modeled without introducing any inaccuracy to the precise evaluation of large power networks. The main challenge relays on determining a rational function capable of producing the same frequency response characteristic of the original circuit that would be replaced by the equivalent. The rational function determines the impedance to be used in the network equivalent. The methodology is based on Evolutionary Algorithms (EAs). Basically, different set of values for the rational function parameters are evaluated. During the convergence process, the EAs modify the values for the function’s parameters suitably, reducing the RMS error between the original frequency response and the one produced by the equivalent circuit. The results obtained through this methodology were compared with those obtained through the “Vector Fitting” methodology, which is the methodology commonly used to solve similar problems. Keywords Network equivalents · Frequency-dependent models · Rational functions · Evolutionary algorithms 1 Introduction Network equivalents are widely used in the evaluation of electric power systems in order to simplify their representaC. F. M. Almeida (B) · N. Kagan ENERQ - USP, Av. Professor Luciano Gualberto, Travessa 3, no 158, Bloco A, Sala A1-21, Cidade Universitária, CEP 05508-900 São Paulo, SP, Brazil e-mail: [email protected]
tion. Thus, only relevant parts of the systems are represented in detail, while others are replaced by electrical equivalent circuits connected at the coupling points. The network equivalent must be able to suitably represent the behavior of parts of the system to which they are related. For this, they should produce responses similar to those parts of the system that they replace whatever is the frequency considered in the performance evaluation study. In the harmonic assessment of power systems, accurate representation of network equivalents becomes even more important because they directly influence the propagation of harmonic distortions. There are two basic types of network equivalents: • Thevenin equivalent circuits; • Norton equivalent circuits. The basic difference in the use of each possible type is the representation of the active element. That is, one can choose an equivalent composed by voltage source (Thevenin equivalent circuit) or an equivalent composed by current sources (Norton equivalent circuit). The determination of the values of the voltage sources used by the Thevenin eq
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